


\section{Other approaches to reference optimization and interpolation}\label{sec:other-approaches}

In this paper, we demonstrate our procedure using the neighbor-average Tikhonov regularization method and simple piecewise-linear interpolation. However, the general idea of creating reference pulses to interpolate between could have many possible implementations. Below, we list several alternative ways of realizing this methodology.

\begin{itemize}
    \item \emph{Nonuniform reference point distribution}. Reference points need not be distributed on a rectangular grid; the Delaunay triangulation method will work just as well with a nonuniform distribution of points. It may be effective to use more densely-packed reference points in certain parts of parameter space to give better interpolations, or less dense points in other areas to improve calibration efficiency.

    \item \emph{Different guess methods for Tikhonov re-optimization}. We used the simple neighbor-average method to generate the initial guesses for our Tikhonov regularization re-optimization approach. However, our general approach could also be applied using guesses generated in some other way. For example, each new reference point guess could be a weighted sum of all reference pulses, with weights determined by some notion of distance between the operations.

    \item \emph{Selective re-optimization}. Some reference points likely do not need to be re-optimized if they already have a high degree of similarity to neighboring reference pulses; computation time could be easily improved by only re-optimizing the points that differ from nearby pulses by more than some amount.

    \item \emph{Higher-order interpolation fits}. Regardless of optimization method, different methods can be used for the interpolation step of the procedure. In this paper, we show results using a simple piecewise-linear interpolation. Alternatively, interpolation could be done using a higher-order model such as a spline or other function.
    
   
    
    \item \emph{Minimal-curvature optimization and interpolation}. Instead of running individual optimizations at each reference point, all reference pulses could be tuned at once in one large optimization. We imagine a cost function consisting of the standard infidelity term, averaged over all reference points, as well as a new term that is proportional to the total curvature (or some other metric) of an interpolation function fitted to the reference point pulses. The interpolating function would then be built into the optimization directly, potentially yielding more accurate results.
\end{itemize}

We do not perform these other interpolation methods for this paper due to the high effectiveness of the simpler linear model and lack of a clearly-motivated choice of a more intricate method. However, our general framework is flexible and will work with any optimization or interpolation method, and we anticipate that more complex methods could potentially yield better results (depending on the system and gate family of interest).

\section{Extension: variable pulse duration}\label{sec:duration-scaling}

In this work, we assumed that all pulses have the same fixed duration, regardless of the target quantum operation. In reality, it is much easier to perform a gate that is close to the identity than a more complex operation such as SWAP. We expect that these easier gates could be accomplished with a shorter duration within the same pulse amplitude bounds, compared to the more complex gates in the family.

Depending on the chosen pulse description, it can be possible to include the duration of the pulse directly as an optimizable quantity. The duration is then an additional element in the pulse vector and can be included in the interpolation.

Alternatively, when constrained to a specific duration as in this work, we expect that some easier operations need not make use of the full allowed amplitude of the pulse. In this case, for some Hamiltonians, it is possible to upscale the amplitude and downscale the duration equivalently to obtain a shorter pulse with the same effect. A new cost function term could be added to the optimizations to penalize the maximum value of the control field to improve the amount of downscaling possible.

\begin{figure*}[t]
    \centering
    \includegraphics[width=\textwidth]{comparison.eps}
    \caption{Infidelities at 2197 test points within the $[0,1]^3$ Cartan coordinate box for 0 to 3 rounds of neighbor-average re-optimization. Average and worst-case infidelities are shown in Table \ref{tab:cartan-infidelities-SI}.}
    \label{fig:comparison-SI}
\end{figure*}


\section{General methodology: coordinated re-optimization}

The methodology for applying our general approach (coordinated re-optimization of individual reference pulses) to an arbitrary family of quantum operations is as follows:

\begin{enumerate}
    \item \emph{Setup.} If using a model-based optimizer, obtain a model of the device, such as a Hamiltonian. Choose a pulse description $\vec \alpha$ (a finite set of variables used to construct each pulse) and a pulse optimization algorithm. Define the parameters of the gate family and determine the space to interpolate within. Sample a number of parameter points $\{p_i\}$ from this space and obtain the corresponding quantum operations. 

    \item \emph{Initial optimization.} Use the optimizer to generate initial reference pulses $\vec \alpha_i$ for each reference point.

    \item \emph{Re-optimization.} For each reference point $p_i$: 
    
    \begin{enumerate}
        \item Calculate some new target pulse $\vec \alpha_{0,i}$ based on the set of existing reference pulses.
        
        \item Re-optimize the reference pulse with this target pulse as the initial guess. Use Tikhonov regularization in the cost function (or some other method) to encourage the final pulse to be close to the target pulse.
    \end{enumerate} 

    Repeat this step as needed.

    \item \emph{Interpolation.} Choose an interpolation function $f:(\widetilde p, \{p_i, \vec \alpha_i\}) \to \vec \alpha_{\widetilde p}$ that calculates interpolated pulse $\vec \alpha_{\widetilde p}$ at parameter-space point $\widetilde p$ given the set of optimized reference points and pulses $\{p_i, \vec \alpha_i\}$.
    
\end{enumerate}

\section{Details on Weyl chamber examples from Main Text}

For all examples, we fix the pulse duration to $\pi$ and parameterize each of the 5 control functions $f(t)$ as a piecewise-constant function of 6 segments, yielding 30 total optimizable parameters. We set the Tikhonov regularization weight $\lambda = 10^{-2}$. The pulse optimizer is run for a maximum of 50 iterations for each individual optimization (or re-optimization), although it often terminates early upon reaching convergence, especially for later rounds of re-optimization where the pulse shapes do not change as drastically. We found this to be sufficient for convergence over multiple re-optimization rounds, and it also avoids unnecessary computation during early rounds of optimization (where the pulse shapes need not be finalized). We use the default Q-CTRL convergence criteria \cite{ball_software_2021}. We use various point granularities to uniformly cover the parameter space of interest; the number of points generated by grids of various granularities (for both the Weyl chamber and the larger parameter-space box used to compare to \cite{sauvage_optimal_2022}) are shown in Table \ref{tab:granularities-SI}.

For the example used in Figures 1 and 2 in the Main Text, we use a reference point granularity of 1/4 and a test point granularity of 1/16. We initialize the reference points with naive pulse optimizations, run three consecutive rounds of neighbor-average re-optimization, and use the initial and final reference pulses to extract the data shown in the two figures. 

To demonstrate the relationship between computational cost and average infidelity (Figure 3 in the Main Text), we use reference point granularities ranging from 1/2 to 1/8 and always use test point granularity of 1/24. We run three rounds of re-optimization for each reference point granularity. After initialization and after each round of re-optimization, we record the average and maximum infidelities achieved over all test points. These values are displayed in Figure 3 of the Main Text. Table \ref{tab:weyl-infidelities-SI} shows specific values for the average and worst-case infidelities achieved after the final round of re-optimization. Increasing reference point granularity appears to significantly improve interpolation quality, but at the cost of more pulse optimizer iterations.


\section{Comparison to existing methods}

In this section, we describe our methodology for comparing the approach presented in this work with that of \cite{sauvage_optimal_2022}, which to the best of our knowledge is the only other work that addresses the problem of pulse generation for multi-parameter continuous sets of gates. It is critical to minimize the computational overhead of this problem to enable a future experimental implementation.

\subsection{Setup}

We specifically compare 3-parameter interpolation for the 2-qubit gate family described by
\begin{equation} \label{eq:cartan-SI}
    U = \exp\Bigg(-i \frac \pi 2\sum_{j=x,y,z} t_j \sigma_j^{(1)} \sigma_j^{(2)}\Bigg)
\end{equation}
where $t_x,t_y,t_z$ are the Cartan coordinates of $U$. This is the same unitary as used in the Main Text for the Weyl chamber example, but for a fair comparison with the results of Ref. \cite{sauvage_optimal_2022}, we extend the parameter bounds to $t_x,t_y,t_z \in [0,1]$. This occupies a volume 24 times larger than the Weyl chamber in parameter space. However, we note that the Weyl chamber alone can already represent any two-qubit operation (up to single-qubit operations), so this extra volume in parameter space only serves to increase computational complexity without adding any practical benefit. 

We discretize each of the five control pulses into 6 piecewise-constant segments of equal duration. As in the main paper, we use the Hamiltonian 

\begin{align}\label{eq:hamiltonian-SI}
    H (t) &= f^{\vec \alpha}_{xx}(t) \sigma_x^{(1)} \sigma_x^{(2)} + \sum\limits_{j=2}^2 f^{\vec \alpha}_{jy}(t) \sigma_y^{(i)} + f^{\vec \alpha}_{jz}(t) \sigma_z^{(i)}
\end{align}
and set the pulse duration to $\pi$.

We pick reference points on a grid in parameter space with spacing 1/6. We find that the optimization and interpolation methods discussed in the Main Text provide good results with little need for fine-tuning.


\subsection{Results}

Interpolation results are shown in Figure \ref{fig:comparison-SI} for various numbers of re-optimization rounds. Infidelities are evaluated on a grid in parameter space with granularity 1/12. Large regions of parameter space appear to already interpolate reasonably well; we attribute this to the constraints we have imposed on the pulses, namely the small number of optimizable parameters (each pulse consisting of only 6 piecewise-constant segments) and the initial Tikhonov regularization that favors low-amplitude pulses.

Using a neural network, Ref. \cite{sauvage_optimal_2022} reports average pulse infidelity of $4 \pm 4 \times 10^{-4}$ while using 51,200 system evolutions to train the network. As shown in Table \ref{tab:cartan-infidelities-SI}, our method reaches an average pulse infidelity of $3.55 \pm 6.72 \times 10^{-4}$ in 34,407 total iterations, which can further be refined to $2.54 \pm 4.33 \times 10^{-4}$ in 41,289 total iterations. Each iteration corresponds to a system evolution.

This demonstrates that our method can obtain similar or better results than previous methods using less computation, with the additional benefits (as discussed in the Main Text) of improved explainability and modularity.

We expect that computation time could be further reduced in several simple ways, if desired. For example, the cost value convergence threshold of the pulse optimizer could be reduced, since the reference pulses are re-optimized several times and intermediate results are thus not required to be perfect. Additionally, as mentioned in the first section of this Supplemental Material, some reference points may not need to be re-optimized as many times as others.
