\section{Introduction}

Topology in quantum many-body systems has been at the forefront of condensed matter research in recent years~\cite{quantum_info_meets_quantum_matter}. Topological invariants allow us to classify the ground states of gapped local Hamiltonians into distinct phases~\cite{ent_spectrum_Pollmann, RevModPhys.88.035005,RevModPhys.89.041004,PhysRevB.83.035107, PhysRevB.84.235128}, characterized by non-local order parameters, and displaying exotic properties, such as anyonic excitations or gapless edge states. A particularly rich phase diagram arises in the presence of symmetries, displaying various symmetry-protected topological (SPT) phases that cannot be characterized in terms of the spontaneous breaking of a global symmetry~\cite{science.1227224, annurev-conmatphys-031214-014740}. Instead, they are characterized by entanglement patterns between subsystems~\cite{ent_spectrum, ent_spectrum_Pollmann}, captured by a topological entanglement entropy~\cite{topent_PhysRevLett.96.110404, topo_ent_entropy}, as well as non-local ``string order''~\cite{string_order, detection_spt_string_order}.

Recently, the concept of SPT phases has been extended from equilibrium systems to non-equilibrium scenarios~\cite{Lavasani2021} in the context of measurement-induced entanglement transitions in quantum random circuits~\cite{review_random_circuits, EPT1,Li_2019,  EPT2, EPT3}. In these quantum circuits, the time evolution is governed by a competition between random unitary gates, spreading information and tending to scramble the system, and repeated local measurements, reducing entanglement. The interplay of these opposing effects leads to dynamical phase transitions between different stationary states:  a highly entangled thermal state characterized by a volume law scaling of sub-system entanglement entropy, and non-thermal area law states~\cite{review_random_circuits, EPT1,Li_2019,  EPT2, EPT3, A5, A7, EPT4, A3, A1, A6, A8, A9, A10, A11, A13, MIPT1, MIPT2, A2, A12, MIPT3,Khemani_PhysRevLett.126.060501,Grover_PRXQuantum.2.040319, Sierant_2022, MIPT4, MIPT5, MIPT6, Khemani_PhysRevX.12.011045}, with different types of measurements generating novel non-equilibrium phases of matter~\cite{Lavasani2021, Sang_2021, Klocke_2022, Lavasani2, Lavasani3, Wampler_2022, SPT_measurement_only}. In particular, in Ref.~\cite{Lavasani2021}, the authors have demonstrated that the area law stationary state can also exhibit SPT order, similarly to the area entangled ground states of gapped Hamiltonians.

These recent advances raise exciting questions about the measurement-induced topological phases in quantum circuits. The SPT phase found in Ref.~\cite{Lavasani2021} emerged in a Clifford quantum random circuit model in the presence of a protecting $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry, and was detected through a topological entanglement entropy. Generalizing this construction to other types of topological phases, as well as finding and classifying the topological phases accessible in out-of-equilibrium systems can offer new insights into the properties of dynamical phase transitions. Another interesting aspect concerns the order parameter of the phase transition. Clifford random circuits have a special structure, allowing to simulate them efficiently by relying on the stabilizer formalism~\cite{AaronsonGottesman}. In this setting, the topological entanglement entropy is well-suited for detecting SPT order in the numerical calculations. However, the topological entanglement entropy is very difficult to access in experimental realizations. For this reason, it is important to identify other, more accessible order parameters. Another open question is the  stability of the SPT phase in the wider class of Haar random circuits.

In this paper, we take the first steps towards answering these questions. We generalize the construction of Ref.~\cite{Lavasani2021} to generate the whole family of generalized cluster models~\cite{generalized_cluster_models, Verresen17}. This extended set of random circuits hosts different types of SPT phases, as well as phases with simultaneous SPT order and spontaneous symmetry breaking (SSB). We also construct a set of non-local string order parameters, and demonstrate that they are capable of distinguishing the different phases realized by the circuits, thereby providing a convenient alternative to topological entanglement entropy that is more accessible both numerically and experimentally~\cite{Sompet_2022}. To this end, we analyze two members of the family of generalized cluster models in detail, by combining simulations in the stabilizer formalism with tensor network methods. First, we focus on the XZX model already examined in Ref.~\cite{Lavasani2021}, and confirm that the string order parameter can be used the determine the full phase diagram. By relying on tensor network simulations, we also show that the phase diagram is remarkably stable against extending the class of unitaries from random Clifford to random Haar, provided that the protecting $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry is respected by the gates. Secondly, we consider the so-called XZZX cluster model, and demonstrate that it hosts a phase with coexisting SPT order and SSB. We determine the full phase diagram of this model within the stabilizer formalism, by evaluating the string order parameters and a connected correlator capturing SSB.


The paper is organized as follows. We first discuss the general theoretical framework in Sec.~\ref{sec:framework}. Here we introduce a set of quantum random circuits, realizing an out-of-equilibrium version of the whole family of generalized cluster models, and we also construct string order parameters capturing the SPT order in the area law stationary states of these circuits. We then turn to the XZX cluster model in Sec.~\ref{sec:xzx}. First, we focus on Clifford random circuits in Sec.~\ref{subsec:clifford}, and we validate the string order parameter proposed before, by using it to obtain the full phase diagram, and comparing it to predictions relying on entanglement entropies from Ref.~\cite{Lavasani2021}. We test the stability of this phase diagram by extending the class of random unitary gates from Clifford to Haar random gates in Sec.~\ref{subsec:haar}. We then demonstrate the coexistence of SPT order with spontaneous symmetry breaking by studying the XZZX cluster model in Sec.~\ref{sec:xzzx}. We summarize our main conclusions in Sec.~\ref{sec:conclusion}.


\section{General framework}\label{sec:framework}

In this section, we construct a family of quantum random circuit models that will be the main focus of this paper. We first review the equilibrium definition of generalized cluster models. Then, relying on the insights gained from Ref.~\cite{Lavasani2021},  we turn to the non-equilibrium scenario and formulate the random circuit models that realize their out-of-equilibrium counterparts. We argue that these models display various dynamical phases with SPT order and/or SSB, and we also construct a set of non-local string order parameters and SSB local order parameters allowing us to determine the full phase diagram.

The family of generalized cluster models in a one-dimensional spin chain~\cite{generalized_cluster_models, Verresen17} is generated by Hamiltonians of the form
\begin{equation}
\label{eq:hamiltonians}
    H_\alpha = -\sum_n X_n\underbrace{Z_{n+1}...Z_{n+\alpha-1}}_{\alpha-1}X_{n+\alpha}, \quad \alpha \geq 1,
\end{equation}
where $X_n$, $Y_n$, and $Z_n$ denote the Pauli matrices at site $n$, and $\alpha$ is a positive integer parametrizing the members of the class. For a given $\alpha$, the model is symmetric under a set of $\alpha$ global symmetries,
\begin{equation}\label{eq:symmetries}
    G_1=\prod_k Z_{\alpha k +1},\, ... ,\, G_{\alpha} = \prod_k Z_{\alpha k + \alpha}.
\end{equation}
Each of the symmetries $G_i$ is a product of $Z$ operators, distanced by $\alpha-1$ sites in the chain. 

All terms in the Hamiltonian $H_\alpha$, the so-called cluster operators $g^\alpha_n$, commute with each other, and thus the ground states are defined by the condition $g_n^\alpha\ket{\psi^\alpha_0} = \ket{\psi^\alpha_0}$ for every $n$. Such ground states realize different types of phases, with SSB and/or SPT order~\cite{generalized_cluster_models, Verresen17}. For example, $\alpha=1$ corresponds to the $\mathbb{Z}_2$ symmetric Ising chain, displaying SSB. The $\alpha=2$ case is the so-called cluster model that realizes an SPT phase, protected by $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry~\cite{Son_2011_spt_cluster, PhysRevA.84.022304_spt_cluster, zeng2016topological}. For $\alpha=3$, Eq.~\eqref{eq:hamiltonians} defines the $XZZX$ cluster model with coexisting SSB and SPT orders~\cite{zeng2016topological, Verresen17}. In general, every odd  $\alpha$ value is characterized by $\mathbb{Z}_2$ symmetry breaking and $\mathbb{Z}_2^{\times (\alpha-1)}$ SPT order (except for $\alpha=1$), while even integers yield pure SPT order protected by a $\mathbb{Z}_2^{\times \alpha}$ symmetry.

\begin{figure}[t!]
    \centering
    \includegraphics[width=0.48\textwidth]{xzx_xzzx_circuits.pdf}
    \caption{A particular realization of the cluster circuit model for (a) $\alpha=2$ and (b) $\alpha=3$, showing the first two time steps for system size $N=6$. The  boxes labelled as $Z$, $XZX$ and $XZZX$ represent projective measurements, whereas the ones with label $U$ denote random unitary gates preserving the symmetries ~\eqref{eq:symmetries}.}
    \label{circuit_models}
\end{figure}


In Ref.~\cite{Lavasani2021}, Lavasani et al. showed that the SPT phase of the $XZX$ cluster model, $\alpha=2$, can be realized in a quantum random circuit by implementing properly designed measurements. Here we generalize this construction to induce by measurements the SPT phases that are realized by the generalized cluster models \eqref{eq:hamiltonians}. To this end, we construct a set of random circuit models in the following way. We consider a chain of $N$ qubits subject to open boundary conditions, with an initial state that can  be an arbitrary eigenstate of the symmetry operators $G_1,...,G_\alpha$, e.g., the trivial product state $\ket{Z=1}^{\otimes N}$. At each step, we update the state by applying a sequence of three different operations:
\begin{enumerate}
    \item With probability $p_t$, we measure a cluster operator $X_iZ_{i+1}...Z_{i+\alpha-1}X_{i+\alpha}$, with $i\in\{1,...,N-\alpha\}$  chosen randomly.
    \item With probability $p_s$, we measure a single-qubit operator $Z_i$ on a random site $i$  of the chain.
    \item With probability $p_u=1-p_s-p_t$, a random unitary preserving the protecting symmetries~\eqref{eq:symmetries} and acting on $\alpha+1$ neighboring qubits is sampled and applied at a random position. The length of these  unitaries, $\alpha+1$, was chosen as the shortest range such that the gates can create  entanglement while preserving the symmetries of the model.
\end{enumerate}
The collection of $N$ such operations form a single \textit{time step} in the time evolution of the system. Figure \ref{circuit_models} illustrates this construction by showing the first two time steps in a particular realization of the circuit for $\alpha=2$ (left) and $\alpha=3$ (right).  We denote the \textit{circuit average} of a quantity $A$, measured in the steady state of the circuit, by $\overline{A}$. We note that in this setting, time average is equivalent to ensemble average, therefore we can choose to perform the averaging  over time steps and/or over circuit realizations.

This family of circuit models realizes three different phases for each $\alpha$, depending on the dominant operation. For large enough probability $p_u$, the unitary evolution dominates, resulting in a volume law phase  with the entanglement entropy of subsystems  scaling with their volume. In contrast, if measurements are applied at a sufficiently high rate, the stationary state is characterized by area law entanglement scaling. In particular, for large enough $p_s$, the $Z$ measurements  tend to collapse the state to  a trivial area law phase, whereas the measurements of cluster operators $X_iZ_{i+1}...Z_{i+\alpha-1}X_{i+\alpha}$ may induce  area law phases with symmetry-protected topological phases, and/or ordered phases with a local order parameter. For example, the case $\alpha=1$ has been studied in Ref.~\cite{Sang_2021}, and was found to realize the out-of-equilibrium counterpart of the SSB order in the ground state of the Ising Hamiltonian for large enough $p_t$, dubbed a spin glass phase. For $\alpha=2$, Lavasani et al. demonstrated the emergence of an SPT phase~\cite{Lavasani2021}.


Before investigating the phase diagram of the first few members of this family of random circuit models in detail, we comment on the order parameters that can distinguish different phases. One indicator that has been successfully applied in previous works is a topological entanglement entropy, readily accessible in numerical simulations for Clifford circuits, relying on the stabilizer formalism. However, in simulations of more general random circuits, for instance in the presence of general Haar unitary gates $U$, as well as in possible experimental realizations, topological entanglement entropies are challenging to access. Therefore, we propose another way to distinguish the different phases, by extending the concept of string order parameters, designed to detect equilibrium SPT phases~\cite{string_order,detection_spt_string_order}, to this out-of-equilibrium scenario. In general, for a state that is invariant under a protecting symmetry $G$ that can be expressed as the product of local unitary operators $\Sigma_i$, $G=\prod_i \Sigma_i$,  a string order parameter corresponding to boundary operators $O^{L/R}$ can be defined as follows~\cite{string_order,detection_spt_string_order},
\begin{equation}\label{eq:string}
    \mathcal{S}_\Sigma^{O^L, O^R} = \lim_{|j-k|\rightarrow\infty} \ev**{O^L(j)\left(\prod_{i=j+1}^{k-1}\Sigma_i\right)O^R(k)}{\psi_0}.
\end{equation}
These string order parameters allow to differentiate topologically distinct states, by choosing the operators $O^{L/R}$ appropriately. Importantly, the bulk part of the string operator is constructed from the symmetry operators $\Sigma_i$. Therefore, for a symmetric state, and for generic operators $O^{L/R}$, the string order parameter  takes a non-zero value and varies smoothly as a function of the parameters of the Hamiltonian.  In order to distinguish topologically distinct phases, the boundary operators $O^{L/R}$ have to be chosen carefully, such that the emerging SPT order gives rise to selection rules that ensure the vanishing of a string order in a certain phase~\cite{detection_spt_string_order}. This method thus allows to detect topological phases through the exact relation $\mathcal{S}_\Sigma^{O^L, O^R}\equiv 0$. Choosing various pairs $O^{L/R}$, such that the resulting string orders $\mathcal{S}_\Sigma^{O^L, O^R}$ vanish in different phases, grants access to the full phase diagram. Convenient string order parameters for the generalized cluster models are given by
\begin{equation}
\label{eq:triv_string}
    \mathcal{S}_{Z}^{\mathds{1}, \mathds{1}}(\alpha) = \lim_{|j-k|\rightarrow\infty} \expval{\prod_{i=j + 1}^{k}Z_{\alpha i}},
\end{equation}
vanishing in the ground state of the cluster Hamiltonian, as well as
\begin{equation}
\small
\label{eq:spt_string}
    \mathcal{S}_{Z}^{X, X}(\alpha) = \lim_{|j-k|\rightarrow\infty} \expval{X_{\alpha j}\left(\prod_{i=j + 1}^{k}Z_{\alpha i-\alpha+1}...Z_{\alpha i-1}\right)X_{\alpha k}},
\end{equation}
yielding zero for a trivial $Z$ product state.

Besides the SPT order, for odd $\alpha$ values the ground state of Hamiltonian~\eqref{eq:hamiltonians} displays SSB.  This type of order can be detected through the connected correlators of  a symmetry breaking local order parameter, defined as
\begin{equation}
\label{eq:local_order_parameter}
    M(\alpha)=X_i\underbrace{(Y_{i+1}X_{i+2})...(Y_{i+\alpha-2}X_{i+\alpha-1})}_{(\alpha+1)/2}.
\end{equation}
We note that for $\alpha=1$, $M(\alpha)$ reduces to the Ising order parameter $X_i$.

While the various order parameters defined in Eqs~\eqref{eq:triv_string}, ~\eqref{eq:spt_string} and ~\eqref{eq:local_order_parameter} are well-suited for determining the phase diagram of the generalized cluster models in equilibrium, they are not directly applicable for the random circuit scenario. The reason for this is that for different realizations of the disordered circuit, the sign of string order parameters and correlation functions fluctuates randomly, yielding a vanishing circuit average in all non-trivial regions of the parameter space. Similarly, the time average of all of these quantities vanishes for any fixed circuit realizations. This property can be understood by noting that all the measured operators have eigenvalues $\pm1$, resulting in a randomly changing sign during the time evolution because of the probabilistic measurement outcomes and the repeatedly applied random unitary gates. 

The vanishing of these circuit averages is also intimately related to the nature of the measurement-induced entanglement transition. This dynamical phase transition is unconventional in the sense that it relies on the properties of individual quantum trajectories instead of the disorder averaged quantum state, and it can only be detected through quantities that are non-linear in the density operator of the system, e.g. through entanglement entropies~\cite{review_random_circuits,Li_2019}. In contrast, the circuit average of the density matrix is a trivial infinite temperature density matrix. Therefore, the rich entanglement structure of individual quantum trajectories remains hidden at the level of the average density matrix, and consequently, at the level of disorder averaged operator averages, such as the string order parameters or connected correlators considered above. 

This difficulty can be overcome by modifying the proposed order parameters in such a way that they become non-linear in the density matrix of the system. This can be easily achieved by considering the time and/or circuit average of the \textit{absolute value} of the string order parameters, $\overline{|\mathcal{S}|}$, with a similar idea applied to the correlator of the local order parameters $M(\alpha)$. We test these proposed order parameters below by benchmarking them against the behavior of the topological entanglement entropy for various circuit models, and demonstrate that they capture the full phase diagram correctly.

Before turning to the numerical simulations, we briefly comment on the special case of $p_u=0$, resulting in models without unitary evolution. We find that circuits, consisting only of projective measurements, share certain universal properties for any value of $\alpha$. In particular, they all display a phase transition between the trivial and the SPT and/or SSB area law phase for $p_s=1/2$, i.e., where the rates of both types of measurements are equal. A proof of this statement, relying on a duality argument, is presented in Appendix~\ref{appendix_A}.

\section{SPT phase in the XZX cluster circuit model}\label{sec:xzx}

In this section, we revisit the SPT phase of the $XZX$ cluster circuit model, by applying the framework presented in Sec.~\ref{sec:framework}. We first focus on Clifford circuits in Sec.~\ref{subsec:clifford}, already studied in Ref.~\cite{Lavasani2021}. In this special set of circuits both topological entanglement entropies and string order parameters are readily accessible by relying on the stabilizer formalism. We benchmark the string order parameters~\eqref{eq:triv_string} and~\eqref{eq:spt_string} by using them to detect the phase transitions, and comparing  the phase diagram to the one obtained from entanglement entropies. We then turn to more general Haar random circuits in Sec.~\ref{subsec:haar}, extending the class of random circuits compared to Ref.~\cite{Lavasani2021}, and testing the stability of the phase diagram against allowing a wider set of unitary gates during the time evolution. Here the stabilizer formalism is no longer applicable. Instead, we rely on efficient tensor network methods to simulate the dynamics, and extract the string order parameters~\eqref{eq:triv_string} and~\eqref{eq:spt_string} in order to determine the full phase diagram. We find that the phase boundaries remain strikingly stable against extending the class of unitary gates, yielding a phase diagram that is essentially identical to the one obtained for Clifford circuits.

\subsection{Time evolution with Clifford unitary gates}\label{subsec:clifford}

\begin{figure}[t!]
    \centering
    \includegraphics{string_SOP_XZX_2.pdf}
    \caption{String order parameters in a Clifford circuit for $\alpha=2$. (a) String operators~\eqref{eq:spt_string} and~\eqref{eq:triv_string} in a finite lattice, characterized by the same bulk operator (light shading), and distinguished by the different boundary operators (dark shading). (b) String order parameters for measurement-only dynamics $p_u=0$, shown as a function of the probability of single qubit measurement $p_s$ in the vicinity of  the self-dual point $p_s=0.5$. String order parameters distinguish two area law phases, one with SPT order (purple) and a trivial one (orange). The vertical line indicates the exact critical point of the phase transition, $p_s=0.5$. (c) String order parameters in the presence of a finite rate of unitary gates, $p_u=0.3$, shown as a function of $p_s$ across the phase boundaries. Unitary gates are Clifford unitaries respecting  the protecting $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry. String order parameters distinguish an SPT (purple) and a trivial (orange) area law phase, separated by a volume law region (green). Numerically determined phase boundaries are indicated by vertical dashed lines. In figures (b) and (c) we used system size $N=1024$.}
    \label{fig:XZX_Clifford}
\end{figure}


In this section, we reproduce the phase diagram of Ref. ~\cite{Lavasani2021}, by relying on the string order parameters instead of a topological entanglement entropy. The special structure of  Clifford unitary gates allows us to efficiently simulate large system sizes, up to $N=1024$ qubits, by applying the stabilizer formalism. This method relies on representing the wave function of the system for a given circuit realization as the eigenstate of $N$ linearly independent (under multiplication) commuting Pauli strings, the so-called stabilizers. Both Clifford unitaries and projective measurements preserve this structure by mapping the stabilizers to another set of $N$ independent commuting Pauli strings, allowing to simulate the time evolution efficiently \cite{AaronsonGottesman}. The circuit averages are then obtained by performing the averaging over the stationary states of  $10^3$ random circuits. For each of these circuits, the qubits are initialized in the state $\ket{Z=-1}^{\otimes N}$, and let to evolve for $2N$ time steps (corresponding to $2N^2$ operations) to reach the steady state. Then, an additional time averaging is performed by evolving the system for another $10^3$ time steps, calculating the desired string expectation values after each of them, and taking the average of the obtained values.


For the finite systems considered here, we choose string operators of length $N/2-1$, located in the middle of the chain, as depicted in Fig.~\ref{fig:XZX_Clifford}a. We also make use of sub-lattice symmetry in the following way. The string illustrated in Fig.~\ref{fig:XZX_Clifford}a, displaced by one site to the right,  will lead to the same circuit average. Therefore, we can improve the convergence of disorder averaging by  averaging our results for the original and shifted strings, for both string order parameters~\eqref{eq:triv_string} and~\eqref{eq:spt_string}.

Before turning to the time evolution in the presence of unitary gates, we first comment on the special case of measurement-only dynamics, $p_u=0$, a  line in parameter space that is the same for Clifford and Haar random circuits. As shown in Fig.~\ref{fig:XZX_Clifford}b, the string order parameters allow to distinguish the two different area law phases in the model. The SPT phase (purple) is characterized by
$\overline{|\mathcal{S}_Z^{X,X}|}>0$ and $\overline{|\mathcal{S}_Z^{\mathds{1},\mathds{1}}|}=0$, while the trivial phase is signalled by $\overline{|\mathcal{S}_Z^{X,X}|}=0$ and $\overline{|\mathcal{S}_Z^{\mathds{1},\mathds{1}}|}>0.$ Based on duality arguments, this transition happens exactly at $p_s=1/2$, in good agreement with our numerical results. We note that at the critical point $p_s=1/2$, both string order parameters should vanish  in the thermodynamic limit. Due to finite size effects, we find a small finite value instead.

Next, we turn to the case involving random Clifford unitary gates, preserving the $\mathbb{Z}_2\cross\mathbb{Z}_2$ symmetry~\eqref{eq:symmetries}. Fig.~\ref{fig:XZX_Clifford}c shows the behavior of the string order parameters as a function of the single-qubit measurement probability $p_s$ for a fixed rate of unitary evolution, $p_u=0.3$. We find that the string operators are still well suited for reconstructing the phase diagram. Besides the area law  SPT (purple) and trivial (orange) phases already seen for measurement-only dynamics, the volume law phase (green) is clearly distinguished by the vanishing of both string order parameters.

Our results demonstrate that string order parameters provide an accessible alternative way to determine the phase boundaries. To account for finite size effects,  we calculate the string order parameters $\overline{|\mathcal{S}|}(N)$ for various system sizes $N$. At the critical point, this quantity is expected to decay towards zero as a power law in $N$, allowing us to implement an extrapolation to the thermodynamic limit, as discussed in Appendix~\ref{appendix_B}. The phase diagram determined with this method is shown in Fig.~\ref{fig:XZX_phase_diagrams}a, and is in perfect agreement with the results of Ref.~\cite{Lavasani2021}.



\subsection{Phase diagram for Haar random unitary gates}\label{subsec:haar}

\begin{figure}[b!]
    \centering
    \includegraphics{MPS_SOP_pu0.3.pdf}
    \caption{String order parameters in the presence of Haar random unitary gates in the $\alpha=2$ cluster circuit model, plotted as a function of single spin measurement probability $p_s$ across the phase boundaries, for fixed $p_u=0.3$. These order parameters distinguish an SPT (purple) and a trivial (orange) area law phase, separated by a volume law phase (green). The vertical dashed lines show the numerically determined phase boundaries, with the grey shadow indicating the uncertainty. We used system size $N=256$ and bond dimension $\chi_{\text{max}}=128$.}
    \label{fig:XZX_MPS_pu0.3_256}
\end{figure}

In this section we check the stability of the phase diagram of the circuit model with respect to a broader class of unitary gates. To this end, we consider the time evolution in the presence of generic random Haar unitaries, with the only restriction that the gates still preserve the protecting symmetries ~\eqref{eq:symmetries}. Thereby, we considerably extend the set of allowed gates in the circuit compared to the special Clifford gates considered earlier. In principle, relaxing this constraint on the structure of gates could lead to a more efficient entanglement generation during the time evolution, and might lead to a much broader region of volume law phase in parameter space. Therefore, we examine the sensitivity of phase boundaries towards such an extension of allowed quantum circuits. As we will demonstrate below, in this particular case the phase diagram remains strikingly stable, and our calculations yield essentially identical results for Clifford and Haar random circuits.

For general Haar random gates, an efficient simulation of the circuit relying on the stabilizer formalism is no longer possible. Instead, we represent the wave function using Matrix Product States (MPS)~\cite{DMRG_MPS, tenpy}, and implement the Time-Evolving Block Decimation algorithm to calculate the dynamics~\cite{TEBD_algorithm}. This method is well-suited for studying the SPT and trivial area law phases~\footnote{The existence of an efficient MPS representation of the area law state is only proved for the case when the Rényi entropy $S_{n<1}$ is bounded by $\log N$. In other cases, it is in general undetermined whether the MPS approximates the state properly, see Appendix~\ref{appendix_C} for more details.}, however, the MPS representation with finite bond dimension breaks down in the volume law phase. Nevertheless, we can still detect the volume law phase through the steadily increasing half-chain entanglement entropy, as well as through the vanishing of both string order parameters, upon increasing system size and bond dimension. We also note that measuring the topological entanglement entropy  is much more demanding both in our MPS simulations and in an experimental setup, therefore, here we solely rely on more accessible string order parameters to determine the phase diagram.

In our MPS simulations we consider system sizes with up to $N=256$ qubits, with maximal bond dimension $\chi_{\text{max}}=128$ (a discussion about convergence with bond dimension can be found in Appendix~\ref{appendix_C}). We note that in the SPT phase it is important to choose a bond dimension $\chi_{\text{max}}$ that is a power of 2, to avoid breaking the symmetry by truncating degenerate Schmidt values, see Appendix~\ref{appendix_D}. We obtain the circuit averages by generating $10^2$ random circuits, with the qubits initialized in the state $\ket{Z=-1}^{\otimes N}$. We also perform an additional time averaging over $10^3$ time steps, after the system has reached the steady state.

\begin{figure}[t!]
    \centering
    \includegraphics[width = 0.47\textwidth]{XZX_phase_diagrams_strings.pdf}
    \caption{Phase diagram determined from string order parameters, (a) with random Clifford unitary gates, (b) with random Haar unitaries, preserving the $\mathbb{Z}_2\cross\mathbb{Z}_2$ symmetry. Numerical uncertainty is indicated by grey shades. The phase diagram is strikingly stable against changing the set of allowed unitaries, and displays an SPT and a trivial area law phase, separated by a volume law region. Clifford results are consistent with the phase diagram obtained in Ref.~\cite{Lavasani2021}, relying on entanglement entropies.}
    \label{fig:XZX_phase_diagrams}
\end{figure}


The string order parameters are shown as a function of $p_s$ in Fig.~\ref{fig:XZX_MPS_pu0.3_256}, for a fixed probability of unitary gates $p_u=0.3$. For a small enough rate of $Z$ measurements $p_s$, our simulation yields string order parameters $\overline{|\mathcal{S}_Z^{X,X}|}>0$ and $\overline{|\mathcal{S}_Z^{\mathds{1},\mathds{1}}|}=0$, confirming the presence of an SPT phase also for random Haar unitary gates (purple region). Similarly, $\overline{|\mathcal{S}_Z^{X,X}|}=0$ and $\overline{|\mathcal{S}_Z^{\mathds{1},\mathds{1}}|}>0$ indicate a trivial area law phase (orange) for large enough $p_s$. We can detect the volume law phase (green) by looking at the half-chain entanglement entropy and observing that it keeps increasing with system size and bond dimension. Also, numerical data shows that both string order parameters converge quickly to 0 with increasing bond dimension in this phase. We note that the position of phase boundaries remains unchanged with respect to the Clifford circuit within numerical precision (cf. to Fig.~\ref{fig:XZX_Clifford}c, noting the different scale on the vertical axis).


Relying on the string order parameters, we can determine the full phase diagram for the Haar random circuit, by performing an extrapolation to the thermodynamic limit similar to the Clifford case (see Appendix~\ref{appendix_B}). The result is shown in Fig.~\ref{fig:XZX_phase_diagrams}b, where it is  compared to the phase diagram of the Clifford circuit, Fig.~\ref{fig:XZX_phase_diagrams}a. The estimated numerical uncertainty of the phase boundaries is indicated by grey shading, with the Haar circuit results  having a larger error due to the  system size limitations of the MPS simulations. Nevertheless, the two phase  diagrams are strikingly similar, suggesting that the restrictions on the structure of unitary gates in the Clifford circuits do not change the phase boundaries within numerical precision.


\section{Coexistence of SPT and SSB}\label{sec:xzzx}

Having validated our approach in the previous section, we now turn to other members of the family of generalized cluster models. In this section, we demonstrate that a coexisting SPT order and SSB can be realized in the area law phase of the circuit models for odd $\alpha$ values. To this end, here we focus on the case with $\alpha=3$, realizing the $XZZX$ circuit model shown in Fig.~\ref{circuit_models}b. The ground state of the corresponding generalized cluster model Hamiltonian ~\eqref{eq:hamiltonians} shows both SPT and SSB orders. As we demonstrate below, the out-of-equilibrium version displays two area law phases: one characterized by coexisting SPT and SSB orders and a trivial phase, separated by a volume law entangled phase. In this section we restrict our attention to Clifford circuits, with the unitary gates preserving the symmetries ~\eqref{eq:symmetries}, allowing us to reach large system sizes by applying the stabilizer formalism. The measurement-only limit of this model has been recently studied in Ref.~\cite{Klocke_2022}.

\begin{figure}[t!]
    \centering
    \includegraphics{XZZX_model_results_2.pdf}
    \caption{Phases realized by the $XZZX$ ($\alpha=3$) circuit model. (a) String operators and  connected correlator of the local order parameter in a finite lattice. The two string operators (top and middle) are characterized by a non-trivial bulk operator (light shading), and boundary operators (dark shading), whereas the local order parameter (bottom) is measured at two distant positions (dark shading) to capture SSB. (b) Order parameters across the two phase transitions versus the probability of single qubit measurement $p_s$  at fixed rate of unitary gates $p_u=0.1$, for system size $N=768$. The two string operators and the correlator (multiplied by 50 for better visibility) reveal an area law phase with coexisting SPT and SSB order (magenta), as well as a trivial area law phase (orange), separated by a volume law phase (green). Vertical lines indicate the numerically determined phase boundaries.   (c) Full phase diagram of the $XZZX$ circuit model obtained via string order parameters.}
    \label{fig:XZZX_model}
\end{figure}


To differentiate the phases realized by this model, we use both the string order parameters of Eqs.~\eqref{eq:triv_string} and ~\eqref{eq:spt_string},  and the local order parameter of Eq.~\eqref{eq:local_order_parameter},  yielding $M_i=X_{i}Y_{i+1}X_{i+2}$ in this case. The SSB ordering can be detected through the correlators of this local order parameter,
\begin{equation}\label{eq:correlator}
    \mathcal{C}_M = \lim_{|j-k|\rightarrow\infty}\expval{M_jM_k}.
\end{equation}
The finite size versions of the string order parameters, as well as the correlation function of Eq.~\eqref{eq:correlator} used for this model, are illustrated in Fig.~\ref{fig:XZZX_model}a. The initial state of the qubits, as well as the procedure for obtaining the circuit averages of various quantities, is the same as the one applied for the XZX model. 

We show the behavior of the circuit averaged string order parameters and the correlator of the local order parameter in Fig.~\ref{fig:XZZX_model}b, for a fixed rate of unitaries $p_u=0.1$, using system size $N=768$. For a small rate of $Z$ measurements $p_s$, the string order parameter $\mathcal{S}_{Z}^{\mathds{1}, \mathds{1}}$ vanishes, while the other string operator $\mathcal{S}_{Z}^{X,X}$ and the correlator $\mathcal{C}_{XYX}$  both take finite values. This indicates an area law phase characterized by the coexistence of SPT and SSB orders (magenta). We note that up to numerical precision, the SPT order and SSB vanish at the same critical point, thus we do not find an area law phase displaying solely SPT or SSB order  in this model. As shown in Ref.~\cite{Klocke_2022} in the limit of measurement-only dynamics, an SPT phase without SSB order can be generated by adding symmetry breaking operations, such as $Y$ measurements, to the circuit. For large single qubit measurement probability $p_s$, we find a trivial area law phase (orange) with vanishing string order parameter $\mathcal{S}_{Z}^{X,X}$ and correlator $\mathcal{C}_{XYX}$, but a finite value for $\mathcal{S}_{Z}^{\mathds{1}, \mathds{1}}$. As before, the two area law region are separated by a volume law phase (green), where all order parameters become zero.

The full phase diagram of this model, extracted from the extrapolation to the thermodynamic limit of the three types of order parameters (as detailed in Appendix~\ref{appendix_B}), is shown in Fig.~\ref{fig:XZZX_model}c. By comparing it to the phase diagram of the $XZX$ model, Fig.~\ref{fig:XZX_phase_diagrams}, we observe that the volume law phase becomes more extended in the $XZZX$ model, and the region with SPT and SSB orders decreases in size with respect to the SPT phase of the $XZX$ circuit. This effect stems from the longer range of the random unitary gates in the $XZZX$ circuits, leading to more efficient entanglement generation. Similarly to the $XZX$ model, the numerical results, as well as the analytical arguments presented in Appendix~\ref{appendix_A}, suggest that the point $p_s=p_c=1/2$ and $p_u=0$ is a tricritical point. 


\section{Conclusion}\label{sec:conclusion}

We have studied the interplay of symmetry-protected topological phases and measurement-induced entanglement transition by introducing a class of quantum random circuit models, consisting of projective measurements and random unitary gates respecting a set of global symmetries. We showed that the circuits in this family display measurement-induced phase transitions between a thermal volume law phase and different non-thermal area law stationary states, and realize the out-of-equilibrium version of all generalized cluster states in their area law phase. Motivated by the string operators used to detect SPT order in equilibrium settings, we have constructed a set of non-equilibrium string order parameters, well suited for revealing SPT order in this class of circuit models, and accessible to numerical simulations and experimental realizations. We benchmarked our framework by studying the string order in the $XZX$ circuit model, and comparing it to the behavior of the topological entanglement entropy in the special case of Clifford unitary gates. We then tested the stability of the phase diagram by relaxing the strong constraint on the structure of unitary gates, and allowing for a wider set of symmetry preserving Haar random unitaries. Relying on MPS simulations, we found that the phase boundaries are remarkably insensitive to extending the class of unitary gates, providing additional evidence that the rich structure observed in Clifford circuits also appears in more generic quantum circuit models. Finally, we demonstrated in the example of the XZZX circuit model that the out-of-equilibrium generalized cluster states can host simultaneous SPT order and SSB, similarly to their equilibrium Hamiltonian counterparts.

Our results pave the way to study topological phases in a wider range of quantum circuit models. In particular, as we demonstrated in this work, the out-of-equilibrium string order parameters are accessible in MPS simulations, allowing us to study the phase diagram of generic Haar random circuits. One of the interesting open questions in this direction concerns the universality class of various entanglement transitions. Whether the phase transition in Clifford and Haar random circuits belongs to the same universality class is an exciting unresolved problem that could be addressed within the framework developed in this paper. Another interesting direction is characterizing all the possible ordered phases that can arise in the area law stationary states of random circuit models. The family of random circuits considered in this paper provides one example for a class of ordered phases, however, the construction could be extended to give a recipe for realizing other types of phases, such as true topological order in higher dimensional circuits. 

\textbf{Acknowledgements}.
This work was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant agreement Nos. 851161 and 771537. F.P. acknowledges the support of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence
Strategy EXC-2111-390814868.  F.P.’s research is part of
the Munich Quantum Valley, which is supported by the
Bavarian state government with funds from the Hightech
Agenda Bayern Plus. I.L. acknowledges  support from the Gordon and Betty Moore Foundation through Grant GBMF8690 to UCSB and from the National Science Foundation under Grant No. NSF PHY-1748958.

\textbf{Data and materials availability.} Data analysis and simulation codes are available on Zenodo upon reasonable request~\cite{Zenodo}.

\begin{appendix}

\section{Phase transition in measurement-only random circuits}
\label{appendix_A}
\setcounter{equation}{0}
\renewcommand{\theequation}{A\arabic{equation}}


In this appendix we focus on random circuits in the measurement-only regime, $p_u=0$, displaying a single phase transition between two topologically distinct area law phases. Here we prove via a duality argument that the critical point, characterized by a logarithmic scaling of the entanglement entropy, is located at $p_s=1/2$ for any value of $\alpha$. We note that our reasoning generalizes the proof from Ref.~\cite{Lavasani2021}, finding the critical point of the measurement-only $XZX$ circuit cluster model, $\alpha=2$, by extending it to all values of $\alpha$.

Consider the circuit model for $\alpha>0$ and periodic boundary conditions, with initial state $\ket{\psi_0} = \ket{0, ..., 0}$, described by the generating set of stabilizers $\{Z_1, Z_2, ..., Z_{N-\alpha}, G_1, ..., G_\alpha\}$. Here we make the symmetries of the model explicit, as stabilizers that remain unaltered throughout the evolution of the circuit. We denote the cluster operators by $g_i = X_{i-\alpha/2}Z_{i-\alpha/2+1}...Z_{i+\alpha/2-1}X_{i+\alpha/2}$, with $i\in \{1,..., N\}$ for $\alpha$ even and $i\in \{1/2, 3/2, ..., N-1/2\}$ for $\alpha$ odd (note that all sums in indices are understood to be modulo $N$). At each step of the circuit we measure $Z_i$ with probability $p_s$ and $g_i$ with probability $1-p_s$. From the Gottesmann-Knill theorem~\cite{measurementsGottesman,AaronsonGottesman}, a stabilizer at any time step of the evolution can be written as a product of $Z$ and $g$ operators.

For each realization of the circuit model with probability $p_s$ of $Z$ measurement, we construct a dual version with probability $1-p_s$ in the following way. For $\alpha$ even, the dual circuit will be defined on the same lattice as the original one, whereas for $\alpha$ odd we introduce a dual lattice, with lattice sites indexed by half integers $i\in \{1/2, 3/2, ..., N-1/2\}$, and perform a mapping to this dual space. First, we set the initial state of the dual circuit to be the one fixed by the stabilizers $\{g_1,...,g_{N-\alpha}, G_1,...,G_\alpha\}$. Then, we substitute every $Z_i$ measurement by a $g_i$ measurement, and vice versa.  Fig.~\ref{fig:duality} shows the duality transformation for a realization of the circuit at $\alpha=2$ and $\alpha=1$. The evolved state of the dual circuit is closely related to the state of the original circuit, as the following Lemma shows.

\begin{figure}[t!]
    \centering
    \includegraphics[width=0.48\textwidth]{duality_circuits.pdf}
    \caption{Duality map between circuit realizations for (a) $\alpha = 2$ and (b) $\alpha = 1$. For $\alpha$ even the mapping connects circuits defined on the same lattice, whereas for $\alpha$  odd we introduce a dual space. In both cases, the initial state of the original circuit is given by the stabilizers $\{Z_1, Z_2, ..., Z_{N-\alpha}, G_1, ..., G_\alpha\}$, and the initial state of the dual circuit is given by stabilizers $\{g_1,...,g_{N-\alpha}, G_1,...,G_\alpha\}$.}
    \label{fig:duality}
\end{figure}
\
\\
\textbf{Lemma}. Let 
\begin{equation}
    s = \left(\prod_{i\in I} Z_i\right)\left(\prod_{j\in J} g_j\right),
\end{equation}
be a stabilizer of the state of the original circuit at the updating step $M$, where $I\subset \{1,..., N\}$ and $J\subset\{1,...,N\}$ for $\alpha$ even and $J\subset\{1/2,...,N/2-1\}$ for $\alpha$ odd. Then, the state of the dual circuit at updating step $M$ is stabilized by the operator
\begin{equation}
    \tilde{s} = \left(\prod_{i\in I} g_i\right)\left(\prod_{j\in J} Z_j\right).
\end{equation}
\\
\\
\textit{Proof}. We prove the lemma by induction. By construction, it is true for the initial state. Suppose it is true at updating step $M$, and now let us apply a new measurement. If the new measurement commutes with all stabilizers, then the state is unchanged, so the claim is true. Let us consider the case where the measured operator does not commute with all the stabilizers. For concreteness, suppose that we measure the operator $Z_i$ in the original circuit (and thus $g_i$ in the dual one). The measurement anticommutes with all the stabilizers that, when expressed as a product of $g$ and $Z$ operators, contain either $g_{i+\alpha/2}$ or $g_{i-\alpha/2}$ (but not both). We denote this set of stabilizers by $\{s_1,...,s_n\}$. When $Z_i$ is measured, this set is updated to $\{Z_i, s_1s_2, ..., s_1s_n\}$, by virtue of the Gottesmann-Knill theorem. In the dual circuit, $g_i$ anticommutes with $Z_{i+\alpha/2}$ and $Z_{i-\alpha/2}$, so all the stabilizers commuting with these are $\{\tilde{s}_1, ..., \tilde{s}_n\}$ due to the induction hypothesis. The updated state is stabilized by the operators $\{g_i, \tilde{s}_1\tilde{s}_2,..., \tilde{s}_1\tilde{s}_n\}$, so for the modified stabilizers the claim is still true ($s_1s_j \rightarrow \tilde{s}_1\tilde{s}_j$). The same argument holds for the case where the measured stabilizer is $g_i$. $\square$

To prove that the phase transition is located at $p_s=1/2$, we apply a duality argument. Below we show that if the state of one of the circuits has area law entanglement at some time step, then so does its dual counterpart. Therefore, the critical point with logarithmic entanglement scaling must coincide with the self-dual point $p_s=1/2$.  Within the stabilizer formalism~\cite{AaronsonGottesman}, the entanglement entropy of a stabilizer state $\ket{\psi}$ for a region $A$ of the chain is given by \cite{stabilizer_entanglement}
\begin{equation}
\label{ent_stabilizer}
    S_A(\ket{\psi})=n_A-\log_2|G_A|.
\end{equation} 
Here $n_A$ is the number of qubits in $A$ and $|G_A|$ is the total number of stabilizers with support within $A$, i.e., of stabilizers that act trivially on qubits outside of $A$. Let $\mathcal{S} = \{s_1,...,s_{n}\}$ denote a generating set of the subgroup $G_A$, with $n=\log_2|G_A|$. Without loss of generality, we can assume that for every site $i\in A$, there are at most two stabilizers in the generating set that start or end at $i$ (i.e., for which the first or last non-trivial Pauli operator is at $i$). Each of the generators $s_i$ can be written as a product of $g$ and $Z$ operators contained in $A$ (for a contiguous region $A$). 

We now consider the set of operators $\Tilde{\mathcal{S}}=\{\tilde{s}_1, ..., \tilde{s}_n\}$, where $Z$ operators are replaced by $g$ and vice versa. By the previous Lemma, these operators are stabilizers of the state of the dual circuit, and one can easily check that they are linearly independent with respect to the product of Pauli strings (since the original set is linearly independent). We define the region $\Tilde{A}=A$ in the $\alpha$ even case and $\Tilde{A} = \{i +1/2| i\in A\}$ in the $\alpha$ odd case. In order to obtain an upper bound for the entanglement entropy of $\Tilde{A}$, we find a lower bound for the number of stabilizers $\Tilde{s}_i$ fully contained inside $\Tilde{A}$. We observe that the transformation $Z_i\rightarrow g_i$ moves the first and last nontrivial operators by $\alpha/2$ positions to the left and to the right, respectively. Therefore, $\Tilde{s}_i$ is still fully contained inside $\Tilde{A}$ if $s_i$ does not have any non-trivial operator closer than $\alpha/2$ to one of the edges of $A$. Since there are at most 2 operators with initial or final non-trivial operator at each site inside $A$, at least $n - 2\alpha$ of the string operators in $\Tilde{\mathcal{S}}$ will be contained in $\Tilde{A}$. Therefore $\log_2|G_{\tilde{A}}| \geq \log_2|G_A| - 2\alpha$. Comparing to equation \eqref{ent_stabilizer} gives
\begin{equation}
    S_{\tilde{A}}\left(\ket{\Tilde{\psi}}\right) \leq S_A(\ket{\psi})+2\alpha,
\end{equation}
proving that the state of the dual circuit $\ket{\Tilde{\psi}}$ indeed obeys area law entanglement scaling for any area law entangled state $\ket{\psi}$ of the original circuit. This completes our proof that the critical point must be self-dual, $p_{s}^{{\rm crit}}=1/2$.


\section{Finite size corrections to string order parameters}
\label{appendix_B}
\setcounter{equation}{0}
\renewcommand{\theequation}{B\arabic{equation}}

To determine the critical lines using string order parameters, we examine the finite size corrections that acquire for various system sizes. This procedure allows us to extrapolate our results to the thermodynamic limit, and to determine the phase boundaries with higher numerical precision. In the thermodynamic limit, the area law phases are distinguished by the vanishing of exactly one of the string order parameters, while both of them become zero in the volume law phase, allowing us to extract the critical lines of the system. In more details, we use the following numerical procedure. For a fixed $p_u$, we calculate the relevant string order parameter for different measurement rates $p_s$ close to the transition, for several system sizes $N$. Then, for each $p_s$, we fit the results with a power law function $f(N)=c N^{-a}+b$. We expect that at the critical point the string order parameter approaches zero as $c N^{-a}$, thus the finite size results can be well fitted  with $a>0$ and $b=0$. Therefore, we identify the critical point with the parameter $p_s$ yielding a good fit with $b=0$. We note that away from the critical point, this fitting of finite size results breaks down, and far enough from the phase boundary the string order parameter might even converge to a constant non-monotonously with $N$. 

\begin{figure}[t!]
    \centering
    \includegraphics{finite_scaling2.pdf}
    \caption{Finite size corrections to string order parameters for the XZX Clifford circuit model. We plot (a)~$\overline{|\mathcal{S}_Z^{X,X}|}$ for various measurement rates $p_s$ across the SPT to volume law transition and (b)~$\overline{|\mathcal{S}_Z^{\mathds{1},\mathds{1}}|}$ across the volume law to trivial area law transition, as a function of system size $N^{-a}$, for rate of Clifford gates $p_u=0.2$. The exponent $a$ is obtained by linear regression at the $p_s$ value that gives the best fit of the form $c N^{-a}$, yielding (a)~$a=1.24$ and (b)~$a=0.86$. The dashed line indicates the linear regression extrapolated to $N\rightarrow\infty$.}
  \label{fig:Cliff_strings_difNs_pu0}
\end{figure}

We illustrate this procedure for the XZX Clifford circuit model in Fig.~\ref{fig:Cliff_strings_difNs_pu0}, where we show the finite size corrections to string order parameters at different single qubit measurement rates $p_s$, for Clifford unitary gate rate $p_u=0.2$. For $\overline{|\mathcal{S}_Z^{X,X}|}$, the best fit of the form $c N^{-a}$ is obtained for $p_s=0.27$, with $a=1.24$, see Fig.~\ref{fig:Cliff_strings_difNs_pu0}a. This result is consistent with the critical $p_s$ obtained based on entanglement entropies, see Ref.~\cite{Lavasani2021}. For slightly higher values of $p_s$, the string order parameter converges to zero even more rapidly, while for $p_s$ below the critical value, it remains finite in the thermodynamic limit. Similarly, Fig.~\ref{fig:Cliff_strings_difNs_pu0}b shows the results for $\overline{|\mathcal{S}_Z^{\mathds{1},\mathds{1}}|}$, predicting a transition at $p_s=0.38$, again consistent with the critical value found in Ref.~\cite{Lavasani2021}. Similar behavior is observed in the other cluster circuit models, both with Clifford and Haar random unitaries. We note that in the latter case, MPS simulations are limited to smaller system sizes,  increasing the numerical uncertainty of the critical point.




\section{Convergence of MPS simulations with bond dimension}
\label{appendix_C}
\setcounter{equation}{0}
\renewcommand{\theequation}{C\arabic{equation}}


In the presence of  Haar random unitary gates, we rely on the time-evolving block decimation algorithm to study the time evolution. In this method, the state of the system  is represented as an MPS with maximal bond dimension $\chi_{\text{max}}$. Such an MPS captures the exact state in area law phases for large enough $\chi_{\text{max}}$, while  volume law phases require a bond dimension that diverges with increasing system size in the thermodynamic limit~\cite{DMRG_MPS,tenpy}. In this appendix, we examine the  convergence of our numerical results with $\chi_{\text{max}}$ for the XZX Haar random circuit model in the different phases, by focusing on the half-chain entanglement entropy $S_{N/2}$ at different points in phase space. 

\begin{figure}[t!]
    \centering
    \includegraphics{bond_dim_convergence.pdf}
    \caption{Half-chain entanglement entropy in the steady state of the XZX Haar random circuit as a function of the maximum bond dimension $\chi_{\text{max}}$, for different system sizes $N$. (a) In a trivial area law phase, a finite  $\chi_{\text{max}}$ independent of the system size is sufficient to ensure convergence. (b) In a volume law phase,  the  $\chi_{\text{max}}$ required to represent the state increases with system size.}
  \label{fig:bond_dim_area_volume}
\end{figure}

Figure~\ref{fig:bond_dim_area_volume} shows the averaged steady-state half-chain entanglement entropy in the trivial area law and volume law phases as a function of maximal bond dimension,  using different system sizes. In the trivial area law phase, Fig.~\ref{fig:bond_dim_area_volume}a, $S_{N/2}$ reaches its steady state value at a bond dimension independent of system size, confirming the proper convergence of our MPS results. In contrast, in the volume law phase, Fig.~\ref{fig:bond_dim_area_volume}b, the entanglement entropy increases logarithmically with bond dimension and reaches larger values for larger system sizes. Fully converged results can only be obtained for small system sizes, and the state of the system is not captured by the MPS ansatz in the thermodynamic limit. We note, however, that besides detecting the volume law phase via $S_{N/2}$, we can also identify this phase from the string order parameters calculated at finite bond dimension, as we have demonstrated above.

Ensuring convergence with bond dimension is more tricky in the SPT phase, since truncating the MPS to bond dimension $\chi_{\text{max}}$ can slightly break the $\mathbb{Z}_2\cross\mathbb{Z}_2$ symmetry of the model. This may happen when the Schmidt decomposition of the state has degenerate singular values, an effect already well-known from the MPS representation of SPT phases in equilibrium~\cite{ent_spectrum_Pollmann}.   For example, truncating the MPS to a sufficiently  small odd $\chi_{\text{max}}$ always breaks the $\mathbb{Z}_2\cross\mathbb{Z}_2$ symmetry for the following reason.  The symmetry implies a 4-fold degenerate entanglement spectrum (see Appendix~\ref{appendix_D}), therefore,  breaking the degeneracy also breaks the symmetry,  affecting  the averaged long-time values of certain quantities. More generally, in the circuit model the entanglement spectrum can show even higher degeneracies at certain time steps, always in powers of 2. For this reason, it is convenient to always choose $\chi_{\text{max}}$  as a power of 2, leading to better conservation of the symmetry during the truncation step in the evolution, even for relatively small bond dimensions. Figs.~\ref{fig:bond_dim_spt}a and b show the time evolution of the circuit-averaged symmetry operator $G_1$ and of the half-chain entanglement entropy $S_{N/2}$, respectively, at the point $p_u=0.1$ and $p_s=0.3$, belonging to the  SPT phase, for various maximal bond dimensions with  a fixed number of qubits $N=100$. The smallest bond dimension, $\chi_{\text{max}}=16$,  is not enough to preserve the symmetry during the time evolution, and we observe that the entanglement entropy is reduced upon breaking the symmetry. However, all of the higher bond dimensions preserve the symmetry, and yield the correct, converged steady state value for the half-chain entanglement entropy. We note that for bond dimensions that are not powers of 2, the symmetry can be broken for higher values of $\chi_{\text{max}}$, yielding an incorrect, reduced $S_{N/2}$. We show the convergence of $S_{N/2}$ with $\chi_{\text{max}}$, only using bond dimensions that are powers of 2, for various system sizes $N$, in Fig.~\ref{fig:bond_dim_spt}c~\footnote{In the case of $\chi_{\text{max}}=16$, here we only average  over time steps where the steady state has already been reached, but the symmetry is not yet broken.}. Similarly to the  trivial area law phase, we observe good convergence to the steady state value at a finite bond dimension independent of $N$.

\begin{figure}[t!]
    \centering
    \includegraphics{bond_dim_convergence_spt.pdf}
    \caption{Convergence in the SPT phase of XZX Haar random circuit. Time evolution of the circuit averaged (a) symmetry operator $G_1$ and (b) half-chain entanglement entropy $S_{N/2}$,  for different bond dimensions in a system of size $N=100$ in the SPT phase, at $p_u=0.1$ and $p_s=0.3$. Truncation can break the symmetry at low bond dimensions, leading to a reduced $S_{N/2}$. (c) Half-chain entanglement entropy as a function of the maximum bond dimension $\chi_{\text{max}}$ for different system sizes $N$ for the same point in the SPT phase.}
  \label{fig:bond_dim_spt}
\end{figure}

We note that in contrast to Clifford random circuits, in the presence of Haar random unitary gates  the position of the phase boundary  between area law and volume law entanglement scaling can vary with the index of the R\'enyi entropy. In particular,  it has been shown \cite{A10} that in a certain class of random quantum circuits, the measurement-induced phase transition between area law and volume law entanglement is located at different critical probabilities for Rényi entropies $n=0$ and $n\geq 1$, with $p_c^{n=0} \geq p_c^{n\geq 1}$. In this case, for measurement probabilities $p_c^{n\geq 1} \leq p \leq p_c^{n=0}$, one needs a bond dimension extensive in system size to exactly describe the state with an MPS, even though the von Neumann entanglement entropy follows an area law scaling. For such states, it can not be  determined whether the MPS representation is able to efficiently approximate the exact state of the system, based on the entanglement scaling alone~\cite{Schuch_2008}. For the family of circuit models studied here, we only relied on the von Neumann entropy and string order parameters to detect the phases, and we have checked the convergence of both these quantities with bond dimension within the area law phases.  Both methods indicated the same critical lines. Further study would be required to check the faithfulness of the MPS representation.

\section{Entanglement spectrum of XZX Haar random circuit}
\label{appendix_D}
\setcounter{equation}{0}
\renewcommand{\theequation}{D\arabic{equation}}

The MPS representation of a state gives direct access to the Schmidt coefficients $\lambda_\alpha$ for any partition of the qubit chain into subsystems $A$ and $B$. These Schmidt values $\lambda_\alpha$ are  defined through
\begin{equation}
    \ket{\psi}_{AB}=\sum_\alpha \lambda_\alpha \ket{\psi_\alpha}_A\otimes\ket{\psi_\alpha}_B,
\end{equation}
where $\{\ket{\psi_\alpha}_{A}\}$ and $\{\ket{\psi_\alpha}_{B}\}$ are orthonormal bases for susbsystems $A$ and $B$, respectively. Therefore, we have access to the entanglement spectrum of the system, defined as $-\ln\lambda_\alpha$, at different points of phase space. In this section, we focus on the entanglement spectrum of the half-chain. We find equivalent results for any other non-trivial partition of the system. 

In a non-equilibrium system, the entanglement spectrum changes with time. Therefore, we illustrate  its structure in the steady state by showing the instantaneous entanglement spectrum at a few selected time steps. The results for the different phases of the model are displayed in Fig.~\ref{fig:entanglement_spectrum}.

\begin{figure}[t!]
    \centering
    \includegraphics{entanglement_spectrum.pdf}
    \caption{Entanglement spectrum of the XZX Haar random circuit model. The instantaneous entanglement spectrum is plotted for selected time steps  in the steady state within the (a) SPT phase ($p_s=0.1$, $p_u=0.1$), (b) trivial phase ($p_s=0.45$, $p_u=0.1$) and (c) volume law phase ($p_s=0.6$, $p_u=0.25$). The entanglement spectrum shows a fourfold degeneracy in the SPT area law phase. The Schmidt values at bottom, yielding the dominant contribution to the entanglement entropy, remain well separated in both area law phases, while forming a dense cluster in the volume law phase. We used  $N=256$ and $\chi_{\text{max}}= 128$.}
    \label{fig:entanglement_spectrum}
\end{figure}

In the SPT area law phase, displayed in Fig.~\ref{fig:entanglement_spectrum}a, we find that each Schmidt coefficient is 4-fold degenerate. A similar fourfold degeneracy of the entanglement spectrum in the SPT phase with $\mathbb{Z}_2\cross\mathbb{Z}_2$ symmetry has been observed in equilibrium quantum matter~\cite{ent_spectrum_Pollmann}, a property that translates to this  non-equilibrium setting. In the trivial area law phase, the fourfold degeneracy is lifted, but the Schmidt values at the bottom, most relevant for the entanglement entropy, remain well separated from the rest, see Fig.~\ref{fig:entanglement_spectrum}b.  Finally, in the volume law phase of Fig.~\ref{fig:entanglement_spectrum}c,  we find a large number of Schmidt values at the bottom of the spectrum, a sign of a highly entangled state. In this case, choosing a larger $\chi_{\text{max}}$ would add relevant Schmidt values, significantly changing the value of the entanglement entropy, as was shown in Appendix~\ref{appendix_C}.
\newpage
\end{appendix}

\bibliographystyle{apsrev4-2}
