\section{Introduction}

Quantum computation offers the promise of tackling classically intractable computational problems with careful manipulation of quantum mechanical systems. However, practical quantum computational devices usually suffer from various sources of error and imprecision, precluding us from obtaining meaningful results from computational processes since the results are dominated by meaningless noise. One of the major challenges in quantum hardware design, fabrication and calibration is to implement the basic elements of quantum computation with very high precision. Not only does this improve the performance of Noisy Intermediate Scale Quantum (NISQ) algorithms~\cite{preskill2018quantum} running on bare physical qubits, but it is also a crucial step in realizing fault-tolerant quantum computation. Among the different basic elements of quantum computation, the two-qubit gate operation is the most error-prone as it acts upon a composite quantum system in which more complex errors can occur. It is therefore a high priority to theoretically design and experimentally realize an ultra-high fidelity two-qubit gate.

Many candidates for which two-qubit gate we should use have been proposed. In superconducting quantum computing, these include the cross-resonance gate~\cite{paraoanu2006microwave,rigetti2010fully,de2010selective}, the dynamically decoupled CZ gate~\cite{bermudez2012robust, zheng2002quantum,guo2018dephasing,abrams2020implementation}, the fSim gates~\cite{arute2019quantum}, the $\mathrm{CPHASE} $  family gates~\cite{barends2014superconducting, kelly2014optimal} and the $\mathrm{\lowercase{i}SWAP} $ family gates~\cite{majer2007coupling,wendin2007quantum,steffen2006measurement,bialczak2010quantum,neeley2010generation,dewes2012characterization, sillanpaa2007coherent}. The square root of the $\mathrm{\lowercase{i}SWAP} $ gate, $$\mathrm{SQ\lowercase{i}SW} \equiv\begin{bmatrix}1&0&0&0\\0&\frac1{\sqrt{2}}&\frac{i}{\sqrt{2}}&0\\0&\frac{i}{\sqrt{2}}&\frac1{\sqrt{2}}&0\\0&0&0&1\end{bmatrix},$$ is in the $\mathrm{\lowercase{i}SWAP} $ family and its properties in compilation was studied in~\cite{arute2020observation, arute2020hartree,bialczak2010quantum,mi2021information, abrams2020implementation, peterson2020fixed, haug2021capacity, goerz2017charting}. Note that we choose this particular square root since it can be physically implemented by halving the $\mathrm{\lowercase{i}SWAP} $ gate. This halving of gate time also implies that $\mathrm{SQ\lowercase{i}SW} $  suffers from less $T_1$ and $T_2$ errors, the usual bottlenecks in gate fidelity, as well as well-known $ZZ$ errors~\cite{arute2020observation,sung2020realization}. It can therefore be realized with higher fidelities. Moreover, there is evidence that the $\mathrm{SQ\lowercase{i}SW} $  gate is still a very useful two-qubit gate. For example, it is a {\em perfect entangler}, that is, it maps a certain product state to a maximally entangled state; indeed, it is the smallest fraction of $\mathrm{\lowercase{i}SWAP} $ with that property\cite{zhang2003geometric}. Despite these hints of promise, few works have systematically studied the information processing capabilities of $\mathrm{SQ\lowercase{i}SW} $ or its detailed experimental implementation, calibration and benchmarking, both of which we aim to accomplish in this work.

To this end, we propose $\mathrm{SQ\lowercase{i}SW} $ as a native two-qubit gate, highlighting its ultra-high-fidelity implementation and its information processing capabilities in several gate and circuit compilation tasks. We show that the $\mathrm{SQ\lowercase{i}SW} $ gate outperforms the incumbent native two-qubit gates such as $\mathrm{CNOT} $ and $\mathrm{\lowercase{i}SWAP} $ on many tasks, both with a higher fidelity and sometimes even a lower gate count. More specifically, the central results of the report can be summarized as follows:

\paragraph{Ultra-high-fidelity implementation.} Taking into account decoherence, stray coupling between qubits, and the instrumental limitations on experimental control parameters, we conduct a detailed numerical simulation showing that the $\mathrm{SQ\lowercase{i}SW} $  gate can be realized in the ultra-high fidelity regime ($\geq 99.9\%$ average gate fidelity). Based on experimentally achievable circuit parameters and decoherence times, we estimate that we can implement the $\mathrm{SQ\lowercase{i}SW} $  gate with an error of about $5\times 10^{-4}$.

\paragraph{Superior information processing capability.} We prove that an arbitrary two-qubit gate can be compiled using at most three uses of the $\mathrm{SQ\lowercase{i}SW} $ gate interleaved with arbitrary single-qubit gates, and we give explicit algorithms compiling an arbitrary gate with the optimal number of $\mathrm{SQ\lowercase{i}SW} $  gates. Moreover, we prove that 79\% of two-qubit gates (under the Haar measure) can be generated using only two uses of the $\mathrm{SQ\lowercase{i}SW} $  gate. As a comparison, only a zero-measure set of two-qubit gates can be generated using two uses of common two-qubit gates such as $\mathrm{CPHASE} $ family gates, $\mathrm{SWAP} $  family gates or the $\mathrm{\lowercase{i}SWAP} $ gate. Furthermore, we prove that in the task of generating certain W-like states, we need strictly less $\mathrm{SQ\lowercase{i}SW} $ gates compared to $\mathrm{\lowercase{i}SWAP} $ or $\mathrm{CNOT} $ gates. These results suggest that while $\mathrm{SQ\lowercase{i}SW} $  is the square root of $\mathrm{\lowercase{i}SWAP} $, its information processing capabilities can rival and even surpass that of $\mathrm{\lowercase{i}SWAP} $ and $\mathrm{CNOT} $ in multiple meaningful aspects.

\paragraph{Calibration and benchmarking.} We develop a framework for efficiently calibrating and benchmarking the $\mathrm{SQ\lowercase{i}SW} $  gate. In particular, we introduce a variant of the randomized benchmarking framework~\cite{emerson2005scalable}, which we call {\em interleaved fully randomized benchmarking (iFRB)}, in order to benchmark the performance of an arbitrary, possibly non-Clifford, gate. The iFRB framework is based on the original proposal for randomized benchmarking using Haar random gates~\cite{emerson2005scalable}, which we refer to as fully randomized benchmarking (FRB), that characterizes the average noise level in a quantum system and the interleaved randomized benchmarking framework (iRB)~\cite{magesan2012efficient} that characterizes the noise level associated to a particular target Clifford gate. Compared to previous approaches, iFRB does not require the target gate to lie in the Clifford group or exhibit any particular group structure. Using the scheme to compile arbitrary two-qubit gates using $\mathrm{SQ\lowercase{i}SW} $, iFRB can readily be used to benchmark the $\mathrm{SQ\lowercase{i}SW} $ gate. In general, it is applicable whenever an efficient compilation scheme of an arbitrary gate is available.

In \cite{peterson2020fixed}, the problem of generating two-qubit gates from a fixed set of native two-qubit gates and arbitrary single qubit gates was systematically studied using the monodromy polytope theory. They showed that the set of two-qubit gates generated with a fixed number of native two-qubit gates can always be characterized as a finite union of polytopes with respect to a certain parameterization. This makes such sets much easier to study, leading to many results regarding the $\mathrm{CPHASE} $ gate family and the $\mathrm{\lowercase{i}SWAP} $ gate family. However, as pointed out in~\cite{peterson2020fixed}, it is not clear how their results can be translated to efficient compilation algorithms in general, or whether such an algorithm exists at all. In our paper, we focus on the $\mathrm{SQ\lowercase{i}SW} $  gate, and provide efficient and explicit compilation algorithms. These algorithms makes the $\mathrm{SQ\lowercase{i}SW} $  gate practically useful in scenarios where arbitrary two-qubit gates, or families of two-qubit gates, are needed.

The rest of the report proceeds as follows. In Section~\ref{sec:motiv}, we show how the $\mathrm{SQ\lowercase{i}SW} $  gate can be physically implemented on a superconducting quantum computing device and provide numerical evidence that it can be performed with ultra-high fidelity ($\geq99.9\%$). Section~\ref{sec:compilation} focuses on $\mathrm{SQ\lowercase{i}SW} $'s information processing capabilities such as compilation and generating entanglement (that is, W-like states). Section~\ref{sec:imp} provides proposals for calibrating and benchmarking the $\mathrm{SQ\lowercase{i}SW} $  gate, including our iFRB technique. We summarize the results and give a brief discussion of future work in Section~\ref{sec:summary}.



\section{$\mathrm{SQ\lowercase{i}SW} $: Ultra-high Fidelity Implementation}
\label{sec:motiv}
In this section, we give numerical evidence that we can implement $\mathrm{SQ\lowercase{i}SW} $ on superconducting platforms with ultra-high fidelity.

Two-qubit gates in the $\mathrm{\lowercase{i}SWAP} $ family can be implemented by
tuning two superconducting qubits with transversal coupling into resonance.
This can be demonstrated in a two-qubit system such as two tunable transmons or tunable fluxoniums capacitively coupled directly or through a bus resonator.
Such an implementation suffers from possible stray longitudinal coupling ($ZZ$ interaction),
inaccurate flux tuning and decoherence\cite{barends2019,ganzhorn2020,sung2020realization}.  Here we show how these errors will affect the fidelity of implementation.

First we will investigate the coherent error in the gate.
Without loss of generality, we consider a two-qubit system with $YY$ coupling~\cite{krantz2019quantum},
with only the lowest two levels of each qubit included. The relevant Hamiltonians near resonance are given by
\begin{align*}
    H_1 & =\left[\begin{matrix} 0 & 0 \\ 0 & \omega\end{matrix}\right]\otimes I_2 ;                                               \\
    H_2 & =I_1 \otimes \left[\begin{matrix} 0 & 0 \\ 0 & \omega+\Delta\end{matrix}\right]                                             ; \\
    H_c & =\frac{1}{4}\left(g_{yy}\sigma_{y1}\otimes\sigma_{y2}+g_{zz}\sigma_{z1}\otimes\sigma_{z2}\right) ;\\
    H   & = H_1 + H_2 + H_c = \left[\begin{matrix}
            g_{zz}/4  & 0                      & 0               & -g_{yy}/4               \\
            0         & \omega+\Delta-g_{zz}/4 & g_{yy}/4        & 0                       \\
            0         & g_{yy}/4               & \omega-g_{zz}/4 & 0                       \\
            -g_{yy}/4 & 0                      & 0               & 2\omega+\Delta+g_{zz}/4  \end{matrix}\right],
\end{align*}
where $H_1$ and $H_2$ are the single-qubit Hamiltonians, $H_c$ is
the coupling term, $H$ is the Hamiltonian of the whole two-qubit system, $\omega$ and $\omega+\Delta$ are frequencies of two qubits, and $g_{yy}$ and $g_{zz}$ are
the corresponding coupling strengths.
$g_{yy}$ is the term that contributes to the $\mathrm{\lowercase{i}SWAP} $ family gate.
$\Delta$ and $g_{zz}$ are introduced to account for inaccurate flux tuning and stray longitudinal coupling, respectively.

We move into the frame rotating with both qubits at frequency $\omega$. The rotating frame transformation $R(t)$ and the rotated Hamiltonian $H^R(t)$ are given by
\begin{align*}
    R(t)   & =\left[\begin{matrix}
            0 & 0 \\ 0 & e^{-i\omega t}\end{matrix}\right]
    \otimes\left[\begin{matrix}0 & 0 \\ 0 & e^{-i\omega t}\end{matrix}\right],                                       \\
    H^R(t) & =R^{\dagger}(t)H(t)R(t) + i\frac{\partial R^{\dagger}(t)}{\partial t}R(t)=
    \left[\begin{matrix}
            g_{zz}/4               & 0               & 0         & e^{-i2\omega t}g_{yy}/4 \\
            0                      & \Delta-g_{zz}/4 & g_{yy}/4  & 0                       \\
            0                      & g_{yy}/4        & -g_{zz}/4 & 0                       \\
            e^{i2\omega t}g_{yy}/4 & 0               & 0         & \Delta+g_{zz}/4
        \end{matrix}\right].
\end{align*}
Assuming $\omega$
is much larger than any other frequency in the Hamiltonian, we can take the rotating wave approximation and eliminate
all fast-oscillation terms with
$e^{i2\omega t}$ since the time-average is approximately zero.
We thus have an approximate time-independent Hamiltonian of the system given by
\begin{align*}
    H_R & \approx
    \left[\begin{matrix}
            g_{zz}/4 & 0               & 0         & 0               \\
            0        & \Delta-g_{zz}/4 & g_{yy}/4  & 0               \\
            0        & g_{yy}/4        & -g_{zz}/4 & 0               \\
            0        & 0               & 0         & \Delta+g_{zz}/4
        \end{matrix}\right].
\end{align*}

The time evolution operator corresponding to this Hamiltonian is
\begin{align*}
    U(t) & = e^{-iH_Rt}.
\end{align*}
It is easy to verify that when all error terms are zero, that is $\Delta=0$ and $g_{zz}=0$,
$U(t)$ is given by the well-known form of $\mathrm{\lowercase{i}SWAP} $ family gate:
\begin{align*}
    U(t)             & = \left[\begin{matrix}
            1 & 0              & 0              & 0 \\
            0 & \cos(\theta)   & -i\sin(\theta) & 0 \\
            0 & -i\sin(\theta) & \cos(\theta)   & 0 \\
            0 & 0              & 0              & 1
        \end{matrix}\right] ;\\
    U(t=\pi/g_{yy})  & = \mathrm{SQ\lowercase{i}SW} ^\dag=\begin{bmatrix}1&0&0&0\\0&\frac1{\sqrt{2}}&-\frac{i}{\sqrt{2}}&0\\0&-\frac{i}{\sqrt{2}}&\frac1{\sqrt{2}}&0\\0&0&0&1\end{bmatrix}                                  ;\\
    U(t=2\pi/g_{yy}) & = \mathrm{\lowercase{i}SWAP} ^\dag=\begin{bmatrix}1&0&0&0\\0&0&-i&0\\0&-i&0&0\\0&0&0&1\end{bmatrix},
\end{align*}
where $\theta\equiv tg_{yy}/4$. Note that we assume above $g_{yy} >0$ since the gate time must be positive. When $g_{yy}<0$, we instead implement $U(t=-\pi/g_{yy})=\mathrm{SQ\lowercase{i}SW} $. Note that $\mathrm{SQ\lowercase{i}SW} $  is equivalent to $\mathrm{SQ\lowercase{i}SW} ^\dag$ under local unitaries: we have $\mathrm{SQ\lowercase{i}SW} ^\dag = (Z\otimes I)\mathrm{SQ\lowercase{i}SW} (Z\otimes I)$. The properties of the two gates are very similar despite minor sign differences in compilation. In the subsequent sections we will consider the $\mathrm{SQ\lowercase{i}SW} $  gate even though $\mathrm{SQ\lowercase{i}SW} ^\dag$ is more common in physical implementations.

In realistic systems $g_{zz}$ and $\Delta$ are generally not zero.
This causes a nonzero error, which we can quantify using the average fidelity $F$ between two unitary matrices $U,V$~\cite{nielsen2002simple}:
\begin{align*}
    F(U,V) & = \frac{\left|\mathrm{Tr}(V^{\dagger}U)\right|^2+d}{d(d+1)}
\end{align*}
where $d=4$ is the dimension of our matrices.
We can estimate the infidelity $E=1-F$ between the simulated physical implementation of $\mathrm{SQ\lowercase{i}SW} ^\dag$ with $U(t=\pi/g_{yy})$
and the ideal $\mathrm{SQ\lowercase{i}SW} ^\dag$ unitary induced by a specific error term by turning on the given error while turning off all others.

We find that the infidelity induced by the $ZZ$ interaction $g_{zz}$ and detuning $\Delta$
can be written as a power series of $g_{zz}/g_{yy}$ and $\Delta/g_{yy}$ respectively,
assuming each error term is small:
\begin{align}
    E_{ZZ}     & \approx \frac{\pi^2}{20}\left(\frac{g_{zz}}{g_{yy}}\right)^2 ;  \nonumber\\
    E_{\Delta} & \approx \frac{8+\pi^2}{10}\left(\frac{\Delta}{g_{yy}}\right)^2. \label{eq:quad_err}
\end{align}
Therefore, the infidelity of $\mathrm{SQ\lowercase{i}SW} $  scales quadratically with the ratio of $ZZ$ to $YY$ coupling and the ratio of $\Delta$ to $YY$ coupling.
We also expect there to be a cross-interaction between these two errors, and this is investigated below.
To include the decoherence of the system, we perform numerical simulations
to estimate the effect of $T_1$ and $T_{\phi}$ on the fidelity.
The time evolution is based on the Lindblad master equations:
\begin{align*}
    \frac{d\rho(t)}{dt} & =-\frac{i}{\hbar}[H(t),\rho(t)]+\sum_{j=1,2}\left(
    \Gamma_{1,j}\mathcal{L}[\sigma_j^-] +
    \frac{1}{2}\Gamma_{\phi,j}\mathcal{L}[\sigma_j^z]
    \right)\rho(t)
\end{align*}
where $\rho(t)$ is the time-dependent density matrix, $H(t)$ is the time-dependent Hamiltonian, $\Gamma_{1,j}$ is the dissipation rate of $j$-th qubit, $\Gamma_{\phi,j}$ is the dephasing rate of $j$-th qubit, and $\mathcal{L}[c]\rho=c\rho c^{\dagger}-\frac{1}{2}(\rho c^{\dagger}c-c^{\dagger}c\rho)$ is the Lindblad superoperator. Note this model includes only the Markovian noise. The gate fidelity is affected little by the non-Markovian noise when the gate time is short~\cite{omalley2015}, so we neglected the non-Markovian noise in the discussion.
We can then compute the average gate fidelity via the process fidelity and $\chi$-matrix as follows~\cite{nielsen2002simple}:
\begin{align*}
    F_p(\chi_1,\chi_2) & =\mathrm{Tr}(\chi_1\chi_2);                                                     \\
    F(\chi_1, \chi_2)  & = \frac{dF_p(\chi_1,\chi_2)+1}{d+1} = \frac{\mathrm{Tr}(\chi_1\chi_2)d+1}{d+1}.
\end{align*}

We choose the following range of parameters, which can be experimentally realized in fabricated fluxonium qubits, to perform the numerical simulations.
For simplicity, we assume $T_1=T_{1,1}=T_{1,2}$ and $T_{\phi}=T_{\phi,1}=T_{\phi,2}$.
\begin{itemize}
    \item $g_{yy}/h=25\mathrm{MHz}$, corresponds to a $\mathrm{SQ\lowercase{i}SW} $  gate time of $20\mathrm{ns}$
    \item $g_{zz}/h\in [-0.6, +0.6]\ \mathrm{MHz}$
    \item $\Delta/h\in [-0.4, +0.4]\ \mathrm{MHz}$
    \item $T_1 \geq 25\mathrm{\mu s}$
    \item $T_{\phi}\geq 25\mathrm{\mu s}$
         
\end{itemize}
We check the effect of nonzero temperature as the thermal population of the first excited state $\ket{e}$
may be large. However, we find no major difference between the infidelity at zero temperature
and that of a typical fluxonium system ($T=50\mathrm{mK}$ and frequency $\omega_{ge}=2\pi\times1\mathrm{GHz}$).

After computing the infidelity via numerical simulation, we perform
a polynomial regression up to 2nd order in the different parameters to identify the error sources (3rd order polynomial regression does not add any new important terms as shown in Fig.~\ref{fig:importance_error}).
In particular, $g_{zz}$, $\Delta$, $\Gamma_1$ and $\Gamma_{\phi}$,
are features in the regression as they are expected to increase the infidelity linearly or quadratically.
The regression accuracy is plotted in Fig. \ref{fig:infidelity_regression}.
The polynomial regression works very well in this case: the root mean square error of the regression is on the order of $10^{-9}$.

To identify the key error sources contributing to the infidelity, we check the permutation feature importance as shown in Fig. \ref{fig:importance_error}.
There are 5 dominant features in the polynomial regression.
The infidelity depends linearly on $\Gamma_{\phi}$ and $\Gamma_1$, which is a general behavior of decoherence.
And the infidelity also depends quadratically on $g_{zz}$ and $\Delta$, which agrees with the symbolic results in~\cref{eq:quad_err}.
Finally, an additional term $g_{zz}\Delta$ emerges, which is the cross-term between $g_{zz}$ and $\Delta$. This term makes it unclear what the error contributions are due to $g_{zz}$ and $\Delta$ individually. In our experiments, $g_{zz}$ is fixed at the design stage and $\Delta$ is microwave tunable, so we can choose an optimal $\Delta$ in the experiments to minimize the infidelity contributed by the terms $c_{\Delta^2}\Delta^2,c_{g_{zz}\Delta}g_{zz}\Delta, c_{g_{zz}^2} g_{zz}^2$. This isolates the contribution to the infidelity from $g_{zz}$. This optimal $\Delta$ will be our operating point. However, the precision of the frequency is limited by our instruments, and there is an error caused by the deviation from the optimal point which we call $\Delta_p$.  Overall, we define the infidelity from each error source as follows:
\begin{itemize}
    \item $c_{\Delta^2}\Delta^2+c_{g_{zz}\Delta}g_{zz}\Delta+ c_{g_{zz}^2} g_{zz}^2$ at the optimal $\Delta$ as the error from the stray coupling $g_{zz}$;
    \item $c_{\Delta^2}\Delta_p^2$ as the error from the instrumental limitation of the frequency $\Delta_p$;
    \item $c_{\Gamma_1}\Gamma_1$ and $c_{\Gamma_{\phi}}\Gamma_{\phi}$ as the error from the decoherence processes.
\end{itemize}
These errors are plotted for both $\mathrm{SQ\lowercase{i}SW} $  and $\mathrm{\lowercase{i}SWAP} $ in Fig.~\ref{fig:importance_error}. Note that because we are working at the optimal $\Delta$, the above errors add up to the total error.

Considering the recent progress on fluxonium fabrication~\cite{nguyen2019} and the precision of the arbitrary wave generator, we compute for a realistic set of parameters  $g_{yy}/h=25\mathrm{MHz}$, $T_1=100\mathrm{\mu s},T_{\phi}=100\mathrm{\mu s},
    g_{zz}/h=-0.3\mathrm{MHz},\Delta_p/h=0.18\mathrm{MHz}$,
    which is what we take as input for the results in~Fig.~\ref{fig:importance_error}, $\mathrm{SQ\lowercase{i}SW} $  can be realized with about $5\times10^{-4}$ infidelity.



\begin{figure}[h]
    \centering
    \includegraphics[width=0.35\textwidth]{lr_gate0-gxx2_full.png}
    \caption{Infidelity via polynomial regression against simulated infidelity of $\mathrm{SQ\lowercase{i}SW} $  gates. The black dashed line is the $x=y$ line for visual effect. The root mean square error of the regression is on the order of $10^{-9}$.
    }
    \label{fig:infidelity_regression}
\end{figure}


\begin{figure}[h]
    \centering
    \includegraphics[width=0.35\textwidth]{permutation_importance_gate0-gxx2.png}
    \includegraphics[width=0.40\textwidth]{error_budget_iswap_sqiswap_type3_gxx2.png}
    \caption{Left: The permutation feature importance for the polynomial regression in Fig.~\ref{fig:infidelity_regression}. Only features with importance $>0.01$ are included in the figure. Right: Comparison of features' contributions to the infidelity of $\mathrm{SQ\lowercase{i}SW} $  and $\mathrm{\lowercase{i}SWAP} $ with the parameters $g_{yy}/h=25\mathrm{MHz}, T_1=100\mathrm{\mu s},T_{\phi}=100\mathrm{\mu s},
    g_{zz}/h=-0.3\mathrm{MHz},\Delta_p/h=0.18\mathrm{MHz}$.}
    \label{fig:importance_error}
\end{figure}



\section{$\mathrm{SQ\lowercase{i}SW} $: Information Processing Capabilities}
\label{sec:compilation}
In this section, we first show some basic mathematical properties of the $\mathrm{SQ\lowercase{i}SW} $  gate and then study the information processing capabilities of the $\mathrm{SQ\lowercase{i}SW} $  gate, in particular its ability to compile two-qubit and higher operations. Specifically, we prove that an arbitrary two-qubit gate can be decomposed into at most three applications of the $\mathrm{SQ\lowercase{i}SW} $  gate interleaved by single qubit rotations and give explicit decompositions for certain families of two-qubit rotations.
The $\mathrm{CNOT} $  gate and the $\mathrm{\lowercase{i}SWAP} $ gate also generate all two-qubit gates with three applications; however we prove that a majority ($\sim79\%$) of two-qubit gates, under the Haar measure, can be generated using only two uses of the $\mathrm{SQ\lowercase{i}SW} $  gate, whereas gates generated by two uses of the $\mathrm{CNOT} $  gate or the $\mathrm{\lowercase{i}SWAP} $ gate only spans zero-measure sets. We lastly prove that $\mathrm{SQ\lowercase{i}SW} $  has an advantage over the $\mathrm{CNOT} $ and $\mathrm{\lowercase{i}SWAP} $  gates in the task of preparing W-like states.

\subsection{Basic mathematical properties}
We summarize some useful mathematical properties of the $\mathrm{SQ\lowercase{i}SW} $  gate. Besides being the square root of the $\mathrm{\lowercase{i}SWAP} $ gate, $\mathrm{SQ\lowercase{i}SW} $  satisfies the following properties:
\begin{itemize}
\item $\mathrm{SQ\lowercase{i}SW} $  lies in the third level of the Clifford hierarchy: just like the $T$ gate and the Controlled-$S$ gate, the $\mathrm{SQ\lowercase{i}SW} $  gate conjugates Pauli matrices to Clifford matrices. Also, it is not in the second level of the Clifford hierarchy, meaning that it itself is not a Clifford gate.
    \item $\mathrm{SQ\lowercase{i}SW} $  is a perfect entangler, that is, it maps a product state into a maximally entangled state. Explicitly,
    $$\mathrm{SQ\lowercase{i}SW}  |01\rangle = \frac{1}{\sqrt{2}}(|01\rangle + i|10\rangle).$$
    \item $\mathrm{SQ\lowercase{i}SW} $  is an excitation number preserving gate, meaning that for all $\theta$, $[\mathrm{SQ\lowercase{i}SW} , Z_\theta\otimes Z_\theta]=0$.
\end{itemize}
To explore further properties, we first introduce some mathematics.

\subsubsection{KAK decomposition and the Weyl chamber}
The KAK decomposition and Weyl chamber provide mathematical tools to characterize two-qubit gates up to single qubit gates. That is, they give the ``non-local'' information of a two-qubit gate. In particular, the KAK decomposition characterizes equivalence classes of two-qubit unitaries, or elements in the group $SU(4)$, under actions by single-qubit rotations in $SU(2) \otimes SU(2)$ before and after. This perspective is particularly useful when we can experimentally regard single qubit local operations as free resources that introduce little error compared to two-qubit gates. We here directly state the results and refer the reader to~\cite{zhang2003geometric,tucci2005introduction} for more detailed expositions.
\begin{thm}[KAK decomposition~\cite{zhang2003geometric}]
For an arbitrary $U\in SU(4)$, there exists a unique $\Vec{\eta}=(x,y,z), \frac{\pi}4\geq x\geq y\geq |z|$, single qubit rotations $A_0,A_1,B_0,B_1\in SU(2)$ and a global phase $g\in\{1,i\}$ such that
$$U = g\cdot\left(A_1\otimes A_2\right)\exp\{i\Vec{\eta}\cdot\Vec{\Sigma}\}\left(B_1\otimes B_2\right),$$
where $\Vec{\Sigma} \equiv [\sigma_X\otimes \sigma_X, \sigma_Y\otimes \sigma_Y,\sigma_Z\otimes \sigma_Z]$. The tuple $(g,\Vec{\eta}, A_0, A_1, B_0, B_1)$ is called the \emph{KAK decomposition} of the unitary $U$.
\end{thm}

Define the magic basis change matrix $M\equiv \frac{1}{\sqrt{2}}\begin{bmatrix}1&0&0&i\\ 0&i&1&0\\0&i&-1&0\\ 1&0&0&-i\end{bmatrix}$. The KAK decomposition theorem can be equivalently stated as follows: $$M^\dag U M = g\cdot A\cdot K\cdot B,$$ where $A,B\in SO(4)$ and
$$K=\begin{bmatrix}e^{i(x-y+z)}&0&0&0\\0&e^{i(x+y-z)}&0&0\\0&0&e^{i(-x-y-z)}&0\\0&0&0&e^{i(-x+y+z)}\end{bmatrix}$$
is a diagonal matrix.

The equivalence class of a unitary $U$ under local unitaries is characterized by the \emph{interaction coefficients} $\eta(U)$, which lives in a 3-dimensional tetrahedron called the \emph{Weyl chamber}\footnote{Note that there are many ways one can choose the canonicalization. Our convention follows the one in \cite{cirq_developers_2021_4586899} and \cite{cross2019validating}.} $$W\equiv\{\pi/4\geq x\geq y\geq |z|\ \textrm{and}\ z\geq 0 \text{ if }x=\frac\pi4\mid(x,y,z)\in\mathbb{R}^3\}.$$ We say that two unitaries $U,V\in SU(4)$ are locally equivalent, or $U\sim V$, if $\eta(U)=\eta(V)$. We give the interaction coefficients for some common gates:
\begin{itemize}
    \item $I:\eta(I)=(0,0,0)$;
    \item $\mathrm{SWAP} : \eta(\mathrm{SWAP} )=(\frac{\pi}4,\frac{\pi}4,\frac{\pi}4)$;
    \item $\mathrm{CNOT} $, CZ$: \eta(\mathrm{CNOT} )=\eta(\textrm{CZ})=(\frac{\pi}4,0,0)$. Note that $\mathrm{CNOT} \sim$ CZ by a local Hadamard conjugation on the target qubit;
    \item $\mathrm{\lowercase{i}SWAP} :\eta(\mathrm{\lowercase{i}SWAP} ) = (\frac{\pi}4,\frac{\pi}4,0)$;
    \item $B$~\cite{zhang2004minimum}$:\eta(B)=(\frac{\pi}4,\frac{\pi}8,0)$;
    \item $\mathrm{SQ\lowercase{i}SW} : \eta(\mathrm{SQ\lowercase{i}SW} ) = (\frac{\pi}8,\frac{\pi}8,0)$.
\end{itemize}
These gates and their positions in the Weyl chamber is given in~Fig.~\ref{fig:weyl}.
\begin{dfn}
Let $L(x,y,z) \equiv \exp\left(i [x,y,z]\cdot \Vec{\Sigma}\right)$ be the canonical element of the equivalence class.
\end{dfn}
We have $U\sim L(\eta(U))$ for all $U\in SU(4)$.


\begin{figure}
    \centering
\begin{tikzpicture}[scale=.5]
\def \tta{ 90.000000000000 }
\def \k{    0.30000000000000 }
\def \l{     7.00000000000000 }
\def 2.7{     5.50000000000000 }
\def \h{     7.00000000000000 }
one
\coordinate (I) at (0,0);
\coordinate (Cnot) at (0,{-\h});
\coordinate (iSwap) at ({-\l*sin(\tta))},
                    {-\h+\l*cos(\tta)});
\coordinate (B) at ({-\l*sin(\tta)/2)},
                    {-\h+\l*cos(\tta)/2});
\coordinate (Swap) at ({-\l*sin(\tta)-2.7*sin(\k*\tta)},{-\h+\l*cos(\tta)+2.7*cos(\k*\tta)});
\coordinate (Swapm) at ({-\l*sin(\tta)+2.7*sin(\k*\tta)},{-\h+\l*cos(\tta)-2.7*cos(\k*\tta)});
\coordinate (SQiSW) at ({-\l*sin(\tta))/2},
                    {(-\h+\l*cos(\tta))/2});
\coordinate (b) at ({-\l*sin(\tta))/2-1.9},
                    {(-\h+\l*cos(\tta))/2+0.6});

\draw[-,thick] (Cnot) --  (Swapm)
                        (I) --  (Swap)
                        (I) -- (Swapm)
                        (I) -- (Cnot)
                        (Swap) --  (Swapm)
                        (I) --  (iSwap);

\draw[dashed,thick] (iSwap) --  (Cnot)
                        (Swap)  -- (Cnot);


\fill[black]  (I) circle [radius=2pt];
\fill[black]    (iSwap) circle [radius=2pt];
\fill[black]  (Cnot) circle [radius=2pt];
\fill[black] (Swap) circle [radius=2pt];
\fill[black] (Swapm) circle [radius=2pt];
\fill[black] (B) circle [radius=2pt];
\fill[red] (SQiSW) circle [radius=2pt];

\draw (I) node [right]           {$I (0,0,0)$}
      (Cnot) node [right]     {$\mathrm{CNOT}  (\frac\pi4,0,0)$}
      (Swap) node [left]           {$\mathrm{SWAP} ^\dag (\frac\pi4,\frac\pi4,\frac{-\pi}4)$}
      (Swapm)  node [left]       {$\mathrm{SWAP}  (\frac\pi4,\frac\pi4,\frac\pi4)$}
      (iSwap)  node [left]            {$\mathrm{\lowercase{i}SWAP}  (\frac\pi4, \frac\pi4, 0)$}
      (B) node [below] {$B$}
      (b) node    {$\mathrm{SQ\lowercase{i}SW}  (\frac{\pi}8,\frac\pi8,0)$};

\end{tikzpicture}
\caption{An illustration of the Weyl chamber and the positions of common gates. Note that $\mathrm{SQ\lowercase{i}SW} $ lies in the midpoint of the identity and $\mathrm{\lowercase{i}SWAP} $. The point $\mathrm{SWAP} ^\dag$ is to be identified with the point $\mathrm{SWAP} $ but is drawn separately for easier visualization.}
\label{fig:weyl}
\end{figure}

\subsubsection{Local invariants and the character polynomial} The KAK decomposition geometrically characterizes the equivalence class of a unitary $U\in SU(4)$; however, it requires diagonalization of matrices and thus can sometimes be difficult to study analytically. Local invariants~\cite{makhlin2002nonlocal} characterizes the equivalence class of the unitary $U$, while still being easy to solve. There are many different choices of local invariants. We choose ours to be the degree-4 polynomial
$$F_U(t)=\det[\Re[M^\dag U M] + t\cdot \Im[M^\dag U M]],$$
where $\Re[\cdot]$ and $\Im[\cdot]$ represent the (element-wise) real and imaginary part of a matrix. We call this the \emph{character polynomial}. To see that the polynomial is locally invariant, we first observe $U\sim V\Leftrightarrow \exists O_1, O_2\in SO(4), O_1M^\dag UMO_2=M^\dag VM$. Then
\begin{align*}
    F_U(t)&=\det[\Re[M^\dag U M] + t\cdot \Im[M^\dag U M]]\\
    &=\det[\Re[O_1M^\dag V MO_2] + t\cdot \Im[O_1M^\dag V MO_2]]\\
    &=\det[O_1\Re[M^\dag V M]O_2 + t\cdot O_1\Im[M^\dag V M]O_2]\\
    &=\det[O_1(\Re[M^\dag V M] + t\cdot \Im[M^\dag VM])O_2]\\
    &=\det[O_1]\det[\Re[M^\dag V M] + t\cdot \Im[M^\dag V M]]\det[O_2]\\
    &=\det[\Re[M^\dag V M] + t\cdot \Im[M^\dag V M]]=F_V(t).\\
\end{align*}

The polynomial is a complete characterization of the equivalence classes as the zeros of the polynomial are $-\cot(x-y+z), -\cot(x+y-z), -\cot(-x-y-z)$ and $-\cot(-x+y+z)$ by evaluating it on the canonical element. Hence, one only needs to check the corresponding character polynomial coefficients in order to determine whether two unitaries are locally equivalent. Furthermore,
$$F_U(i) = \det[M^\dag U M] = 1, F_U(-i) = \det[M^\dag U^* M]=1,$$
since $U,U^*\in SU(4)$, leaving the character polynomial with three free coefficients. For $U$ with interaction coefficients $(x, y,z)$, we have
$$F_U(t)=(t^2+1)(Ct^2+Bt+A)-t^2,$$
where
\begin{align}
    A&=\cos(x+y-z)\cos(x-y+z)\cos(-x-y-z)\cos(-x+y+z),\label{eq:charpoly_A}\\
    B&=-\sin2x\sin2y\sin2z,\label{eq:charpoly_B}\\
    C&=\sin(x+y-z)\sin(x-y+z)\sin(-x-y-z)\sin(-x+y+z).\label{eq:charpoly_C}
\end{align}

\subsubsection{Effective target size}
An interesting way to quantify how easy it is to realize a two-qubit gate with quantum control is its effective target size, as put forth in~\cite{watts2013metric}. Intuitively, the effective target size is the invariant volume of the region around a two-qubit gate that corresponds to a small perturbation of its interaction coefficients. We show in this section that the effective target size of $\mathrm{SQ\lowercase{i}SW} $ is larger than that of $\mathrm{CNOT} $ and $\mathrm{\lowercase{i}SWAP} $, having a target size that scales with the perturbation better than any other common two-qubit gate, save for the $B$ gate.

Let $U \in SU(4)$ and its interaction coefficients $\eta(U) = (x,y,z) \in W$. Furthermore, let $\mathcal{U}$ be the neighborhood of $\eta(U)$ given by a box with edge length $a$ centered on $\eta(U)$ and with sides parallel to the $x,y,z$ axes. Then, the effective target size of $U$ is defined as
\begin{align*}
T(U) \equiv \int_{(SU(2) \otimes SU(2)) \times \mathcal{U} \times (SU(2) \otimes SU(2))} d\mu = \int_{\mathcal{U}} d\mu_W
\end{align*}
where $d\mu$ is the Haar measure over $SU(4)$ and
\begin{align}
    d\mu_W&\equiv M_W(x,y,z) dx\land dy\land dz \nonumber\\
    &=\frac{3}\pi[\cos 2x\cos 4y+\cos 2y\cos 4z+\cos 2z\cos 4x-\cos 2x\cos 4z-\cos 2y\cos 4x-\cos 2z\cos 4y ]dx\land dy\land dz \label{eq:haar_weyl}
\end{align}
is the normalized Haar measure over $W$.

We note that the effective target size is the same for mirror gates (differ by a $\mathrm{SWAP} $) since its definition is symmetric under exchanging which qubits we deem as the first and the second. Now, the Weyl coordinates of $\mathrm{SQ\lowercase{i}SW} $  is $(\pi/8, \pi/8, 0)$. Its mirror gate has coordinates (e.g. see~\cite{cross2019validating})
\begin{align*}
    (\pi/4 - 0, \pi/4-\pi/8, 1\times(\pi/8 - \pi/4)) & = (\pi/4, \pi/8, -\pi/8)\\
    & \sim (-\pi/4, \pi/8, -\pi/8)\\
    & \sim (\pi/4, \pi/8,\pi/8),
\end{align*}
where the first equivalence follows by subtracting $\pi/2$ from the first coordinate and the second follows from flipping the signs of the first and third coordinates~\cite{tucci2005introduction}. Hence, the effective target size of $\mathrm{SQ\lowercase{i}SW} $ is the same as that of its mirror gate $(\pi/4, \pi/8, \pi/8)$ which is given~\footnote{Note that the convention in~\cite{watts2013metric} for the Weyl coordinates is $2$ times that of ours. } in (47) of~\cite{watts2013metric}:
\begin{align*}
    T(\mathrm{SQ\lowercase{i}SW} )&  = \frac{1}{2\pi} [3 \cos (a) - 3 \cos( 3a)  - 4a \sin (3a)  ] \\
    & = 4 a^4/\pi + O(a^6) \quad \text{as $a \to 0$.}
\end{align*}
This is a larger area than that of $\mathrm{CNOT} $ and $\mathrm{\lowercase{i}SWAP} $~\cite{watts2013metric}:
\begin{align*}
    T(\mathrm{CNOT} ) & = T(\mathrm{\lowercase{i}SWAP} ) \\
    & =  \frac{1}{2\pi} [8a+7a \cos(3a) -15a \cos(a)-9\sin(3a) +12\sin(2a)+3\sin(a)]\\
    & = 4a^5/\pi + O(a^7) \quad \text{as $a \to 0$.}
\end{align*}
Note that the effective target sizes of $\mathrm{CNOT} $ and $\mathrm{\lowercase{i}SWAP} $ are the same since they are mirror gates up to local equivalence.


\subsection{Compiling two-qubit gates into $\mathrm{SQ\lowercase{i}SW} $ and single-qubit rotations}
We first study the problem of compiling arbitrary two-qubit gates. In particular, we prove the following theorem.
\begin{thm}
Every two-qubit unitary can be expressed by at most 3 $\mathrm{SQ\lowercase{i}SW} $  gates interleaved by single qubit gates.
\end{thm}

The proof of the theorem will consist of two steps. We first completely characterize the set $W(S_2)$ of all two-qubit gates that can be generated using only 2 uses of the $\mathrm{SQ\lowercase{i}SW} $  gate. We second show how to decompose a gate outside of $W(S_2)$ into one use of $\mathrm{SQ\lowercase{i}SW} $  and one use of a gate in $W(S_2)$. This completes the proof. We end by providing an explicit decomposition algorithm.

\subsubsection{Weyl chamber region spanned by two $\mathrm{SQ\lowercase{i}SW} $  gates}

We now study the region in the Weyl chamber that can be generated by two $\mathrm{SQ\lowercase{i}SW} $  gates interleaved with single qubit rotations. This will later help us establish compilation schemes of arbitrary two-qubit gates using $\mathrm{SQ\lowercase{i}SW} $  and single qubit rotations.

\begin{lem}
The Weyl chamber region spanned by two $\mathrm{SQ\lowercase{i}SW} $  gates is the region described by the inequalities $\frac{\pi}{4}\geq x\geq y\geq |z| \land x\geq y+|z|  .$
\label{lem:p}
\end{lem}

\begin{proof}
Denote the subset of the Weyl chamber spanned by two $\mathrm{SQ\lowercase{i}SW} $  gates $W(S_2)$ and $$W'\equiv\left\{\frac{\pi}{4}\geq x\geq y\geq |z| \land x\geq y+|z| \mid (x,y,z)\in W\right\}.$$ The proof proceeds in two steps: We first prove that $W'\subset W(S_2)$ by giving an analytical solution to the interleaving single qubit rotations for a general element in $W'$, and then prove that $W(S_2)\subset W'$ by investigating the character polynomial coefficients associated to a general element in $W(S_2)$.
\begin{description}
\item[$W'\subset W(S_2)$:] We prove this statement constructively by giving explicit analytical forms for the interleaving single qubit rotations: For $(x,y,z)\in W'$, consider the following gate in $W(S_2)$:
$$U(\alpha, \beta, \gamma)\equiv S\cdot\left(\begin{bmatrix}e^{i\gamma}\cos\alpha/2 & i\sin\alpha/2 \\ i\sin\alpha/2 & e^{-i\gamma}\cos\alpha/2\end{bmatrix}\otimes \begin{bmatrix}\cos\beta/2 & i\sin\beta/2 \\ i\sin\beta/2 & \cos\beta/2 \end{bmatrix}\right)\cdot S,$$
where
\begin{align}
    \alpha &= \arccos\left(\cos 2x-\cos 2y+\cos 2z+2\sqrt{C}\right), \label{eq:alpha_xyz}\\
    \beta &= \arccos\left(\cos 2x-\cos 2y+\cos 2z-2\sqrt{C}\right),\\
    \gamma &= \arccos\left(\mathrm{sgn} z\cdot\sqrt{\frac{4\cos^2x\cos^2z\sin^2y}{4\cos^2x\cos^2z\sin^2y+\cos2x\cos2y\cos2z}}\right).
\end{align}
Here we define $\mathrm{sgn}(z)=1$ if $z\geq 0$ and is otherwise $-1$.
Note that $C$ in terms of $x,y,z$ was given in~\cref{eq:charpoly_C}. One can verify that all operations including the square root and the inverse cosine functions are legal when $(x,y,z)\in W'$, and one can also verify that the interaction coefficient associated to $U(\alpha, \beta, \gamma)$ is indeed $(x,y,z)$ by comparing the coefficients in the character polynomial.
\item[$W(S_2)\subset W'$:] Up to local equivalence, a general element in $W(S_2)$ can be parameterized by six parameters:
$$
         U(\alpha, \beta,\gamma_1, \gamma_2, \delta_1,\delta_2)
        =  S\cdot \left(
        \begin{bmatrix}
        e^{i\gamma_1}\cos \alpha  & e^{i\delta_1}\sin\alpha \\
        -e^{-i\delta_1}\sin\alpha & e^{-i\gamma_1}\cos \alpha
        \end{bmatrix}\otimes
        \begin{bmatrix}
        e^{i\gamma_2}\cos\beta  &  e^{i\delta_2}\sin\beta\\
        -e^{-i\delta_2}\sin\beta & e^{-i\gamma_2}\cos\beta
        \end{bmatrix}\right)\cdot S,$$
where $S$ is shorthand for SQiSW. The corresponding coefficient $C$ in the character polynomial associated to it is then
\begin{align}
    C & =\frac{1}{16}(\cos2\alpha-\cos2\beta)^2.
    \label{eq:c_positive}
\end{align}
Combining~\cref{eq:c_positive} with with~\cref{eq:charpoly_C}, we have
$$C = \sin(x+y-z)\sin(x-y+z)\sin(-x-y-z)\sin(-x+y+z)\geq 0,$$
where $(x,y,z)=\eta(U(\alpha, \beta,\gamma_1, \gamma_2, \delta_1,\delta_2))$. Since $(x,y,z)\in W$ ensures that $\sin(x+y-z),\sin(x+y+z)\geq 0$, we know that $$\sin(x-y+z)\sin(x-y-z)\geq 0\Rightarrow |z|\leq x-y$$
when $\frac{\pi}4\geq x\geq y\geq |z|\geq0$.
Combining this constraint with the ones from $W$ gives us $$\frac{\pi}{4}\geq x\geq y\geq |z| \land x\geq y+|z| \Rightarrow W(S_2)\subset W'.$$

\end{description}
\end{proof}

In~Fig.~\ref{fig:s2_region} we show the region $W' \subset W$. We also show a schematic of how an element of $W(S_2)$ can be decomposed in~Fig.~\ref{fig:red}.

\begin{figure}
    \centering
\begin{tikzpicture}[scale=.5]
\def \tta{ 90.000000000000 }
\def \k{    0.30000000000000 }
\def \l{     7.00000000000000 }
\def 2.7{     5.00000000000000 }
\def \h{     7.0000000000000 }

\coordinate (I) at (0,0);
\coordinate (Cnot) at (0,{-\h});
\coordinate (iSwap) at ({-\l*sin(\tta))},
                    {-\h+\l*cos(\tta)});
\coordinate (Swap) at ({-\l*sin(\tta)-2.7*sin(\k*\tta)},{-\h+\l*cos(\tta)+2.7*cos(\k*\tta)});
\coordinate (Swapm) at ({-\l*sin(\tta)+2.7*sin(\k*\tta)},{-\h+\l*cos(\tta)-2.7*cos(\k*\tta)});
\coordinate (SQiSW) at ({-\l*sin(\tta))/2},
                    {(-\h+\l*cos(\tta))/2});
\coordinate (mSQiSWm) at ({(-\l*sin(\tta)-2.7*sin(\k*\tta))/2},{-\h+(\l*cos(\tta)+2.7*cos(\k*\tta))/2});

\coordinate (m) at ({(-\l*sin(\tta)-2.7*sin(\k*\tta))/2+0.4},{-\h+(\l*cos(\tta)+2.7*cos(\k*\tta))/2-1.2});
\coordinate (mSQiSW) at ({(-\l*sin(\tta)+2.7*sin(\k*\tta))/2},{-\h+(\l*cos(\tta)-2.7*cos(\k*\tta))/2});


 \filldraw[color=red, fill opacity = 0.5]
  (I) -- (Cnot) -- (mSQiSWm);
  \filldraw[color=red, fill opacity = 0.5]
  (I) -- (iSwap) -- (mSQiSWm);
  \filldraw[color=red, fill opacity = 0.5]
  (I) -- (iSwap) -- (mSQiSW);
\filldraw[color=red, fill opacity = 0.5]
  (I) -- (Cnot) -- (mSQiSW);
 \filldraw[color=red, fill opacity = 0.5]
  (mSQiSWm) -- (Cnot) -- (mSQiSW) -- (iSwap);

\draw[-,thick] (Cnot) --  (Swapm)
                        (I) --  (Swap)
                        (I) -- (Swapm)
                        (I) -- (Cnot)
                        (Swap) --  (Swapm)
                        (I) --  (iSwap);

\draw[dashed,thick] (iSwap) --  (Cnot)
                        (Swap)  -- (Cnot);

\draw[dotted, thick] (iSwap) --  (mSQiSW)
                        (iSwap)  -- (mSQiSWm)
                        (I) -- (mSQiSW)
                        (I) -- (mSQiSWm);


\fill[black]  (I) circle [radius=2pt];
\fill[black]    (iSwap) circle [radius=2pt];
\fill[black]  (Cnot) circle [radius=2pt];
\fill[black] (Swap) circle [radius=2pt];
\fill[black] (Swapm) circle [radius=2pt];
\fill[black] (mSQiSW) circle [radius=2pt];
\fill[black] (mSQiSWm) circle [radius=2pt];
\fill[red] (SQiSW) circle [radius=2pt];

\draw (I) node [right]           {$I$}
      (Cnot) node [right]     {$\mathrm{CNOT} $}
      (Swap) node [left]           {$\mathrm{SWAP} ^\dag$}
      (Swapm)  node [left]       {$\mathrm{SWAP} $}
      (iSwap)  node [left]            {$\mathrm{\lowercase{i}SWAP} $}
      (SQiSW) node [above left]           {$\mathrm{SQ\lowercase{i}SW} $}
      (mSQiSW) node [right] {$(\frac{\pi}4,\frac{\pi}8,\frac{\pi}8)$}
      (m) node {$(\frac{\pi}4,\frac{\pi}8,\frac{-\pi}8)$};
\end{tikzpicture}
\caption{The region $W'=W(S_2)$ spanned by 2 $\mathrm{SQ\lowercase{i}SW} $ gates. It is a pyramid with vertices $I$, $\mathrm{CNOT} $, $(\frac{\pi}{4},\frac{\pi}{8},\frac{\pi}{8})$, $\mathrm{\lowercase{i}SWAP} $, and $(\frac{\pi}{4},\frac{\pi}{8},-\frac{\pi}{8})$.}
\label{fig:s2_region}
\end{figure}

\begin{figure}
    \begin{tikzpicture}
    \centering
\node {
    \begin{quantikz}
             & \gate[style={fill=red!50}, 2]{U\in S_2} &  \qw \\
             & \qw & \qw
             \end{quantikz}
             $\sim$\begin{quantikz}
              & \gate[2]{\mathrm{SQ\lowercase{i}SW} } & \gate[1]{R_z(\gamma)R_x(\alpha)R_z(\gamma)}   &\gate[2]{\mathrm{SQ\lowercase{i}SW} }  &\qw  \\
              & \qw & \gate[1]{R_x(\beta)} & \qw&\qw
             \end{quantikz}};
             \end{tikzpicture}
    \caption{Illustration of decomposition of a two-qubit gate in $W(S_2)$ into two $\mathrm{SQ\lowercase{i}SW} $  gates up to local equivalence. The special form of the interleaving single qubit gates are due to the proof that the three parameters $\alpha, \beta, \gamma$ running over $[0,2\pi]$ are sufficient to generate the whole region $W'$.}
    \label{fig:red}
\end{figure}

\subsubsection{Decomposing arbitrary two-qubit gate into $\leq 3$ $\mathrm{SQ\lowercase{i}SW} $  gates}

We now consider unitaries whose interaction coefficients lie outside of the region $W'$. Those gates include the SWAP family $(x,x,\pm x)$, the Sycamore fSim gates and so on. We show below that a third $\mathrm{SQ\lowercase{i}SW} $  gate is sufficient to span the whole Weyl chamber.


Given that all gates in the Weyl chamber region $W'$ can be generated using 2 $\mathrm{SQ\lowercase{i}SW} $  gates by Lemma~\ref{lem:p}, it suffices to prove the following.
\begin{lem}
For all $(x,y,z)\in W\setminus W'$, $L(x,y,z)$ can be generated with one use of some $L(x',y',z')$ and one $L(\frac{\pi}8,\frac{\pi}8,0)\sim \mathrm{SQ\lowercase{i}SW} $, where $(x',y',z')\in W'$.
\end{lem}
\begin{proof}

Before proceeding to the proof, we first visualize the constraints imposed by region $W$ and $W'$ in terms of the eigenphases $\{a_0,a_1,a_2,a_3\}$ of $L(x,y,z)$, where
\begin{align*}
    a_0 &= x+y-z,\\
    a_1 &= x-y+z,\\
    a_2 &= -x+y+z,\\
    a_3 &= -x-y-z.
\end{align*}
The constraint that $(x,y,z)\in W$ can be equivalently stated as
$$a_0\geq a_1\geq a_2\geq a_3, \sum_ia_i=0, a_0+a_1\leq \frac{\pi}2.$$
It can be deduced that $a_0\geq 0\geq a_3$. $(x,y,z)\in W'$ imposes an additional constraint:
$$a_0\geq a_1\geq{\color{red}0}\geq  a_2\geq a_3, \sum_ia_i=0, a_0+a_1\leq \frac{\pi}2.$$

Assuming that $z= \frac12(a_1+a_2) \geq  0$ (the other case can be reduced to this by observing that $\mathrm{SQ\lowercase{i}SW} \sim \mathrm{SQ\lowercase{i}SW}  ^\dag$ and $L(x,y,z)\sim L^\dag(x,y,-z)$), $(x,y,z)\in W\setminus W'$ indicates that this additional constraint is violated via the sign violation $a_2>0$. We show that the following is true: we can always select $a_i, a_j, i\neq j$, append on them phases $\frac{\pi}4,-\frac{\pi}4$ such that $b_0\geq b_1\geq b_2\geq b_3$ being the sorted permutation of $(a_i+\frac{\pi}{4}, a_j-\frac{\pi}{4}, a_k, a_l), \{i,j,k,l\}\in\{0,1,2,3\}$ satisfies
$$b_0\geq b_1\geq0\geq  b_2\geq b_3, \sum_ib_i=0, b_0+b_1\leq \frac{\pi}2.$$

This indicates that there is a way of decomposing $L(x,y,z)$ to $L(x',y',z')$ associated to the eigenphases $(b_0,b_1,b_2,b_3)$ and $L(\frac{\pi}8,\frac{\pi}8,0)$. Explicitly, we argue via the following two cases. We also give a visual argument in~Fig.~\ref{fig:visual_arg}.

\begin{enumerate}
    \item $x\leq \frac{\pi}8$. Then $a_2\leq x\leq \frac{\pi}{8}$. One can take $$\{b_0,b_1\}=\left\{a_0+\frac{\pi}4, a_1\right\}, \{b_2,b_3\}=\textrm{sort}\left\{a_2-\frac{\pi}{4},a_3\right\},$$
    where ``$\textrm{sort}$'' means the set is sorted in descending order. One has $a_0+\frac{\pi}{4}\geq  a_1\geq 0, a_2-\frac{\pi}4, a_3\leq 0, a_0+a_1+\frac{\pi}4=2x+\frac{\pi}4\leq\frac{\pi}2$.
    \item $x>\frac{\pi}8.$ Then $a_3=-2x-a_2<-\frac{\pi}4$ and $a_2\leq x\leq \frac{\pi}{4}$. One can take
     $$\{b_0,b_1\}=\left\{a_0, a_1\right\}, \{b_2,b_3\}=\textrm{sort}\left\{a_2-\frac{\pi}4,a_3+\frac{\pi}4\right\}.$$
     One has $a_0\geq a_1\geq 0, a_2-\frac{\pi}4, a_3+\frac{\pi}4\leq 0, a_0+a_1\leq\frac{\pi}2$.
\end{enumerate}
Note that when an \emph{eigenphase crossing} happens, i.e.\ that $a_2-\frac\pi4<a_3$ in case 1 and $a_2-\frac\pi4<a_3+\frac\pi4$ in case 2, additional single qubit gates need to be applied to switch the two-qubit unitary to the canonical form for compilation purposes, see Algorithm~\ref{alg:decomp}.
\begin{figure}
\begin{subfigure}{\textwidth}
\centering
    \begin{tikzpicture}[scale=1,cap=round,>=latex]
    \def 2.{3.0}
    \def 2{2}
    \def 4.0{4.0}
    \def 2.7{4.0}
    \def 25.0{20.0}
    \coordinate (x) at ({2.*cos(25.0)}, {2.*sin(25.0)});
    \coordinate (mx) at ({2.*cos(25.0)}, {-2.*sin(25.0)});
    \coordinate (a0) at ({2.*cos(35)}, {2.*sin(35)});
    \coordinate (a1) at ({2.*cos(5)}, {2.*sin(5)});
    \coordinate (a2) at ({2.*cos(10)}, {-2.*sin(10)});
    \coordinate (a3) at ({2.*cos(30)}, {-2.*sin(30)});
    \coordinate (pi) at ({2.*cos(45)}, {2.*sin(45)});
    \coordinate (mpi) at ({2.*cos(45)}, {-2.*sin(45)});
       
        \draw[->] (-2.7,0) -- (2.7,0) node[right,fill=white] {$0$};
        \draw[->] (0,-2.7) -- (0,2.7) node[above,fill=white] {$\frac\pi2$};

       
        \draw[thick] (0,0) circle(2.);
        \draw[gray, dotted, thick] (0,0) -- (2.7,2.7);
        \draw[gray, dotted, thick] (0,0) -- (2.7,-2.7);
        \draw (0,0) -- (a0);
        \draw (0,0) -- (a1);
        \draw (0,0) -- (a2);
        \draw (0,0) -- (a3);
        \draw[gray, dashed, thick] (0,0) -- (2.7,{2.7*tan(25.0)});
        \draw[gray, dashed, thick] (0,0) -- (2.7,{-2.7*tan(25.0)});
        \fill[black]  (x) circle [radius=1pt];

        \fill[black]  (mx) circle [radius=1pt];
        \fill[black]  (a0) circle [radius=2pt];
        \fill[black]  (a1) circle [radius=2pt];
        \fill[black]  (a2) circle [radius=2pt];
        \fill[black]  (a3) circle [radius=2pt];

        \draw (a0) node [right]     {$a_0$}
      (a1) node [right]           {$a_1$}
      (a2)  node [right]       {$a_2$}
      (a3)  node [right]            {$a_3$}
      ({4.0*cos(25.0)},{4.0*sin(25.0)}) node [fill=white]           {$x$}
      ({4.0*cos(25.0)},{-4.0*sin(25.0)}) node [fill=white] {$-x$}
      ({4.0*cos(45)},{4.0*sin(45)}) node [fill=white] {$\frac{\pi}4$}
      ({4.0*cos(45)},{-4.0*sin(45)}) node [ fill=white] {$-\frac\pi4$};

    \draw[very thick] ({cos(5)},{sin(5)}) arc [start angle=5, end angle=35, radius=1];
    \draw[very thick] ({0.95*cos(10)},{-0.95*sin(10)}) arc [start angle=-10, end angle=-30, radius=0.95];
    \draw[very thick] ({1.05*cos(10)},{-1.05*sin(10)}) arc [start angle=-10, end angle=-30, radius=1.05];
    \end{tikzpicture}
    \caption{Eigenphases without sign violation of $a_2$.}
    \label{fig:case0}
    \end{subfigure}
     \begin{subfigure}{0.45\textwidth}
    \centering
    \begin{tikzpicture}[cap=round,>=latex]
    \def 2.{2}
    \def 2{2}
    \def 4.0{4.0}
    \def 2.7{2.7}
    \def 25.0{20.0}
   
   
    \coordinate (a0) at ({2.*cos(25)}, {2.*sin(25)});
    \coordinate (a0p) at ({2.*cos(70)}, {2.*sin(70)});
    \coordinate (a1) at ({2.*cos(15)}, {2.*sin(15)});
    \coordinate (a2) at ({2.*cos(10)}, {2.*sin(10)});
    \coordinate (a2p) at ({2.*cos(35)}, {-2.*sin(35)});
    \coordinate (a3) at ({2.*cos(50)}, {-2.*sin(50)});
    \coordinate (pi) at ({2.*cos(45)}, {2.*sin(45)});
    \coordinate (mpi) at ({2.*cos(45)}, {-2.*sin(45)});
       
        \draw[->] (-2.7,0) -- (2.7,0) node[right,fill=white] {$0$};
        \draw[->] (0,-2.7) -- (0,2.7) node[above,fill=white] {$\frac\pi2$};

       
        \draw[thick] (0,0) circle(2.);
        \draw[cyan] (0,0) -- (a0);
        \draw[cyan, dashed] (0,0) -- (a0p);
        \draw (0,0) -- (a1);
        \draw[red] (0,0) -- (a2);
        \draw[red, dashed] (0,0) -- (a2p);
        \draw (0,0) -- (a3);
        \fill[cyan]  (a0) circle [radius=2pt];
        \fill[cyan]  (a0p) circle [radius=2pt];
        \fill[black]  (a1) circle [radius=2pt];
        \fill[red]  (a2) circle [radius=2pt];
        \fill[red]  (a2p) circle [radius=2pt];
        \fill[black]  (a3) circle [radius=2pt];

        \draw (a0) node [right]     {\color{cyan} $a_0$}
      (a0p) node [right]     {\color{cyan} $a'_0$}
      (a1) node [right]           {$a_1$}
      (a2)  node [right]       {\color{red} $a_2$}
      (a2p)  node [right]       {\color{red} $a'_2$}
      (a3)  node [right]            {$a_3$};
    \draw[->, cyan, very thick] ({(2.+0.6)*cos(25)},{(2.+0.6)*sin(25)}) arc [start angle=25, end angle=70, radius={2.+0.5}];
    \draw[->, red, very thick] ({(2.+0.5)*cos(10)},{(2.+0.5)*sin(10)}) arc [start angle=10, end angle=-35, radius={2.+0.5}];
    \end{tikzpicture}
    \caption{$a_2$ has a sign violation and $x\leq \frac{\pi}8$.}
    \label{fig:case1}
    \end{subfigure}
    ~
        \begin{subfigure}{.45\textwidth}
    \centering
                \begin{tikzpicture}[cap=round,>=latex]
    \def 2.{2.}
    \def 2{2}
    \def 4.0{4.0}
    \def 2.7{2.7}
    \def 25.0{25.0}
    \coordinate (x) at ({2.*cos(25.0)}, {2.*sin(25.0)});
    \coordinate (mx) at ({2.*cos(25.0)}, {-2.*sin(25.0)});
    \coordinate (a0) at ({2.*cos(30)}, {2.*sin(30)});
    \coordinate (a1) at ({2.*cos(20)}, {2.*sin(20)});
    \coordinate (a2) at ({2.*cos(10)}, {2.*sin(10)});
    \coordinate (a2p) at ({2.*cos(35)}, {-2.*sin(35)});
    \coordinate (a3) at ({2.*cos(60)}, {-2.*sin(60)});
    \coordinate (a3p) at ({2.*cos(15)}, {-2.*sin(15)});
    \coordinate (pi) at ({2.*cos(45)}, {2.*sin(45)});
    \coordinate (mpi) at ({2.*cos(45)}, {-2.*sin(45)});
       
        \draw[->] (-2.7,0) -- (2.7,0) node[right,fill=white] {$0$};
        \draw[->] (0,-2.7) -- (0,2.7) node[above,fill=white] {$\frac\pi2$};

       
        \draw[thick] (0,0) circle(2.);
        \draw (0,0) -- (a0);
        \draw[cyan, dashed] (0,0) -- (a3p);
        \draw (0,0) -- (a1);
        \draw[red] (0,0) -- (a2);
        \draw[red, dashed] (0,0) -- (a2p);
        \draw[cyan] (0,0) -- (a3);
        \fill[cyan]  (a3) circle [radius=2pt];
        \fill[cyan]  (a3p) circle [radius=2pt];
        \fill[black]  (a1) circle [radius=2pt];
        \fill[red]  (a2) circle [radius=2pt];
        \fill[red]  (a2p) circle [radius=2pt];
        \fill[black]  (a0) circle [radius=2pt];

        \draw (a0) node [right]     {$a_0$}
      (a3p) node [right]     {\color{cyan} $a'_3$}
      (a1) node [right]           {$a_1$}
      (a2)  node [right]       {\color{red} $a_2$}
      (a2p)  node [right]       {\color{red} $a'_2$}
      (a3)  node [right]            {\color{cyan} $a_3$};
    \draw[->, cyan, very thick] ({(2.+0.6)*cos(60)},{-(2.+0.6)*sin(60)}) arc [start angle=-60, end angle=-15, radius={2.+0.8}];
    \draw[->, red, very thick] ({(2.+0.5)*cos(10)},{(2.+0.5)*sin(10)}) arc [start angle=10, end angle=-35, radius={2.+0.5}];
    \end{tikzpicture}
    \caption{$a_2$ has a sign violation and $x> \frac{\pi}8$.}
    \label{fig:case2}
    \end{subfigure}
    \caption{Illustration of the eigenphases $a_0\geq a_1\geq a_2\geq a_3$. Being in $W$ requires that $x\leq\frac{\pi}{4}$, and $a_0, a_1$ and $a_2, a_3$ lie symmetrically with respect to $x$ and $-x$ respectively.~Fig.~\ref{fig:case0} corresponds to no sign violation of $a_2$ and hence can be generated using 2 $\mathrm{SQ\lowercase{i}SW} $  gates.~Fig.~\ref{fig:case1} and~Fig.~\ref{fig:case2} are possible value assignments corresponding to the two eigenphase modifications corresponding to case 1 and 2 in the proof, respectively. }
    \label{fig:visual_arg}
\end{figure}

\begin{figure}
    \centering

        \begin{subfigure}{0.3\textwidth}
    \centering
\begin{tikzpicture}[scale=0.5]
\def \tta{ 90.000000000000 }
\def \k{    0.30000000000000 }
\def \l{     7.00000000000000 }
\def 2.7{     5.00000000000000 }
\def \h{     7.0000000000000 }

\coordinate (I) at (0,0);
\coordinate (Cnot) at (0,{-\h});
\coordinate (iSwap) at ({-\l*sin(\tta))},
                    {-\h+\l*cos(\tta)});
\coordinate (Swap) at ({-\l*sin(\tta)-2.7*sin(\k*\tta)},{-\h+\l*cos(\tta)+2.7*cos(\k*\tta)});
\coordinate (Swapm) at ({-\l*sin(\tta)+2.7*sin(\k*\tta)},{-\h+\l*cos(\tta)-2.7*cos(\k*\tta)});
\coordinate (SQiSW) at ({-\l*sin(\tta))/2},
                    {(-\h+\l*cos(\tta))/2});
\coordinate (mSQiSWm) at ({(-\l*sin(\tta)-2.7*sin(\k*\tta))/2},{-\h+(\l*cos(\tta)+2.7*cos(\k*\tta))/2});
\coordinate (mSQiSW) at ({(-\l*sin(\tta)+2.7*sin(\k*\tta))/2},{-\h+(\l*cos(\tta)-2.7*cos(\k*\tta))/2});
\coordinate (SQSW) at ({(-\l*sin(\tta)+2.7*sin(\k*\tta))/2},{(-\h+\l*cos(\tta)-2.7*cos(\k*\tta))/2});
\coordinate (SQSWm) at ({(-\l*sin(\tta)-2.7*sin(\k*\tta))/2},{(-\h+\l*cos(\tta)+2.7*cos(\k*\tta))/2});
\coordinate (mSQSW) at ({(-\l*sin(\tta)+2.7*sin(\k*\tta))/4},{(-\h+\l*cos(\tta)-2.7*cos(\k*\tta))/4});
\coordinate (mSQSWm) at ({(-\l*sin(\tta)-2.7*sin(\k*\tta))/4},{(-\h+\l*cos(\tta)+2.7*cos(\k*\tta))/4});
\coordinate (mmSQiSWm) at ({(-\l*sin(\tta)-2.7*sin(\k*\tta))/4},{-\h/2+(\l*cos(\tta)+2.7*cos(\k*\tta))/4});
\coordinate (mmSQiSW) at ({(-\l*sin(\tta)+2.7*sin(\k*\tta))/4},{-\h/2+(\l*cos(\tta)-2.7*cos(\k*\tta))/4});

\filldraw[color=green, fill opacity=0.5]
(SQSWm) -- (SQiSW) -- (mmSQiSWm);
\filldraw[color=green, fill opacity=0.5]
(SQSWm) -- (SQiSW) -- (iSwap) -- (Swap);
\filldraw[color=green, fill opacity=0.5]
(iSwap) -- (SQiSW) -- (mmSQiSWm) -- (mSQiSWm);
\filldraw[color=green, fill opacity=0.5]
(SQSWm) -- (Swap) -- (mSQiSWm) --  (mmSQiSWm);
\filldraw[color=green, fill opacity=0.5]
(Swap) -- (iSwap) -- (mSQiSWm);

 \filldraw[color=red, fill opacity = 0.5]
  (mSQiSWm) -- (Cnot) -- (mSQiSW) -- (iSwap);
 \filldraw[color=red, fill opacity = 0.5]
  (I) -- (Cnot) -- (mSQiSWm);
  \filldraw[color=red, fill opacity = 0.5]
  (I) -- (iSwap) -- (mSQiSWm);
  \filldraw[color=red, fill opacity = 0.5]
  (I) -- (iSwap) -- (mSQiSW);
\filldraw[color=red, fill opacity = 0.5]
  (I) -- (Cnot) -- (mSQiSW);

 \filldraw[color=green, fill opacity=0.5]
(SQSW) -- (SQiSW) -- (mmSQiSW);
\filldraw[color=green, fill opacity=0.5]
(SQSW) -- (SQiSW) -- (iSwap) -- (Swapm);
\filldraw[color=green, fill opacity=0.5]
(iSwap) -- (SQiSW) -- (mmSQiSW) -- (mSQiSW);
\filldraw[color=green, fill opacity=0.5]
(SQSW) -- (Swapm) -- (mSQiSW) --  (mmSQiSW);
\filldraw[color=green, fill opacity=0.5]
(Swapm) -- (iSwap) -- (mSQiSW);

\draw[-,thick] (Cnot) --  (Swapm)
                        (I) --  (Swap)
                        (I) -- (Swapm)
                        (I) -- (Cnot)
                        (Swap) --  (Swapm)
                        (I) --  (iSwap);

\draw[dashed,thick] (iSwap) --  (Cnot)
                        (Swap)  -- (Cnot);

\draw[dotted, thick] (iSwap) --  (mSQiSW)
                        (iSwap)  -- (mSQiSWm)
                        (I) -- (mSQiSW)
                        (I) -- (mSQiSWm)
                        (SQiSW) -- (SQSW)
                        (SQiSW) -- (SQSWm)
                        (SQiSW) -- (mmSQiSW)
                        (SQiSW) -- (mmSQiSWm)
                        (SQSW) -- (mmSQiSW)
                        (SQSWm) -- (mmSQiSWm);


\fill[black]  (I) circle [radius=2pt];
\fill[black]    (iSwap) circle [radius=2pt];
\fill[black]  (Cnot) circle [radius=2pt];
\fill[black] (Swap) circle [radius=2pt];
\fill[black] (Swapm) circle [radius=2pt];
\fill[black] (mSQiSW) circle [radius=2pt];
\fill[black] (mSQiSWm) circle [radius=2pt];
\fill[black] (mmSQiSW) circle [radius=2pt];
\fill[black] (mmSQiSWm) circle [radius=2pt];
\fill[red] (SQiSW) circle [radius=2pt];

\end{tikzpicture}
\caption{$x>\frac\pi8$.}
    \end{subfigure}~
    \begin{subfigure}{0.3\textwidth}
    \centering
\begin{tikzpicture}[scale=0.5]
\def \tta{ 90.000000000000 }
\def \k{    0.30000000000000 }
\def \l{     7.00000000000000 }
\def 2.7{     5.00000000000000 }
\def \h{     7.0000000000000 }

\coordinate (I) at (0,0);
\coordinate (Cnot) at (0,{-\h});
\coordinate (iSwap) at ({-\l*sin(\tta))},
                    {-\h+\l*cos(\tta)});
\coordinate (Swap) at ({-\l*sin(\tta)-2.7*sin(\k*\tta)},{-\h+\l*cos(\tta)+2.7*cos(\k*\tta)});
\coordinate (Swapm) at ({-\l*sin(\tta)+2.7*sin(\k*\tta)},{-\h+\l*cos(\tta)-2.7*cos(\k*\tta)});
\coordinate (SQiSW) at ({-\l*sin(\tta))/2},
                    {(-\h+\l*cos(\tta))/2});
\coordinate (mSQiSWm) at ({(-\l*sin(\tta)-2.7*sin(\k*\tta))/2},{-\h+(\l*cos(\tta)+2.7*cos(\k*\tta))/2});
\coordinate (mSQiSW) at ({(-\l*sin(\tta)+2.7*sin(\k*\tta))/2},{-\h+(\l*cos(\tta)-2.7*cos(\k*\tta))/2});
\coordinate (SQSW) at ({(-\l*sin(\tta)+2.7*sin(\k*\tta))/2},{(-\h+\l*cos(\tta)-2.7*cos(\k*\tta))/2});
\coordinate (SQSWm) at ({(-\l*sin(\tta)-2.7*sin(\k*\tta))/2},{(-\h+\l*cos(\tta)+2.7*cos(\k*\tta))/2});
\coordinate (mSQSW) at ({(-\l*sin(\tta)+2.7*sin(\k*\tta))/4},{(-\h+\l*cos(\tta)-2.7*cos(\k*\tta))/4});
\coordinate (mSQSWm) at ({(-\l*sin(\tta)-2.7*sin(\k*\tta))/4},{(-\h+\l*cos(\tta)+2.7*cos(\k*\tta))/4});


  \filldraw[color=cyan, fill opacity = 0.5]
  (I) -- (iSwap) -- (Swap);
    \filldraw[color=cyan, fill opacity = 0.5]
  (I) -- (iSwap) -- (mSQiSWm);
    \filldraw[color=cyan, fill opacity = 0.5]
  (I) -- (mSQiSWm) -- (Swap);
    \filldraw[color=cyan, fill opacity = 0.8]
  (mSQiSWm) -- (iSwap) -- (Swap);


 \filldraw[color=red, fill opacity = 0.5]
  (mSQiSWm) -- (Cnot) -- (mSQiSW) -- (iSwap);
 \filldraw[color=red!60, fill opacity = 0.5]
  (I) -- (Cnot) -- (mSQiSWm);
  \filldraw[color=red, fill opacity = 0.5]
  (I) -- (iSwap) -- (mSQiSWm);
  \filldraw[color=red, fill opacity = 0.5]
  (I) -- (iSwap) -- (mSQiSW);
\filldraw[color=red, fill opacity = 0.5]
  (I) -- (Cnot) -- (mSQiSW);

\draw[-,thick] (Cnot) --  (Swapm)
                        (I) --  (Swap)
                        (I) -- (Swapm)
                        (I) -- (Cnot)
                        (Swap) --  (Swapm)
                        (I) --  (iSwap);

\draw[dashed,thick] (iSwap) --  (Cnot)
                        (Swap)  -- (Cnot);

\draw[dotted, thick]
                        (iSwap) --  (mSQiSW)
                        (iSwap)  -- (mSQiSWm)
                        (I) -- (mSQiSW)
                        (I) -- (mSQiSWm);


\fill[black]  (I) circle [radius=2pt];
\fill[black]    (iSwap) circle [radius=2pt];
\fill[black]  (Cnot) circle [radius=2pt];
\fill[black] (Swap) circle [radius=2pt];
\fill[black] (Swapm) circle [radius=2pt];
\fill[red] (SQiSW) circle [radius=2pt];

\end{tikzpicture}
\caption{$z<0$.}
    \end{subfigure}~
    \begin{subfigure}{0.3\textwidth}
    \centering
\begin{tikzpicture}[scale=0.5]
\def \tta{ 90.000000000000 }
\def \k{    0.30000000000000 }
\def \l{     7.00000000000000 }
\def 2.7{     5.00000000000000 }
\def \h{     7.0000000000000 }

\coordinate (I) at (0,0);
\coordinate (Cnot) at (0,{-\h});
\coordinate (iSwap) at ({-\l*sin(\tta))},
                    {-\h+\l*cos(\tta)});
\coordinate (Swap) at ({-\l*sin(\tta)-2.7*sin(\k*\tta)},{-\h+\l*cos(\tta)+2.7*cos(\k*\tta)});
\coordinate (Swapm) at ({-\l*sin(\tta)+2.7*sin(\k*\tta)},{-\h+\l*cos(\tta)-2.7*cos(\k*\tta)});
\coordinate (SQiSW) at ({-\l*sin(\tta))/2},
                    {(-\h+\l*cos(\tta))/2});
\coordinate (mSQiSWm) at ({(-\l*sin(\tta)-2.7*sin(\k*\tta))/2},{-\h+(\l*cos(\tta)+2.7*cos(\k*\tta))/2});
\coordinate (mSQiSW) at ({(-\l*sin(\tta)+2.7*sin(\k*\tta))/2},{-\h+(\l*cos(\tta)-2.7*cos(\k*\tta))/2});
\coordinate (SQSW) at ({(-\l*sin(\tta)+2.7*sin(\k*\tta))/2},{(-\h+\l*cos(\tta)-2.7*cos(\k*\tta))/2});
\coordinate (SQSWm) at ({(-\l*sin(\tta)-2.7*sin(\k*\tta))/2},{(-\h+\l*cos(\tta)+2.7*cos(\k*\tta))/2});
\coordinate (mSQSW) at ({(-\l*sin(\tta)+2.7*sin(\k*\tta))/4},{(-\h+\l*cos(\tta)-2.7*cos(\k*\tta))/4});
\coordinate (mSQSWm) at ({(-\l*sin(\tta)-2.7*sin(\k*\tta))/4},{(-\h+\l*cos(\tta)+2.7*cos(\k*\tta))/4});
\coordinate (mmSQiSWm) at ({(-\l*sin(\tta)-2.7*sin(\k*\tta))/4},{-\h/2+(\l*cos(\tta)+2.7*cos(\k*\tta))/4});
\coordinate (mmSQiSW) at ({(-\l*sin(\tta)+2.7*sin(\k*\tta))/4},{-\h/2+(\l*cos(\tta)-2.7*cos(\k*\tta))/4});



\filldraw[color=violet, fill opacity=0.5]
(I) -- (SQiSW) -- (mmSQiSWm);
\filldraw[color=violet, fill opacity=0.5]
(I) -- (mSQSWm) -- (mmSQiSWm);
\filldraw[color=violet, fill opacity=0.5]
(mSQSWm) -- (SQiSW) -- (mmSQiSWm);
\filldraw[color=violet, fill opacity=0.5]
(SQSWm) -- (SQiSW) -- (mmSQiSWm);
\filldraw[color=violet, fill opacity=0.5]
(SQSWm) -- (iSwap) -- (mSQiSWm);
\filldraw[color=violet, fill opacity=0.5]
(SQSWm) -- (mmSQiSWm) -- (mSQiSWm);
\filldraw[color=violet, fill opacity=0.5]
(SQSWm) -- (mSQiSWm) -- (mmSQiSWm);
\filldraw[color=violet, fill opacity=0.5]
(iSwap) -- (SQiSW) -- (mmSQiSWm) -- (mSQiSWm);

 \filldraw[color=red, fill opacity = 0.5]
  (mSQiSWm) -- (Cnot) -- (mSQiSW) -- (iSwap);
 \filldraw[color=red, fill opacity = 0.5]
  (I) -- (Cnot) -- (mSQiSWm);
  \filldraw[color=red, fill opacity = 0.5]
  (I) -- (iSwap) -- (mSQiSWm);
  \filldraw[color=red, fill opacity = 0.5]
  (I) -- (iSwap) -- (mSQiSW);
\filldraw[color=red, fill opacity = 0.5]
  (I) -- (Cnot) -- (mSQiSW);

 \filldraw[color=violet, fill opacity=0.5]
(I) -- (mSQSWm) -- (SQiSW) -- (mSQSW);
\filldraw[color=violet, fill opacity=0.5]
(SQSW) -- (SQSWm) -- (iSwap);
 \filldraw[color=violet, fill opacity=0.5]
(I) -- (SQiSW) -- (mmSQiSW);
\filldraw[color=violet, fill opacity=0.5]
(I) -- (mSQSW) -- (mmSQiSW);
\filldraw[color=violet, fill opacity=0.5]
(mSQSW) -- (SQiSW) -- (mmSQiSW);
\filldraw[color=violet, fill opacity=0.5]
(SQSW) -- (SQiSW) -- (mmSQiSW);
\filldraw[color=violet, fill opacity=0.5]
(SQSW) -- (iSwap) -- (mSQiSW);
\filldraw[color=violet, fill opacity=0.5]
(SQSW) -- (mmSQiSW) -- (mSQiSW);
\filldraw[color=violet, fill opacity=0.5]
(SQSW) -- (mSQiSW) -- (mmSQiSW);
\filldraw[color=violet, fill opacity=0.5]
(iSwap) -- (SQiSW) -- (mmSQiSW) -- (mSQiSW);


\draw[-,thick] (Cnot) --  (Swapm)
                        (I) --  (Swap)
                        (I) -- (Swapm)
                        (I) -- (Cnot)
                        (Swap) --  (Swapm)
                        (I) --  (iSwap);

\draw[dashed,thick] (iSwap) --  (Cnot)
                        (Swap)  -- (Cnot);

\draw[dotted, thick] (iSwap) --  (mSQiSW)
                        (iSwap)  -- (mSQiSWm)
                        (I) -- (mSQiSW)
                        (I) -- (mSQiSWm)
                        (SQiSW) -- (SQSW)
                        (SQiSW) -- (SQSWm)
                        (SQiSW) -- (mmSQiSW)
                        (SQiSW) -- (mmSQiSWm)
                        (SQSW) -- (mmSQiSW)
                        (SQSWm) -- (mmSQiSWm)
                        (SQSWm) -- (iSwap)
                        (SQSW) -- (iSwap)
                        (mSQiSW) -- (SQSW)
                        (mSQiSWm) -- (SQSWm)
                        (SQiSW) -- (mSQSW)
                        (mmSQiSW) -- (mSQSW)
                        (SQiSW) -- (mSQSWm)
                        (mmSQiSWm) -- (mSQSWm);


\fill[black]  (I) circle [radius=2pt];
\fill[black]    (iSwap) circle [radius=2pt];
\fill[black]  (Cnot) circle [radius=2pt];
\fill[black] (Swap) circle [radius=2pt];
\fill[black] (Swapm) circle [radius=2pt];
\fill[black] (mSQiSW) circle [radius=2pt];
\fill[black] (mSQiSWm) circle [radius=2pt];
\fill[black] (mmSQiSW) circle [radius=2pt];
\fill[black] (mmSQiSWm) circle [radius=2pt];
\fill[red] (SQiSW) circle [radius=2pt];

\end{tikzpicture}
\caption{Eigenphase crossing:\\ $y+|z|<\frac{\pi}8, x\leq\frac\pi8$ or \\ $ y+|z|<\frac{\pi}4, x>\frac\pi8 $.}
    \end{subfigure}\\
    \begin{subfigure}{\textwidth}
        \begin{tikzpicture}
\node {\begin{quantikz}
             &\gate{{\color{cyan}Z}\cdot {\color{violet}R_x(\frac\pi2)}\cdot {\color{cyan}Z}} &\gate[style={fill=red!50},2]{L(x',y',z')} &\gate{{\color{cyan}Z}\cdot {\color{violet}R_x(-\frac\pi2)}\cdot {\color{green}R_z(\frac{\pi}2)}\cdot R_y(-\frac{\pi}2)\cdot {\color{cyan}Z}} &\gate[2]{\mathrm{SQ\lowercase{i}SW} } & \gate{{\color{cyan}Z}\cdot R_y(\frac{\pi}2)\cdot {\color{green}R_z(-\frac\pi2)}\cdot {\color{cyan}Z}} &\qw  \\
             &\gate{{\color{violet}R_x(-\frac{\pi}2)}}&\qw &\gate{{\color{violet}R_x(\frac\pi2)}\cdot {\color{green}R_z(-\frac{\pi}2)}\cdot R_y(\frac{\pi}2)}&\qw&\gate{ R_y(-\frac{\pi}2)\cdot {\color{green}R_z(\frac\pi2)}}&\qw
             \end{quantikz}};
             \end{tikzpicture}
             \caption{}
    \end{subfigure}
\caption{Visualization of the full compilation scheme. When a gate is outside of the region $W'$, there are eight cases corresponding to different circuit compilations, indicated by three inequalities. (a) $x\stackrel{?}{>}\pi/8$, the $>$ case indicated in green. This corresponds to the two cases considered in the proof. (b) $z\stackrel{?}{<}0$, the $<$ case indicated in blue. The proof only deals with the case $z>0$, where the case $z<0$ follows similarly with appropriate inversions. (c) Whether there is \emph{eigenphase crossing}, i.e. whether the order of $a_0,\ldots, a_3$ is preserved after the phase modification. The purple region shows when it isn't, and a corresponding correction needs to be made in order to transform the gate to its canonical form. (d) The corresponding modifications are the green $R_z$-conjugations around the $\mathrm{SQ\lowercase{i}SW} $  gate, the violet $R_x$-conjugations around the $L(x',y',z')$ in red, and the cyan $Z$ gates on the first qubit.}
\label{fig:s3}
\end{figure}


\end{proof}
\subsubsection{Decomposition algorithms for two-qubit gates into $\mathrm{SQ\lowercase{i}SW} $  gates}

The full decomposition algorithm for an arbitrary two-qubit gate into sequences of single qubit rotations and the $\mathrm{SQ\lowercase{i}SW} $  gate is summarized in~\cref{alg:decomp} and visualized in~Fig.~\ref{fig:s3}. We also list compilation schemes of some common two-qubit gates or gate families into $\mathrm{SQ\lowercase{i}SW} $  below and summarize the results in~Fig.~\ref{fig:special}.

In this section and throughout the rest of the paper, we use $||$ to denote the concatenation of two quantum gates. For example, $A||B$ represents a composite quantum gates where $A$ is applied before $B$, resulting an overall operation of $B\cdot A$.

\begin{algorithm}[t]
  \caption{Decomposing an arbitrary two-qubit gate into a sequence of single qubit rotations and the $\mathrm{SQ\lowercase{i}SW} $  gate.}
  \label{alg:decomp}
   \begin{algorithmic}[1]
   \Procedure{Decomp}{U} \Comment{Decompose $U$ into single qubit gates and the $\mathrm{SQ\lowercase{i}SW} $  gate}
   \State $g,(x,y,z),A_1,A_2,B_1,B_2\leftarrow$\textsc{KAKDecomp}($U$)\Comment{$U = g\cdot (A_1\otimes A_2)L(x,y,z)(B_1\otimes B_2)$}
   \If{$|z|\leq x-y$} \Comment{2 $\mathrm{SQ\lowercase{i}SW} $  gates needed}
   \State $C_1,C_2\leftarrow $\textsc{InterleavingSIngleQubitRotations}($x,y,z$)
   \State $V\leftarrow \mathrm{SQ\lowercase{i}SW}  (C_1\otimes C_2)\mathrm{SQ\lowercase{i}SW} $
   \State $g',(x,y,z),D_1,D_2,E_1,E_2\leftarrow$\textsc{KAKDecomp($V$)}\Comment{$L(x,y,z)=(1/g')(D_1^\dag\otimes D_2^\dag)\mathrm{SQ\lowercase{i}SW} (C_1\otimes C_2)\mathrm{SQ\lowercase{i}SW} (E_1^\dag\otimes E_2^\dag)$}
   \State \Return $E_1^\dag B_1\otimes E_2^\dag B_2||\mathrm{SQ\lowercase{i}SW} ||C_1\otimes C_2||\mathrm{SQ\lowercase{i}SW} ||A_1D_1^\dag \otimes A_2D_2^\dag $ \Comment{Global phases $g,g'$ omitted}
   \Else
   \State $(x',y',z'),F_1,F_2,G_1,G_2,H_1,H_2\leftarrow$\textsc{Canonicalize}($x,y,z$) \State\Comment{$L(x,y,z)=(F_1\otimes F_2)L(x',y',z')(G_1\otimes G_2)\mathrm{SQ\lowercase{i}SW} (H_1\otimes H_2), x',y',z'\in W'$}

   \State $C_1,C_2\leftarrow $\textsc{InterleavingSIngleQubitRotations}($x',y',z'$)
   \State $V\leftarrow \mathrm{SQ\lowercase{i}SW}  (C_1\otimes C_2)\mathrm{SQ\lowercase{i}SW} $
   \State $g',(x',y',z'),D_1,D_2,E_1,E_2\leftarrow$\textsc{KAKDecomp($V$)}\Comment{$L(x',y',z')=(1/g')(D_1^\dag\otimes D_2^\dag)\mathrm{SQ\lowercase{i}SW} (C_1\otimes C_2)\mathrm{SQ\lowercase{i}SW} (E_1^\dag\otimes E_2^\dag)$}
   \State\Return $H_1B_1\otimes H_2B_2||\mathrm{SQ\lowercase{i}SW} ||E_1^\dag G_1\otimes E_2^\dag G_2||\mathrm{SQ\lowercase{i}SW} ||C_1\otimes C_2||\mathrm{SQ\lowercase{i}SW} ||A_1F_1D_1^\dag\otimes A_2F_2D_2^\dag$
   \EndIf
   \EndProcedure
   \Procedure{InterleavingSingleQubitGates}{x,y,z} \Comment{Output the single qubit rotations given the interaction coefficients $(x,y,z)\in W'$ when sandwiched by two $\mathrm{SQ\lowercase{i}SW} $  gates}
   \State $C\leftarrow \sin(x+y-z)\sin(x-y+z)\sin(-x-y-z)\sin(-x+y+z)$
   \State $\alpha\leftarrow \arccos(\cos 2x-\cos 2y+\cos 2z+2\sqrt{C})$
   \State $\beta\leftarrow \arccos(\cos 2x-\cos 2y+\cos 2z-2\sqrt{C})$
   \State $\gamma\leftarrow \arccos(\mathrm{sgn} z\cdot \sqrt{\frac{4\cos^2x\cos^2z\sin^2y}{4\cos^2x\cos^2z\sin^2y+\cos2x\cos2y\cos2z}})$
   \State \Return $R_z(\gamma)R_x(\alpha)R_z(\gamma), R_x(\beta)$
   \EndProcedure
   \Procedure{Canonicalize}{x,y,z} \Comment{Decompose an arbitrary gate into one $\mathrm{SQ\lowercase{i}SW} $  and one $L(x',y',z')$ where $(x',y',z')\in W'$ and output the coefficients $(x',y',z')$ and the interleaving single qubit rotations}
   \State$A_0\leftarrow I, A_1\leftarrow I, B_1\leftarrow R_y(-\frac\pi2), B_2\leftarrow R_y(\frac\pi2), C_1\leftarrow R_y(\frac\pi2), C_2\leftarrow R_y(-\frac\pi2), s\leftarrow \mathrm{sgn}(z), x'\leftarrow x, y'\leftarrow y, z'\leftarrow |z|$
   \If{$x>\frac\pi8$}
   \State $y'\leftarrow y'-\frac\pi8,z'\leftarrow z'-\frac\pi8,B_1\leftarrow R_z(\frac\pi2)B_1, B_2\leftarrow R_z(-\frac\pi2)B_2, C_1\leftarrow C_1R_z(-\frac\pi2), C_2\leftarrow C_2R_z(\frac\pi2)$
   \Else
   \State $x'\leftarrow x'+\frac\pi8,z'\leftarrow z'-\frac\pi8$
   \EndIf
   \If{$|y'|<|z'|$}\Comment{Eigenphase crossing}
   \State$y',z'\leftarrow -z',-y',A_1\leftarrow R_x(\frac\pi2), A_2\leftarrow R_x(-\frac\pi2), B_1\leftarrow R_x(-\frac{\pi}2)B_1, B_2\leftarrow R_x(\frac\pi2)B_2$
   \EndIf
   \If {$s<0$}
   \State$z'\leftarrow -z',A_1\leftarrow ZA_1Z, B_1\leftarrow ZB_1Z, C_1\leftarrow ZC_1Z$
   \EndIf
   \State \Return $(x',y',z'), A_1, A_2, B_1, B_2, C_1, C_2$
   \EndProcedure
   \end{algorithmic}
\end{algorithm}



Before proceeding, note that $[R_z(\alpha)\otimes R_z(\alpha), \mathrm{SQ\lowercase{i}SW}  ]=0$ for all $\alpha$. This introduces gauge freedom in compilation of the circuit and enables us to choose the single qubit gates with the simplest form in our compilation.



\begin{description}
\item[Special orthogonal gates $(x,y,0)$] All gates locally equivalent to special orthogonal gates in $SO(4)$, i.e. gates that lie in the I--$\mathrm{CNOT} $--$\mathrm{\lowercase{i}SWAP} $~plane can be generated with two $\mathrm{SQ\lowercase{i}SW} $  gates. Moreover, the expressions of $\alpha, \beta, \gamma$ can be simplified as
\begin{align*}
    \alpha&=0,\\
    \beta&=2\arccos \sqrt{\cos 2x+2\sin^2y},\\
    \gamma &= \arccos\sqrt{\frac{4\cos^2x\sin^2y}{\cos2x+2\sin^2 y}}.\\
\end{align*}
Therefore, one can check by applying the gauge freedom that
$$L(x,y,0)\sim \mathrm{SQ\lowercase{i}SW}  \cdot (I\otimes B)\cdot \mathrm{SQ\lowercase{i}SW} ,$$
where
$$B=R_z(\gamma)R_x(\beta)R_z(\gamma)=\begin{bmatrix}2\cos x\sin y-i\sqrt{\cos 2x\cos 2y}&i\sqrt{\cos 2y-\cos 2x}\\ i\sqrt{\cos 2y-\cos 2x}& 2\cos x\sin y+i\sqrt{\cos 2x\cos 2y}\end{bmatrix}.$$

In the case of special orthogonal gates, the single qubit corrections can be solved analytically. Let
\begin{align*}
    \xi &\equiv -\arcsin\left(\sin y\cdot \sqrt{\frac{2\cos 2x}{\cos2x+\cos2y}}\right),\\
    \phi &\equiv \arccos\left(-\cos y\cdot \sqrt{\frac{2\cos 2x}{\cos2x+\cos2y}}\right),\\
    \psi &\equiv-\arccos\left(\cot(x)\tan(y)\right).
\end{align*}

Then
\begin{align*}
    L(x,y,0)=&(R_z(\xi)\otimes R_z(\phi)R_x(\psi))\cdot \mathrm{SQ\lowercase{i}SW}  \cdot(I\otimes B)\cdot \mathrm{SQ\lowercase{i}SW}  \cdot(R_z(\xi)\otimes R_x(\psi)R_z(\phi-\pi)).
\end{align*}
Specific examples of special orthogonal gates include:
\begin{itemize}
    \item The $\mathrm{CPHASE} $  ~family $(x,0,0)$, where $B=ZR_y(2\arcsin(2\sqrt{2}\sin x))$;
    \item The super-controlled gate family\cite{ye2004super} $(\frac{\pi}4, y, 0)$, where $B=R_x(2\arccos(2\sqrt{2}\sin y))$;
    \item The $\mathrm{\lowercase{i}SWAP} $ ~family $(x,x,0)$, where $B=R_z( 4x-\frac\pi)$.
\end{itemize}

\item [Improper orthogonal gates $(\frac{\pi}4,y,z)$] Half of the improper orthogonal gate family can be generated by 2 $\mathrm{SQ\lowercase{i}SW} $  gates when $y+|z|\leq \frac{\pi}4$. In this case, one has
\begin{align*}
    \alpha&=\arccos\left(\cos 2z-\cos 2y+\sqrt{\frac{\cos 4z+\cos 4y}{2}}\right),\\
    \beta&=\arccos \left(\cos 2z-\cos 2y-\sqrt{\frac{\cos 4z+\cos 4y}{2}}\right),\\
    \gamma &= 0.\\
\end{align*}

Therefore,
$$L(\frac{\pi}4,y,z)\sim \mathrm{SQ\lowercase{i}SW}  \cdot (R_x(\alpha)\otimes R_x(\beta))\cdot \mathrm{SQ\lowercase{i}SW}  .$$
In this case we can also explicitly solve for the single qubit corrections. Let
$$\phi = -\arccos\sqrt{\frac{1+\tan (y-z)}2},\psi = \arccos\sqrt{\frac{1+\tan(y+z)}2},$$
then
$$L(\frac{\pi}4,y,z)=(R_x(\phi+\psi)\otimes R_z(\frac\pi2)R_x(\phi-\psi))\cdot \mathrm{SQ\lowercase{i}SW}  \cdot (R_x(\alpha)\otimes R_x(\beta))\cdot \mathrm{SQ\lowercase{i}SW}  \cdot (R_x(\phi+\psi)\otimes R_x(\phi-\psi)R_z(-\frac\pi2)).$$

Decomposition for the other half of the improper orthogonal gates can be obtained by first decomposing it into one $\mathrm{SQ\lowercase{i}SW} $ {} and a gate in $W(S_2)$, then decomposing the gate in $W(S_2)$ by observing that it is an improper orthogonal gate.


\begin{figure}
    \centering
    \begin{tikzpicture}
\node[scale=0.8] {
    \begin{quantikz}
             & \gate[style={fill=pink!50}, 2]{L(x,y,0)} &  \qw \\
             & \qw & \qw
             \end{quantikz}
             $=$\begin{quantikz}
             &\gate{R_z(\xi)}&\gate[2]{\mathrm{SQ\lowercase{i}SW} }  & \qw&\gate[2]{\mathrm{SQ\lowercase{i}SW} } &\gate{R_z(\xi)}&\qw  \\
             &\gate{R_z(\phi)R_x(\psi)}&\qw &\gate[1]{R_z(-\gamma) R_x(\beta) R_z(-\gamma)}& \qw& \gate{R_x(\psi)R_z(\phi-\frac\pi2)}&\qw
             \end{quantikz}};
             \end{tikzpicture}
    \begin{tikzpicture}
\node[scale=0.8]{
    \begin{quantikz}
             & \gate[style={fill=pink!50}, 2]{L(x,0,0)} &  \qw \\
             & \qw & \qw
             \end{quantikz}
             $=$\begin{quantikz}
             &\qw&\gate[2]{\mathrm{SQ\lowercase{i}SW} }  & \qw&\gate[2]{\mathrm{SQ\lowercase{i}SW} } &\qw&\qw  \\
             &\gate{R_z(\sin^{-1}\tan x)R_x(-\frac\pi2)}&\qw &\gate[1]{ZR_y(2\sin^{-1}\sqrt{2}\sin x)} & \qw& \gate{R_x(-\frac\pi2)R_z(\sin^{-1}\tan x-\pi)}&\qw
             \end{quantikz}};
             \end{tikzpicture}
                 \begin{tikzpicture}
\node[scale=0.8] {
    \begin{quantikz}
             &\gate[style={fill=pink!50}, 2]{L(\frac{\pi}4,y,0)}&\qw \\
             & \qw & \qw
             \end{quantikz}
             $=$\begin{quantikz}
             & \qw&\gate[2]{\mathrm{SQ\lowercase{i}SW} }  & \qw&\gate[2]{\mathrm{SQ\lowercase{i}SW} } &   \qw&\qw  \\
             &\gate{R_z(\frac\pi2)R_x(-\cos^{-1}\tan y)}&\qw &\gate[1]{R_x(2\cos^{-1}\sqrt{2}\sin y)} & \qw&\gate{R_x(-\cos^{-1}\tan y)R_z(-\frac\pi2)}&\qw
             \end{quantikz}};
             \end{tikzpicture}
                 \begin{tikzpicture}
\node[scale=0.8] {
    \begin{quantikz}
             & \gate[style={fill=pink!50}, 2]{L(x,x,0)} &  \qw \\
             & \qw & \qw
             \end{quantikz}
             $=$\begin{quantikz}
             &\qw&\gate[2]{\mathrm{SQ\lowercase{i}SW} }  & \gate{R_z(2x)}&\gate[2]{\mathrm{SQ\lowercase{i}SW} } &\qw&\qw  \\
             &\qw&\qw &\gate{R_z(-2x+\pi)} & \qw&\gate{R_z(-\pi)}&\qw
             \end{quantikz}};
             \end{tikzpicture}
    \begin{tikzpicture}
\node[scale=0.8] {$y+|z|\leq\frac{\pi}4\Rightarrow$
    \begin{quantikz}
             & \gate[style={fill=pink!50}, 2]{L(\frac{\pi}4,y,z)} &  \qw \\
             & \qw & \qw
             \end{quantikz}
             $=$\begin{quantikz}
             &\gate{R_x(\phi+\psi)}&\gate[2]{\mathrm{SQ\lowercase{i}SW} }  & \gate{R_x(\alpha)}&\gate[2]{\mathrm{SQ\lowercase{i}SW} } &\gate{R_x(\phi+\psi)}&\qw  \\
             &\gate{R_z(\frac\pi2)R_x(\phi-\psi)}&\qw &\gate[1]{R_x(\beta)} & \qw&\gate{R_x(\phi-\psi)R_z(-\frac\pi2)}&\qw
             \end{quantikz}};
             \end{tikzpicture}
    \caption{Summary of compilations of common two-qubit gates into $\mathrm{SQ\lowercase{i}SW} $  gates}
    \label{fig:special}
\end{figure}


\end{description}
\subsubsection{Comparison with other two-qubit gates}
It can be observed from the visualization in~Fig.~\ref{fig:s2_region} that $W'$ takes up $1/2$ of the entire Weyl chamber, similar to the set of all perfect entanglers~\cite{zhang2003geometric}. However, the measure in the Weyl chamber does not reflect the Haar measure of the unitary group $SU(4)$~\cite{watts2013metric}. Indeed, the probability that a Haar random element in $SU(4)$ can be decomposed with two $\mathrm{SQ\lowercase{i}SW} $  gates can be calculated as $$\int_{(x,y,z)\in W'}d\mu_W=\frac78-\frac4{15\pi}\approx 79\%.$$ It is well known that the B gate with interaction coefficients $(\frac{\pi}{4},\frac{\pi}{8},0)$ spans the whole Weyl chamber with only two uses~\cite{zhang2004minimum}. It is also well known that many two-qubit gates, including the CNOT gate, the iSWAP gate and the other gates in the super controlling gate family, generates the whole Weyl chamber with three uses\cite{zhang2004optimal}. We find that $\mathrm{SQ\lowercase{i}SW} $ lies in-between: although it cannot generate the whole Weyl chamber, it generates a unitary subset of a \emph{nonzero measure}. Although this holds for general two-qubit gates\cite{peterson2020fixed}, we show that two uses of standard gates such as $\mathrm{CNOT} $, $\mathrm{\lowercase{i}SWAP} $ or the $\mathrm{SWAP} $  family actually generate a subset of the Weyl chamber with zero measure, even though three uses of either gate span the whole Weyl chamber.

\begin{prp}
For any two gates $U_1,U_2\in SU(4)$ in the $\mathrm{CPHASE} $  ~gate family $\eta(U_1)=(x_1,0,0), \eta(U_2)=(x_2,0,0)$ and any two single qubit gates $A_1,A_2\in SU(2)$, define the gate
$$V\equiv U_1(A_1\otimes A_2)U_2.$$
Then the last element in $\eta(V)$ is always zero. Equivalently, $V$ must lie in the I-$\mathrm{CNOT} $-$\mathrm{\lowercase{i}SWAP} $ ~plane in the Weyl chamber.
\end{prp}

\begin{proof}
Let $\eta(V)=(x',y',z')$. It can be checked from the characteristic polynomial $F_V(t)$ that the corresponding polynomial coefficient
$$B=\sin 2x'\sin2y'\sin2z'=0,$$
regardless of how $U_1,U_2,A_1,A_2$ are chosen. This indicates that $z'=0$ given $\frac\pi4\geq x\geq y\geq |z|$.
\end{proof}

Two $\mathrm{CPHASE} $ family gates only generates a two dimensional submanifold  because they are $U(1)$-covariant, or ``leaky''\cite{koponen2006discrete,peterson2020fixed}; we have
$$[R_Z(\theta_1)\otimes R_Z(\theta_2)] \mathrm{diag}(1,1,e^{-i\phi},e^{i\phi}) = \mathrm{diag}(1,1,e^{-i\phi},e^{i\phi})[R_Z(\theta_1)\otimes R_Z(\theta_2) ]$$
for all $\theta_1,\theta_2$ and $\phi$. By commuting the Z-rotations, the interleaving single qubit gates, which can each be decomposed into a $Z-X-Z$ sequence of rotations, can only generate a two-dimensional manifold in the Weyl chamber, as illustrated in~Fig.~\ref{fig:cphase}.

\begin{figure}
    \centering
    \begin{tikzpicture}
\node {
    \begin{quantikz}
             & \gate[2]{\mathrm{CPHASE} (\phi_1)} & \gate[1]{R_z(\alpha_1)R_x(\beta_1)R_z(\gamma_1)}   &\gate[2]{\mathrm{CPHASE} (\phi_2)}  &\qw  \\
              & \qw & \gate[1]{R_z(\alpha_2)R_x(\beta_2)R_z(\gamma_2)} & \qw&\qw
             \end{quantikz}
             $\sim$\begin{quantikz}
             & \gate[2]{\mathrm{CPHASE} (\phi_1)} & \gate[1]{R_x(\beta_1)}   &\gate[2]{\mathrm{CPHASE} (\phi_2)}  &\qw  \\
              & \qw & \gate[1]{R_x(\beta_2)} & \qw&\qw
             \end{quantikz}};
             \end{tikzpicture}
    \caption{Decomposing each single qubit gate as a $Z-X-Z$ sequence of rotations and commuting the Z-rotations with the $\mathrm{CPHASE} $ gates, it can be seen that two $\mathrm{CPHASE} $  gates only generate a two dimensional submanifold of the Weyl chamber.}
    \label{fig:cphase}
\end{figure}

By making use of the properties of mirror gates, we can extend this result to gates on the $\mathrm{\lowercase{i}SWAP} -\mathrm{SWAP} $ line as well. The results are visualized in~Fig.~\ref{fig:c2}.

\begin{cor}
For any two gates $U_1,U_2\in SU(4)$ such that$$\eta(U_1)=(\frac{\pi}4,\frac{\pi}4,x_1), \eta(U_2)=(\frac{\pi}4,\frac{\pi}4,x_2)$$ and any two single qubit gates $A_1,A_2\in SU(2)$, define the gate
$$V\equiv U_1(A_1\otimes A_2)U_2.$$
Then the last element in $\eta(V)$ is always zero.
\end{cor}

\begin{cor}
For any two gates $U_1,U_2\in SU(4)$ such that $$\eta(U_1)=(\frac{\pi}4,\frac{\pi}4,x_1), \eta(U_2)=(x_2,0,0)$$ and any two single qubit gates $A_1,A_2\in SU(2)$, define the gate
$$V\equiv U_1(A_1\otimes A_2)U_2.$$
Then the first element in $\eta(V)$ is always $\frac\pi4$, i.e.\ the gate $V$ always lies inside the $\mathrm{CNOT} -\mathrm{\lowercase{i}SWAP} -\mathrm{SWAP} $ plane of the Weyl chamber.
\end{cor}
\begin{figure}
    \centering
\begin{tikzpicture} [scale=.5]
\def \tta{ 90.000000000000 }
\def \k{    0.30000000000000 }
\def \l{     6.00000000000000 }
\def 2.7{     4.00000000000000 }
\def \h{     6.0000000000000 }

\coordinate (I) at (0,0);
\coordinate (Cnot) at (0,{-\h});
\coordinate (iSwap) at ({-\l*sin(\tta))},
                    {-\h+\l*cos(\tta)});
\coordinate (Swap) at ({-\l*sin(\tta)-2.7*sin(\k*\tta)},{-\h+\l*cos(\tta)+2.7*cos(\k*\tta)});
\coordinate (Swapm) at ({-\l*sin(\tta)+2.7*sin(\k*\tta)},{-\h+\l*cos(\tta)-2.7*cos(\k*\tta)});
\coordinate (SQiSW) at ({-\l*sin(\tta))/2},
                    {(-\h+\l*cos(\tta))/2});


 \filldraw[color=red!50, fill opacity = 0.5]
  (I) -- (Cnot) -- (iSwap);
  \filldraw[color=green!50, fill opacity = 0.5]
  (Cnot) -- (Swap) -- (Swapm);

\draw[-] (Cnot) --  (Swapm)
                        (I) --  (Swap)
                        (I) -- (Swapm)
                        (I) --  (iSwap);

\draw[-,very thick, blue] (I)--(Cnot)
                    (Swap) -- (Swapm);

\draw[dashed] (iSwap) --  (Cnot)
                        (Swap)  -- (Cnot);




 \draw[-, very thick,green] let \p0=(Cnot),\p1=(Swap) in
         ([xshift=-.05cm]\x0,\y0) --  ([xshift=-.05cm]\x1,\y1);
\draw[-, very thick,green] let \p0=(Swapm),\p1=(Swap) in
         ([xshift=.05cm]\x0,\y0) --  ([xshift=.05cm]\x1,\y1);
 \draw[-, very thick,green] let \p0=(Cnot),\p1=(Swapm) in
 ([xshift=-.05cm]\x0,\y0) --  ([xshift=-.05cm]\x1,\y1);

\draw[-, very thick,red] let \p0=(I),\p1=(iSwap) in
         ([xshift=.05cm]\x0,\y0) --  ([xshift=.05cm]\x1,\y1);
\draw[-, very thick,red] let \p0=(iSwap),\p1=(Cnot) in
         ([yshift=.05cm]\x0,\y0) --  ([yshift=.05cm]\x1,\y1);
 \draw[-, very thick,red] let \p0=(I),\p1=(Cnot) in
 ([xshift=-.05cm]\x0,\y0) --  ([xshift=-.05cm]\x1,\y1);

\fill[blue]  (I) circle [radius=2pt];
\fill[blue]    (iSwap) circle [radius=2pt];
\fill[blue]  (Cnot) circle [radius=2pt];
\fill[blue] (Swap) circle [radius=2pt];
\fill[blue] (Swapm) circle [radius=2pt];

\draw (I) node [right]           {$I$}
      (Cnot) node [right]     {$\mathrm{CNOT} $}
      (Swap) node [left]           {$\mathrm{SWAP} ^\dag$}
      (Swapm)  node [left]       {$\mathrm{SWAP} $}
      (iSwap)  node [left]            {$\mathrm{\lowercase{i}SWAP} $};
\end{tikzpicture}
\caption{The area spanned by 2 $\mathrm{CPHASE} $  ~family gates or their mirror gates. Two gates on the I--$\mathrm{CNOT} $ line, or the $\mathrm{SWAP} ^\dag$--$\mathrm{SWAP} $ line, spans the red area, whereas one gate on each line spans the green area.}
\label{fig:c2}
\end{figure}

We can also prove a result for the case when we have two gates in the $\mathrm{SWAP} $  family. This result is visualized in~Fig.~\ref{fig:swap2}.

\begin{prp}
For any two gates $U_1,U_2\in SU(4)$ such that $$\eta(U_1)=(x_1,x_1,x_1), \eta(U_2)=(x_2,x_2,x_2)$$ and any two single qubit gates $A_1,A_2\in SU(2)$, define the gate
$$V\equiv U_1(A_1\otimes A_2)U_2.$$
Then $(x',y',z')\equiv \eta(V)$ must satisfy $y'=x'$ or $y'=|z'|$. Equivalently, $V$ must either lie in the $I-\mathrm{CNOT} -\mathrm{SWAP} $ plane, or the $I-\mathrm{SWAP} ^\dag-\mathrm{SWAP} $ plane in the Weyl chamber.
\end{prp}

\begin{proof}
Let $\eta(V)=(x',y',z')$. It can be checked that the character polynomial $F_V(t)$ has a zero with multiplicity two:
$$(t\cdot \sin(x_1+x_2)+\cos(x_1+x_2))^2~\mid~F_V(t),$$
regardless of how $U_1,U_2,A_1,A_2$ are chosen. This indicates that there must be at least one equality in the inequalities $x'+y'-z'\geq x'-y'+z'\geq -x'+y'+z'\geq -x'-y'-z'$, or equivalently, one equality in $x'\geq y'\geq |z'|$.
\end{proof}

\begin{figure}
    \centering
\begin{tikzpicture} [scale=.5]
\def \tta{ 90.000000000000 }
\def \k{    0.30000000000000 }
\def \l{     6.00000000000000 }
\def 2.7{     4.00000000000000 }
\def \h{     6.0000000000000 }

\coordinate (I) at (0,0);
\coordinate (Cnot) at (0,{-\h});
\coordinate (iSwap) at ({-\l*sin(\tta))},
                    {-\h+\l*cos(\tta)});
\coordinate (Swap) at ({-\l*sin(\tta)-2.7*sin(\k*\tta)},{-\h+\l*cos(\tta)+2.7*cos(\k*\tta)});
\coordinate (Swapm) at ({-\l*sin(\tta)+2.7*sin(\k*\tta)},{-\h+\l*cos(\tta)-2.7*cos(\k*\tta)});
\coordinate (SQiSW) at ({-\l*sin(\tta))/2},
                    {(-\h+\l*cos(\tta))/2});



  \filldraw[color=green!50, fill opacity = 0.5]
  (I) -- (Swap) -- (Cnot);
\filldraw[color=green!50, fill opacity = 0.5]
  (I) -- (Cnot) -- (Swapm);
  \filldraw[color=red!50, fill opacity = 0.5]
  (I) -- (Swap) -- (Swapm);

\draw[-] (Cnot) --  (Swapm)
                        (I) --  (Cnot)
                        (Swap) -- (Swapm)
                        (I) --  (iSwap);


 \draw[-, very thick,green] let \p0=(I),\p1=(Swap) in
         ([yshift=-.1cm]\x0,\y0) --  ([yshift=-.1cm]\x1,\y1);
\draw[-, very thick,green] let \p0=(I),\p1=(Cnot) in
         ([xshift=-.05cm]\x0,\y0) --  ([xshift=-.05cm]\x1,\y1);
 \draw[-, very thick,green] let \p0=(Cnot),\p1=(Swap) in
 ([xshift=.05cm]\x0,\y0) --  ([xshift=.05cm]\x1,\y1);

 \draw[-, very thick,green] let \p0=(I),\p1=(Swapm) in
         ([xshift=.05cm]\x0,\y0) --  ([xshift=.05cm]\x1,\y1);
\draw[-, very thick,green] let \p0=(I),\p1=(Cnot) in
         ([xshift=-.05cm]\x0,\y0) --  ([xshift=-.05cm]\x1,\y1);
 \draw[-, very thick,green] let \p0=(Cnot),\p1=(Swapm) in
 ([xshift=-.05cm]\x0,\y0) --  ([xshift=-.05cm]\x1,\y1);
\draw[dashed] (iSwap) --  (Cnot)
                        (Swap)  -- (Cnot);
\draw[-, very thick,red] let \p0=(I),\p1=(Swap) in
         ([yshift=-.05cm]\x0,\y0) --  ([yshift=-.05cm]\x1,\y1);
\draw[-, very thick,red] let \p0=(Swapm),\p1=(Swap) in
         ([xshift=.05cm]\x0,\y0) --  ([xshift=.05cm]\x1,\y1);
 \draw[-, very thick,red] let \p0=(I),\p1=(Swapm) in
 ([xshift=-.05cm]\x0,\y0) --  ([xshift=-.05cm]\x1,\y1);


\draw[-,very thick, blue] (I)--(Swap)
                    (I) -- (Swapm);

\fill[blue]  (I) circle [radius=2pt];
\fill[black]    (iSwap) circle [radius=2pt];
\fill[black]  (Cnot) circle [radius=2pt];
\fill[blue] (Swap) circle [radius=2pt];
\fill[blue] (Swapm) circle [radius=2pt];

\draw (I) node [right]           {$I$}
      (Cnot) node [right]     {$\mathrm{CNOT} $}
      (Swap) node [left]           {$\mathrm{SWAP} ^\dag$}
      (Swapm)  node [left]       {$\mathrm{SWAP} $}
      (iSwap)  node [left]            {$\mathrm{\lowercase{i}SWAP} $};
\end{tikzpicture}
\caption{The area spanned by 2 $\mathrm{SWAP} $  family gates. Two gates on the $\mathrm{SWAP} ^\dag$--I--$\mathrm{SWAP} $ line spans either the red area (I--$\mathrm{SWAP} $--$\mathrm{SWAP} ^\dag$) or the green area (I--$\mathrm{SWAP} $--$\mathrm{CNOT} $ or I--$\mathrm{SWAP} ^\dag$--$\mathrm{CNOT} $).}
\label{fig:swap2}
\end{figure}


\subsection{Compilation of circuits with more than two qubits}
To further demonstrate the information processing superiority of $\mathrm{SQ\lowercase{i}SW} $, we prove a linear separation between the number of gates needed to generate an $n$-qubit W-like state from the product state $\ket{0}^{\otimes n}$ using the $\mathrm{CNOT} $ gate and the $\mathrm{SQ\lowercase{i}SW} $ gate. Considering the corresponding family of circuits with the $\mathrm{SQ\lowercase{i}SW} $  gates, our proof extends to a linear separation between the gate counts in the task of compiling this family of circuits using the $\mathrm{SQ\lowercase{i}SW} $  gate and the $\mathrm{CNOT} $  gate. (Note that $\mathrm{\lowercase{i}SWAP} $ is equivalent to $\mathrm{CNOT} $ for this purpose, since they are mirror gates up to local equivalence.) Throughout we consider single qubit rotations as free resources.

The $n$-qubit W state is defined as
$$\ket{W_n} \equiv \frac{1}{\sqrt{n}}\sum_{i=1}^n \left(|0\rangle^{\otimes(i-1)}\otimes |1\rangle\otimes |0\rangle^{\otimes(n-i)}\right).$$
An interesting property of the W state is that it is robust against the disposal of qubits~\cite{dur2001multipartite}: even after tracing out any subset of $n-2$ qubits, the marginal state of the remaining two qubits is still an entangled state. In contrast, the most common multipartite generalization of maximally entangled states, the $n$-qubit GHZ state $\ket{GHZ_n} = \left(|0\rangle^{\otimes n}+|1\rangle^{\otimes n}\right)/{\sqrt{2}}$, does not satisfy this condition when $n\ge 3$. This special property of the W state can be abstracted as follows:
\begin{dfn}
An $n$-partite state $\ket{\Psi}$ is \emph{W-like} if the marginal state on any two subsystems is an entangled bipartite state.
\end{dfn}

It is easy to see that $n$-qubit states of the form
\begin{align}
\sum_{i=1}^n \alpha_i \left(|0\rangle^{\otimes(i-1)}\otimes |1\rangle\otimes |0\rangle^{\otimes(n-i)}\right), \alpha_i > 0, \sum_{i=i}^n|\alpha_i|^2 = 1
\label{eq:w-like}
\end{align}
are special cases of  W-like states. Now, in order to generate any $n$-qubit W-like state from $\ket{0}^{\otimes n}$, it takes at least $n-1$ two-qubit gates, since the generating circuit as a graph needs to be at least connected. Surprisingly, $n-1$ $\mathrm{SQ\lowercase{i}SW} $  gates is also sufficient to generate a particular $W$-like state of the form in~\cref{eq:w-like}: it can be verified that
$$\mathrm{SQ\lowercase{i}SW} _{n-1, n}\cdots\mathrm{SQ\lowercase{i}SW} _{2, 3}\mathrm{SQ\lowercase{i}SW} _{1, 2}X_1\ket{0}^{\otimes n}$$
is a state of the above form where $\alpha_i = 2^{-i/2}$ for $i\le n-1$, and $\alpha_n = \alpha_{n-1}$.

In contrast, we show that two-qubit gates that are equivalent to a diagonal gate up to local unitaries, such as $\mathrm{CNOT} \sim$ CZ, are ill-suited for generating any W-like state. We have the following result.

\begin{thm}
\label{thm:cnot}
An $n$-qubit W-like state cannot be generated using single qubit gates and less than $\frac{15n-3}{14}$ $\mathrm{CNOT} $  gates.
\end{thm}
\begin{proof}
The proof is somewhat involved and we give it in Appendix~\ref{sec:wstate}.
\end{proof}
This is evidence that $\mathrm{SQ\lowercase{i}SW} $ has better information processing capabilities beyond just compiling two-qubit gates, but it is unclear how general a statement can be made.

\subsection{Numerical Experiments}
The information processing capabilities of $\mathrm{SQ\lowercase{i}SW} $ we have proved all point to its superiority in actual experimental realizations. To strengthen this claim, in this section we conduct a series of numerical experiments comparing $\mathrm{SQ\lowercase{i}SW} $ to $\mathrm{\lowercase{i}SWAP} $ with respect to different metrics under a noisy setting. For simplicity we assume a simple depolarizing noise model.

\subsubsection{Fidelity of Compiling Two-Qubit Gates}
In our first experiment we compare $\mathrm{SQ\lowercase{i}SW} $ to $\mathrm{\lowercase{i}SWAP} $ by computing the fidelity of generating arbitrary two-qubit gates in a noisy setting. An arbitrary two-qubit gate is compiled using $\mathrm{SQ\lowercase{i}SW} $ according to~\cref{alg:decomp} and using $\mathrm{\lowercase{i}SWAP} $ according to~\cite{ye2004super}. We consider a simple error model: each gate is followed by a depolarizing channel with error rate $p_{\text{iswap}} = 2p_{\text{sqisw}} = 0.005$, $p_{\text{single}}=0.0005$. For the two-qubit gates, each of the qubits undergo a depolarizing channel of the corresponding error rate.

As the family of all two-qubit unitaries $\mathrm{SU}(4)$ has $15$ real degrees of freedom, we choose one element from each Weyl chamber coordinate, appending it with randomly chosen single-qubit gates. The results we find show that the errors are dominated by the two-qubit gates. We use an interleaved version of Fully Randomized Benchmarking, or iFRB~\cite{kong2021fully} to compute the fidelity value for each Weyl chamber coordinate (please refer to~\cref{subsec:frb} for more details on iFRB). The corresponding results are shown in~\cref{fig:frb_wc}.
\begin{figure}
    \centering
    \includegraphics[width=0.7\textwidth]{frb_wc.png}
    \caption{iFRB fidelity value projected onto the Weyl chamber. Data points are taken where $\eta_x$ are multiples of $\pi/20$, and $\eta_y$ and $\eta_z$ are multiples of $\pi/60$. Each data point is collected using iFRB on a gate with the Weyl chamber coordinate, with a randomly chosen set of single-qubit operations applied before and after. For demonstration, we consider a simple error model: each gate is followed by a depolarizing channel with error rate $p_{\text{iswap}} = 2p_{\text{sqisw}} = 0.005$, $p_{\text{single}}=0.0005$. It can be seen from the figure that the effects of the randomly chosen single-qubit operations are negligible as the predominant error sources are the two-qubit gates.}
    \label{fig:frb_wc}
\end{figure}

It can be seen from the figure that, under this particular noise model, all gates can be compiled using $\mathrm{SQ\lowercase{i}SW} $ with an error rate below $1.8\%$. Meanwhile, although gates in the $I$-$\mathrm{CNOT} $-$\mathrm{\lowercase{i}SWAP} $ plane can be compiled with 2 applications of the $\mathrm{\lowercase{i}SWAP} $ gate, reaching an error rate of about $2\%$, general gates requiring $3$ applications of the $\mathrm{\lowercase{i}SWAP} $ gate has error rate about $3\%$. This significant difference indicates an appreciable advantage to using $\mathrm{SQ\lowercase{i}SW} $ for compiling quantum algorithms.

\subsubsection{Achievable Quantum Volume}
Quantum volume~\cite{cross2019validating} is a measure of the largest random quantum circuit of equal width and depth that a quantum computer can successfully implement. It is a all-around measure, taking into account gate fidelities, expressibility of native gate sets, quality of compilers, and even qubit connectivity. We conduct numerical experiments computing the quantum volume that directly compares using $\mathrm{SQ\lowercase{i}SW} $ to using $\mathrm{\lowercase{i}SWAP} $ as the native two-qubit gate, ceteris paribus, under different noise levels and connectivities. Note that unlike compiled two-qubit gate fidelity, this compares the gates in a multi-qubit setting beyond just two qubits.

For the sake of being self-contained, we repeat here the definition of quantum volume. Given the number of qubits and depth $d$, we generate a random circuit of the form shown in~\cref{fig:qv_circuit}.
\begin{figure}
    \centering
    \includegraphics[width = 0.7\textwidth]{qv_circuit.pdf}
    \caption{A random circuit used to evaluate quantum volume. The number of qubits and the number of cycles of permutation plus random two-qubit gates are both $d$.}
    \label{fig:qv_circuit}
\end{figure}
The $SU(4)$ box indicates a Haar-random two-qubit unitary, while the $\pi$ box indicates a uniformly randomly chosen permutation. The circuit as a whole defines an overall unitary $U \in SU(2^d)$. We first numerically compute the probability distribution over bit strings $x \in \{0,1\}^d$ measured if we implement $U$ on $\ket{0}^{\otimes d}$:
\begin{align*}
    p_U(x) \equiv \vert \bra{x} U \ket{0}^{\otimes d} \vert^2.
\end{align*}
Using this, we can define the \textit{heavy outputs} as the bit strings whose probability is higher than the median:
\begin{align*}
    H_U \equiv \{ x \in \{0,1\}^d \vert p_U(x) > p_\text{med} \},
\end{align*}
where $p_\text{med}$ is the median of the probabilities of the bit strings. Next, we compute the probability distribution obtained using imperfect gates, compilation and such: $q_U(x)$~\footnote{Note that this is possible for a numerical simulation, but in a real experimental settings a sampling approach is necessary. }. We then define
\begin{align*}
    h_U \equiv \sum_{x \in H_U} q_U(x).
\end{align*}
We average over unitaries $U$ according to the above distribution to obtain
\begin{align*}
    h_d \equiv \int_U dU h_U.
\end{align*}
The quantum volume is defined as
\begin{align*}
    V_Q \equiv 2^{\max\{ d  \vert h_d > \frac 2 3\}}.
\end{align*}

In our particular case, we generate $q_U(x)$ as follows. As before, an arbitrary two-qubit gate is compiled using $\mathrm{SQ\lowercase{i}SW} $ according to~\cref{alg:decomp} and using $\mathrm{\lowercase{i}SWAP} $ according to~\cite{ye2004super}. The two-qubit gates are then subject to depolarizing noise with rate $p_\mathrm{\lowercase{i}SWAP}  = 2 p_\mathrm{SQ\lowercase{i}SW} $, and the single-qubit gates with rate $5\times 10^{-4}$.

We now show the results of our numerical experiments, conducted using the quantum volume module in Cirq~\cite{cirq_developers_2021_4586899}. $h_d$ is approximated by averaging over 1001 different $U$ for increasing $d$ (We find numerically that $h_U$ is approximately the same for all 1001 samples.), under different noise levels and connectivities. In~\cref{fig:qv_all}, a complete graph is assumed and~\cref{fig:qv_chain} assumes a 1-D chain graph. The chain case is done by computing a list of $\mathrm{SWAP} $ gates needed to implement the random permutations using the corresponding function in the quantum volume module of Cirq. Finally, in the figures we only compute even $d$ for simplicity --- the odd values show a similar trend. We see that for all the error rates we consider, $\mathrm{SQ\lowercase{i}SW} $ clearly outperforms $\mathrm{\lowercase{i}SWAP} $, consistently achieving a higher quantum volume. This indicates that simply changing from $\mathrm{\lowercase{i}SWAP} $ to $\mathrm{SQ\lowercase{i}SW} $ can appreciably change the quantum volume boasted by a quantum computer.
\begin{figure}
    \centering
    \includegraphics[width=0.7\textwidth]{alltoall.png}
    \caption{$h_d$ as a function of $d$ for different depolarizing noise rates assuming we use $\mathrm{SQ\lowercase{i}SW} $ or $\mathrm{\lowercase{i}SWAP} $ as our native two-qubit gate. We assume here a complete connectivity graph.}
    \label{fig:qv_all}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[width=0.7\textwidth]{chain.png}
    \caption{$h_d$ as a function of $d$ for different depolarizing noise rates assuming we use $\mathrm{SQ\lowercase{i}SW} $ or $\mathrm{\lowercase{i}SWAP} $ as our native two-qubit gate. We assume here a 1-D chain connectivity graph.}
    \label{fig:qv_chain}
\end{figure}

\section{$\mathrm{SQ\lowercase{i}SW} $: Calibration and Benchmarking}
\label{sec:imp}
In this section, we provide detailed experimental procedures for calibrating and benchmarking the $\mathrm{SQ\lowercase{i}SW} $  gate for a superconducting circuit architecture. In particular, we give a variant of the interleaved randomized benchmarking framework, which we call iFRB for interleaved fully randomized benchmarking, in order to benchmark non-Clifford target gates. We then apply the new framework to the benchmarking of the $\mathrm{SQ\lowercase{i}SW} $  gate, which is not in the Clifford group. Throughout this section we will need to consider compiling circuits with $\mathrm{SQ\lowercase{i}SW} $  and arbitrary single qubit gates. As noted in~\cite{abrams2020implementation}, $\mathrm{SQ\lowercase{i}SW} $  does not commute with single qubit $Z$ rotations, and therefore common techniques to implement $Z$ rotations such as frame updates~\cite{mckay2017efficient} are not applicable. To address this, we use the single qubit compiling scheme in \cite{chen2021compiling}, which is applicable for any two-qubit gate. We choose to use this instead of specialized schemes such as that of~\cite{abrams2020implementation} to make our framework more general beyond just the $\mathrm{SQ\lowercase{i}SW} $  gate~\footnote{Note that the approach of~\cite{abrams2020implementation} assumes a parametric gate implementation, while we mostly consider the standard flux tuning implementation, which does not allow the additional phase shift. }. Indeed, our iFRB framework can be applied to any two-qubit gate.

\subsection{Calibration of the $\mathrm{SQ\lowercase{i}SW} $ gate}
\label{sec:calib}
When physically implementing the $\mathrm{SQ\lowercase{i}SW} $  gate, there is a variety of phase errors that can occur from different processes, including the phase accumulation due to each individual qubit's static Hamiltonian ($-\frac{\omega}{2} Z$, where $\omega$ is the qubit frequency), phase acquired during flux tuning, and even two-qubit phase errors via unwanted $ZZ$ coupling. Since we know the phase errors for each $\mathrm{SQ\lowercase{i}SW} $ in the quantum circuit before compilation, we can correct for the phase errors at compile time by compiling the single qubit gates in the circuit with the appropriate corrections.

The procedure is as follows. For generality, we consider a two-qubit gate $U$ in the $\mathrm{\lowercase{i}SWAP} $ family, that is, of the form
\begin{align*}
    U = \left[\begin{matrix}1 & 0 & 0 & 0\\0 &  \cos\theta & -i \sin\theta & 0\\0 & -i  \sin\theta & \cos\theta & 0\\0 & 0 & 0 & 1 \end{matrix}\right].
\end{align*}
Common sources of single qubit phase errors transform it into~\cite{arute2020observation}
\begin{align*}
    U' =\left[\begin{matrix}1 & 0 & 0 & 0\\0 & e^{-i(\xi + \zeta)} \cos\theta & -i e^{-i(\xi-\chi)} \sin\theta & 0\\0 & -i e^{-i(\xi+\chi)} \sin\theta & e^{-i(\xi-\zeta)} \cos\theta & 0\\0 & 0 & 0 & e^{-i2\xi}\end{matrix}\right].
\end{align*}
Note we omit the two-qubit phase error~\footnote{This error is denoted as $\phi$ in~\cite{arute2020observation}. } since that cannot be corrected by single qubit gates. The relationship between $U$ and $U'$ can be summarized as
\begin{align*}
    U' = (Z_{\lambda_+} \otimes Z_{\lambda_-}) U (Z_\kappa \otimes Z_{-\kappa}),
\end{align*}
where $\lambda_\pm \equiv -\xi \pm (\zeta-\chi)/2$ and $\kappa \equiv (\zeta+\chi)/2$.

We now give the overall phase correcting compilation algorithm. Consider a quantum circuit consisting of arbitrary single qubit gates and an $\mathrm{\lowercase{i}SWAP} $ family gate $U$. We can represent this as a directed graph $G =(V,E)$ with vertices corresponding to qubits and edges corresponding to instances of $U$ between pairs of qubits. We also impose a total order on the edges corresponding to the order in which the $U$ gates are implemented (for simultaneous gates we assign an arbitrary total order). We give the pseudocode for the algorithm in~\cref{alg:phase_corr}, where throughout we will refer to different compilation schemes named in \cite{chen2021compiling} to compile arbitrary single qubit gates and for conciseness we denote $P(\sigma, \phi) \equiv Z_{-\phi} X_\sigma Z_\phi$.

\begin{algorithm}[H]
\caption{Phase correction compilation algorithm}
\label{alg:phase_corr}
\begin{algorithmic}[1]
\Procedure{PhaseCorrect}{$G = (V,E)$} \Comment{Compile single qubit gates to correct phase errors}
\For{$e = (v_0, v_1) \in E$}
\State $U_0, U_1 \leftarrow $ single qubit gates immediately prior to $e$ on $v_0, v_1$ \Comment{$U_0, U_1$ can be identity}
\State $Z_{C_0}, Z_{C_1} \leftarrow$ phase carried on qubits $v_0, v_1$
\State Compile $U_0$ using virtual $Z$ compilation scheme $Z_{\theta_0} X_{\pi/2} Z_{\phi_0} X_{\pi/2} Z_{\omega_0} $
\State Compile $U_1$ using three-pulse compilation scheme $P(\pi/2,\theta_1) P(\pi, \phi_1) P(\pi/2, \omega_1)$
\State Implement $P(\pi/2,\phi_0+\omega_0-F_0) P(\pi/2, \omega_0-F_0)$ on $v_0$ \Comment{In terms of Paulis, this is $Z_{-\phi_0-\omega_0+F_0} X_{\pi/2} Z_{\phi_0} X_{\pi/2} Z_{\omega_0-F_0}$}
\State $\lambda_+, \lambda_-, \kappa \leftarrow$ single qubit phase errors of $e$
\State $x \leftarrow \kappa-\phi_0-\omega_0+F_0-\theta_0$  \Comment{This is the extraneous phase on $v_0$ }
\State Implement $P(\pi/2, -x-\kappa+\theta_1) P(\pi, (2\phi_1-x-\kappa-F_1)/2) P(\pi/2, \omega_1-F_1)$ on $v_1$ \Comment{In terms of Pauli's this is $Z_{x + \kappa -\theta_1} X_{\pi/2} Z_{\theta_1-\phi_1} X_\pi Z_{\phi_1-\omega_1} X_{\pi/2} Z_{\omega_1-F_1} = Z_{x+\kappa} P(\pi/2,\theta_1) P(\pi, \phi_1) P(\pi/2, \omega_1) Z_{-F_1}$}
\State Implement $e$ \Comment{$\mathrm{SQ\lowercase{i}SW} $ with corresponding phase errors}
\State Phases carried on $v_0, v_1 \leftarrow Z_{x+\lambda_+}, Z_{x+\lambda_-}$
\EndFor
\EndProcedure
\end{algorithmic}
\end{algorithm}



\subsection{Benchmarking of the $\mathrm{SQ\lowercase{i}SW} $ gate}

We now give a variant of the interleaved randomized benchmarking (iRB) framework~\cite{magesan2012efficient} for benchmarking the $\mathrm{SQ\lowercase{i}SW} $  gate. The iRB framework was first proposed to benchmark the average fidelity of a target gate given the ability to implement arbitrary Clifford gates with high fidelity. However, under the iRB framework, the target gate, i.e.\ the gate to be benchmarked, needs to be Clifford too. For this reason, the iRB framework is usually used on two-qubit gates such as the $\mathrm{\lowercase{i}SWAP} $ gate or the $\mathrm{CNOT} $  gate, but not on non-Clifford gates such as the Controlled-$S$ gate , much of the fSim gate family, the matchgates\cite{helsen2020matchgate}, or $\mathrm{SQ\lowercase{i}SW} $. Our variant, called {\em interleaved fully randomized benchmarking (iFRB)}, relies on the efficient implementation of Haar random gates as the reference gate set. Compared to Clifford-based iRB, the iFRB scheme is readily applicable to benchmarking of arbitrary quantum gates (not necessarily Clifford) and especially useful when benchmarking on a small quantum system where efficiency of implementing Haar random gates is not an issue.

\subsubsection{Fully randomized benchmarking}
\label{subsec:frb}
Before introducing iFRB, we first briefly recall the vanilla randomized benchmarking (RB) and the iRB frameworks. Randomized benchmarking~\cite{emerson2005scalable} was first proposed to study the amplitude of the gate-independent, time-independent, average noise level in a quantum system, while isolating out the errors caused by imperfect state preparation and measurement (SPAM error). The experimental protocol goes as follows: for a $d$-dimensional quantum system, choose an appropriate gate sequence length $m$, choose $m$ gates $U_1,\cdots, U_m$ i.i.d.\ from the Haar random distribution and compute $U_{m+1}$ the recovery gate such that $U_{m+1}U_m\cdots U_1=I$ assuming there are no errors. The gate sequence $U_1||U_2||\dots ||U_m||U_{m+1}$ is then performed on an initial state $|0\rangle$ and subsequently measured in the computational basis. A \emph{survival probability} $p_{m,j}$ of measuring $0$ is estimated from repeated experiments with sequence $j$ of length $m+1$. By averaging over many different sequences with the same length and collecting data over several different gate sequence lengths, one can extract the average gate fidelity  $r=1-(1-u)(d-1)/d$, where $d$ is the dimension of the system ($2^n$ for an $n$-qubit system) and $u$ is obtained from fitting the curve
$$\mathbb{E}_j[p_{m,j}]=A\cdot u^m+B.$$
Here, the parameters $A$ and $B$ are supposed to capture the SPAM error, leaving $u$ represent solely the imperfection in gate implementation.
The requirement for Haar randomness was subsequently found to be unnecessary and unscalable to larger sized quantum systems and is relaxed to a unitary 2-design, most commonly a uniform distribution on the Clifford gates\cite{knill2008randomized}. To avoid confusion, we refer to the Haar random randomized benchmarking as fully randomized benchmarking (FRB) and the Clifford-based one RB.

After the first proposal of FRB, follow-up works flourished~\cite{magesan2012efficient, knill2008randomized, emerson2007symmetrized, gambetta2012characterization, wallman2015robust, wallman2016robust, garion2021experimental}, most of which were based on the Clifford gate set. In particular, interleaved randomized benchmarking~\cite{magesan2012efficient} was proposed to benchmark the average fidelity of a specific gate, referred to as the \emph{target gate}, with the hope to exclude not only the SPAM error, but the errors of other gates in a gate set. An iRB experiment consists of two parts. The first part is an ordinary RB protocol on a gate set, referred to as the \emph{reference gate set}. The second part performs RB with interleaved sequences. Given a target gate $T$, a random gate sequence of length $m$, $U_1,U_2,\cdots U_m$, is generated i.i.d.\ from the Clifford group, but the final recovery gate is chosen to be $U'_{m+1}=(T\cdot U_m\cdot T\cdot\dots\cdot T\cdot U_2\cdot T\cdot U_1)^\dag$, and the gate sequence $U_1||T||U_2||T||\dots||T||U_{m}||T||U'_{m+1}$ is then performed as in the RB experiment. A different error quantity $v$ can be calculated from the decay rate of the average survival probability with respect to the sequence length, similar to $u$ in the ordinary RB experiment. The average fidelity of the target gate can then be calculated as $$r_T=1-\frac{(1-v/u)(d-1)}{d}.$$


In order to be able to carry out the iRB experiment, it is crucial that the final recovery gate, $U'_{m+1}$, lies in the reference gate set, i.e.\ the Clifford group. Although this holds when the target gate itself lies in the Clifford group, this is not true for many common gates. That the iRB framework cannot be applied for non-Clifford gates is a serious outstanding issue as a non-Clifford gate is necessary for universal quantum computing, by the Gottesman-Knill Theorem~\cite{gottesman1998heisenberg}. Several alternatives have been proposed, including choosing different finite groups other than the Clifford group~\cite{garion2021experimental}. Altogether different benchmarking experiments were also proposed~\cite{arute2019quantum, reagor2018demonstration}, but these alternatives either rely on extensive algebraic studies of the target gate or lack a rigorous theoretical framework for analyzing their interpretation and applicability.

To resolve this issue thorugh iFRB, we simply apply iRB, except that instead of using random Clifford gates, we return to the original FRB proposal by using Haar random gates. As there are no restrictions on the recovery gate, iFRB applied to any gate. Since Haar random gates are trivially a unitary 2-design, iFRB carries the same theoretical guarantees of RB and iRB that the noise is gate-independent. For the more general Markovian and possibly gate-dependent noises, whether the iFRB framework works as expected requires further investigation.

\begin{figure}
    \centering
    \begin{subfigure}{\textwidth}
    \begin{tikzpicture}
\node {
    \begin{quantikz}
             \lstick{\ket{0}} &\gate[style={fill=green!20}]{C_1}&\gate[style={fill=green!20}]{C_2}  & \ \ldots \ \qw&\gate[style={fill=green!20}]{C_m}& \gate[style={fill=blue!20}]{C_{m+1}}& \meter{0}
             \end{quantikz}};\end{tikzpicture} \\\begin{tikzpicture}

             \node{

             \begin{quantikz}
             \lstick{\ket{0}} &\gate[style={fill=green!20}]{C'_1}&\gate[style={fill=yellow!20}]{T_1}&\gate[style={fill=green!20}]{C'_2}  &\gate[style={fill=yellow!20}]{T_2} &\ \ldots \ \qw &\gate[style={fill=green!20}]{C'_m}&\gate[style={fill=yellow!20}]{T_m}& \gate[style={fill=blue!20}]{C'_{m+1}}& \meter{0}
             \end{quantikz}};
             \end{tikzpicture}
             \caption{Clifford-based iRB. $C_1,\dots, C_m$ and $C'_1,\dots, C'_m$ are taken i.i.d. uniformly from the Clifford group $\mathcal{C}_d$.}
    \end{subfigure}
    \begin{subfigure}{\textwidth}
    \begin{tikzpicture}
\node {
    \begin{quantikz}
             \lstick{\ket{0}} &\gate[style={fill=orange!20}]{U_1}&\gate[style={fill=orange!20}]{U_2} &\ \ldots \ \qw &\gate[style={fill=orange!20}]{U_m}& \gate[style={fill=blue!20}]{U_{m+1}}& \meter{0}
             \end{quantikz}};\end{tikzpicture}\\\begin{tikzpicture}
\node{

             \begin{quantikz}
             \lstick{\ket{0}} &\gate[style={fill=orange!20}]{U'_1}&\gate[style={fill=yellow!20}]{T}&\gate[style={fill=orange!20}]{U'_2}  &\gate[style={fill=yellow!20}]{T}&\ \ldots \ \qw &\gate[style={fill=orange!20}]{U'_m}&\gate[style={fill=yellow!20}]{T}& \gate[style={fill=blue!20}]{U'_{m+1}}& \meter{0}
             \end{quantikz}};
             \end{tikzpicture}
             \caption{Interleaved Fully Randomized Benchmarking. $U_1,\cdots, U_m$ and $U'_1,\dots, U'_m$ are taken i.i.d. Haar randomly from $SU(d)$.}
    \end{subfigure}
    \caption{Illustration of the Clifford-based iRB and the iFRB.}
\end{figure}

The biggest challenge of implementing such a fully randomized scheme is of course the efficient generation and implementation of arbitrary rotations. It is long known that the complexity of implementing Haar random gates grows exponentially with respect to the number of qubits\cite{knill1995approximation}. However, FRB/iFRB can still be a very useful tool to benchmark single qubit or two-qubit gates, or even unitaries acting on a small number of qubits. In superconducting systems, 1-qubit FRB/iFRB is readily realized via the virtual Z compilation scheme~\cite{mckay2017efficient}. On two-qubit systems, the FRB/iFRB framework requires the efficient generation of arbitrary two-qubit unitaries from native two-qubit gates. Luckily, for many families of gates, such as the super-controlling gates\cite{ye2004super} and the $\mathrm{SQ\lowercase{i}SW} $  gate, an efficient decomposition of arbitrary two-qubit gates into an optimal number of native two-qubit gate exists. Hence we can realize two-qubit FRB/iFRB in such cases.

\begin{algorithm}[H]
  \caption{Generating iFRB sequences}
  \label{alg:randgen}
   \begin{algorithmic}[1]
   \Procedure{GenRandSeq}{m,t} \Comment{Generate a random FRB sequence of length $m$, interleaved with $\mathrm{SQ\lowercase{i}SW} $ iff $t$ is True}
   \State $U\leftarrow I$
   \State $S\leftarrow \varepsilon$
   \For{$i=1,\dots, m-1$}
   \State $U_i, S_i\leftarrow$\textsc{GenRandGate}()
   \If{t == True}
   \State $U\leftarrow \mathrm{SQ\lowercase{i}SW}  \cdot U_i\cdot U $
   \State $S\leftarrow S || S_i||\mathrm{SQ\lowercase{i}SW} $
   \Else
   \State $U\leftarrow U_i\cdot U $
   \State $S\leftarrow S || S_i$
   \EndIf
   \EndFor
   \State\Return  $S ||$ \textsc{Decomp}($U^\dag$)\Comment{Append the recovery gate}
   \EndProcedure
   \Procedure{GenRandGate}{} \Comment{Generate a Haar random $SU(4)$ gate and its corresponding decomposition into $\mathrm{SQ\lowercase{i}SW} $  sequence}
   \State $U\leftarrow$\textsc{GenHaarSU(4)} \Comment{Generate a Haar random element in $SU(4)$}
   \State \Return $U,$ \textsc{Decomp}($U$)
   \EndProcedure
   \end{algorithmic}
\end{algorithm}



\section{Summary and future work}

In this report, we investigated the properties of the square-root-iSWAP gate, or $\mathrm{SQ\lowercase{i}SW} $  for short. Being the matrix square root of the $\mathrm{\lowercase{i}SWAP} $ gate, the $\mathrm{SQ\lowercase{i}SW} $  gate can be readily implemented on any quantum computing platform with the ability to implement the $\mathrm{\lowercase{i}SWAP} $ gate. Moreover, taking only roughly half of the time of the $\mathrm{\lowercase{i}SWAP} $ gate, the $\mathrm{SQ\lowercase{i}SW} $  gate can be implemented with ultra-high fidelity. At the same time, it surprisingly still possesses powerful compilation capabilities. The $\mathrm{SQ\lowercase{i}SW} $  gate outperforms both the $\mathrm{\lowercase{i}SWAP} $ and the $\mathrm{CNOT} $ gates in terms of W-like state generation with respect to gate count and in other more standard compilation tasks if the higher fidelity of $\mathrm{SQ\lowercase{i}SW} $  is taken into account.

Our systematic study of  $\mathrm{SQ\lowercase{i}SW} $ supports the use of the gate as a native two-qubit gate in both NISQ and fault-tolerant quantum computing. For example, the $\mathrm{SQ\lowercase{i}SW} $  gate is advantageous in compiling Haar random qubit gates compared to the more conventional $\mathrm{\lowercase{i}SWAP} $ or $\mathrm{CNOT} $  gates, which could be useful for variational algorithms requiring a continuous family of quantum gates. Moreover, it takes the same number of $\mathrm{SQ\lowercase{i}SW} $ and $\mathrm{\lowercase{i}SWAP} $ gates to compile a $\mathrm{CNOT} $  gate and a $\mathrm{SWAP} $  gate respectively. Taking into account the higher fidelity of $\mathrm{SQ\lowercase{i}SW} $, this provides evidence that the $\mathrm{SQ\lowercase{i}SW} $  gate is more suitable than the $\mathrm{\lowercase{i}SWAP} $ gate in realizing certain quantum error correction tasks. However, the analysis of the compilation properties of the $\mathrm{SQ\lowercase{i}SW} $  gate is far from thorough, as its applicability to specific NISQ algorithms or fault-tolerant computing tasks remains unclear.

The fully randomized benchmarking / interleaved FRB schemes are general schemes for benchmarking arbitrary gates, assuming that the gate can be implemented efficiently. Although such schemes face the same scale-up issues of ordinary Clifford-based RB, it is still useful in characterizing and benchmarking operations on a few qubits, in particular single-qubit and two-qubit rotations. Just like the arbitrary single-qubit gate compilation schemes in~\cite{chen2021compiling}, the FRB scheme needs to be verified in experiment. From a theoretical perspective, it is also important to investigate the exponential decay signature of the FRB experiment under different gate noises. A recent work~\cite{helsen2020general} proved that exponential decay, or a linear combination of exponential decays, can be observed when the implementation of the gate is not too far away from an ideal implementation. However, that result is based on the fact that one works with a finite gate set, a different scenario than the FRB, where Haar random arbitrary gates are needed. We leave it to future work to study the mathematical soundness of the FRB protocol.

\begin{acknowledgments}
We would like to thank Tenghui Wang, Casey Duckering and Jun Zhang for insightful discussions.  DD would like to thank God for all of His provisions. LK is supported by NSF grant CCF-1452616.
\end{acknowledgments}

\label{sec:summary}
