# Symmetry enriched phases of quantum circuits

Yimu Bao  Department of Physics, University of California, Berkeley,
California 94720, USA  Soonwon Choi  Department of Physics, University of
California, Berkeley, California 94720, USA  Center for Theoretical Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Ehud Altman
Department of Physics, University of California, Berkeley, California 94720,
USA  Materials Science Division, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA

###### Abstract

Quantum circuits consisting of random unitary gates and subject to local
measurements have been shown to undergo a phase transition, tuned by the rate
of measurement, from a state with volume-law entanglement to an area-law
state. From a broader perspective, these circuits generate a novel ensemble of
quantum many-body states at their output. In this paper, we characterize this
ensemble and classify the phases that can be established as steady states.
Symmetry plays a nonstandard role in that the physical symmetry imposed on the
circuit elements does not on its own dictate the possible phases. Instead, it
is extended by dynamical symmetries associated with this ensemble to form an
enlarged symmetry. Thus, we predict phases that have no equilibrium
counterpart and could not have been supported by the physical circuit symmetry
alone. We give the following examples. First, we classify the phases of a
circuit operating on qubit chains with $\mathbb{Z}_{2}$ symmetry. One striking
prediction, corroborated with numerical simulation, is the existence of
distinct volume-law phases in one dimension, which nonetheless support true
long-range order. We furthermore argue that owing to the enlarged symmetry,
this system can in principle support a topological area-law phase, protected
by the combination of the circuit symmetry and a dynamical permutation
symmetry. Second, we consider a Gaussian fermionic circuit that only conserves
fermion parity. Here the enlarged symmetry gives rise to a $U(1)$ critical
phase at moderate measurement rates and a Kosterlitz-Thouless transition to
area-law phases. We comment on the interpretation of the different phases in
terms of the capacity to encode quantum information. We discuss close
analogies to the theory of spin glasses pioneered by Edwards and Anderson as
well as crucial differences that stem from the quantum nature of the circuit
ensemble.

##  I Introduction

Experiments with quantum circuits designed as platforms for quantum
information processing have recently shown remarkable advances and present a
new class of many body systems Landsman  _et al._ (2019); Arute  _et al._
(2019); Zhong  _et al._ (2020). Generic quantum circuits consist of two
distinct components: unitary evolution that generates entanglement and
scrambles information and measurements or couplings to a noisy environment
that irreversibly reveal or destroy the encoded information. The interplay of
these components can lead to novel types of emergent phenomena, which bear on
the circuit’s capacity to offer a computational advantage over classical
devices Arute  _et al._ (2019); Zhong  _et al._ (2020).

A case in point is the newly discovered phase transition in the dynamics of
unitary circuits subject to local measurements. As the rate of measurements
exceeds a certain threshold, the steady state of the circuit changes from a
highly entangled volume-law state to an area-law state Skinner  _et al._
(2018); Li  _et al._ (2018); Chan  _et al._ (2019). The establishment of a low
entanglement state at high measurement rates can be viewed in terms of the
standard picture of quantum collapse of the wave function due to the repeated
local measurements. On the other hand, the essential mechanism that protects
the wave function from collapse at low measurement rates is understood from
the viewpoint of quantum error correction: generic unitary gates scramble
quantum information and encode it in nonlocal degrees freedom, thereby
affording partial protection of information from the deleterious effects of
measurements. Increasing the measurement rate degrades this encoding and
reduces the capacity of the dynamics to keep information, the so-called
_quantum channel capacity_ , until it vanishes at the phase transition point
Choi  _et al._ (2019); Gullans and Huse (2019); Fan  _et al._ (2020); Ippoliti
_et al._ (2020).

In his famous essay “More is Different”, Anderson remarked that “it is only
slightly overstating the case to say that physics is the study of symmetry”
Anderson (1972). So, it may not be too surprising that the measurement-induced
transition (MIPT) in the entanglement entropy has an appealing theoretical
description in terms of spontaneously broken symmetry Bao  _et al._ (2020);
Jian  _et al._ (2020a); Nahum  _et al._ (2021). What is somewhat
unconventional, however, is that the symmetry in question is not a physical
symmetry of the circuit elements, but is rather a consequence of the ensemble
of quantum states that the circuit generates at its output.

Each member state in this circuit ensemble corresponds to a particular
sequence, or history, of measurement outcomes and appears with probability
assigned by the Born rules. The symmetry arises because distinct features of
the ensemble states can be seen only in nonlinear moments of the density
matrix, like the entanglement entropy or fluctuations of observables between
the different histories, while simple averages over observables give a trivial
result. The time dependence of these moments can be expressed through the
evolution of $n>1$ identical copies of the density matrix. Such dynamics,
generated by unitary gates and measurements, has a $\mathcal{S}_{n}$ symmetry
to permutation among $n$ forward and, separately, $n$ backward propagating
branches (i.e. both ket and bra wave functions). This is the symmetry, which
is spontaneously broken in the MIPT. We note that the independent left and
right replica symmetries arise from the need to describe a quantum state using
a density matrix rather than a probability distribution. This is a crucial
distinction from the replica symmetry breaking in classical spin glasses. We
make further remarks on this point in the discussion section.

A systematic description of the MIPT has been obtained by mapping the circuit
with Haar random unitary gates and measurements to a statistical mechanics
model with $(\mathcal{S}_{n}\times\mathcal{S}_{n})\rtimes\mathbb{Z}_{2}$
symmetry Hayden  _et al._ (2016); Nahum  _et al._ (2018); Vasseur  _et al._
(2018); Zhou and Nahum (2019); Bao  _et al._ (2020); Jian  _et al._ (2020a);
Nahum  _et al._ (2021). The extra $\mathbb{Z}_{2}$ symmetry is inherited from
a symmetry of the time evolution to hermitian conjugation of all copies of the
density matrix. In these studies, the random unitary gates were assumed to be
uniformly distributed over the Haar measure. With the purpose of understanding
more structured circuits, it is natural to ask what other phases may arise if
physical symmetries or other constraints are imposed on it.

In this paper we elucidate how physical symmetries of the circuit elements
combine with the intrinsic dynamical symmetries discussed above to determine
the classification of the steady state phases. In the time evolution of $n$
copies of the density matrix the physical circuit symmetry $G$ is replicated
to the $n$ forward and $n$ backward propagating branches. Conjugating a group
element acting in one copy by a permutation in $\mathcal{S}_{n}$ transforms it
to the corresponding group element in another copy. Thus, the dynamics of $n$
copies has the enlarged symmetry $\mathcal{G}^{(n)}=[(G^{\otimes
n}\rtimes\mathcal{S}_{n})\times(G^{\otimes
n}\rtimes\mathcal{S}_{n})]\rtimes{\mathbb{Z}}_{2}^{\mathbb{H}}$. Here
$\mathbb{Z}_{2}^{\mathbb{H}}$ acts like an anti-unitary time-reversal
symmetry. We shall see that for broad classes of circuit architecture the
actual effective symmetry that determines the phase structure can be
simplified. To substantiate the classification of phases, we develop an exact
map of the dynamics of replicated density matrices in a broad class of quantum
circuits to the imaginary time evolution under an effective quantum
Hamiltonian that inherits the symmetry $\mathcal{G}^{(n)}$ (see Section II).
The possible steady states of the circuit ensemble correspond to the ground
state phases of this effective Hamiltonian, which thus transcend a naive
classification by the physical circuit symmetries. Note that having a volume-
law state in the circuit does not imply that the corresponding effective
Hamiltonian has a ground state with volume-law entanglement. Rather the
volume-law entanglement of the circuit state translates to a certain boundary
operator, which roughly speaking measures the free energy of a spacetime
domain wall.

We offer an information theoretic interpretation of the phases of the circuit
ensemble as distinct patterns of information encoding in the circuit, which
remains well-defined in the replica limit $n\to 1$. Nonetheless, we argue and
provide numerical evidence that for certain purposes a model with $n=2$
replicas represents the true effective symmetries of the system and thus gives
qualitatively correct predictions.

We demonstrate and explore these ideas on two examples representing different
classes of circuits. The first is a circuit with $\mathbb{Z}_{2}$ symmetry
operating on a chain of qubits. The second example is a Gaussian fermionic
circuit, which conserves only the $\mathbb{Z}_{2}$ fermion parity. For the
rest of this introduction, we provide a brief overview of the main results and
insights obtained from studying the two examples and then lay out the general
organization of the paper.

###  I.1 Overview: quantum circuits with $\mathbb{Z}_{2}$ symmetry

In Section III, we investigate the phases of quantum circuits invariant under
a global $\mathbb{Z}_{2}$ symmetry generated by the parity operator
$\hat{\pi}=\prod_{j}Z_{j}$. Thus, the single-qubit measurements of $Z_{i}$ or
two-qubit measurements of $X_{i}X_{i+1}$, which commute with $\hat{\pi}$ are
allowed, while single-qubit measurements of $X_{i}$ are not. Similarly, all
unitary gates must also commute with $\hat{\pi}$.

It was previously demonstrated, based on a specific one-dimensional model Sang
and Hsieh (2020), that circuits with $\mathbb{Z}_{2}$ symmetry can exhibit at
least two distinct area-law entangled phases in which the $\mathbb{Z}_{2}$
symmetry is either preserved or spontaneously broken. The former is stabilized
by on-site measurements of $Z_{i}$, while the latter is driven by measurements
of $X_{i}X_{i+1}$. The existence of a broken symmetry state in this system
agrees with the intuition that area-law entangled states are akin to quantum
ground states, which may spontaneously break a physical $\mathbb{Z}_{2}$
symmetry in one dimension.

One of our main results, however, is that circuits with $\mathbb{Z}_{2}$
symmetry generally admit a much richer phase structure derived from the
enlarged dynamical symmetry. This includes multiple area-law and volume-law
phases, characterized by distinct broken symmetry order parameters, which
could not have been established in presence of the circuit symmetry alone.
Especially unique to this far-from-equilibrium setting is the establishment of
states with volume-law entanglement entropy, which nonetheless exhibit
spontaneously broken circuit symmetries in a one-dimensional system. In the
same vein, we point out the possibility of establishing a new type of symmetry
protected topological (SPT) phase Pollmann  _et al._ (2010); Chen  _et al._
(2011) as a steady state of the quantum circuit evolution, where the
protecting symmetry is the enlarged dynamical symmetry rather than the
physical circuit symmetry. In our case, the $\mathbb{Z}_{2}$ circuit symmetry
alone would not be sufficient to support an SPT phase as a ground state.
However, such a state could be protected by a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ subgroup of the enlarged dynamical
symmetry. Here the second $\mathbb{Z}_{2}$ symmetry originates from the
permutation symmetry of forward branches.

We consider a number of physical probes that can distinguish between the
symmetry enriched phases of the circuit. A useful probe of broken symmetry in
such a system, which was also used in Ref. Sang and Hsieh (2020), is the long
distance Edwards-Anderson (EA) correlation function

$\chi_{EA}(i,j)=\overline{\sum_{m}p_{m}\langle X_{i}X_{j}\rangle^{2}}.$ (1)

Here the sum is over the quantum trajectories of the circuit wave function,
corresponding to different sequences $m$ of measurement outcomes obtained with
probabilities $p_{m}$. The overline represents averaging over different
unitary gates from an ensemble of quantum circuits. In the conclusions, we
will discuss the intriguing connection between this problem and the theory of
spin glasses pioneered by Edwards and Anderson, who also introduced this order
parameter in the 1970s Edwards and Anderson (1975).

Another useful diagnostic of symmetry is the subsystem parity variance

$\Pi(A)=\overline{\sum_{m}p_{m}\langle\prod_{j\in A}Z_{j}\rangle^{2}},$ (2)

expected to be nondecaying in the limit of a large subsystem size when the
state of typical trajectories has a well-defined parity. At a superficial
level, this probe appears to be a “disorder parameter” dual to the EA
correlations. However, we shall see that this duality does not hold in the
enlarged symmetry space, thus allowing phases with coexisting long-range EA
correlations and nondecaying parity variance.

Complementary to these physical observables, we also consider information
theoretic quantities, such as Fisher information obtained by measurement
outcomes about specific perturbations to the initial state. Such information
theoretic probes allow to characterize the phases in terms of the flow of
information in the circuit rather than steady state equal time properties such
as $\chi_{EA}$ or $\Pi(A)$.

A crucial point is that all of these diagnostics must be defined as second (or
even higher) moments in the ensemble of trajectories. For example, a simple
average of order parameter correlation functions, i.e. $\overline{\langle
X_{i}X_{j}\rangle}$ vanishes because the broken symmetry state is
characterized by random $X$ orientations in a given trajectory due to random
measurement outcomes. Evaluating second moments necessarily involves the
dynamics of two replica copies, which should therefore be viewed as
fundamental to the physics of these systems and not merely an averaging trick.
Thus, we devote much of our attention to classifying phases in a model with
two replicas ($n=2$).

We note, however, that to ensure proper averaging over the ensemble one needs
to invoke auxiliary replicas and ultimately take the the replica limit $n\to
1$. While we do not explicitly construct models for $n>2$ to facilitate such
an extrapolation, we show that, apart from the coexistence phases that feature
both nonvanishing $\chi_{EA}$ and $\Pi(A)$, the phases predicted for $n=2$
have natural extensions to higher replicas. Furthermore, when two phases can
be distinguished by physical diagnostics in the two-replica model, this
distinction holds for $n\geq 2$ and is thus expected to extrapolate to the
replica limit.

In Section III.3, we provide numerical evidence for the phase structure
predicted by the two-replica effective theory. The model we consider consists
of two chains where chain 1 is unrestricted by symmetry, while operations on
chain 2 obey a global $\mathbb{Z}_{2}$ symmetry. In other words, only qubits
of chain 2 carry a global $\mathbb{Z}_{2}$ charge. We show that the volume-law
phase in this system can host more than two distinct broken symmetry phases,
confirming that the symmetry dictating the phase structure is larger than the
$\mathbb{Z}_{2}$ circuit symmetry. The coexistence of long-range quantum
orders with volume-law entanglement presents a sharp contrast with thermal
states, which exclude such order in one-dimensional systems.

The fact that the volume-law phase can exhibit long-range order naturally
raises the question if topological states and the ensuing edge modes can also
remain protected in volume-law entangled steady states of quantum circuits.
The results of the two-chain model suggest an affirmative answer. We could
replace the qubits in chain 2 with Majorana fermions conserving the
$\mathbb{Z}_{2}$ fermion parity (still coupled to the qubits in chain 1).
Under a Jordan-Wigner transformation this model maps to the one we computed
numerically. The broken symmetry phase of the qubit circuit translates to a
topological phase with edge zero modes and volume-law entanglement in the
fermionic model.

###  I.2 Overview: Gaussian fermionic circuits

In Section IV we investigate Gaussian fermionic circuits with unitary gates
and measurements that only conserve the $\mathbb{Z}_{2}$ fermion parity, but
not fermion number. The elements in these Gaussian circuits preserve the
gaussianity of the wave function Terhal and DiVincenzo (2002). In particular,
we consider unitary gates generated by a (time-dependent) Hamiltonian that is
quadratic in fermion operators and measurements of quadratic observables
associated with rank one projectors, such as measuring the local parity. These
systems provide a second example of the role played by the enlarged dynamical
symmetry in dictating the phase structure.

The effect of local Gaussian measurements on free fermions was first discussed
by Cao et al. Cao  _et al._ (2018), who argued that these systems cannot
sustain a volume-law state for any nonvanishing measurement rate (see also
Fidkowski  _et al._ (2020)). This can be understood as resulting from the
absence of scrambling in Gaussian circuits, which therefore cannot protect
quantum information (or equivalently entanglement) from measurements through
nonlocal encoding Choi  _et al._ (2019); Gullans and Huse (2019). Recent
numerical results on monitored free fermion dynamics indicated, however, a
possible measurement-induced phase transition from a critical phase, with
entanglement entropy scaling logarithmically in system sizes, to a strict
area-law phase Alberton  _et al._ (2020). Such a transition is also found in a
specific fermion model via a mapping to classical loop models Sang  _et al._
(2021). We note that critical states of nonunitary fermion models (though not
measurement circuits) were studied in Refs. Chen  _et al._ (2020); Liu  _et
al._ (2020); Jian  _et al._ (2020b) as well as in measurement-only models
Nahum and Skinner (2020).

The framework developed in this paper naturally captures the instability of
the volume-law state and the emergence of a critical phase. Both properties
stem from the effective symmetry associated with the dynamics of two copies of
the system, which is enlarged with respect to the physical symmetry of the
circuit.

We consider circuits with a $\mathbb{Z}_{2}$ fermion parity symmetry,
described using two Majorana operators on each physical site $j$,
$\gamma_{2j-1}$ and $\gamma_{2j}$. The local observables measured in the
circuit are the fermion parities on sites
$\hat{\pi}_{s,j}=-\mathrm{i}\gamma_{2j-1}\gamma_{2j}$ and the fermion parity
on bonds connecting two nearest neighbor sites
$\hat{\pi}_{b,j}=-\mathrm{i}\gamma_{2j}\gamma_{2j+1}$. These operators are
also the generators of all the unitary gates.

Due to the Gaussian constraint and fermion commutation relations, the enlarged
dynamical symmetry of this model is different from that of qubit circuits. In
the absence of measurements, the unitary evolution of $n$ replica copies of a
density matrix exhibits a global $O(2n)\rtimes\mathbb{Z}_{2}^{\mathbb{T}}$
corresponding to Bogoliubov rotations of Majorana operators between the $2n$
branches. $\mathbb{Z}_{2}^{\mathbb{T}}$ is a time-reversal like symmetry that
squares to $-1$. This is the same symmetry that emerges in the symplectic
class of fermion systems in a random potential Ryu  _et al._ (2007); Fu and
Kane (2012). There are two important differences, however. First, the replica
limit in this dynamical problem is $n\to 1$, while it is $n\to 0$ in the
Anderson localization problem. Second, the measurements break the symmetry
down to $[O(n)\times O(n)]\rtimes\mathbb{Z}_{2}^{\mathbb{T}}$.

As in the qubit circuits, we focus on the dynamics of the minimal model with
two replica copies, which is mapped to imaginary time evolution generated by a
quantum Hamiltonian. In absence of measurements, averaging over a purely
unitary circuit of Gaussian fermion gates gives rise to an effective
ferromagnetic spin-1 Hamiltonian with $O(3)$ symmetry. The quantum ferromagnet
allows establishment of long-range order, which translates to volume-law
entanglement, even in a one dimensional system.

With non-vanishing rate of measurements the symmetry of the effective model is
reduced to $U(1)\rtimes(\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{\mathbb{T}})$,
which cannot support a broken symmetry state. For moderate measurement rate
this leads to the establishment of a critical ground state of
$H_{\text{eff}}$. The sub-system purity $\exp(-S_{A}^{(2)})$ of the circuit
state translates to a boundary correlation function, which decays as a power
law in the critical phase of $H_{\text{eff}}$ with a decay exponent that
decreases continuously with increasing measurement rate. Accordingly, the
entanglement entropy scales as $\log|A|$ with a continuously varying pre-
factor.

The two-replica theory predicts that the critical phase ends with a
Kosterlitz-Thouless transition that takes the system into one of two area-law
phases. The circuits diagnostics that allow to distinguish between the two
area-law phases are the subsystem site and bond parity variances:

$\Pi_{a,A}\equiv\overline{\sum_{m}p_{m}\langle\prod_{j\in
A}\hat{\pi}_{a,j}\rangle_{m}^{2}}.$ (3)

Here the subscript $a=s,b$ stands for site or bond.

In the area-law phase with dominant site measurements, the subsystem site
parity variance is non-decaying, whereas the bond parity variance decays
exponentially. If the bond measurements dominate, then the bond parity
variance is non-decaying. In terms of the effective ground state description,
these two states correspond to a trivial and a Haldane SPT phase,
respectively. The transition between the two area-law phases with the same
characteristics has been discussed in the context of measurement-only dynamics
of a Majorana chain Nahum and Skinner (2020); Lang and Büchler (2020).

The fact that all phases can be diagnosed by quantities that are second
moments over the ensemble of quantum trajectories hints at that a model of two
replica copies captures the correct emergent symmetries in the problem.
Nonetheless, appropriate averaging of $\Pi_{a,A}$ requires introducing
auxiliary replicas followed by extrapolation to the physical replica limit
$n\to 1$. This may alter the critical behavior.

In Section IV.5, we compare the predictions of the effective two-replica
theory to exact numerical simulations of the dynamics in the Gaussian
fermionic circuits, which can be done on relatively large systems. We find
good agreements with the expected properties of the critical phase, the two
area-law phases and even the Kosterlitz-Thouless transition. In particular,
the correlation length we extract from the numerical simulations shows a
divergence that matches well with the hallmark exponential form of the
Kosterlitz-Thouless transition $\xi\sim\exp(A/\sqrt{p-p_{c}})$. On the other
hand the two replica model does not capture the observed critical behavior
associated with the direct transition between the two area-law phases.

###  I.3 Organization of the paper

The rest of the paper is organized as follows. In Section II, we explain the
basic formalism and introduce a broad class of circuits that allows a mapping
of the replicated dynamics to effective quantum ground state problems. We
discuss the basic structure and symmetry of the model and lay out the
dictionary for translating between the quantities measured in the physical
circuit and the corresponding operators in the effective ground state problem.
In Section III, we apply the framework to classify the phases of a random
circuit model with $\mathbb{Z}_{2}$ symmetry. We then demonstrate numerically
that some of the new phases are found in the phase diagram of a concrete
stabilizer circuit model. In Section IV, we consider Gaussian fermionic
circuits as outlined above. Section V presents a broader discussion and
summary of the results. In particular, we remark on connections between the
quantum circuit ensemble discussed in this paper and the theory of spin
glasses pioneered by Edwards and Anderson Edwards and Anderson (1975). Close
analogies exist both at the level of emergent symmetries and in aspects of
information theory. We shall also comment on fundamental differences between
the problems associated with the quantum and dynamical nature of the circuit.

##  II Framework

In this section, we introduce a framework for mapping the time evolution of an
ensemble of quantum trajectories to an effective ground state problem. We
identify the symmetry of this problem as an extension of the physical circuit
symmetry by intrinsic dynamical symmetries.

###  II.1 States and operators in duplicated Hilbert spaces

We consider the dynamics of quantum systems undergoing unitary evolution
interspersed by projective measurements. The outcome of each projective
measurement is probabilistic, determined by the usual Born rules. This leads
to stochastic dynamics of the unnormalized density matrix of the system, which
for a particular sequence of measurement outcomes is given by

$\displaystyle\tilde{\rho}_{m}(t)=U_{k}\dots
U_{2}P_{m_{1}}U_{1}\rho_{0}U_{1}^{\dagger}P_{m_{1}}U_{2}^{\dagger}\dots
U_{k}^{\dagger}.$ (4)

Here $\rho_{0}$ is the initial state, $U_{j}$ are the set of unitary
evolution, and $P_{m_{j}}$ are projection operators associated with
measurement outcome $m_{j}$. Given $U_{j}$, the sequence of measurement
outcomes $m=(m_{1},m_{2},\cdots)$ defines the trajectory of the wave function,
which occurs with the probability $p_{m}=\mathrm{tr}\tilde{\rho}_{m}$. The set
of normalized quantum states $\rho_{m}\equiv\tilde{\rho}_{m}/p_{m}$ and their
assigned probabilities $p_{m}$ form the measurement ensemble of the circuit.
We are interested in certain steady-state properties of this ensemble.

Recent works have shown that the steady states of quantum circuits with
unitary gates and measurements can exhibit measurement-induced phase
transitions, characterized by non-analyticities in various information
theoretic quantities such as entanglement entropy, the global purity, and the
Fisher information Li  _et al._ (2018); Skinner  _et al._ (2018); Li  _et al._
(2019); Choi  _et al._ (2019); Gullans and Huse (2019); Bao  _et al._ (2020);
Jian  _et al._ (2020a). These phase transitions have no signature in the
averaged density matrix over quantum trajectories
$\rho=\sum_{m}p_{m}\rho_{m}=\sum_{m}\tilde{\rho}_{m}$. In fact, it can be
shown that $\rho$ approaches the maximally mixed state in the models
considered in the literature Li  _et al._ (2018); Skinner  _et al._ (2018);
Bao  _et al._ (2020).

To characterize phase transitions in the measurement ensemble, one needs to
consider the statistics of the ensemble, which is encoded in the time
evolution of multiple copies of the trajectory density matrix, that is
$\tilde{\rho}_{m}^{\otimes n}$. In what follows, it will be convenient to view
these objects as state vectors in a duplicated Hilbert space
$\mathcal{H}^{(n)}=(\mathcal{H}\otimes\mathcal{H}^{*})^{\otimes n}$. Thus, for
example, the replicated un-normalized trajectory density matrix is denoted by

$\displaystyle\mbox{$|\tilde{\rho}_{m}^{(n)}\rangle\rangle$}\equiv\tilde{\rho}_{m}^{\otimes
n}.$ (5)

This state vector is evolved linearly by unitary and projection operators

$\displaystyle\mathcal{U}^{(n)}_{i}\equiv(U_{i}\otimes U_{i}^{*})^{\otimes
n},\;\;\;\mathcal{M}^{(n)}_{m_{i}}\equiv(P_{m_{i}}\otimes P_{m_{i}})^{\otimes
n},$ (6)

defined through their action on the replicated density matrix

$\displaystyle\mathcal{U}^{(n)}_{i}\mbox{$|\tilde{\rho}_{m}^{(n)}\rangle\rangle$}$
$\displaystyle\equiv\left(U_{i}\tilde{\rho}_{m}U_{i}^{\dagger}\right)^{\otimes
n},$ (7)
$\displaystyle\mathcal{M}^{(n)}_{m_{i}}\mbox{$|\tilde{\rho}_{m}^{(n)}\rangle\rangle$}$
$\displaystyle\equiv\left(P_{m_{i}}\tilde{\rho}_{m}P_{m_{i}}\right)^{\otimes
n}.$ (8)

If the state in the duplicated Hilbert space is not factorizable, we use the
linearity of the operators to define the actions of unitary gates and
projections.

We will show that pertinent properties of the circuit ensemble are encoded in
ensemble states defined as a sum over the replicated trajectory-states

$\displaystyle\mbox{$|\tilde{\rho}^{(n)}\rangle\rangle$}\equiv\sum_{m}\mbox{$|\tilde{\rho}_{m}^{(n)}\rangle\rangle$}.$
(9)

This un-normalized ensemble state is very convenient because it undergoes a
linear time evolution

$|\tilde{\rho}^{(n)}(t)\rangle\rangle$ $\displaystyle=$
$\displaystyle\mathcal{V}^{(n)}(t)\mbox{$|\tilde{\rho}^{(n)}_{0}\rangle\rangle$}$
$\displaystyle\equiv$
$\displaystyle\sum_{m}\mathcal{U}^{(n)}_{t}\cdots\mathcal{M}^{(n)}_{m_{2}}\mathcal{U}^{(n)}_{2}\mathcal{M}^{(n)}_{m_{1}}\mathcal{U}^{(n)}_{1}\mbox{$|\tilde{\rho}^{(n)}_{0}\rangle\rangle$}.$

Later in this section, we present a class of models, in which this time
evolution is exactly mapped to an imaginary time evolution generated by an
effective quantum Hamiltonian, so that

$\displaystyle\mbox{$|\tilde{\rho}^{(n)}(t)\rangle\rangle$}=e^{-tH^{(n)}_{\text{eff}}}\mbox{$|\rho^{(n)}_{0}\rangle\rangle$}.$
(11)

In these models, the properties of the ensemble state
$|\tilde{\rho}^{(n)}(t)\rangle\rangle$ in late times are faithfully encoded in
the ground state of $H^{(n)}_{\text{eff}}$.

A key step in mapping of moments in the trajectory ensemble of the circuit to
properties of the effective ground state is to translate the trace operation
in the replicated Hilbert space to the state-vector formalism. For example, we
will need the trace $(\mathrm{tr}\tilde{\rho}_{m})^{n}$ in order to normalize
the trajectory states. In the state-vector notation, these traces can be
expressed as an inner product with a reference state

$\displaystyle\mbox{$\langle\langle\mathcal{I}|$}=\mathds{1}^{\otimes{n}}=\sum_{\\{\tau_{\ell}\\}}\mbox{$\langle\langle\tau_{1}\tau_{1},\tau_{2}\tau_{2},\ldots,\tau_{n}\tau_{n}|$},$
(12)

where each pair of $\tau_{\ell}$ labels the ket and bra state of copy $\ell$.
Now, with the inner product between states in the replicated Hilbert space
defined as
$\mbox{$\langle\langle\chi|\sigma\rangle\rangle$}\equiv\mathrm{tr}(\chi^{\dagger}\sigma)$
we can write the trace
$(\mathrm{tr}\tilde{\rho}_{m})^{n}=\mathrm{tr}(\tilde{\rho}_{m}^{\otimes
n})=\mbox{$\langle\langle\mathcal{I}|\tilde{\rho}_{m}^{(n)}\rangle\rangle$}$.

The simplest quantities that involve higher moments of the density matrix are
the average $k$-th moments of an observable $\hat{O}$ over the measurement
ensemble and circuit realizations

$O_{k}=\overline{\sum_{m}p_{m}\left[\frac{\text{tr}(\hat{O}\tilde{\rho}_{m})}{\text{tr}\tilde{\rho}_{m}}\right]^{k}}.$
(13)

For example, when dealing with circuits with a global $\mathbb{Z}_{2}$
symmetry (see Section III) it is natural to consider the fluctuations of a
local order parameter $\hat{O}=X_{i}$, which is odd under the symmetry. In
this case the object $O_{2}=\sum_{m}p_{m}\langle X_{i}\rangle_{m}^{2}$ is an
Edwards-Anderson type order parameter that can detect the broken symmetry in
individual trajectories Edwards and Anderson (1975). More precisely, we’ll be
interested in the operator $\hat{O}=X_{i}X_{j}$ that gives rise to the EA
correlations $O_{2}=\sum_{m}p_{m}\langle X_{i}X_{j}\rangle_{m}^{2}$. Another
example of an operator of interest is the $\mathbb{Z}_{2}$ parity on subsystem
$A$, $\hat{O}=\prod_{j\in A}Z_{j}$. In this case,
$O_{2}=\sum_{m}p_{m}\langle\prod_{j\in A}Z_{j}\rangle_{m}^{2}$ measures the
parity variance on $A$.

The quantities given in Eq. (13) can be obtained from a replica limit. To
formally eliminate the denominator in Eq. (13), we introduce a replica index
$n$ to write $p_{m}^{1-k}=\lim_{n\to 1}p_{m}^{n-k}$. Substituting into (13),
we can express the $k$-th moment as a limit $n\to 1$ of the sequence of
replica quantities

$\displaystyle O_{k}^{(n)}$
$\displaystyle=\frac{\text{tr}\left(\mathcal{O}^{(k)}\tilde{\rho}^{\otimes
n}\right)}{\text{tr}\tilde{\rho}^{\otimes
n}}=\frac{\mbox{$\langle\langle\mathcal{I}|$}\mathcal{O}^{(k)}\mbox{$|\tilde{\rho}^{(n)}\rangle\rangle$}}{\mbox{$\langle\langle\mathcal{I}|\tilde{\rho}^{(n)}\rangle\rangle$}},$
(14)

where we defined

$\displaystyle{\mathcal{O}}_{k}^{(n)}\equiv\left[\bigotimes_{i=1}^{k}(\hat{O}\otimes\mathds{1})\right]\;\otimes\;(\mathds{1}\otimes\mathds{1})^{\otimes
n-k}.$ (15)

The denominator in Eq. (14) is added to ensure that the quantities
$O_{k}^{(n)}$ correspond to normalized boundary correlations in the effective
model. It is exactly equal to unity in the replica limit $n\to 1$.

Subsystem purities can also be expressed in the same framework. The average
$k$-th purity of a subsystem $A$ is given by

$\mu_{k,A}=\overline{\sum_{m}p_{m}{\text{tr}(\tilde{\rho}_{A,m}^{k})\over(\text{tr}\tilde{\rho}_{m})^{k}}}.$
(16)

We can express it in the form (14) by choosing the operator
$\mathcal{O}^{(k)}$ to be the subsystem cyclic permutation operator (of kets)

$\displaystyle\mathcal{C}_{\ell,A}^{(k)}$
$\displaystyle=\sum_{\\{\alpha_{i}\\}}\bigotimes_{i=1}^{k}\left(\mbox{$|\alpha_{i+1}\rangle$}\mbox{$\langle\alpha_{i}|$}\otimes\mathds{1}\right),$
(17)

where $\alpha_{k+1}\equiv\alpha_{1}$ is assumed and the subscript $\ell$
implies acting on the left of the density matrices (i.e. permutation of kets).
$|\alpha_{i}\rangle$ runs over all basis states of subsystem $A$, while
$\mathcal{C}_{\ell,A}^{(k)}$ acts as the identity outside of subsystem $A$.
That is, we have

$\mu_{k,A}^{(n)}=\frac{\mbox{$\langle\langle\mathcal{I}|$}\mathcal{C}_{\ell,A}^{(k)}\mbox{$|\tilde{\rho}^{(n)}\rangle\rangle$}}{\mbox{$\langle\langle\mathcal{I}|\tilde{\rho}^{(n)}\rangle\rangle$}},$
(18)

and $\mu_{k,A}=\lim_{n\to 1}\mu_{k,A}^{(n)}$.

In much of this paper we study the behavior of second moments in a two replica
model (i.e. $k=2,n=2$). It is therefore worth writing the moments explicitly
in this special case

$O_{2}^{(2)}=\frac{\mbox{$\langle\langle\mathcal{I}|$}\mathcal{O}^{(2)}\mbox{$|\tilde{\rho}^{(2)}\rangle\rangle$}}{\mbox{$\langle\langle\mathcal{I}|\tilde{\rho}^{(2)}\rangle\rangle$}}=\frac{\sum_{m}p_{m}^{2}\langle\hat{O}\rangle_{m}^{\,\,2}}{\sum_{m}p_{m}^{2}},$
(19)

and similarly

$\mu_{2,A}^{(2)}=\frac{\sum_{m}p_{m}^{2}\mathrm{tr}(\rho_{A,m}^{2})}{\sum_{m}p_{m}^{2}}.$
(20)

We see that in the two replica model the trajectories are weighted by a
distorted probability distribution
$p_{m}^{(2)}=p_{m}^{2}/(\sum_{m^{\prime}}p_{m^{\prime}}^{2})$.

Finally, for completeness, we discuss the computation of the von Neumann
entanglement entropy within the state-vector formalism. For this purpose it is
useful to treat the classical measurement device $M$ as a part of the extended
system, as we have shown in Ref. Bao  _et al._ (2020). In this picture, the
randomness of the measurement outcomes are encoded in the correlation between
the system and measurement device.

The average von Neumann entanglement entropy of subsystem $A$ over the
possible measurement outcomes is

$\displaystyle
S_{A}=\sum_{m}p_{m}S_{A,m}=\sum_{m}-p_{m}\mathrm{tr}(\rho_{A,m}\log\rho_{A,m}).$
(21)

In the extended system, we can express this average as the conditional entropy

$S_{A}=S(A|M)\equiv S_{MA}-S_{M},$ (22)

where the measurement device is characterized by a diagonal density matrix
$\rho_{M}=\delta_{mm^{\prime}}p_{m}$. In this formulation, we can further
include an average over the random unitary gates and express the average
conditional von Neumann entropy $\overline{S_{A}}$ as a limit of “conditional
Rényi entropies” Bao  _et al._ (2020),

$\displaystyle S^{(n)}_{A}$
$\displaystyle=\frac{1}{1-n}\log\left(\,\overline{\mathrm{tr}\rho_{MA}^{n}}\,\right)-\frac{1}{1-n}\log\left(\,\overline{\mathrm{tr}\rho_{M}^{n}}\,\right)$
$\displaystyle=\frac{1}{1-n}\log\left(\frac{\overline{\sum_{m}p_{m}^{n}\mathrm{tr}(\rho_{A,m}^{n})}}{\overline{\sum_{m}p_{m}^{n}}}\right).$
(23)

The properly averaged von Neumann entropy is restored in the replica limit:
$\overline{S_{A}}=\lim_{n\rightarrow 1}S_{A}^{(n)}$. The conditional Rényi
entropies have a simple expression within our formalism as the logarithm of a
boundary matrix element. As in the case of purity, the relevant operator is
the permutation $\mathcal{C}_{\ell,A}^{(n)}$ of forward propagating
trajectories (kets) within subsystem $A$:

$\displaystyle
S_{A}^{(n)}=\frac{1}{1-n}\log\frac{\langle\langle\mathcal{I}|\mathcal{C}_{\ell,A}^{(n)}|\tilde{\rho}^{(n)}\rangle\rangle}{\langle\langle\mathcal{I}|\tilde{\rho}^{(n)}\rangle\rangle}.$
(24)

Note that Eq. (II.1) implies that each of the conditional Rényi entropies
involves weighting of the subsystem purities by the outcome probabilities to
the $n$-th power. Thus, they are not identical to the Rényi entropies averaged
over the measurement ensemble. However, as noted above, the von Neumann
entropy obtained in the replica limit, is correctly weighted by the outcome
probabilities. In this paper, we mainly focus on $n=2$ model, the conditional
Rényi entropy and purity are related by $e^{-S^{(2)}_{A}}=\mu_{2,A}^{(2)}$.

###  II.2 Enlarged dynamical symmetry

The dynamics $\mathcal{V}(t)$ of the quantum state in the duplicated Hilbert
space exhibits an enlarged symmetry, which is an extension of the physical
symmetry of the circuit by the dynamical symmetries inherent to the time
evolution of identical copies of the density matrix. In general, the dynamical
symmetry does not commute with the physical circuit symmetry, and the
combination of the two produces the enlarged symmetry of $\mathcal{V}(t)$. We
shall see that this enlarged symmetry dictates the possible steady states of
the circuit.

We say that the circuit has a physical symmetry $G$ if all the unitary gates
$U_{i}$ and the projectors $P_{m_{i}}$ implementing the measurements commute
with the symmetry generators $g\in G$. In this paper, for simplicity, we
assume the symmetry group $G$ has a unitary representation and do not consider
the anti-unitary circuit symmetries. Note that the circuit symmetry is
important for characterizing the steady state, however, unlike in purely
unitary dynamics it does not give rise to a conserved quantity enforced at the
level of individual trajectories. Given an initial state with indefinite
quantum numbers associated with the symmetry, projective measurements may lead
to the collapse of wave functions, such that quantum amplitudes or
probabilities in each symmetry sector changes. Once the system has a definite
set of quantum numbers, then the measurements cannot change them.

In the dynamics of $n$ replica copies, the physical circuit symmetry $G$ is
duplicated to each of the forward and backward evolving branches, leading to a
symmetry group $G^{\otimes n}\times G^{\otimes n}$. We call the first and
second $G^{\otimes n}$ the _left_ and _right_ circuit symmetry. In addition,
the time evolution $\mathcal{V}(t)$ of $n$ copies of the density matrix is
invariant to permutation among the different copies of the identical circuit
elements ($U_{i}$ and $P_{m_{i}}$) operating on the density matrix from the
left and, independently, to permutation of the circuits elements acting from
the right. The operators representing the left and right permutation symmetry
are given respectively by

$\displaystyle\mathcal{C}_{\ell,\xi}$
$\displaystyle=\sum_{\\{\alpha_{a}\\}}\bigotimes_{a}\left(\mbox{$|\alpha_{\xi(a)}\rangle$}\mbox{$\langle\alpha_{a}|$}\otimes\mathds{1}\right),$
$\displaystyle\mathcal{C}_{r,\xi}$
$\displaystyle=\sum_{\\{\alpha_{a}\\}}\bigotimes_{a}\left(\mathds{1}\otimes\mbox{$|\alpha_{a}\rangle$}\mbox{$\langle\alpha_{\xi(a)}|$}\right),$
(25)

where $\xi\in\mathcal{S}_{n}$ is a member of the permutation group of $n$
elements. Here, we omit the superindex for $n$ replica copies for the
simplicity of notation. The left cyclic permutation $\mathcal{C}_{\ell}$,
which is a special case of $\mathcal{C}_{\ell,\xi}$, was already introduced in
Eq. (17). The left and right permutation symmetries are independent from one
another since $[\mathcal{C}_{\ell,\xi},\mathcal{C}_{r,\eta}]=0$ for any
$\xi,\eta\in\mathcal{S}_{n}$. These permutation symmetries transform the
replicated $G^{\otimes n}$ circuit symmetries into each other giving rise to
$G^{\otimes n}\rtimes\mathcal{S}_{n}$ invariance, independently for the
forward and backward branches. The group extension $G^{\otimes
n}\rtimes\mathcal{S}_{n}$ is also known as the wreath product
$G\wr\mathcal{S}_{n}$.

Finally, there is one more dynamical symmetry associated with hermiticity of
the density matrix, which the time evolution induced by $\mathcal{V}(t)$ must
preserve. The operator representing this symmetry is the hermitian conjugation

$\displaystyle\mathbb{H}:c_{\alpha\beta\gamma\delta}\mbox{$|\alpha\rangle$}\mbox{$\langle\beta|$}\otimes\mbox{$|\gamma\rangle$}\mbox{$\langle\delta|$}\mapsto
c_{\alpha\beta\gamma\delta}^{*}\mbox{$|\beta\rangle$}\mbox{$\langle\alpha|$}\otimes\mbox{$|\delta\rangle$}\mbox{$\langle\gamma|$},$
(26)

which is anti-linear (anti-unitary) and of order $2$, such that
$\mathbb{H}^{2}=\mathds{1}$. We denote this symmetry by
$\mathbb{Z}_{2}^{\mathbb{H}}$. The physical circuit symmetries combine with
the left and right permutation symmetries and hermiticity to give the complete
dynamical symmetry

$\displaystyle\mathcal{G}^{(n)}=\left[\left(G^{\otimes
n}\rtimes\mathcal{S}_{n}\right)\times\left(G^{\otimes
n}\rtimes\mathcal{S}_{n}\right)\right]\rtimes\mathbb{Z}_{2}^{\mathbb{H}}.$
(27)

Equivalently, $\mathcal{G}^{(n)}$ can be written in terms of the wreath
product $\mathcal{G}^{(n)}=G\wr\mathcal{S}_{n}\wr\mathbb{Z}_{2}^{\mathbb{H}}$.

We make a few remarks. While Eq. (27) is generically valid, the symmetry of
$\mathcal{V}(t)$ can be even larger in the presence of additional constraints.
For example, in Section IV, we consider a fermionic circuit that preserves the
Gaussianity of the fermionic wave function, which leads to a continuous
symmetry. In addition, while $\mathcal{G}^{(n)}$ itself is very large, the
physically relevant symmetry can be a smaller subgroup of $\mathcal{G}^{(n)}$,
depending on the details of the model. For instance, if $\mathcal{V}$ is
averaged over a specific random circuit ensemble, the averaging process can
constrain the state to be in faithful representations only of a subgroup of
$\mathcal{G}^{(n)}$ thus reducing the available options for symmetry breaking.

We also note that our analysis only concerns the symmetry of $\mathcal{V}(t)$
and does not address additional physical constraints such as the complete
positivity of quantum channels.

###  II.3 Effective quantum Hamiltonian

Figure 1:  Mapping the dynamics of random quantum circuits to the imaginary
time evolution with an effective Hamiltonian. (a) A quantum circuit with
layers of random on-site $Z$-rotations $U_{Z}=e^{-\mathrm{i}\theta_{i}Z_{i}}$.
The inter-layer elements include measurements and unitary gates satisfying
conditions needed for a hermitian effective Hamiltonian (see text). (b) The
effective time evolution of a doubled density matrix is effected by four
copies of the quantum circuit, two corresponding to forward propagation and
two backward. (c,d) Averaging over the random $Z$ rotations projects on the
reduced site Hilbert space $\mbox{$|m,s\rangle\rangle$}_{i}$, which can be
interpreted as quantum spin states. The states at successive time steps are
connected by the transfer matrix $\mathcal{T}(\delta t)$, which is generated
by an effective Hamiltonian $H_{\text{eff}}$ in the limit $\delta t\to 0$.

In this section, we introduce an exact mapping of the dynamics
$\mathcal{V}(t)$ in a broad class of qubit circuits to imaginary time
evolution generated by an effective Hamiltonian $H_{\text{eff}}$, which
inherits the enlarged dynamical symmetry. Thus, the long-time steady states of
the quantum circuits can be characterized by ground states of
$H_{\text{eff}}$.

The structure of the quantum circuits we consider is shown in Fig. 1(a). Each
“time step” $\delta t$ is subdivided into layers. The first layer in each time
step is made of single-site random unitary phase rotations (i.e. rotation
about the $Z$-axis). This is not a necessary step in our scheme, but we shall
see that averaging over the random phases simplifies the effective model by
projecting onto a reduced Hilbert space. The next layer consists of near
identity unitary gates, such as
$U_{ij}=\exp\left(-\mathrm{i}\theta^{\alpha\beta}_{ij}\tau^{\alpha}_{i}\tau^{\beta}_{j}\right)$,
where $\tau^{\alpha}_{j}$ with $\alpha=0,\ldots,3$ represent the identity and
Pauli operators on the qubit. The random couplings $\theta^{\alpha\beta}_{ij}$
are drawn from a symmetric Gaussian distribution with variance
$\text{var}(\theta_{ij}^{\alpha\beta})\equiv J_{\alpha\beta}\delta t$. This
layer also contains projective measurements of Pauli string operators
$M_{\alpha}$, applied with probability $p_{\alpha}=\Gamma_{\alpha}\delta t$.

The replicated density matrix $|\tilde{\rho}^{(n)}\rangle\rangle$ is evolved
by the operator $\mathcal{V}(t)$, consisting of the replicated circuit
elements. We obtain a transfer matrix by averaging the evolution by one time
step over the “circuit ensemble”, namely the distribution of unitary gates and
the probabilities $p_{\alpha}$ of applying measurement operators. The latter
are also replicated on the $n$ forward and $n$ backward branches [e.g. the
case of $n=2$ is depicted in Fig. 1(b)].

For an infinitesimal time step, $\delta t\to 0$, the transfer matrix takes the
form $\mathcal{T}=\overline{\mathcal{V}(\delta t)}=\exp({-\delta
tH_{\text{eff}}})$. Hermiticity of $\mathcal{T}$ (and thus also of
$H_{\text{eff}}$) is ensured if the distribution of the coupling constants in
the unitary gates is symmetric about zero. In the long-time evolution, the
steady state
$\mbox{$|\tilde{\rho}^{(n)}\rangle\rangle$}=\lim_{t\to\infty}e^{-tH_{\text{eff}}}\mbox{$|\rho_{0}\rangle\rangle$}$
is given by the ground state of $H_{\text{eff}}$.

For the circuits considered in this section, the effective Hamiltonian
$H_{\text{eff}}$ has a set of mutually commuting local integrals of motion.
These are the local parity operators
$\mathcal{X}_{j}=\prod_{a=1}^{n}(X_{a}X_{\bar{a}})_{j}$,
$\mathcal{Y}_{j}=\prod_{a=1}^{n}(Y_{a}Y_{\bar{a}})_{j}$, and
$\mathcal{Z}_{j}=\prod_{a=1}^{n}(Z_{a}Z_{\bar{a}})_{j}$ (see appendix A). The
subscript $a$ and $\bar{a}$ denote the Pauli operators acting on the $a$-th
forward and backward branch, respectively. The eigenstates of $H_{\text{eff}}$
are labeled by the eigenvalues of the $\mathcal{X}_{j}$, $\mathcal{Y}_{j}$,
and $\mathcal{Z}_{j}$ on all sites.

We will only need to characterize the ground state of $H_{\text{eff}}$ in the
even sector of the local parities in order to compute the physical diagnostics
in Eq. (13) and (II.1). The reason is that the reference state
$\langle\langle\mathcal{I}|$ has definite local parities on all sites,
$\mathcal{X}_{j}=\mathcal{Z}_{j}=+1$ and $\mathcal{Y}_{j}=(-1)^{n}$.
Meanwhile, all the duplicated operators associated with diagnostics considered
in this paper, e.g. $\mathcal{O}_{2}^{(2)}=(X\otimes\mathds{1})^{\otimes 2}$,
preserve the local parities. Therefore, only the ground state in the same
local parity sector as the reference state $|\mathcal{I}\rangle\rangle$ has a
nonvanishing contribution to the matrix element in Eq. (14) and (24).

We now turn to construct the effective Hamiltonian explicitly for the case of
$n=2$ replicas. Generalizing to higher replicas $n\geq 3$ is straightforward.
Consider first the single-site phase rotation at the entry to each time step.
Averaged over the random phase, the two copy evolution through this layer
takes the form

$\displaystyle\overline{e^{-i\theta_{j}Z_{j}}\otimes
e^{i\theta_{j}Z_{j}}\otimes e^{-i\theta_{j}Z_{j}}\otimes
e^{i\theta_{j}Z_{j}}}$ $\displaystyle={1\over
2\pi}\int_{0}^{2\pi}d\theta_{j}e^{-\mathrm{i}\theta_{j}\left(Z_{j1}-Z_{j\overline{1}}+Z_{j2}-Z_{j\overline{2}}\right)}$
$\displaystyle=\sum_{k=1}^{6}\mbox{$|\varphi_{k}\rangle\rangle$}\mbox{$\langle\langle\varphi_{k}|$}.$
(28)

The averaging over the identical rotation angle of the four branches yields a
delta function that implements a projection onto the six-dimensional subspace
defined by vanishing of $Z_{j1}+Z_{j2}-Z_{j\overline{1}}-Z_{j\overline{2}}$. A
convenient basis for this subspace, labeled by $|m,s\rangle\rangle$, is given
explicitly by

$|1,\pm\rangle\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\mbox{$|1\rangle$}\mbox{$\langle
1|$}\otimes\mbox{$|0\rangle$}\mbox{$\langle
0|$}\pm\mbox{$|0\rangle$}\mbox{$\langle
0|$}\otimes\mbox{$|1\rangle$}\mbox{$\langle 1|$}\right)$
$|0,\pm\rangle\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\mbox{$|1\rangle$}\mbox{$\langle
1|$}\otimes\mbox{$|1\rangle$}\mbox{$\langle
1|$}\pm\mbox{$|0\rangle$}\mbox{$\langle
0|$}\otimes\mbox{$|0\rangle$}\mbox{$\langle 0|$}\right)$
$|-1,\pm\rangle\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\mbox{$|0\rangle$}\mbox{$\langle
1|$}\otimes\mbox{$|1\rangle$}\mbox{$\langle
0|$}\pm\mbox{$|1\rangle$}\mbox{$\langle
0|$}\otimes\mbox{$|0\rangle$}\mbox{$\langle 1|$}\right).$ (29)

The quantum numbers $s_{i}=\pm 1$ on site $i$ happen to be the eigenvalues of
the local integrals of motion $\mathcal{X}_{i}$ (and of $\mathcal{Y}_{i}$). As
discussed above, for $n=2$, the reference state $|\mathcal{I}\rangle\rangle$
is in the even sector of local parities. Hence, we need to consider only the
even sector $s_{i}=+1$. This leaves us with three states on every site
$m=-1,0,1$, which can be represented by a spin-1 degree of freedom.

The left permutation (swap) symmetry $\mathcal{S}_{2}$ has a simple action on
the basis (29)

$\displaystyle\mathcal{C}_{\ell}:\mbox{$|m,s\rangle\rangle$}\leftrightarrow\mbox{$|-m,s\rangle\rangle$}.$
(30)

Thus, spontaneous breaking of the $\mathcal{S}_{2}$ permutation symmetry will
appear as onset of a spontaneous $z$-magnetization of the spin-1 degrees of
freedom.

We now turn to construct the transfer matrix operating between the states of
the reduced Hilbert space

$\displaystyle\mbox{$\langle\langle\\{m,s\\}|$}\mathcal{T}\mbox{$|\\{m^{\prime},s^{\prime}\\}\rangle\rangle$}=\mbox{$\langle\langle\\{m,s\\}|$}\prod_{\nu}\mathcal{M}_{\nu}\prod_{\mu}\mathcal{U}_{\mu}\mbox{$|\\{m^{\prime},s^{\prime}\\}\rangle\rangle$}.$
(31)

Here, $\mathcal{U}_{\mu}$ and $\mathcal{M}_{\nu}$ represent the averaged
unitary gates and measurements acting on the duplicated state. The averaging
is over both the random couplings in the unitary gates and the probabilistic
application of the measurements. The indices $\mu,\nu$ run over all different
unitary gates and measurements in the layers of the circuit that are part of a
single time step $\delta t$. All of these operators are designed to be close
to unity. The deviation from unity after averaging is of order $\delta t$.

The averaged measurement operators take the form

$\mathcal{M}_{\nu}=(1-\Gamma_{\nu}\delta t)\mathds{1}^{\otimes
4}+\Gamma_{\nu}\delta t\sum_{m_{\nu}=\pm}P_{m_{\nu}}^{\otimes 4},$ (32)

where $\Gamma_{\nu}$ may be viewed as the rate at which a measurement of type
$\nu$ is performed, and $P_{\pm}=(1\pm M_{\nu})/2$ is the projection onto the
$\pm$ eigenstate of the Pauli string operator $M_{\nu}$.

The unitary gates operating on qubits are of the general form:
$U_{j}=\exp(-\mathrm{i}\theta_{j}^{\alpha\beta}\tau_{j}^{\alpha}\tau_{j+1}^{\beta})$,
where $\tau^{\alpha}_{j}$ denote local Pauli operators, including the
identity. That is $\tau^{0}_{i}\equiv\mathds{1}_{j}$, $\tau^{1}_{j}\equiv
X_{j}$ etc. Averaging over the Gaussian distribution of
$\theta_{j}^{\alpha\beta}$ with variance $J_{\alpha\beta}\,\delta t$ gives:

$\displaystyle\mathcal{U}_{j}$ $\displaystyle=\overline{U_{j}\otimes
U_{j}^{*}\otimes U_{j}\otimes U_{j}^{*}}$ (33) $\displaystyle=e^{-\frac{\delta
tJ_{\alpha\beta}}{2}\left(\tau_{j,1}^{\alpha}\tau_{j+1,1}^{\beta}-\tau_{j,\overline{1}}^{\alpha}\tau_{j+1,\overline{1}}^{\beta}+\tau_{j,2}^{\alpha}\tau_{j+1,2}^{\beta}-\tau_{j,\overline{2}}^{\alpha}\tau_{j+1,\overline{2}}^{\beta}\right)^{2}}.$

For an infinitesimal $\delta t$, the transfer matrix (31) takes the form
$\mathcal{T}=1-H_{\mathcal{M}}\delta t-H_{\mathcal{U}}\delta t+O(\delta
t^{2})$. The effective Hamiltonian
$H_{\text{eff}}=H_{\mathcal{M}}+H_{\mathcal{U}}$ can be read from Eqs. (32)
and (33) projected to the reduced Hilbert space of three states per site.

In Appendix B we exemplify a detailed construction of an effective Hamiltonian
for the two replica dynamics of a simple circuit with $\mathbb{Z}_{2}$
symmetry. The result is a spin-1 model with
$D_{4}\times\mathbb{Z}_{2}^{\mathbb{H}}$ symmetry. In the next section, we
discuss the possible steady-state phases of circuits with $\mathbb{Z}_{2}$
symmetry more generally. We present direct simulation of a concrete quantum
circuit, which exhibits a subset of the possible steady-state phases.

##  III Qubit circuit with $\mathbb{Z}_{2}$ symmetry

We now discuss how the phase structure of a quantum circuit with measurements
is enriched by imposing a physical $\mathbb{Z}_{2}$ symmetry on the circuit
elements. To derive general results, we go beyond the circuit models discussed
in Section II.3. In particular, we do not enforce a projection to a reduced
six-dimensional local Hilbert space.

In Section III.1 below, we present the general symmetry analysis of the
problem to identify the allowed phases and determine what are their sharp
signatures in the physical probes of the circuit state. In Section III.3, we
present the results of numerical simulations of a concrete circuit model with
$\mathbb{Z}_{2}$ symmetry. The results demonstrate the establishment of at
least two of the possible area-law phases and three of the distinct volume-law
phases in the steady state phase diagram of this circuit model.

###  III.1 Phases protected by enlarged symmetry

The circuits we consider in this section operate on a one-dimensional array of
qubits. Unitary gates in the circuit are generated by Pauli operators with
random couplings drawn from symmetric distributions as considered in Section
II. All circuit operations, unitary gates and measurements, commute with a
global $\mathbb{Z}_{2}$ parity operator defined on a single chain

$\displaystyle\hat{\pi}=\prod_{j=1}^{L}Z_{j}.$ (34)

To make the model more general, we allow additional chains of qubits, which
are not transformed by the above parity operator. Thus, operations applied to
qubits on the additional chains are not restricted by symmetry.

We first focus on the phases that arise in the simplest case of two replica
copies (i.e. $n=2$). A model of two copies gives a natural framework to
address fluctuations that are second moments of observables over the
trajectory ensemble. This is not an exact approach because without
extrapolating to the replica limit $n\to 1$ the weighting of trajectories is
distorted. To validate key aspects of the phase structure predicted by the
two-replica model we use direct simulations of a quantum circuit model. In
particular we confirm the existence of new phases enabled by the enlarged
symmetry.

According to the analysis presented in Section II.2, the enlarged symmetry of
the two-replica model should be $\mathcal{G}^{(2)}=(D_{4}\times
D_{4})\rtimes\mathbb{Z}_{2}^{\mathbb{H}}$. The left (right) $D_{4}$ group
consist of the $\mathbb{Z}_{2}$ symmetries of the two copies on the forward
(backward) branch, compounded with the $\mathcal{S}_{2}$ permutation symmetry
between the two copies, i.e.
$D_{4}=(\mathbb{Z}_{2}\times\mathbb{Z}_{2})\rtimes\mathcal{S}_{2}$. The
$\mathbb{Z}_{2}$ symmetries of a single copy and branch are generated by
product operators which we denote as $\prod_{j}(ZIII)_{j}$ (copy 1 forward
branch), $\prod_{j}(IZII)_{j}$ (copy 1 backward), $\prod_{j}(IIZI)_{j}$ (copy
2 forward) and $\prod_{j}(IIIZ)_{j}$ (copy 2 backward).

The effective symmetry that determines the phase structure can be reduced and
simplified by accounting for additional constraints. Specifically, in our
case, the effective Hamiltonian conserves a set of local parities
$\mathcal{X}_{j}=(XXXX)_{j}$, $\mathcal{Y}_{j}=(YYYY)_{j}$ and
$\mathcal{Z}_{j}=(ZZZZ)_{j}$ (see Appendix A). This leads to reduction of the
effective symmetry $G^{(2)}$ as follows. First, the fact that the operators
$\mathcal{X}_{j}$ have non-vanishing expectation values on every site implies
breaking of all four single-branch $\mathbb{Z}_{2}$ symmetries already at the
outset because the $\mathcal{X}_{j}$ anti-commutes with these symmetries.
Second, as noted before, we seek the ground state in the even sector of
$\mathcal{X}_{j}$, $\mathcal{Y}_{j}$ and $\mathcal{Z}_{j}$ on every site. The
exchange symmetry $\mathcal{S}_{2}^{X}$, generated by the product of left and
right permutation $\mathcal{C}_{\ell}\mathcal{C}_{r}$, acts trivially in this
sector (see Appendix C). Therefore it cannot be broken and we can quotient it
out. Third, we can also quotient out the symmetry $\prod_{j}\mathcal{Z}_{j}$
because it is a product of the local parities, which cannot be broken.

The above considerations leave us with an effective global symmetry
$\mathcal{G}_{\text{eff}}^{(2)}=D_{4}\times\mathbb{Z}_{2}^{\mathbb{H}}$, where
$D_{4}=(\mathbb{Z}_{2}^{\Pi_{L}}\times\mathbb{Z}_{2}^{\Pi_{1}})\rtimes\mathcal{S}_{2}$.
Here, $\mathbb{Z}_{2}^{\Pi_{L}}$ and $\mathbb{Z}_{2}^{\Pi_{1}}$ are generated
by $\hat{\Pi}_{L}=\prod_{j}(ZIZI)_{j}$ and
$\hat{\Pi}_{1}=\prod_{j}(ZZII)_{j}$, respectively. In Appendix B, we
explicitly derive the effective Hamiltonian for a concrete model,
demonstrating the $D_{4}\times\mathbb{Z}_{2}^{\mathbb{H}}$ symmetry.

The Hermitian conjugate $\mathbb{H}$ in the even parity sector is given by the
complex conjugate $\mathcal{K}$ as shown in Appendix C and commutes with the
rest symmetry generators in $\mathcal{G}^{(2)}_{\text{eff}}$. Hence,
$\mathbb{H}$ can be broken independently. Moreover, the reference state
$\langle\langle\mathcal{I}|$ and
$\mbox{$\langle\langle\mathcal{I}|$}\mathcal{O}_{2}$ associated with the
physical diagnostics are symmetric under Hermitian conjugate, making them
blind to its breaking. Therefore, in this paper, we do not distinguish phases
by the hermiticity breaking and consider the phases allowed by the $D_{4}$
symmetry.

We are now in a position to classify the possible steady states for the case
of $n=2$ through their correspondence to ground states of an effective
Hamiltonian with $D_{4}$ symmetry. This includes broken symmetry states,
characterized by the residual subgroups of $D_{4}$, as well as SPT phases
protected by the $D_{4}$ effective symmetry. The results of this analysis are
summarized in Table 1. We shall also discuss the physical diagnostics in
quantum circuits that can distinguish different phases. In Appendix D we
comment on the extension to higher replicas.

Entropy | Name | Residual symmetry | $\Pi^{(2)}(A)$ | $\chi_{EA}^{(2)}(i,j)$ | $\Phi^{(2)}$
---|---|---|---|---|---
Area law | Symmetric phase (trivial/SPT) | $(\mathbb{Z}_{2}^{\Pi_{L}}\times\mathbb{Z}_{2}^{\Pi_{1}})\rtimes\mathcal{S}_{2}$ | const. | $\to 0$ | 0
Coexistence phase (trivial/SPT) | $\mathbb{Z}_{2}^{\Pi_{L}}\times\mathcal{S}_{2}$ | const. | const. | 0
Broken symmetry phase | $\mathcal{S}_{2}$ | $\to 0$ | const. | const.
| Composite phase | $\mathbb{Z}_{4}^{\mathcal{C}_{\ell}\Pi_{1}}$ | const. | const. | 0
Volume law | Symmetric phase | $\mathbb{Z}_{2}^{\Pi_{L}}\times\mathbb{Z}_{2}^{\Pi_{1}}$ | const. | $\to 0$ | 0
Featureless phase | $\mathbb{Z}_{2}^{\Pi_{1}}$ | $\to 0$ | $\to 0$ | const.
Coexistence phase I | $\mathbb{Z}_{2}^{\Pi_{1}\Pi_{L}}$ | const. | const. | 0
Coexistence phase II | $\mathbb{Z}_{2}^{\Pi_{L}}$ | const. | const. | 0
Broken symmetry phase | $\emptyset$ | $\to 0$ | const. | const.
Composite phase | $\mathbb{Z}_{2}^{\mathcal{C}_{\ell}\Pi_{L}}$ | $\to 0$ | const. | 0
Table 1: Possible phases of a qubit circuit with $\mathbb{Z}_{2}$ symmetry and
their diagnostics. Four area-law and six volume-law phases are characterized
by distinct residual symmetries given in the second column. The generators of
the respective $\mathbb{Z}_{2}$ subgroups are specified in superscript.
$\hat{\Pi}_{L}=\prod_{j}(ZIZI)_{j}$ is the parity in the two forward branches.
$\hat{\Pi}_{1}=\prod_{j}(ZZII)_{j}$ is the parity in the first copy. For each
phase we give the long distance limit of the parity variance $\Pi^{(2)}(A)$
and EA correlation $\chi_{EA}^{(2)}(i,j)$ as well as the value of the Fisher
information “order parameter” $\Phi^{(2)}=1-\frac{1}{2}F^{(2)}$ in the bulk.
The phases we observe numerically in the model studied in Section III.3 are
shaded gray.

####  III.1.1 “Conventional” area-law phases

We start from a working definition of the area-law regime as ground states of
the effective Hamiltonian that preserve the $\mathcal{S}_{2}$ permutation
symmetry generated by $\mathcal{C}_{\ell}$.111The left and right permutation
$\mathcal{C}_{\ell}$ and $\mathcal{C}_{r}$ are identified because the exchange
operation $\mathcal{C}_{\ell}\mathcal{C}_{r}$ acts as an identity matrix in
the symmetric sector of the local symmetries. Indeed in these states domain
walls of the $\mathcal{S}_{2}$ symmetry are condensed. Therefore the sub-
system swap operator that inserts such domain walls on either side of the sub-
system is non-decaying implying non-decaying purity or $O(1)$ entanglement
entropy. However, we shall see that the working definition needs to be
modified slightly to account for one possible area-law state with broken
$\mathcal{S}_{2}$ symmetry.

Under the working definition the different area-law states are characterized
by distinct residual subgroups of $D_{4}$ that contain $\mathcal{S}_{2}$. Let
us start from the most symmetric state, which retains the full $D_{4}$
symmetry. We first ask what are the “charges” (order parameters) that can
condense and are invariant under the $\mathcal{S}_{2}$ permutation symmetry.

The first option is to condense the local charge
$\mathcal{Q}_{1,j}\equiv(XXII)_{j}$ together with $(IXXI)_{j}$, which is
related to $\mathcal{Q}_{1,j}$ by the $\mathcal{S}_{2}$ symmetry. If these two
charges condense, then the charges $(IIXX)_{j}$ and $(XIIX)_{j}$ must also
condense because of the definite local parities $\mathcal{X}_{j}$ in the
ground state. Condensing these charges breaks the $\hat{\Pi}_{L}$ and
$\hat{\Pi}_{1}$ symmetry, leaving a residual global symmetry $\mathcal{S}_{2}$
in addition to the local symmetries
$\mathcal{X}_{j},\mathcal{Y}_{j},\mathcal{Z}_{j}$. We call this phase the
broken symmetry area-law phase.

The second route for symmetry breaking, starting from the symmetric phase, is
to condense the charges $(XIXI)_{j}$ and $(IXIX)_{j}$. This condensate breaks
the symmetry $\hat{\Pi}_{1}$, but leaves intact the symmetry $\hat{\Pi}_{L}$,
which commutes with the order parameter. We shall refer to this phase as the
“coexistence phase” for reasons that will become clear below. From the
coexistence phase, the system can further break the residual symmetry by
condensing $\mathcal{Q}_{1,j}$ and its symmetry related charges defined above.
This leads to the broken symmetry area-law phase discussed above. We note that
charges containing an odd number of Pauli-$X$ matrices, such as $XIII$ and
$XXXI$, cannot condense because they anti-commute with the local parities
$\mathcal{Y}_{j}$ and $\mathcal{Z}_{j}$, which cannot be broken.

Let us discuss the physical interpretation of the area-law phases outlined
above and how they can be distinguished by diagnostics of the circuit. We
consider properties of the output state, such as the subsystem entanglement
entropy, the Edwards-Anderson correlation function
$\chi_{EA}\equiv\sum_{m}p_{m}\langle X_{i}X_{j}\rangle_{m}^{2}$ and the
subsystem parity variance
$\Pi(A)\equiv\sum_{m}p_{m}\langle\hat{\pi}(A)\rangle_{m}^{2}=\sum_{m}p_{m}\langle\prod_{j\in
A}Z_{j}\rangle_{m}^{2}$. As discussed in Section II, these diagnostics map to
different normalized matrix elements between the ground state and a reference
state.

Our measure of the purity and entanglement entropy is obtained from the
overlap

$e^{-S^{(2)}(A)}=\frac{\mbox{$\langle\langle\mathcal{I}|$}\prod_{j\in
A}{\mathcal{C}}_{\ell,j}\mbox{$|\psi_{gs}\rangle\rangle$}}{\mbox{$\langle\langle{\mathcal{I}}|$}\psi_{gs}\rangle\rangle}.$
(35)

Here the operator acting on $|\psi_{gs}\rangle\rangle$ creates a pair of
domain walls of the left permutation (swap) symmetry at the two edges of the
region $A$. If the ground state is symmetric under swap, then these domain
walls are condensed, implying $\prod_{j\in
A}{\mathcal{C}}_{\ell,j}\mbox{$|\psi_{gs}\rangle\rangle$}\approx\lambda\mbox{$|\psi_{gs}\rangle\rangle$}$
for a long region $A$, leading to a length independent (area-law) entanglement
entropy.

The parity variance in region $A$ is similarly obtained by the matrix element
of a nonlocal operator that creates domain walls of the $\mathbb{Z}_{2}$
symmetry generated by $\hat{\Pi}_{L}$,
$\Pi^{(2)}(A)=\mbox{$\langle\langle{\mathcal{I}}|$}\prod_{j\in
A}(ZIZI)_{j}\mbox{$|\psi_{gs}\rangle\rangle$}/\mbox{$\langle\langle{\mathcal{I}}|\psi_{gs}\rangle\rangle$}$.
Finally, the Edwards-Anderson order parameter in the circuit maps to the the
following matrix element in the effective model
$\Psi_{EA}^{(2)}(i)=\mbox{$\langle\langle{\mathcal{I}}|$}(XIXI)_{i}\mbox{$|\psi_{gs}\rangle\rangle$}/\mbox{$\langle\langle{\mathcal{I}}|\psi_{gs}\rangle\rangle$}$.
Strictly speaking, we need to consider the long-range EA correlation
$\chi^{(2)}_{EA}(i,j)=\mbox{$\langle\langle{\mathcal{I}}|$}(XIXI)_{i}(XIXI)_{j}\mbox{$|\psi_{gs}\rangle\rangle$}/\mbox{$\langle\langle{\mathcal{I}}|\psi_{gs}\rangle\rangle$}$
to detect the spontaneous symmetry breaking because the initial state time
evolution and reference state are all symmetric with respect to
$\hat{\Pi}_{1}$. To simplify the explanations below, however, we will assume
the presence of an infinitesimal symmetry breaking field and thus refer to
establishment of an EA order parameter.

To understand the behavior of these objects in the different phases, we need
to determine, besides the symmetry of the ground state, also that of the
reference state $\langle\langle\mathcal{I}|$ in Eq. (12). It is easy to check
that $\langle\langle\mathcal{I}|$ is invariant under the
$\mathbb{Z}_{2}^{\Pi_{1}}$ symmetry generated by
$\hat{\Pi}_{1}=\prod_{j}(ZZII)_{j}$, the $\mathcal{S}_{2}^{X}$ exchange
symmetry, and the $\mathcal{Z}_{2}^{\mathbb{H}}$ hermiticity symmetry, while
breaking all other global symmetries. In particular, the reference state
$\langle\langle\mathcal{I}|$ is a condensate of the charges $(XXII)_{j}$,
$(IIXX)_{j}$, $(YYII)_{j}$ and $(IIYY)_{j}$. For example, we have
$\mbox{$\langle\langle\mathcal{I}|$}(XXII)_{j}=\mbox{$\langle\langle\mathcal{I}|$}$.

Consider first the signatures of the symmetric phase, which retains the full
$D_{4}$ symmetry. Invariance with respect to $\hat{\Pi}_{L}$ means that the
domain wall associated with this symmetry is condensed in the ground state,
$\prod_{j\in A}(ZIZI)_{j}\mbox{$|\psi_{gs}\rangle\rangle$}\approx
c\mbox{$|\psi_{gs}\rangle\rangle$}$. Thus, the subsystem parity variance
$\Pi^{(2)}(A)$ remains constant independent of the size of $A$. The EA order
parameter, on the other hand, vanishes because both the ground state and the
reference state are even under $\hat{\Pi}_{1}$, whereas $XIXI$ is an odd
operator with respect to this symmetry (anti-commutes with $ZZII$).

Next, consider the broken symmetry phase, obtained by condensing
$\mathcal{Q}_{1,j}=(XXII)_{j}$ and its symmetry related charges. When
calculating the EA order parameter $\Psi^{(2)}_{EA}(i)$ we can replace
$(XIXI)_{i}$ by $(IXXI)_{i}=(XXII)_{i}(XIXI)_{i}$ because the reference state
is an eigenstate of $(XXII)_{i}$ with eigenvalue $1$. Hence, we expect a
nonzero value of $\Psi^{(2)}_{EA}(i)$ owing to the condensation of
$(IXXI)_{i}$ in the ground state $|\psi_{gs}\rangle\rangle$. At the same time,
the parity variance $\Pi^{(2)}(A)$ is expected to decay exponentially. The
subsystem parity $\prod_{j\in A}(ZIZI)_{j}$ acts nontrivially on the broken
symmetry ground state $|\psi_{gs}\rangle\rangle$, creating a domain of a
symmetry related ground state within region $A$. Since
$\langle\langle\mathcal{I}|$ is a uniform broken symmetry state, its
normalized overlap with the transformed region is exponentially decaying,
$\Pi^{(2)}(A)\sim e^{-L_{A}/\xi}$ for $\xi\ll L_{A}\ll L$.222The broken
symmetry state is strictly speaking a cat-like superposition of the two
distinct broken symmetry states, leading to $\Pi^{(2)}(A)\sim
e^{-L_{A}/\xi}+e^{-(L-L_{A})/\xi}$ under periodic boundary condition.

The symmetric and broken symmetry phases outlined above are the same as those
discussed in Ref. Sang and Hsieh (2020) and observed there in simulations of a
concrete model. The physical intuition for the establishment of these phases
is clear. In the broken symmetry phase, dominant nearest neighbor $XX$
measurements collapse the individual trajectories to a near product state with
qubits polarized in random directions along $X$. As in a glassy phase, such
random $X$ orientations are detected by averaging the square of individual
trajectory expectation values $\Psi_{EA}=\sum_{m}p_{m}\langle
X_{i}\rangle_{m}^{2}$. At the same time, the parity of a long region $A$ has a
vanishing expectation value $\langle\hat{\pi}(A)\rangle_{m}=\langle\prod_{j\in
A}Z_{j}\rangle_{m}\to 0$ in every trajectory. In the symmetric phase, dominant
single-qubit $Z$ measurements collapse individual trajectories to near product
states with definite, yet random parity on each qubit. This is detected by
taking the variance over trajectories of the parity in region $A$:
$\sum_{m}p_{m}\langle\hat{\pi}(A)\rangle_{m}^{2}\to\text{const}$. At the same
time, the EA order parameter vanishes in the symmetric phase because $\langle
X_{i}\rangle_{m}=0$ in every trajectory.

Besides these two phases, we have alluded to the possible existence of
another, more peculiar phase, in which the symmetry is partially broken by
condensing just the charges $XIXI$ and $IXIX$. Such a condensate implies the
nonvanishing EA order parameter $\Psi_{EA}^{(2)}(i)\to\text{const}.$ At the
same time, the parity symmetry generated by $\hat{\Pi}_{L}$ remains intact,
implying nonvanishing parity variance on a long subsystem
$\Pi^{(2)}(A)\to\text{const}$.

The coexistance of a nonvanishing EA order parameter with a non-decaying
parity variance, allowed within the effective model, may seem paradoxical at
first sight when considering the wave functions of individual trajectories.
The EA order parameter must originate from trajectories $m$ that exhibit
nonvanishing $\langle X_{i}\rangle_{m}\neq 0$. The parity expectation value
$\langle\hat{\pi}(A)\rangle_{m}$ must vanish in these symmetry breaking
trajectories. Nonetheless, because we have an ensemble of trajectories, the EA
order parameter and non-vanishing parity variance can coexist in separate
trajectories. In theory, such a phase is established if symmetry breaking and
symmetric trajectories both appear with nonvanishing probability in the
ensemble. It is left as an open question if there are physical obstructions
towards realizing such a coexistence phase.

Having considered area-law states which preserve the $\mathcal{S}_{2}$
symmetry, we now show that this is not a necessary condition for getting a
phase with area-law entanglement. Specifically, consider breaking the $D_{4}$
symmetry while preserving a $\mathbb{Z}_{4}$ subgroup generated by the
composite operator $\hat{\Pi}_{1}\mathcal{C}_{\ell}$. This state, which we
term th “composite area-law phase”, can be obtained from the symmetric phase
by condensing the operator $(YZXI)_{j}$, symmetrized over the residual
$\mathbb{Z}_{4}$ subgroup. The key to calculate the entanglement entropy in
this state is to note that the reference state is an eigenstate of
$\hat{\Pi}_{1,j}=(ZZII)_{j}$ with eigenvalue $1$. This allows to make the
replacement
$\mbox{$\langle\langle\mathcal{I}|$}\mathcal{C}_{\ell,j}=\mbox{$\langle\langle\mathcal{I}|$}\hat{\Pi}_{1,j}\mathcal{C}_{\ell,j}$
when calculating the subsystem purity to obtain

$\displaystyle
e^{-S^{(2)}(A)}=\frac{\mbox{$\langle\langle\mathcal{I}|$}\prod_{j\in
A}\left(\hat{\Pi}_{1,j}\mathcal{C}_{\ell,j}\right)\mbox{$|\psi_{gs}\rangle\rangle$}}{\mbox{$\langle\langle\mathcal{I}|\psi_{gs}\rangle\rangle$}}\to\text{const}.$
(36)

The subsystem purity tends to a constant for large $L_{A}$ because the
operator appearing in the matrix element creates domain walls of the unbroken
$\mathbb{Z}_{4}$ symmetry at the two boundaries of $A$.

Besides area-law entanglement, the “composite phase” is also characterized by
non-decaying EA correlations and subsystem parity variance. To see the former
we use the fact that $\langle\langle{\mathcal{I}}|$ is an eigenstate of
$(ZZII)_{j}$ to replace the operator $(XIXI)_{j}$ with
$\mathrm{i}(YZXI)_{j}=(ZZII)_{j}(XIXI)_{j}$ in the EA correlation

$\chi_{EA}^{(2)}(i,j)=\frac{\mbox{$\langle\langle\mathcal{I}|$}\mathrm{i}(YZXI)_{i}\mathrm{i}(YZXI)_{j}\mbox{$|\psi_{gs}\rangle\rangle$}}{\mbox{$\langle\langle\mathcal{I}|\psi_{gs}\rangle\rangle$}}\to\text{const}.$
(37)

This is non-decaying because the operator $(YZXI)_{j}$ overlaps with the order
parameter of the state as discussed above. The parity variance is non-decaying
because the symmetry $\hat{\Pi}_{L}=(\hat{\Pi}_{1}\mathcal{C}_{\ell})^{2}$ is
preserved.

We note that the residual $\mathbb{Z}_{4}$ symmetry enforces an exact relation
between different EA correlations, which distinguishes the composite area-law
phase from the coexistence phase. To see this we consider a distinct EA
correlation $\chi^{\prime}_{EA}(i,j)\equiv\sum_{m}p_{m}\langle
X_{i}Y_{j}\rangle_{m}^{2}$. The corresponding quantity in the two-replica
model can be written as
$\chi^{\prime(2)}_{EA}(i,j)=\mbox{$\langle\langle\mathcal{I}|$}\mathrm{i}(YZXI)_{i}(-\mathrm{i})(XZYI)_{j}\mbox{$|\psi_{gs}\rangle\rangle$}/\mbox{$\langle\langle\mathcal{I}|\psi_{gs}\rangle\rangle$}.$
Here, we replace $(YIYI)_{j}$ with
$(-\mathrm{i})(XZYI)_{j}=(ZZII)_{j}(YIYI)_{j}$. In the composite phase, we
have an exact relation $\chi^{(2)}_{EA}=\chi^{\prime(2)}_{EA}$ because charges
$\mathrm{i}(YZXI)$ and $(-\mathrm{i})(XZYI)$ related by the conjugation of
$\hat{\Pi}_{1}\mathcal{C}_{\ell}$ are condensed in the ground state with the
same amplitude. However, in the coexistence phase,
$\chi^{\prime(2)}_{EA}(i,j)$ and $\chi^{(2)}_{EA}(i,j)$ are both nonvanishing
but generally of different values. Further reducing the
$\mathbb{Z}_{4}^{\mathcal{C}_{\ell}\Pi_{1}}$ symmetry will give rise to
volume-law phases discussed in the subsection III.1.3.

####  III.1.2 SPT area-law phases

In this section, we point to the possibility of establishing symmetry
protected topological (SPT) area-law phases protected by the effective $D_{4}$
symmetry. To identify and characterize such phases, we employ the decorated
domain wall picture of Ref. Chen  _et al._ (2014). Starting from the broken
symmetry area-law phase, symmetry can be restored by condensing the domain
walls of the symmetry $\hat{\Pi}_{L}$. For the ensuing phase to be distinct
from the trivial symmetric phase the domain walls must condense only while
bound to charges of an independent symmetry. In this case the only option is
the $\mathcal{S}_{2}$ permutation symmetry. Note that condensing the
$\mathcal{S}_{2}$ charges bound to a domain wall, does not lead to breaking of
the $\mathcal{S}_{2}$ symmetry in this case because the condensed object is
nonlocal. Indeed, a dual description of the same phase is a condensate of the
$\mathcal{S}_{2}$ domain walls bound to the charges of the symmetry
$\hat{\Pi}_{L}$. The state established in this way is a fully symmetric SPT
phase, protected by the $\mathbb{Z}_{2}^{\Pi_{L}}\times\mathcal{S}_{2}$
symmetry. Starting from this SPT phase we can obtain another one by condensing
the charges $XIXI$ and $IXIX$, which commute with the protecting symmetry.
This gives the “coexistence” SPT phase.

We emphasize that the enlarged dynamical symmetry is essential for
establishing these topological phases as the $\mathbb{Z}_{2}$ symmetry of the
physical circuit does not, on its own, allow an SPT phase in a one-dimensional
system. This is different from the SPT state obtained in a measurement-only
model in Ref. Lavasani  _et al._ (2021), which is protected by the physical
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry of the measurement operators.

Let us confirm that these are indeed area-law phases, by considering the
boundary overlap of the operator that creates a pair of $\mathcal{S}_{2}$
domain walls, $\mathcal{D}_{i_{\ell}}$ and $\mathcal{D}_{i_{r}}$, at the ends
of the region $A$ (35), i.e.
$\mbox{$\langle\langle{\mathcal{I}}|$}\mathcal{D}_{i_{\ell}}\mathcal{D}_{i_{r}}\mbox{$|\psi_{gs}\rangle\rangle$}$,
where $\mathcal{D}_{i}=\prod_{j\leq i}\mathcal{C}_{\ell,j}$. Unlike in the
trivial phase, the $\mathcal{S}_{2}$ domain walls are not individually
condensed in the effective ground state. So, one might be tempted to conclude
that the overlap decays exponentially, implying a volume-law state. However,
the area law is saved by the fact that the reference state
$\langle\langle\mathcal{I}|$ breaks the $\hat{\Pi}_{L}$ symmetry, i.e. it is a
condensate of the respective charge $\mathcal{Q}_{1,j}$. This allows us to
extract such charges from the condensate and write

$\displaystyle e^{-S^{(2)}(A)}$
$\displaystyle=\frac{\mbox{$\langle\langle{\mathcal{I}}|$}\mathcal{D}_{i_{\ell}}\mathcal{D}_{i_{r}}\mbox{$|\psi_{gs}\rangle\rangle$}}{\langle\langle{\mathcal{I}}\mbox{$|\psi_{gs}\rangle\rangle$}}$
(38)
$\displaystyle=\frac{\mbox{$\langle\langle{\mathcal{I}}|$}(\mathcal{Q}_{1,i_{\ell}}\mathcal{D}_{i_{\ell}})(\mathcal{Q}_{1,i_{r}}\mathcal{D}_{i_{r}})\mbox{$|\psi_{gs}\rangle\rangle$}}{\langle\langle{\mathcal{I}}\mbox{$|\psi_{gs}\rangle\rangle$}}\to\text{const}.$

In the last line, we used the fact that the $\mathcal{S}_{2}$ domain wall
coupled to the $\hat{\Pi}_{L}$ charge (i.e. $\mathcal{Q}_{1,j}$) is condensed
in the ground state.

Finally we note that these SPT phases cannot be distinguished from their
trivial counterparts using the bulk probes discussed above. We can show that
the subsystem parity variance $\Pi_{A}^{(2)}$ is non-decaying using the dual
argument to that used for the purity. $\Pi_{A}^{(2)}$ is given as a boundary
matrix element of the $\hat{\Pi}_{L}$ string operator $\prod_{j\in
A}\hat{\Pi}_{L,j}$, which creates a pair of domain walls on the two ends of
subsystem $A$. While these domain walls are not individually condensed, a
bound state of a $\hat{\Pi}_{L}$ domain wall and an $\mathcal{S}_{2}$ charge
is condensed. Since $\langle\langle{\mathcal{I}}|$ breaks the
$\mathcal{S}_{2}$ symmetry, when calculating the matrix element we can always
extract $\mathcal{S}_{2}$ charges $\langle\langle{\mathcal{I}}|$ and attach to
the domain walls. This implies non-decaying subsystem parity. Note that this
argument implies that the parity string is identical to the string order
parameter due to the broken symmetry in the reference state. Thus string order
parameters are also ruled out as diagnostics to distinguish the SPT and
trivial phases.

Nor do the EA correlations able to distinguish the SPT phases from the trivial
ones. The EA correlations decay exponentially in the symmetric SPT phase, as
in the trivial state, because both the ground state and the reference state
$\langle\langle{\mathcal{I}}|$ are symmetric under $\hat{\Pi}_{1}$, while the
EA order parameter is odd under this symmetry. In the “coexistence” SPT phase
the order parameter $(XIXI)_{j}$ is condensed leading to nondecaying EA
correlations, as in the trivial coexistence phase.

We conclude that the SPT phases cannot be distinguished by bulk probes. In
Section III.2 we show that a different type of diagnostic, when applied to the
edge of the system, allows to distinguish between the SPT and trivial phases.

####  III.1.3 Volume-law phases

Based on the criterion we developed above for establishing an area-law phase,
we expect that a volume-law phase is established if the $\mathcal{S}_{2}$
permutation symmetry is broken along with all composite symmetries
$g_{\mathcal{I}}\,\mathcal{C}_{\ell}$, where $g_{\mathcal{I}}$ is a symmetry
present in the reference state $\langle\langle\mathcal{I}|$. One example of
such composite symmetry is $\hat{\Pi}_{1}\mathcal{C}_{\ell}$.

To identify the volume-law phases, we start from the fully symmetric area-law
phase with the effective $D_{4}$ symmetry and determine the possible charges
that can be condensed to reduce the symmetry in a way that $\mathcal{S}_{2}$
and required composite symmetries are broken.

The most straightforward option to break symmetry is to condense the
$\mathcal{S}_{2}$ charges that are neutral under the parity symmetries
$\hat{\Pi}_{1}$ and $\hat{\Pi}_{L}$. This gives the symmetric volume-law phase
with a residual symmetry
$\mathbb{Z}_{2}^{\Pi_{L}}\times\mathbb{Z}_{2}^{\Pi_{1}}$. Due to the presence
of the symmetries $\hat{\Pi}_{L}$ and $\hat{\Pi}_{1}$, the EA order and parity
variances behave exactly as they do in the symmetric area-law phase.

One way to further break symmetry is to condense the pair of charges
$(XXII)_{j}$ and $(IIXX)_{j}$, which are related by the local conserved
parities $\mathcal{X}_{j}$. A second way is to condense the charges
$(XIIX)_{j}$ and $(IXXI)_{j}$, similarly related by $\mathcal{X}_{j}$. Recall
that in the area-law phase the first pair of charges was related to the second
pair by the $\mathcal{S}_{2}$ permutation symmetry, which is broken in the
volume-law phase. Thus, the phase transition, which in the area-law phase
involved simultaneous condensation of all four charges, generically splits
into two transitions. Which intermediate phase is established between the
symmetric and the broken symmetry volume-law states depends on which pair of
charges condenses first.

Yet another way to break symmetry, starting from the symmetric volume-law
phase, is to condense the charges $XIXI$ and $IXIX$ to establish a third
distinct intermediate phase. These three routes of breaking successive
symmetries starting from
$\mathbb{Z}_{2}^{\Pi_{L}}\times\mathbb{Z}_{2}^{\Pi_{1}}$ are summarized as
follows

$\displaystyle 1)\,$
$\displaystyle\mathbb{Z}_{2}^{\Pi_{L}}\times\mathbb{Z}_{2}^{\Pi_{1}}$
$\displaystyle\rightarrow\leavevmode\nobreak\ \mathbb{Z}_{2}^{\Pi_{1}}$
$\displaystyle\rightarrow\leavevmode\nobreak\ \emptyset,$ $\displaystyle 2)\,$
$\displaystyle\mathbb{Z}_{2}^{\Pi_{L}}\times\mathbb{Z}_{2}^{\Pi_{1}}$
$\displaystyle\rightarrow\leavevmode\nobreak\ \mathbb{Z}_{2}^{\Pi_{1}\Pi_{L}}$
$\displaystyle\rightarrow\leavevmode\nobreak\ \emptyset,$ $\displaystyle 3)\,$
$\displaystyle\mathbb{Z}_{2}^{\Pi_{L}}\times\mathbb{Z}_{2}^{\Pi_{1}}$
$\displaystyle\rightarrow\leavevmode\nobreak\ \mathbb{Z}_{2}^{\Pi_{L}}$
$\displaystyle\rightarrow\leavevmode\nobreak\ \emptyset.$ (39)

We note that these phases are not necessarily realized in a given model, and
one may be able to make physical arguments why certain phases can be hard or
even impossible to realize in the quantum circuit.

Route 1 is realized if, starting from the symmetric volume-law phase, $XXII$
and $IIXX$ are condensed first. Because this condensate breaks the
$\hat{\Pi}_{L}$ and $\hat{\Pi}_{1}\hat{\Pi}_{L}$ symmetry, the subsystem
parity variance decays exponentially in this phase. Furthermore, we cannot
create the EA order parameter $XIXI$ out of any pair of condensed charges.
Hence, the EA order also vanishes. We therefore term this phase the
featureless volume-law phase. It is worth pointing out that this state has
exactly the same symmetry breaking pattern as the maximally mixed state
$\mbox{$|{\mathcal{I}}\rangle\rangle$}=\mathds{1}^{\otimes 2}$. Hence, we can
view this state as being smoothly connected to a conventional thermal state.
From this phase, the system can go into the broken symmetry volume-law phase
by condensing the charges $XIIX$ and $IXXI$.

Route 2 is realized if the charges $XIIX$ and $IXXI$ are the first to condense
starting from the symmetric volume-law phase, leaving the symmetry generated
by $\hat{\Pi}_{1}\hat{\Pi}_{L}$ intact. The ensuing intermediate phase
exhibits both long-range EA order and nondecaying parity variance. The former
is seen by writing the EA order parameter as the product $XIXI=(XXII)(IXXI)$.
The first factor is condensed in $\langle\langle{\mathcal{I}}|$, whereas the
second factor is condensed in $|\psi_{gs}\rangle\rangle$. The latter is seen
by factoring
$\hat{\Pi}_{L}(A)=\hat{\Pi}_{1}(A)(\hat{\Pi}_{1}(A)\hat{\Pi}_{L}(A))$. We call
this phase the volume-law coexistence phase I. The broken symmetry phase is
reached from here by further condensing the charges $XXII$ and $IIXX$

Finally, route 3 is realized, starting from the symmetric volume-law phase, by
condensing the charges $XIXI$ and $IXIX$. As in the area-law case, this again
gives rise to coexistence of long-range EA order and nondecaying parity
variance. We thus term this phase the volume-law coexistence phase II.

There is yet one more volume-law phase, obtained from the symmetric area-law
phase by breaking the $\mathcal{S}_{2}$ symmetry while retaining a composite
$\mathcal{Z}_{2}$ symmetry generated by $\mathcal{C}_{\ell}\hat{\Pi}_{L}$.
This is achieved by condensing the order parameter $ZYXI$ symmetrized over the
residual symmetry $\mathcal{C}_{\ell}\hat{\Pi}_{L}$ and the local symmetries.
This “composite” volume-law phase, as we term it, exhibits an exponentially
decaying parity variance because the ground state breaks the $\hat{\Pi}_{L}$
symmetry along with all the composite symmetries
$g_{\mathcal{I}}\,\hat{\Pi}_{L}$. At the same time, there is a long-range EA
order, which can be seen by factoring $XIXI=(YYII)\mathrm{i}(ZYXI)$. The first
factor is condensed in $\langle\langle\mathcal{I}|$. The second is a component
of the order parameter of this phase. Further reducing the residual symmetry
leads to the broken symmetry volume-law phase.

###  III.2 Fisher Information as a probe of circuit states

So far we have considered equal-time properties of the output state, which
translate to boundary matrix elements in the effective model. An alternative
diagnostic approach, which is more natural to implement experimentally,
focuses on the probability distribution of measurement outcomes obtained along
the time evolution. In Ref. Bao  _et al._ (2020), we showed that the
sensitivity of the measurement outcome distribution to changes in the initial
state is a direct probe of the emergence of an encoding state. This
sensitivity to initial conditions, is measured by a quantity known as the
Fisher information in the distribution. The Fisher information is depressed
below its maximal attainable value upon entering the volume-law phase,
indicating that information about the perturbation to the initial state
remains partially hidden and thus protected from the measurements. Within the
effective model of random unitary circuits, the behavior of the Fisher
information is a direct probe of spontaneous breaking of the $\mathcal{S}_{n}$
permutation symmetries established in the volume-law phases Bao  _et al._
(2020).

The Fisher information probe can be adapted to characterize the symmetry
enriched phases of the circuit. Specifically, we consider the sensitivity of
the measurement outcome distribution to a symmetry breaking perturbation of
the initial state. We will show that the corresponding Fisher information
allows to distinguish between different broken symmetry states and even
identify SPT phases.

Let us start by giving a precise definition of the probing scheme. In the
first, initialize the system in a pure state with definite parity
$\mbox{$|\psi_{0}\rangle$}=\bigotimes_{j=1}^{L}\mbox{$|0\rangle$}$. The next
step is to apply a weak parity changing perturbation at site $i$,
$U_{\theta,i}=e^{-\mathrm{i}\theta X_{i}}$. Then operate the hybrid circuit on
this state. We denote by $p_{\theta,m}$ the distribution of measurement
outcomes resulting from the initial state perturbed by $U_{\theta}$, dropping
the site index $i$ for simplicity. The Fisher information, which quantifies
the sensitivity of the measurement outcome distribution to the perturbation
$\theta$ is defined as

$\displaystyle
F=\frac{\mathrm{d}^{2}}{\mathrm{d}\theta^{2}}D_{KL}(p_{0,m}||p_{\theta,m})\Big{|}_{\theta=0}.$
(40)

$D_{KL}$, the Kullback-Leibler (KL) divergence, measures the
distinguishability of two probability distributions
$D_{KL}(p_{0,m}||p_{\theta,m})=\sum_{m}p_{0,m}\log(p_{0,m}/p_{\theta,m})$. The
first order derivative of $D_{KL}$ is zero as $D_{KL}$ is non-negative and
vanishes at $\theta=0$.

In close analogy with the von-Neumann entropy we analyze KL divergence by
defining a replica sequence

$\displaystyle
D^{(n)}\equiv\frac{1}{1-n}\log\left(\frac{\sum_{m}p_{0,m}p_{\theta,m}^{n-1}}{\sum_{m}p_{0,m}^{n}}\right),$
(41)

where $D_{KL}$ is recovered in the replica limit $n\to 1$, i.e.
$D_{KL}=\lim_{n\to 1}D^{(n)}$. The $n=2$ replica quantity $D^{(2)}$ can be
formulated in our framework as

$\displaystyle D^{(2)}$
$\displaystyle=\lim_{t\to\infty}-\log\left(\frac{\mbox{$\langle\langle\mathcal{I}|$}e^{-tH_{\text{eff}}}\mathcal{U}_{\theta,X}\mbox{$|\psi_{0}\rangle\rangle$}}{\mbox{$\langle\langle\mathcal{I}|$}e^{-tH_{\text{eff}}}\mbox{$|\psi_{0}\rangle\rangle$}}\right)$
$\displaystyle=-\log\left(\frac{\mbox{$\langle\langle\psi^{\mathcal{I}}_{gs}|$}\mathcal{U}_{\theta,X}\mbox{$|\psi_{0}\rangle\rangle$}}{\mbox{$\langle\langle\psi^{\mathcal{I}}_{gs}|\psi_{0}\rangle\rangle$}}\right),$
(42)

where
$\mbox{$|\psi_{0}\rangle\rangle$}=\bigotimes_{j=1}^{L}\mbox{$|0000\rangle\rangle$}$
represents the initial state $|\psi_{0}\rangle$ in the duplicated Hilbert
space $\mathcal{H}^{(2)}$, and
$\mathcal{U}_{\theta,X}=\mathds{1}\otimes\mathds{1}\otimes U_{\theta,i}\otimes
U_{\theta,i}^{*}$ is the unitary operation in $\mathcal{H}^{(2)}$ generated by
the single-qubit rotation $U_{\theta,i}$ acting on one copy of the density
matrix. The ground state
$\mbox{$\langle\langle\psi^{\mathcal{I}}_{gs}|$}=\lim_{t\to\infty}\mbox{$\langle\langle\mathcal{I}|$}e^{-tH_{\text{eff}}}$
is obtained from long imaginary time evolution of the reference state. We
distinguish this state by the superscript $\mathcal{I}$ from the ground state
obtained by evolving the initial state $|\psi_{0}\rangle\rangle$. There is a
subtle difference between these two states in the broken symmetry phase.
Because the reference state $\langle\langle\mathcal{I}|$ has broken
symmetries, the ground state evolved from it can break symmetries explicitly.
By contrast, the state $|\psi_{gs}\rangle\rangle$ in the broken symmetry phase
is a macroscopic superposition of the two broken-symmetry states (i.e. a cat
state).

From the KL divergence, we define the $n$-th Fisher information as
$F^{(n)}\equiv\mathrm{d}^{2}_{\theta}D^{(n)}\rvert_{\theta=0}$. Thus, the
second Fisher information $F^{(2)}$ is given by a boundary matrix element

$\displaystyle F^{(2)}$
$\displaystyle\equiv\mathrm{d}_{\theta}^{2}D^{(2)}=2\frac{\mbox{$\langle\langle\psi^{\mathcal{I}}_{gs}|$}1-\mathcal{O}_{X,i}\mbox{$|\psi_{0}\rangle\rangle$}}{\mbox{$\langle\langle\psi^{\mathcal{I}}_{gs}|\psi_{0}\rangle\rangle$}}$
$\displaystyle=2\left(1-\frac{\mbox{$\langle\langle\psi^{\mathcal{I}}_{gs}|$}\widetilde{\mathcal{O}}_{X,i}\mbox{$|\psi_{0}\rangle\rangle$}}{\mbox{$\langle\langle\psi^{\mathcal{I}}_{gs}|\psi_{0}\rangle\rangle$}}\right).$
(43)

Here $\mathcal{O}_{X,i}=[(IIXX)_{i}+(XXII)_{i}]/2$ and in the last equality we
claim that $\mathcal{O}_{X,i}$ can be replaced with the operator
$\widetilde{\mathcal{O}}_{X,i}=[(IIXX)_{i}+(XXII)_{i}+(XIIX)_{i}+(IXXI)_{i}]/2$
without changing the Fisher information. This is because $XIIX$ and $IXXI$ are
odd under the symmetry $\hat{\Pi}_{1}$, whereas $\langle\langle\mathcal{I}|$,
$|\psi_{0}\rangle\rangle$ and $\exp(-tH_{\text{eff}})$ are even. Therefore,
the matrix element of these components vanishes.333Note that even if the
ground state spontaneously breaks the $\hat{\Pi}_{1}$ symmetry, it manifests
in $\langle\langle\psi^{\mathcal{I}}_{gs}|$ through emergence of a cat state
because the reference state is symmetric under $\hat{\Pi}_{1}$.

For convenience, we define the “order parameter”

$\Phi^{(2)}\equiv 1-{1\over
2}F^{(2)}=\frac{\mbox{$\langle\langle\psi^{\mathcal{I}}_{gs}|$}\widetilde{\mathcal{O}}_{X,i}\mbox{$|\psi_{0}\rangle\rangle$}}{\mbox{$\langle\langle\psi^{\mathcal{I}}_{gs}|\psi_{0}\rangle\rangle$}}.$
(44)

Because $\widetilde{\mathcal{O}}_{X,i}$ is symmetric under both left and right
swap, while it is odd under $\hat{\Pi}_{L}$, the order parameter can detect
breaking of circuit symmetries that anti-commute with $XXII$ and $IIXX$, while
it is insensitive to breaking of the $\mathcal{S}_{2}$ permutation symmetry.

It is important to note that this behavior is special to circuits with
physical $\mathbb{Z}_{2}$ symmetry. In particular, the replacement of
$\mathcal{O}_{X,i}$ with $\widetilde{\mathcal{O}}_{X,i}$ can be made only if
$H_{\text{eff}}$ inherits the Ising symmetry $\hat{\Pi}_{1}$ from the circuit.
To define a probe that differentiates between area-law and volume-law phases
as in Ref. Bao  _et al._ (2020), we would need to consider the Fisher
information, which measures the sensitivity of the measurement outcomes to a
different perturbation, which respects the circuit symmetries. Note also that
the matrix element $\Phi^{(2)}$ is necessarily positive in the broken symmetry
state because $\langle\langle\psi^{\mathcal{I}}_{gs}|$ breaks the symmetry in
a definite way inherited from $\langle\langle{\mathcal{I}}|$, which is a
positive weight superposition of symmetric and anti-symmetric states.

The behavior of $\Phi^{(2)}$ in the different phases is summarized in Table 1.
From the information theoretical perspective, $\Phi^{(2)}>0$ (i.e. Fisher
information below its maximal value) means that the information about the
perturbation to the initial state remains partially hidden from the
measurement results.

The Fisher information can furthermore allow detection of the SPT area-law
phases by directly probing edge states. Recall that the reference state
$\langle\langle{\mathcal{I}}|$ provides symmetry breaking initial conditions
to the imaginary time evolution
$\mbox{$\langle\langle\mathcal{I}|$}e^{-tH_{\text{eff}}}\to\mbox{$\langle\langle\psi^{\mathcal{I}}_{gs}|$}$.
In the SPT phase, the broken symmetry can propagate to infinite time by
coupling to the edge zero mode. This implies the order parameter has a spatial
dependence, and $\Phi^{(2)}_{i}>0$ for $i$ near the edge, but decaying
exponentially into the bulk of the qubit chain. It is also possible to probe
the broken permutation symmetry at the edge of the SPT phase by measuring the
Fisher information associated with a perturbation that respects the physical
circuit symmetry.

Crucially, this detection scheme for the edge modes of the SPT phase is
efficient and scalable in the sense that the number of repetitions of the
experiment needed to gather enough data does not grow with the system size.
Since we are probing a gapped state with a finite correlation length $\xi$ in
space and time, the natural expectation is that it is enough to produce the
Fisher information from a partial measurement record including only a finite
space-time bubble of radius $r\gg\xi$ away from the perturbation. Indeed,
within the effective theory, discarding measurement results creates a bulk
field that breaks the $\mathcal{S}_{2}$ symmetry in the discarded region
outside of the bubble Bao  _et al._ (2020). The effect of these symmetry
breaking conditions on the perturbed site in the bubble decays exponentially
with $r/\xi$, leading to a correspondingly small effect on the Fisher
information. Thus, the required number of repetitions is expected to scale as
$e^{\xi}$, allowing an efficient detection.

###  III.3 Numerical demonstration of transitions between volume-law phases

Figure 2:  (a) Two chain model investigated numerically. Internal dynamics of
chain-1 is generated by purely unitary two-qubit gates, while that of chain-2
by pure projective measurements (of $Z_{i}$ or $X_{i}X_{i+1}$). The two chains
are coupled by random unitary gates applied with probability $q$. All gates
and measurements commute with the $\mathbb{Z}_{2}$-parity symmetry of chain 2.
(b) Circuit diagram for the model. Every time step consists of three types of
operations: two-qubit unitary gates within chain-1 (blue boxes) are randomly
drawn from the Clifford group; measurements in $X_{i}X_{i+1}$\- or
$Z_{i}$-basis are performed on chain-2 (green boxes with rounded corners) with
the probabilities $r$ and $1-r$, respectively; and the inter-chain gates
(orange bonds) are randomly drawn from the Clifford elements preserving
$Z_{i}$ on the chain-2 and applied to the circuit with the probability $q$.

Having classified the possible phases of quantum circuits enriched by the
physical $\mathbb{Z}_{2}$ circuit symmetry, we now demonstrate that parts of
this phase structure is realized in a concrete circuit model illustrated in
Fig. 2. We simulate the circuit directly without relying on replica tricks.
The results therefore give further confidence that the effect of the enlarged
dynamical symmetry persists to the physical replica limit $n\to 1$.

Figure 3:  (a) Phase diagram of the two chain model. $q$ is the inter-chain
coupling. $r$ is the probability of measuring $X_{i}X_{i+1}$ on chain 2
(versus $1-r$ of measuring $Z_{i}$). The red crosses indicate phase boundaries
extracted from the numerics. All three phases exhibit volume-law entanglement.
(b) Numerical simulation results. Top panels: Half-chain entanglement entropy
normalized by volume. Insets: system size dependence of the entanglement
entropy density at points $r=0.6,0.8,0.9$ (indicated by arrows) indicates
convergence to a non-vanishing value as $L\to\infty$. Bottom: Behavior of the
Edwards-Anderson correlation function $\chi_{EA}(L/2)$ (blue diamonds) and
parity variance $\Pi(A=L/2)$ (red crosses) as a function of $r$ for two values
of $q$. The nearly overlapping lines correspond to different system sizes
$L=70,110,180,280,450$.

Our model consists of two coupled $L$-qubit chains in a ladder geometry,
undergoing a hybrid dynamics that consists of unitary gates and projective
measurements. We restrict the set of unitary gates to the elements of the two-
qubit Clifford group, which allows efficient simulation of large systems
Aaronson and Gottesman (2004); Chandran and Laumann (2015). This restriction
should not affect any qualitative features of our analysis as Clifford gates
are statistically indistinguishable from Haar random unitary gates up to third
moments, i.e. Clifford group forms unitary 3-design Webb (2015).

The circuit dynamics can be summarised as follows. First, all unitary gates
and measurements respect the global $\mathbb{Z}_{2}$ symmetry generated by
$\hat{\pi}=\prod_{j}Z_{j}$, where $Z_{j}$ are the Pauli-$Z$ operators acting
on the qubit at site $j$ in chain-2. Thus, the qubits of chain-1 are neutral
with respect to the $\mathbb{Z}_{2}$ symmetry. The time evolution internal to
chain-1 is generated purely by unitary gates, which are unrestricted by
symmetry. The time evolution internal to chain-2 is generated purely by
measurements: with probability $r$ a pair of qubits at sites $j$ and $j+1$ are
measured in the $X_{j}X_{j+1}$-basis or, with probability $1-r$ the qubit at
site $j$ is measured in the $Z_{j}$-basis. These measurements are denoted by
the green boxes in the figure. Finally, a coupling between the two chains is
implemented by two-qubit unitary gates on the rungs of the ladder, applied
with probability $q$ per gate at each time step. These gates, represented by
orange rungs in Fig. 2, are drawn uniformly from one of the 384 elements in
the Clifford group that commute with $Z_{j}$ in the chain-2. Thus, the
parameter $q$ controls the unitary coupling strength between two chains, and
$r$ tunes the ratio between different kinds of measurements on chain-2. The
system is initialized in the state
$\mbox{$|\psi_{0}\rangle$}=\mbox{$|0\rangle$}^{\otimes 2L}$, and we are
interested in its steady-state properties.

We can easily understand the steady-state behavior of our model in the
decoupled chains limit $q=0$. In this case, chain-1 evolves as a random
unitary circuit, in which the entanglement entropy of any subsystem grows
linearly until it saturates to its maximum Nahum  _et al._ (2017); von
Keyserlingk  _et al._ (2018). In contrast, chain-2 undergoes measurement-only
dynamics. Numerical simulations and analytical arguments in Ref. Sang and
Hsieh (2020) have shown that this leads to two area-law phases separated by a
phase transition at $r_{c}=1/2$. For $r<r_{c}$, chain-2 is in the symmetric
area-law phase characterized by nondecaying parity variance and exponentially
decaying EA correlations. Chain-2 with $r>r_{c}$ realizes the broken symmetry
area-law phase with long-range EA correlations and exponentially decaying
parity variance.

When the two decoupled chains are considered as a single system, they
trivially realize a volume-law entanglement scaling (due to chain-1), which
undergoes $\mathbb{Z}_{2}$ symmetry breaking phase transition (due to chain-2)
at $r=r_{c}$. Within our formalism these two volume-law phases are represented
by gapped ground states of an effective Hamiltonian. Thus, they are expected
to be robust to (at least) weak coupling between the chains.444By the same
token, the phases are also robust to any other small perturbation to the
circuit, which respects the symmetry. Furthermore, we can use the symmetry
analysis of Section III.1 to predict which phases would be realized.

Since chain-1 has broken $\mathcal{S}_{2}$ symmetry, its leading effect at
weak coupling $q\ll 1$ is to exert an effective $\mathcal{S}_{2}$-symmetry
breaking field on chain-2, which does not affect the properites associated
with the parity symmetry. Thus, from the symmetry perspective the state
established for $r<r_{c}$ at an infinitesimal coupling is identical to the
symmetric volume-law phase (row 5 in Table 1). Similarly, the state
established for $r>r_{c}$ at an infinitesimal coupling is identical to the
broken symmetry volume-law phase (row 9 in Table 1).

Finally, coupling to the symmetry breaking field produced by chain-1 is
relevant at the critical point separating the two area-law phases of chain-2.
As discussed in subsection III.1.1, the critical point at $q=0$ is
characterized by condensation of the charges $\mathcal{Q}_{1}=XXII$ and
$\mathcal{Q}_{2}=IIXX$ together with $\mathcal{Q}_{1}^{\prime}=IXXI$ and
$\mathcal{Q}_{2}^{\prime}=XIIX$. Because the two pairs of charges are
connected by the $\mathcal{S}_{2}$ symmetries, they must condense together
when these symmetries are present. However, once we turn on the field breaking
the $\mathcal{S}_{2}$ symmetries, the critical point will generically split
into two critical points, one involving condensation $\mathcal{Q}_{1}$ and
$\mathcal{Q}_{2}$ and the other involving condensation of
$\mathcal{Q}_{1}^{\prime}$ and $\mathcal{Q}_{2}^{\prime}$. This gives rise to
an intermediate phase in which only one of the two pairs of charges is
condensed (the featureless phase in row 6 or the coexistence phase I in row 7
of Table 1). Which intermediate phase is realized depends on the charges that
acquire a lower energy, which in turn may depend on the details of the model.

The persistence of the two distinct volume-law phases and the emergence of an
intermediate phase between them in the coupled-chain model are nontrivial
predictions of the effective theory, which we now turn to test numerically. To
this end, we simulate the time evolution of systems with length $L\leq 450$.
We evaluate $\Pi(A)$, $\chi_{EA}$, and the volume-normalized half-system
entanglement entropy, $S(L/2)/(L/2)$ averaged over $\sim 500$ random circuit
realizations. To focus on the long-range and late-time behaviors, we use
$i_{\ell}\approx L/3$ and $i_{r}\approx 2L/3$ for evaluating
$\chi_{EA}(i_{\ell},i_{r})$ and $\Pi(A)$ (as the left and right edges of $A$)
and we evaluate all quantities at time $T\approx 3L$.555More precisely, we
choose $T=3L+1-a$ with $a=0$ or $a=1$ depending on whether $L$ or $L+2$ is a
multiple of $4$, respectively. We always choose even $L$. This choice is
introduced in order to avoid artifacts associated the gates acting across the
half-chain boundaries at the very last time step. We did similar
regularizations for the choice of $i$ and $j$ for $\Pi(A)$ and $\chi_{EA}$.

The results of our numerical simulations are summarized in Fig. 3. For the
decoupled chains ($q=0$), we find, as expected, a single phase transition
tuned by the ratio $r$ of bond to site measurements. Upon increasing the
inter-chain coupling to $q=0.3$ and $q=0.5$, we observe a splitting into two
critical points and emergence of an intermediate phase. As seen in the bottom
of panel (b), the intermediate phase is characterized by vanishing of both the
the EA correlations and the parity variance, consistent with the featureless
phase in row 6 of Table 1. Physically, this may be interpreted as a result of
the unitary inter-chain coupling countering the effect of the $X_{i}X_{i+1}$
measurements, favoring the state without long-range order. We see that for
sufficiently large $q$ there is no broken symmetry state for any value of $r$.
At the same time, the unitary coupling $q$ involving a single-site $Z_{i}$
operator on chain 2 commutes with the sub-system parity. Therefore introducing
this coupling does not change the point at which non-decaying sub-system
parity variance is established. It would be interesting to check if coupling
the chains by various measurements that preserve the $\mathbb{Z}_{2}$ symmetry
instead of unitary gates could favor a volume-law coexistence state (row 7 in
Table 1) as the intermediate phase.

The top panels of Fig. 3(b) confirm that all the observed phases are indeed
characterized by volume-law scaling of the entanglement entropy. For large
values of $r$, we observe a slow drift of the entropy-density $S(L/2)/(L/2)$
as a function of system sizes [inset of Fig. 3(b)]. We attribute this drift to
the additive contribution $S_{0}$ of chain-2 to $S(L/2)$. Indeed, plotted as a
function of $1/L$, the entanglement entropy density shows a clear linear
dependence, $S(L/2)/(L/2)=2S_{0}/L+s$ with a finite entropy density $s>0$ in
the limit $L\rightarrow\infty$ (the dotted lines in the inset).

The establishment of long-range order in a one-dimensional system with
extensive entropy, may appear, at first sight, contradictory to theorems
prohibiting such order at any non-vanishing temperature. But, this is not a
thermal system.

As we mentioned above, an appropriate description of the volume-law phase at
weak coupling ($q\ll 1$) is to replace chain-1 by an effective field that
breaks the $\mathcal{S}_{2}$ permutation symmetries. In the physical circuit,
this corresponds to coupling chain-2 to an effective infinite temperature bath
in place of chain-1. We are then interested in the trajectories of the mixed
state in chain-2, connected by coupling $q$ to an infinite temperature bath
and subject to local measurements. For each individual trajectory, the
measurements act as couplings to zero temperature bath, wherein the
measurement projection collapse the local wave function into a definite
quantum state with zero entropy. Because of this coupling to two very
different baths, each trajectory is very far from being in thermal equilibrium
and establishment of long-range order is not prohibited.

One of the nontrivial results of this section is that measurements can
facilitate the establishment of quantum order in one dimension, even in states
with volume-law entanglement entropy. We may further argue that this
protection extends also to topological order and edge states. To this end, we
note that our model can be also interpreted as the system of Majorana fermions
on chain-2 via Jordan-Wigner transformation. On-site $Z$ measurements map to
measurement of the Majorana parity $\mathrm{i}\gamma_{2i-1}\gamma_{2i}$ on odd
bonds, whereas the $XX$ measurements map to measurements of parity on even
bonds $\mathrm{i}\gamma_{2i}\gamma_{2i+1}$. Chain-1 is coupled to chain-2 with
unitary coupling $q$ to the Majorana bond parities. The broken symmetry
volume-law phase of the qubit system (phase III in Fig. 3) translates to a
topological volume-law state with free Majorana edge modes. The symmetric
volume-law phase (phase I in Fig. 3) translates to a volume-law phase which is
distinct from a thermal state by having subsystem fermion-parity variance that
is nondecaying with subsystem size. Finally, phase II in Fig. 3 is smoothly
connected to the conventional thermal phase of the Majorana system, where
there is no long-range correlation and the fermion parity of a subsystem
fluctuates.

##  IV Gaussian fermionic circuits

We turn to investigate hybrid quantum circuits that operate on fermionic
degrees of freedom rather than on qubits. Specifically we consider quadratic
gates and measurements that preserve the Gaussianity of the fermionic wave
function. Such circuits cannot sustain a volume-law state for any finite rate
of measurements Cao  _et al._ (2018); Fidkowski  _et al._ (2020). However,
recent numerical work indicated that a critical phase with entanglement that
scales as $\log(L)$ is established at low measurement rates Alberton  _et al._
(2020); Sang  _et al._ (2021).

Here, we provide a simple description of this critical phase using the
framework developed in Section II to map the dynamics to an effective ground
state problem. While we describe the general structure of the theory for any
number of replicas $n$, we carry out a more detailed study only of the case of
$n=2$. This theory predicts a measurement-induced Kosterlitz-Thouless (KT)
transition from the critical phase into two area-law phases, which are
distinguished by the sign of the vortex fugacity. These are identified as a
trivial and a topological area-law phase. We map pertinent properties of the
circuit dynamics to universal boundary operators in this theory, thus
facilitating a detailed comparison to exact numerical simulations of the
circuit. The numerical results, obtained for systems of size $L\leq 160$, are
consistent with a KT transition.

Before proceeding we note that the two area-law phases, a topological and a
trivial phase have been previously discussed in the context of measurement
only models Nahum and Skinner (2020); Lang and Büchler (2020). Furthermore,
Sang et al. Sang  _et al._ (2021) argued that adding a specific set of gates
to the measurement only model, allows a mapping to the completely packed loop
model with crossings Nahum  _et al._ (2013), which also hosts a critical state
known as the Goldstone phase. We comment on the connections of our results to
these loop model predictions at the end of the section.

###  IV.1 Model

The general structure of the circuits we consider here is shown in Fig. 4.
Circuit elements are operating on a one-dimensional chain of Majorana modes
$\gamma_{2j-1}$ and $\gamma_{2j}$. We view such pairs of “Majorana sites”
$(2j-1,2j)$ as making up a single “physical site” $j$, hosting a single
complex fermion $f_{j}=(\gamma_{2j-1}+\mathrm{i}\gamma_{2j})/2$ on site $j$.
For simplicity, we take the initial state to be the vacuum of all fermionic
modes, i.e. $f_{j}|\text{vac}\rangle_{f}=0\leavevmode\nobreak\
\leavevmode\nobreak\ \forall j$.

Figure 4: Gaussian fermionic circuit. Each time step $\delta t$ consists of
sequential application of one layer of site unitary gates, bond unitary gates,
site measurements, and bond measurements. The quadratic site and bond parity
measurements are applied with probabilities $\Gamma_{s}\delta t$ and
$\Gamma_{b}\delta t$ respectively.

The unitary elements of the circuit are generic Gaussian gates operating on
the nearest neighbor Majorana pairs

$\displaystyle U_{s,j}=e^{\theta_{s,j}\gamma_{2j-1}\gamma_{2j}},\quad
U_{b,j}=e^{\theta_{b,j}\gamma_{2j}\gamma_{2j+1}}.$ (45)

We call $U_{s,j}$ the site unitary and $U_{b,j}$ the bond unitary as they
operate on physical sites and bonds, respectively. These gates are depicted as
gray and yellow boxes in Figure 4.

Similar to the qubit circuits discussed in the previous section, drawing the
couplings $\theta_{s,j}$ and $\theta_{b,j}$ from symmetric distributions
ensures a mapping to imaginary time evolution with an effective Hermitian
Hamiltonian. Here, we shall draw the bond coupling $\theta_{b,j}$ from a
Gaussian distribution $N(0,\sigma^{2}_{J})$ with zero mean and variance
$\sigma^{2}_{J}$, while the site coupling $\theta_{s,j}$ is taken from a
uniform distribution over $[0,2\pi)$. We note that this specific choice of the
distributions is made to simplify the exact mapping and the resulting
effective Hamiltonian, and it is not crucial for our general discussion in
this section.

After each layer of unitary gates in the circuit, measurements are made on a
certain fraction of the fermion parity operators defined on sites and bonds:

$\displaystyle\hat{\pi}_{s,j}=-\mathrm{i}\gamma_{2j-1}\gamma_{2j},\quad\hat{\pi}_{b,j}=-\mathrm{i}\gamma_{2j}\gamma_{2j+1}.$
(46)

Measurements of the site parity are performed with a probability $p_{s}$ at
each time step and those of bond parity with probability $p_{b}$. Upon the
measurement of $\hat{\pi}_{a,j}$ with $a\in\\{s,b\\}$, the quantum state
evolves according to the Born rules: projection onto a definite parity
eigenstate $P_{\pm}|\Psi\rangle/\sqrt{\langle\Psi|P_{\pm}|\Psi\rangle}$ with
probability $\langle\Psi|P_{\pm}|\Psi\rangle$, where
$P_{\pm}=(1\pm\hat{\pi}_{a,j})/2$.

We investigate the quantum state at the output of this circuit after a long
time evolution of $N_{T}\gg 1$ steps. As before, we are interested in
quantities that are nonlinear in the system density matrix, such as the
entanglement entropy of a subsystem $A$ or the variance of fermion parities
taken over the ensemble of trajectories:

$\Pi_{a,A}\equiv\overline{\sum_{m}p_{m}\Big{\langle}\prod_{j\in
A}\hat{\pi}_{a,j}\Big{\rangle}^{2}_{m}}.$ (47)

The sum is over different trajectories (runs of the circuit) characterized by
a specific set of measurement outcomes $m$ and the angled brackets represent a
quantum expectation value in the output state of that trajectory.

###  IV.2 Enlarged Symmetry

The entanglement entropy and other quantities that depend on high moments of
observables can be derived from the average replicated density matrix
$|\tilde{\rho}^{(n)}\rangle\rangle$. Thus, it is important to identify the
symmetry of the dynamics of $n$ replicas, which will allow us to classify
distinct steady state phases. Due to the special structure of Gaussian
fermionic circuits, this symmetry is different from the dynamical symmetry
identified for the qubit circuits in Section II.2.

To facilitate the analysis, we introduce second quantized operators acting in
the duplicated Fock space. The Majorana operators $\gamma_{j}$ acting on the
wave-function of the physical circuit are extended to
$\gamma_{j,\alpha\sigma}$, where $\alpha\in\\{1,2,\dots,n\\}$ indicates the
replica on which this Majorana operates and
$\sigma\in\\{\uparrow,\downarrow\\}$ indicates forward and backward branch of
the evolution, respectively. Note that the mapping from tensor product
operators, such as
$\gamma_{j}\otimes\mathds{1}\otimes\mathds{1}\otimes\mathds{1}$ or
$\mathds{1}\otimes\gamma_{j}\otimes\mathds{1}\otimes\mathds{1}$ to the second
quantized operators in the duplicated Hilbert space requires extra phase
factors to ensure anti-commutation relations between all fermionic operators.
The precise mapping is given in Appendix E.

The evolution of the replicated density matrix in a single time step of the
fermionic circuit in Fig. 4 is given by

$\displaystyle\mbox{$|\tilde{\rho}^{(n)}(t+\delta
t)\rangle\rangle$}=\mathcal{M}^{(n)}_{b}\mathcal{M}^{(n)}_{s}\mathcal{U}_{b}^{(n)}\mathcal{U}_{s}^{(n)}\mbox{$|\tilde{\rho}^{(n)}(t)\rangle\rangle$}.$
(48)

As discussed in Section II, $\mathcal{M}^{(n)}_{s/b}$ is a linear operation on
the unnormalized quantum state $|\tilde{\rho}^{(n)}\rangle\rangle$ generated
by a layer of site/bond measurements performed with probabilities $p_{s/b}$.

We first focus on the the dynamical symmetry in the case of pure unitary
evolution, i.e. when $p_{s}=p_{b}=0$. The generators of the unitary time steps
take the form of decoupled quadratic Majorana Hamiltonians in the duplicated
Fock space:

$\displaystyle h_{s}^{(n)}$
$\displaystyle=\sum_{\alpha=1}^{n}\sum_{j=1}^{L}\mathrm{i}\theta_{s,j}\left(\gamma_{2j-1,\alpha\uparrow}\gamma_{2j,\alpha\uparrow}-\gamma_{2j-1,\alpha\downarrow}\gamma_{2j,\alpha\downarrow}\right),$
$\displaystyle h_{b}^{(n)}$
$\displaystyle=\sum_{\alpha=1}^{n}\sum_{j=1}^{L}\mathrm{i}\theta_{b,j}\left(\gamma_{2j,\alpha\uparrow}\gamma_{2j+1,\alpha\uparrow}-\gamma_{2j,\alpha\downarrow}\gamma_{2j+1,\alpha\downarrow}\right).$
(49)

Using the Majorana operators of the $2n$ chains, it is natural to construct a
local $so(2n)$ algebra on every Majorana site $\ell$, generated by
$\Gamma^{\alpha\sigma,\beta\sigma^{\prime}}_{\ell}=\frac{\mathrm{i}}{4}[\gamma_{\ell,\alpha\sigma},\gamma_{\ell,\beta\sigma^{\prime}}]$.
From these local objects, we construct a global $so(2n)$ algebra with elements

$\Gamma^{\alpha\sigma,\beta\sigma^{\prime}}=\sum_{j=1}^{L}\left({\sigma\sigma^{\prime}}\Gamma^{\alpha\sigma,\beta\sigma^{\prime}}_{2j-1}+\Gamma^{\alpha\sigma,\beta\sigma^{\prime}}_{2j}\right),$
(50)

which commute with the generators (49) of the unitary dynamics. Note that here
and below $\sigma=\uparrow$ ($\sigma=\downarrow$) is identified with
$\sigma=+1$ ($\sigma=-1$).

The $\Gamma$ matrices generate proper rotations among the $2n$ species of
Majorana fermions. In addition we have the single-branch fermion parity
symmetries
$\hat{\Pi}_{\alpha,\sigma}=\prod_{j=1}^{L}\mathrm{i}\gamma_{2j-1,\alpha,\sigma}\gamma_{2j,\alpha,\sigma}$,
which constitute improper rotations, e.g. $\hat{\Pi}_{1,\uparrow}$ that maps
$\gamma_{j,1,\uparrow}\mapsto-\gamma_{j,1,\uparrow}$. Thus, altogether the
purely unitary dynamics on $n$ copies have an $O(2n)$ symmetry.

In addition, the Hamiltonian admits a fermionic time-reversal symmetry
$\mathbb{T}$, under which
$\mathbb{T}\gamma_{\alpha,\uparrow}\mathbb{T}^{-1}=\gamma_{\alpha,\downarrow}$,
$\mathbb{T}\gamma_{\alpha,\downarrow}\mathbb{T}^{-1}=-\gamma_{\alpha,\uparrow}$,
and $\mathbb{T}\,\mathrm{i}\,\mathbb{T}^{-1}=-\mathrm{i}$. We conclude that
the unitary dynamics exhibits the symmetry
$O(2n)\rtimes\mathbb{Z}_{2}^{\mathbb{T}}$. Note that the effective time-
reversal symmetry is not exactly the Hermiticity symmetry, which appears as a
nonlocal transformation in the fermionic duplicated Hilbert space (see
Appendix E).

Adding measurements reduces the symmetry. Only the subset of the generators
(50) having $\sigma=\sigma^{\prime}$ commute with the measurement operators.
These generators, supported only on either the forward or the backward
branches, form a $so(n)\oplus so(n)$ algebra and give rise to a $O(n)\times
O(n)$ symmetry. Together with fermionic time reversal, the symmetry of the
$n$-copy unitary evolution with measurements is $[O(n)\times
O(n)]\rtimes\mathbb{Z}_{2}^{\mathbb{T}}$.

It is worth pointing out that the dynamical symmetry of the pure unitary
evolution is the same as the static symmetry in the case of free fermions in
the symplectic class AII Ryu  _et al._ (2007); Fu and Kane (2012). Moreover,
the addition of measurements here breaks the $O(2n)$ symmetry in the same way
as adding a non-Hermitian quadratic coupling does in the free fermion problem.
There is, however, an important difference in taking the replica limit, which
in our case is $n\to 1$, whereas it is $n\to 0$ for the disordered fermion
ground state problem.

We note on passing that a measurement-only model (i.e. no unitary gates) also
has an enlarged dynamical symmetry $O(2n)\rtimes\mathbb{Z}_{2}^{\mathbb{T}}$
like the purely unitary model. In this case, the $O(2n)$ symmetry generators
are given as simple zero momentum sums of the local generators. That is, the
first term in Eq. (50) does not have an additional sign
$\sigma\sigma^{\prime}$.

Similar to the case of the qubit circuits considered in Section III, the
effective global symmetry is reduced when we account for the presence of local
integrals of motion. Specifically, the fermion gates and measurements commute
with all the local operators
$\mathcal{R}_{\ell}=\prod_{\alpha=1}^{n}\mathrm{i}\gamma_{\ell,\alpha\uparrow}\gamma_{\ell,\alpha\downarrow}$
that measure the total parity of fermions from all species at Majorana site
$\ell$ (see Appendix A). Thus $|\tilde{\rho}^{(n)}\rangle\rangle$ evolves to a
ground state of $H_{\text{eff}}$ with definite eigenvalues of
$\mathcal{R}_{\ell}$. Such a state necessarily breaks the single branch parity
symmetries that anti-commute with the $\mathcal{R}_{\ell}$.

Having removed the improper rotations generated by the single-branch parities,
we are left with the effective symmetry
$SO(2n)\rtimes\mathbb{Z}_{2}^{\mathbb{T}}$ for the purely unitary circuit.
With measurements, the effective symmetry is reduced to
$\mathcal{G}^{(n)}_{f,\text{eff}}\equiv(SO(n)\times
SO(n))\rtimes(\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{\mathbb{T}})$, where the
first $\mathbb{Z}_{2}$ is generated by fermion parity in both the forward and
backward branch of the first copy, i.e.
$\hat{\Pi}_{1}=\hat{\Pi}_{1\uparrow}\hat{\Pi}_{1\downarrow}$. Finally, we note
that the global symmetry
$\prod_{\alpha=1}^{n}\hat{\Pi}_{\alpha}=\prod_{\ell}\mathcal{R}_{\ell}$, which
is part of $\mathcal{G}^{(n)}_{f,\text{eff}}$, cannot be broken because it is
a product of local integrals of motion.

We now elaborate on how this scheme applies to the case of two replicas. The
dynamical $O(2)\times O(2)$ symmetry in presence of measurements, implies two
conserved $U(1)$ charges, which we can write as the total occupation numbers
of complex fermion modes

$N_{\sigma}=\sum_{j=1}^{L}N_{\sigma,j}=\sum_{j=1}^{L}c^{\dagger}_{2j-1,\sigma}c_{2j-1,\sigma}+c^{\dagger}_{2j,\sigma}c_{2j,\sigma},$
(51)

The complex fermions are not local to a single copy of the system and are
rather defined as superpositions of Majorana modes on the two copies:

$\displaystyle
c_{2j-1,\uparrow}=\frac{\gamma_{2j-1,1\uparrow}+\mathrm{i}\gamma_{2j-1,2\uparrow}}{2},c_{2j,\uparrow}=\frac{\gamma_{2j,2\uparrow}-\mathrm{i}\gamma_{2j,1\uparrow}}{2},$
$\displaystyle
c_{2j-1,\downarrow}=\frac{\gamma_{2j-1,1\downarrow}-\mathrm{i}\gamma_{2j-1,2\downarrow}}{2},c_{2j,\downarrow}=\frac{\gamma_{2j,2\downarrow}+\mathrm{i}\gamma_{2j,1\downarrow}}{2}.$
(52)

The $U(1)$ symmetries are supplemented by the single branch $\mathbb{Z}_{2}$
parity symmetries to give the two copies of $O(2)=U(1)\rtimes\mathbb{Z}_{2}$.
These parity transformations act on the two conserved fermion species
independently as particle-hole transformations, namely
$c_{j,\sigma}\mapsto(-1)^{j}c_{j,\sigma}^{\dagger}$. Together with the time-
reversal we have the full symmetry $\mathcal{G}^{(2)}_{f}\equiv[O(2)\times
O(2)]\rtimes\mathbb{Z}_{2}^{\mathbb{T}}$.

For purely unitary dynamics, the symmetry is enlarged to
$O(4)\rtimes\mathbb{Z}_{2}^{\mathbb{T}}=(SO(4)\rtimes\mathbb{Z}_{2})\rtimes\mathbb{Z}_{2}^{\mathbb{T}}$.
The $SO(4)$ symmetry has a useful and intuitive representation in terms of two
$SU(2)$ symmetries, which stems from the isomorphism $SO(4)\cong[SU(2)\times
SU(2)]/\mathbb{Z}_{2}$. One of the $SU(2)$ symmetries is associated with
rotations between the forward and backward branches, labeled as up and down
spins. We call this the $\Sigma$ spin symmetry. The other $SU(2)$ symmetry is
a charge “$\eta$-symmetry” Yang and Zhang (1990). The respective generators of
the spin and charge symmetries can be written explicitly as

$\displaystyle\Sigma_{j}^{+}$ $\displaystyle\equiv
c_{2j-1,\uparrow}^{\dagger}c_{2j-1,\downarrow}+c_{2j,\uparrow}^{\dagger}c_{2j,\downarrow},$
(53a) $\displaystyle\Sigma_{j}^{-}$
$\displaystyle\equiv\left(\Sigma_{j}^{+}\right)^{\dagger},$ (53b)
$\displaystyle\Sigma_{j}^{z}$
$\displaystyle\equiv\frac{1}{2}\sum_{\sigma}\sigma\left(c_{2j-1,\sigma}^{\dagger}c_{2j-1,\sigma}+c_{2j,\sigma}^{\dagger}c_{2j,\sigma}\right),$
(53c) $\displaystyle\eta_{j}^{+}$ $\displaystyle\equiv-
c_{2j-1,\uparrow}^{\dagger}c_{2j-1,\downarrow}^{\dagger}+c_{2j,\uparrow}^{\dagger}c_{2j,\downarrow}^{\dagger},$
(53d) $\displaystyle\eta_{j}^{-}$
$\displaystyle\equiv\left(\eta_{j}^{+}\right)^{\dagger},$ (53e)
$\displaystyle\eta_{j}^{z}$
$\displaystyle\equiv\frac{1}{2}\sum_{\sigma}\left(c_{2j-1,\sigma}^{\dagger}c_{2j-1,\sigma}+c_{2j,\sigma}^{\dagger}c_{2j,\sigma}-1\right).$
(53f)

Here, the ladder operators
$\Sigma^{\pm}_{j}=\Sigma^{x}_{j}\pm\mathrm{i}\Sigma^{y}_{j}$ and
$\eta^{\pm}_{j}=\eta^{x}_{j}\pm\mathrm{i}\eta^{y}_{j}$. The $SU(2)\times
SU(2)$ is quotient by $\mathbb{Z}_{2}$ because the total $\eta$ and $\Sigma$
spins must add to an integer spin representation to give the required $SO(4)$
symmetry. Finally, the $SO(4)$ symmetry is supplemented by the time-reversal
symmetry and the single branch parity symmetry to complete the full
$O(4)\rtimes\mathbb{Z}_{2}^{\mathbb{T}}$ symmetry.

When acting on the spin and $\eta$-operators, the single-branch fermion parity
transformation exchanges two sets of $SU(2)$ generators, i.e.
$\Sigma_{j}^{\pm,z}\mapsto\eta_{j}^{\pm,z}$ and
$\eta_{j}^{\pm,z}\mapsto\Sigma_{j}^{\pm,z}$, and the time reversal flips the
sign of spin operators, while keeping charge $\eta$-operators invariant, i.e.
$\mathbb{T}\Sigma_{j}^{\pm,z}\mathbb{T}^{-1}=-\Sigma_{j}^{\pm,z}$ and
$\mathbb{T}\eta_{j}^{\pm,z}\mathbb{T}^{-1}=\eta_{j}^{\pm,z}$.

As explained above, the ground state of the effective Hamiltonian must break
the single-branch symmetries and thus exhibits a reduced effective symmetry.
For purely unitary dynamics, the full symmetry
$O(4)\rtimes\mathbb{Z}_{2}^{\mathbb{T}}$ is thus reduced to
$SO(4)\rtimes\mathbb{Z}_{2}^{\mathbb{T}}$. With measurements, the reduced
symmetry is $\mathcal{G}^{(2)}_{f,\text{eff}}\equiv(U(1)\times
U(1))\rtimes(\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{\mathbb{T}})$. Here the first
$\mathbb{Z}_{2}$ is generated by the particle-hole transformation $\mathbb{P}$
for both conserved fermion species, which maps
$\mathbb{P}\eta_{j}^{\pm,z}\mathbb{P}^{-1}\mapsto-\eta_{j}^{\mp,z}$ and
$\mathbb{P}\Sigma_{j}^{\pm,z}\mathbb{P}^{-1}\mapsto-\Sigma_{j}^{\mp,z}$.

###  IV.3 Effective 1D quantum Hamiltonian

We now carry out the program developed in Section II to map the dynamics of
the averaged replicated density matrix $|\tilde{\rho}^{(2)}\rangle\rangle$ to
effective imaginary time evolution with a one-dimensional quantum Hamiltonian.
Almost everything proceeds exactly as detailed in Section II for qubit
circuits. Here we elaborate on distinct features in our fermionic model
related to the enlarged dynamical symmetry.

As a first step in the program, we consider averaging over probabilistic
measurements and bond unitary gates. The site unitary gates, designed in this
model to project onto a reduced Hilbert space, will be considered later. It is
straightforward to integrate over the Gaussian distribution of the bond
coupling $\theta_{b,j}$ to obtain the averaged bond unitary acting on the
doubled density matrix