# Phase transitions in $(2+1)$D subsystem-symmetric monitored quantum circuits

Cole Kelson-Packer<EMAIL_ADDRESS>Center for Quantum Information and
Control, Department of Physics and Astronomy, University of New Mexico,
Albuquerque, New Mexico 87106, USA  Akimasa Miyake<EMAIL_ADDRESS>Center for
Quantum Information and Control, Department of Physics and Astronomy,
University of New Mexico, Albuquerque, New Mexico 87106, USA

###### Abstract

The interplay of unitary evolution and projective measurements is a modern
interest in the study of many-body entanglement. On the one hand, the
competition between these two processes leads to the recently-discovered
measurement-induced phase transition (MIPT). On the other, measurement-based
quantum computation (MBQC) is a well-known computational model studying how
measurements simulate unitary evolution utilizing the entanglement of special
resources such as the 2D cluster state. The entanglement properties enabling
MBQC may be attributed to symmetry-protected topological (SPT) orders,
particularly subsystem symmetric (SSPT) orders. It was recently found that the
1D cluster state may be associated with an SPT phase in random circuits
respecting a global $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry, and
furthermore that all phase transitions in this scenario belong to the same
universality class. As resources with greater computational power feature
greater symmetry, it is fruitful to investigate further any relationship
between levels of symmetry in MIPTs and MBQC. In this paper we investigate
MIPTs on a torus with three levels of symmetry-respecting unitary evolution
interspersed by measurements. Although we find two area-law phases and one
volume-law phase with distinct entanglement structures for each ensemble, the
phase transition from the volume-law phase to the area-law phase associated
with the 2D cluster state has variable correlation length exponent $\nu$.
Whereas $\nu\approx 0.90$ for unconstrained Clifford unitaries and $\nu\approx
0.83$ for globally-symmetric Cliffords, subsystem-symmetric Cliffords feature
a much smaller value $\nu\approx 0.38$. It is speculated that the hierarchy of
distinct transitions seen in these random monitored quantum circuit models
might have consequences for computational universality in MBQC.

##  I Introduction

Quantum computers, besides offering prospective advantages in certain
computational tasks [1, 2], and the ability to realize various classically
impossible effects [3, 4, 5], also provide potential tunable platforms for
investigating various many-body phenomena. It is well-established that random
monitored circuits, wherein both unitary operations and mid-circuit
measurements are carried out on a set of qubits, provide novel insights into
the dynamics of quantum entanglement via the measurement induced phase
transition (MIPT). The phenomenology of this transition has been the subject
of many recent works [6, 7, 8, 9, 10, 11, 12], and several reviews are
available [13, 14, 15].

The central fact is that the competing local processes of entanglement-
producing unitaries and disentangling measurements are balanced at a nonzero
measurement rate. Along one line of intuition, this is surprising; for two
subsystems of a pure state, only unitaries crossing their boundary can produce
entanglement between them, whereas measurements may have a deleterious effect
anywhere. According to this perspective any finite projection rate should
destroy entanglement in the thermodynamic limit [16]. While this logic follows
through in certain systems [17], remarkably more often than not a stable
“volume-law” entanglement phase exists in addition to a low entanglement
“area-law” counterpart [6, 8, 9].

The importance of these random monitored circuits and more generally the MIPT
is that their phenomenology is thought to reflect behavior of generic
monitored systems, just as random matrix theory qualitatively describes heavy
atomic nuclei [13, 18]. By tuning the setup more specialized scenarios may be
investigated. One such generalization introduces additional stabilizer
measurements. If these are stabilizers of states of particular interest, then
one may probe general properties of the classes such states belong to. For
example, $(2+1)$D random circuits with toric code measurements feature a phase
diagram with a stable critical region as a consequence of the robustness of
the associated stabilizer state [19]. Similarly, a symmetry-protected
topological (SPT) phase has been realized in $(1+1)$D by considering
stabilizers of the 1D cluster state and limiting the unitaries applied to
those respecting a global $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry [20].

One result of this last study is that the critical exponents of the transition
to the SPT phase match $(1+1)$D percolation, with correlation length exponent
$\nu\approx 4/3$ in particular. Although diagnostics like the topological
entanglement entropy differentiate the SPT phase from the trivial phase, all
transitions in the phase diagram belong to the same universality class. This
is true regardless of whether the transition moves from the volume-law phase
to the SPT phase, or to the trivial entanglement phase. In this sense the
transition would seem insensitive to some properties of the 1D cluster state.

A particularly important property is the fact that the 1D cluster state is a
computational resource for measurement-based quantum computing (MBQC). MBQC is
an alternative to the traditional gate model based on the measurement
protocols carried out on special resource states [21, 22, 23]. While the 1D
cluster state is not a resource for universal computation, it can act as a
quantum wire, teleporting states encoded into its edges by adaptive
measurements. Most quantum states cannot be used for even this purpose;
rather, it is a consequence of the special entanglement structure of the 1D
cluster state emerging from an inherent $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
symmetry [24, 25]. For universal computation the requisite symmetry structure
is even more particular; not only do states with little entanglement lack the
necessary correlations to achieve this effect, but even highly entangled
states fail to be useful [26, 27]. It is an intriguing question to what extent
this hierarchy of symmetry/utility in MBQC has a counterpart in MIPT – that
is, to what extent the nature of the transitions are affected by symmetry.

To this end we consider $(2+1)$D random monitored quantum circuits featuring –
in addition to the usual single-qubit measurements – stabilizer measurements
of the 2D cluster state. We also employ three different ensembles of unitary
dynamics. As defined later, these unitary ensembles consist of 5-qubit local
Cliffords, with no constraints (“Clifford”), global symmetry-protected
topological constraints (“SPT Clifford”), and subsystem symmetry-protected
topological constraints (“SSPT Clifford”), which are associated respectively
with the symmetries of the 2D $L\times L$ lattice as
$\mathds{I}\subset\mathbb{Z}_{2}\times\mathbb{Z}_{2}\subset\mathbb{Z}_{2}^{2L-1}$.
Using entanglement diagnostics to detect the transition, we find that the
critical exponents of the transition – and thus the associated universality
classes – differ between these ensembles.

In particular, we find that while the Clifford ensemble features transitions
between both area-law phases and the volume-law phase with the same critical
exponent as $(2+1)$D percolation, the SSPT Clifford ensemble exhibits new
exponents differing from both each other and percolation. This suggests that
the SSPT circuit – respecting the SSPT symmetry associated with universal
measurement-based quantum computation – is at least sensitive to more
particular characteristics of the states it produces.

Following this Introduction, Sec. II provides background for our own setting
explicated in Subsec. III.1. Then, after the numerical methods adumbrated in
Subsec. III.2, our results are delivered in Subsec. III.3, which are split
into further subsubsections for each unitary ensemble considered. Discussion
and conclusion are covered in Sec. IV. Our appendix includes elaborations of
various minutiae, including further details of our numerical methodology,
elucidation of order parameters, and analysis of properties at the pure-
measurement critical point.

##  II Background

Random monitored quantum circuits present a novel setting for investigating
generic many-body quantum phenomena where both the effects of repeated
measurements and unitary evolution are important. When measurements are
applied at some rate $p$, a so-called measurement-induced phase transition
(MIPT) is observed between an area-law phase of low entanglement at high
measurement rate $p$ and a volume-law entanglement regime of high entanglement
for low $p$ [7, 8, 9]. Although measurements typically have a disentangling
effect wherever they are applied, they are constrained to be local. Unitary
dynamics, meanwhile, tend to conspire to hide information in long-range
correlations. The transition between these two phases reflects whether or not
generic circuits effectively “protect” quantum correlations encoded in the
initial state [11, 12]. Variations – such as additional measurement types or
constraints on the unitaries applied – leads to the emergence of further novel
phases [20, 19, 28, 29, 30, 31].

To understand the basic phenomenology of these circuits it is useful to map
them to more standard mathematical models. For the case of $p=0$, where only
pure unitary dynamics are in play, many of the observed entanglement
properties can be related to the famous Kardar-Parisi-Zhang universality class
[32]. Entanglement growth has a parallel in ballistic deposition, operator
spreading follows exclusion processes, and bounds on entanglement between
regions is echoed by polymers in random energy environments. This last
relationship is especially pertinent for nonzero $p$, as the problem changes
into first passage percolation – another standard mathematical model with
well-studied phase transitions [33]. In particular, the correlation length
critical exponent of percolation in 2D, $\nu=4/3$, matches critical exponents
found for the measurement induced phase transition in $(1+1)$D reasonably well
[6, 8]. The intuition of these mappings is furthermore supported by work
exactly mapping certain select models to classical models with percolation
exponents, such as the Potts model [11, 34].

Good numerical agreement complements these analytical results. It is perhaps
surprising that in spite of the esoteric, nonequilibrium nature of this
transition, many standard numerical tools may still be fruitfully applied. In
particular, the finite-size scaling ansatz [35, 36, 37] retains its utility.
This workhouse uses the often-vindicated hypothesis that, near a transition
point $p_{c}$, an order parameter $\Delta$ follows a certain function of the
tuning parameter $p$ and system length $L$:

$\displaystyle\Delta(L,p)\approx L^{\gamma}F((p-p_{c})L^{1/\nu}),$ (1)

where $F$ is some master function and $\nu$ and $\gamma$ are universal
exponents. By finding a good fit for numerical values of the diagnostic
$\Delta$, $\nu$ and $\gamma$ may be estimated. The tuning parameter $p$ in
this case refers in our context to some measurement rate. As for the order
parameter $\Delta$, an important fact to keep in mind is that the transition
is most readily seen on the level of nonlinear properties of individual
quantum trajectories. For example, a transition emerges when the average of
the entanglement of individual states $\overline{S_{vN}(\rho)}$ is considered,
but disappears for the entanglement of the average state $S_{vN}(\bar{\rho})$.
This is because the entropy is nonlinear in the state – that is, because
$\overline{S_{ent}(\rho)}\neq S_{ent}(\overline{\rho})$. Many useful
diagnostics of the transition, such as the topological- or ancilla
entanglement entropies [38, 39, 40, 10] are also nonlinear. This makes
experimental detection of the transition highly challenging [41, 42, 43].

“The” topological entanglement entropy in fact refers to a class of related
metrics. The essential idea is this: from locality, the von Neumann entropy of
a subsystem $A$ is expected to obey an area-law with subleading corrections,
$S_{A}\sim|\partial A|-\Gamma$. Topological entanglement entropies $S_{top}$
are constructed by taking special combinations of subsystems and summing their
entropies together in such a way as to isolate the subleading $\Gamma$. This
remainder is thought to reflect long-range entanglement properties. Taking
sufficiently large system sizes washes out short range effects, leaving behind
a robust measure of phenomena of interest.

A common version is the 7-term topological entanglement entropy, also known as
the tripartite mutual information [38, 44]. For subregions $A$, $B$, and $C$
of a system – such as for a torus, as depicted in Fig. 1 – this may be defined
as

$\displaystyle S_{top}\equiv S_{AB}+S_{BC}+S_{AC}-S_{A}-S_{B}-S_{C}-S_{ABC}.$
(2)

Figure 1:  Our regions chosen for evaluating the topological entanglement
entropy $S_{top}$ are cylinders on a torus. This can be represented as
rectangles on a 2D plane with opposite sides identified.

$S_{top}$ is particularly valuable when considering generalizations of the
random monitored circuit setup involving exotic measurements and phases. If
the measurements are of stabilizers of a state with a distinguishable value of
$S_{top}$, then an associated circuit phase may be readily distinguished.
Taking the toric code $S_{top}$ as an example, the 4-term variant of $S_{top}$
(related to strong subadditivity) and 7-term version related above yield
values of $1$ or $2$ respectively. This differs from the $0$ of simple product
states, and also from the extensive scaling of states generically produced by
Haar unitary dynamics [19].

Another guiding example is the 1D cluster state defined on an open chain [20].
Where the toric code realizes a genuine topological phase, the 1D cluster
state is a symmetry-protected topological (SPT) state. In 1D such states are
classified by the projective representations of the eponymous symmetry group
acting on the boundary – that is to say, by the second group cohomology [45].
The associated phase in a random monitored quantum circuit also exhibits SPT
behavior; in such a scenario, $S_{top}$ detects the edge degrees of freedom,
again providing a value distinguishing the trivial phase from a more exotic
SPT phase. The ancilla entanglement entropy also couples to these boundary
degrees of freedom, facilitating the long-time storage of encoded edge qubits.

The cluster state behavior is particularly notable given the state’s
relationship with measurement-based quantum computation (MBQC). MBQC is a
well-known alternative to the gate model of quantum computation. Instead of
evolving a state through a circuit and enacting entangling operations, one
begins with a special entangled “resource state” and simulates a circuit via
evolution with adaptive local measurements. A particular class of resource
states are the so-called “graph states,” which are states defined on a graph,
with qubits prepared in the $|{+}\rangle$ state on vertices, and edges
denoting qubits between which controlled-Z operators are enacted.
Equivalently, graph states can be said to be the joint eigenstate of the
commuting operators $\sigma_{x}^{\alpha}\bigotimes\limits_{\beta\in
N(\alpha)}\sigma_{z}^{\beta}$, where $\alpha,\beta$ label vertices and
$N(\alpha)$ denote the graphical neighbors of $\alpha$. We will work with a
slightly different convention, where a Hadamard is acted upon every qubit
after this setup, exchanging the roles of the Pauli operators $X$ and $Z$.

The power of resource states is often attributable to underlying symmetries,
whose presence reflect a kind of irreducible entanglement structure [46].
Cluster states, defined on graphs of regular lattices, exhibit such
symmetries. The 1D cluster state, for example, features
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ global symmetry, the factors flipping
spins on even/odd sites. This symmetry enables the state to act as a 1D wire,
facilitating the teleportation of an encoded edge qubit by measurement [24].
Similarly, the 2D cluster state, enjoying an extensive number of subsystem
$\mathbb{Z}_{2}$ line symmetries, belongs to a universal phase for MBQC [47].

The 1D cluster state was investigated in a random monitored quantum circuit
context by Lavasani et al. [20]. In the spirit of the cluster state belonging
to a nontrivial SPT class, their circuits preserve
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry. Their three-qubit unitaries,
beyond belonging to the Clifford class that maps Pauli operators to other
Paulis, furthermore conserves certain local parity operators. They found three
distinct phases in their scenario, including a special “SPT” phase in addition
to the usual volume-law and trivial entanglement phases. Notably, they also
found that all phase transitions appear to belong to the same universality
class, with correlation length critical exponents of $\nu\approx 4/3$ matching
$(1+1)$D percolation theory.

This is curious; even though the phase transition recognizes that the SPT
phase differs somehow from the trivial phase, the transition in itself does
not reflect the difference in computational usefulness. In a sense, the
transition between volume and trivial phases is “the same” as the transition
between the volume and SPT phases. That is to say, the transition recognizes a
difference in entanglement structure apparently disjoint from a possible
accompanying computational transition.

The question, then, arises as to whether a transition in resource power
distinct from percolation can arise in random monitored quantum circuits. We
might be hopeful, since the 1D cluster state is not a universal resource,
whereas its 2D and 3D counterparts feature novel qualities and more elaborate
symmetry structure. Many other resource states belong to special SPT classes
with distinguished global symmetries [25, 46, 24, 48, 49, 50], but an
important characteristic of the 2D cluster state is the presence of additional
subsystem symmetries [47]. Investigation of this additional find-grained
symmetry has led to reveal the SPT phases which are universal for MBQC [47,
51, 52, 53]. Consideration of this also provides additional constraints on
random monitored circuits, and inspires the setting of our work.

##  III $(2+1)$D Symmetric circuit

###  III.1 Setting

We study $(2+1)$D random monitored circuits with two different measurement
types and draw from select unitary ensembles. Our system is defined
periodically on a torus, and following the style of [20] our operations
applied one at a time, rather than the more standard brick-pattern (see Fig.
2). At each time step we apply one of the following operations with their
corresponding probabilities:

$\displaystyle U$ 5-Qubit Clifford $\displaystyle p_{u},$ $\displaystyle
M^{z}$ Local Z Measurement $\displaystyle p^{z}_{M},$ $\displaystyle M^{s}$
Cluster Stabilizer Measurement $\displaystyle p^{s}_{M}.$

These probabilities sum to unity. After the operation is selected any system
qubit is chosen with equal probability to be the center of the operation. If
the $Z$-basis measurement is the operation, then the qubit is measured. For
the 5-qubit unitary or cluster stabilizer measurement the qubit is chosen as
the center around which either operation is applied in a “+”-shaped stencil,
as depicted in Fig. 3.

Figure 2:  Example 3D (LABEL:fig:exa) and overhead view (LABEL:fig:exb) of the
kind of circuits we consider. Two types of measurements are applied, one on a
single site in the $Z$ basis and another in a “+” shape measuring a stabilizer
of the 2D cluster state. 5-qubit unitaries are also applied in a “+” shape.
One operation is applied every time step $t$. Figure 3:  The three types of
operations performed by our random circuits. Each symbol stands in for a
qubit. On the lower left are local computational basis “Z” measurements. The
upper right has measurements of 2D cluster state-like stabilizers. The upper
left features 5-qubit Clifford unitaries.

We consider three ensembles of unitaries in our circuits. These ensembles are
required to satisfy no symmetry requirement, a global symmetry requirement,
and a subsystem symmetry requirement respectively, with associated groups
$\mathds{I}\subset\mathbb{Z}_{2}\times\mathbb{Z}_{2}\subset\mathbb{Z}_{2}^{2L-1}$
for the 2D $L\times L$ square lattice (See the section 3.4 of Ref. [53] for an
elucidation of the difference between the line symmetry
$\mathbb{Z}_{2}^{2L-1}$ and the so-called cone symmetry
$\mathbb{Z}_{2}^{2L}$). Recall that for more familiar settings of second-order
phase transitions, such as the classical Ising model, the Hamiltonian
possesses some symmetry that can be spontaneously broken, such as
$\mathbb{Z}_{2}$. As the Hamiltonian generates dynamics, it follows that time
evolution under it observes the same symmetry. We can similarly say that a
class of circuit dynamics whose constituent unitaries all respect some
symmetry group possesses that group as a symmetry.

A given circuit in our setting makes use of only one ensemble of unitaries.
Including measurements, we have three phase diagrams for each ensemble, each
with three probabilities. To access large system sizes via the stabilizer
formalism (reviewed in Appendix A.1) [54, 55] all ensembles are subgroups of
the 5-qubit Clifford group. As our primary interest is the entanglement
structure of pure states at various times, we may safely neglect stabilizer
signs and the concomitant computational complexities.

Our first ensemble consists of the entire 5-qubit Clifford group, the set of
all unitaries normalizing 5-qubit Pauli operators, which we say possesses the
trivial symmetry group $\mathds{I}$. The second ensemble is generated
similarly, and comprised of symmetric 5-qubit Cliffords that respect the
global $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ checkerboard symmetry enjoyed by
the 2D cluster state on a torus. This is depicted in Fig. LABEL:fig:syma, and
emulative of the symmetry considered in Ref. [20]. Their reduction to
“+”-shaped stencils is shown in Fig. LABEL:fig:resa. Finally, the last
ensemble has Cliffords respecting the $\mathbb{Z}^{2L-1}$ subsystem symmetries
of the toric 2D cluster state. These symmetries are depicted in the top
subfigures of Fig. LABEL:fig:symb, with their reduced stencils in Fig.
LABEL:fig:resb. Although the former two ensembles are too large to store and
generated “on the fly” as needed, the latter ensemble of unitaries is small
enough to be generated and stored in its entirety. More details on generation
are covered in Appendix A.2.

Figure 4:  The global $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ “checkerboard”
(Fig. LABEL:fig:syma) and subsystem $\mathbb{Z}^{\otimes 2L-1}_{2}$ “diagonal”
(Fig. LABEL:fig:symb) symmetries respected by Clifford ensembles. The system
here consists of $4\times 4$ qubits on a periodic lattice, making the diagonal
symmetries wrap. Figure 5:  Local stencils of Pauli operators that a unitary
applied on the center qubit must respect for the global “checkerboard” (Fig.
LABEL:fig:resa) and “diagonal” subsystem (Fig. LABEL:fig:resb) symmetries.
These are local reductions of the symmetries depicted in Fig. 4.

###  III.2 Numerical Methods

To reach large system sizes while still being able to describe highly
entangled states we employ the stabilizer formalism [54, 55]. All unitaries
used are accordingly necessarily of Clifford type. We have written our own C99
code capable of running several simulations in parallel with the aid of a
Python wrapper. Most matrix manipulations are vectorized, with the binary bits
of stabilizer matrices being stored in 256-bit integer vectors to permit rapid
binary operations. As phase factors are of little interest for our study, we
ignore them, saving on the traditional complications of determined
measurement-outcomes.

We detect transitions using three diagnostics: the topological-, ancilla-, and
dumbbell entanglement entropies ($S_{top}$, $S_{anc}$, and $S_{dumb}$) [38,
39, 40, 10, 56]. $S_{top}$ is our primary workhorse. We make use of the 7-term
variant in Eq. 2 with regions defined as in Fig. 1 along quasi-1D cylinders.
We have furthermore cross-checked our analysis with other geometries, such as
diagonal subregions, as well as a more traditional “ring” setup. Appendix B.1,
besides discussing the finer details of our implementation, also shows that
the critical points and exponents match within numerical error for these other
geometries. The ancilla entanglement entropy $S_{anc}$ has its value as a
geometrically-independent comparative sanity check, and our implementation is
discussed in Appendix B.2.

This question of geometry is relevant, due to the effect of so-called spurious
topological entanglement entropy [56, 57], wherein subsystem symmetries may
induce additional correlations that the topological entanglement entropy
defined for special regions may be sensitive to. The dumbbell entanglement
entropy [56], which is a diagnostic of subsystem symmetry, turns this
inconvenience into a tool. We employ it to detect the transition in the case
of pure measurement dynamics (i.e. when $p_{u}=0$), where neither the
topological entanglement entropy nor the ancilla entanglement entropy are
effective. The details are contained in Appendix B.3.

For our finite-size scaling analysis to retrieve details of our transitions we
employ three different curve-fitting schemes. The first method, “Nearest,”
seeks to make a hypothesized master curve connecting adjacent scaled points
along the master curve as smooth as possible. The second, “Multilevel,”
compares not with just the nearest scaled points, but all the nearest scaled
points for all different system sizes. Finally, the last, “Polynomial,” is as
the name suggests a high-order polynomial fit quantified by the residual. For
sufficiently well-behaved data and enough datapoints the three typically
agree, although the second would seem the most robust, while the last produces
the smoothest plots. A more in-depth coverage of this topic is found in
Appendix C.

Figure 6:  Phase diagrams for the unconstrained Clifford ensemble
(LABEL:fig:regphase) and global symmetry-constrained (LABEL:fig:sptphase)
ensembles in terms of the probabilities of applying 5-qubit Cliffords $p_{u}$
and computational basis $Z$ measurements $p_{M}^{z}$. Phase transitions occur
at black dots, interpolations between which form the phase boundaries. Both
diagrams are qualitatively similar, with small area-law entanglement regions
due to the scrambling power of the 5-qubit Clifford ensembles.

###  III.3 Results

####  III.3.1 Unconstrained Clifford Group

Our unconstrained Clifford group refers to the group of all 5-qubit Cliffords,
whose symmetry group is trivial. The entire phase landscape is shown in Fig.
LABEL:fig:regphase. There are three phases: a trivial area-law phase, a
cluster area-law phase, and one volume-law phase. As a consequence of 5-qubit
Cliffords consisting primarily of entangling unitaries, the volume region is
considerably larger than the other two.

The critical behavior along the $p_{M}^{s}=0$ line (where no cluster
stabilizers are measured) is expected [7], and has been numerically shown [58]
(with some prior controversy [59, 60]), to exhibit behavior akin to the 3D
bond percolation transition. In particular, the correlation critical exponent
is expected to take on the value $\nu\approx 0.88$. It is maybe not surprising
that this critical behavior persists throughout the phase diagram – that is,
even when we introduce cluster stabilizer measurements.

We observe such a similar critical exponent along the $p_{M}^{z}=0$ line. The
data, along with the fitting landscape and the finite size scaling associated
with the minimum thereof, is depicted in Fig. 7 below, where $\nu\approx
0.90$. This is not surprising for the reason that the essential phenomena seen
in the one-operation-per-time-step scheme we employ do not differ from more
standard maximally scrambling setup. In these latter setups, however, the
unitary subgroup averaging property may be used to replace single-qubit
measurements by another operator sharing the same locality as the applied
unitaries. Specifically, since the Clifford unitary taking (in the Heisenberg
picture) the operator $Z$ to the cluster state stabilizer $XXZXX$ belongs to
the set of all 5-qubit Cliffords, it follows that we may heuristically
exchange the roles of measuring either operator and expect qualitatively
similar phenomena. It is important to note, however, that this argument cannot
be applied in the SPT or SSPT case, as symmetry demands that all permitted
unitaries take the $Z$ operator to itself.

We also include in this subsubsection analysis of the pure measurement
dynamics, which, being in the limit where no unitaries are applied, is the
same for all ensembles. The data is depicted in Fig. 8. Numerical collapse
shows the critical point to be at about $p_{c}\approx 0.5$, which a duality
argument similar to [20, 19] suggests should be exactly $p_{c}=0.5$. The
critical exponent, meanwhile, is $\nu\approx 0.85$, which again matches
$(2+1)$D percolation. Further investigation into the properties of this pure
measurement critical phase are found in Appendix D. We thus have evidence that
the criticality class is constant along all phase boundaries for this
unconstrained class of unitaries.

Figure 7:  Data/analysis using the average 7-term topological entanglement
entropy $\bar{S}_{top}$ with quasi-1D cylindrical geometry for the cluster-to-
volume transition for the unconstrained Clifford ensemble. (LABEL:fig:7a):
collapsed data as a function of the probability of applying a unitary
operation $p_{u}$ (unscaled data inset). (LABEL:fig:7b): objective landscape
for the collapse using a polynomial fitting method (slice at $p_{u}=p_{c}$
inset). The minimum is found at the critical point $p_{c}\approx 0.068$ and
critical exponent $\nu\approx 0.90$. Figure 8:  Data/analysis using the
average dumbbell entanglement entropy $\bar{S}_{dumb}$ for pure measurement
dynamics. No unitaries are employed, so this is the same for every ensemble.
(LABEL:fig:8a): collapsed data as a function of the probability of performing
a local computational basis measurement $p_{s}$ (unscaled data inset).
(LABEL:fig:8b): objective landscape determining the collapse (slice at
$p_{s}=p_{c}$ inset). The minimum is at critical point/exponent $p_{c}\approx
0.5$/$\nu\approx 0.84$.

####  III.3.2 SPT Clifford Group

The SPT Clifford group considered consists of all 5-qubit Cliffords that
respect the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ checkerboard symmetry group
of the 2D cluster state. This is akin to a restriction placed on 3-qubit
Cliffords on symmetric $(1+1)$D circuits, which has a phase characterized by
measurements of 1D cluster state stabilizers [20]. The global symmetries in
question preserved by 5-qubit unitaries are depicted in Fig. LABEL:fig:syma,
with associated stencils in Fig. LABEL:fig:resa.

Our results in this case are similar to Ref. [20]. We find that although three
distinct phases exist, the critical exponent remains close to that of $(2+1)$D
percolation, with $\nu\approx 0.84$. The phase diagram, presented in Fig.
LABEL:fig:sptphase, is quite similar to the unrestricted ensemble above. The
primary visual difference is slight growth in the size of the trivial area-law
phase, owing to the unitary ensemble considered being somewhat less entangling
by virtue of imposed symmetry restrictions.

####  III.3.3 SSPT Clifford Group

Figure 9:  Phase diagram for the SSPT Clifford ensemble in terms of the
probability of applying a 5-qubit unitary $p_{u}$ and a computational basis
$Z$ measurement $p_{M}^{z}$. The three phases are separated by boundary lines
determined by interpolating critical points obtained numerically at the black
dots. Compared to Fig. LABEL:fig:regphase and Fig. LABEL:fig:sptphase, the
trivial and cluster area-law regions are considerably larger, and the critical
exponents generally differ.

The SSPT Clifford group is constituted by the 5-qubit Cliffords preserving the
$\mathbb{Z}^{2L-1}_{2}$ line symmetries of the 2D cluster state. This
includes, as a subgroup, the symmetries respected by the SPT ensemble, and so
this constraint is strictly stronger. Examples of the line symmetries are
depicted in Fig. LABEL:fig:symb, while the local stencils are shown in Fig.
LABEL:fig:resb. For the 2D cluster state on open boundaries, as with global
symmetry of the 1D cluster state, these symmetries act projectively on edge
modes, and protect the boundary degeneracy [61]. Although our dynamics take
place on the torus, we nevertheless preserve these symmetries, and hence
expect to preserve certain properties of the resource.

The phase landscape is depicted in Fig. 9. The qualitative features of the
three regions have noticeably changed compared to the previous two ensembles;
the area-law regions have expanded considerably, owing to our constraints
having made the unitaries considered less entangling. The trivial entanglement
phase in particular has grown, as all permissible unitaries must preserve a
Pauli $Z$ acting on the central qubit.

We find that the critical exponent along the $p^{z}_{M}=0$ line differs
significantly from the percolation exponent. Through several diagnostics, and
three separate fitting routines, we consistently find $\nu\approx 0.39$.
Example data is depicted in Fig. 10 – more may be found in Appendix B.1.
Comparable values of $\nu$ are found along the phase boundary, terminating at
the $p_{u}=0,p^{z}_{M}=0.5$ measurement-only critical point. The volume-to-
trivial-area-law transition for this ensemble is presented in terms of the
ancilla entanglement in Fig. 11, with $\nu\approx 0.52$. Both critical
exponents differ considerably from the $(2+1)$D percolation universality class
with its characteristic $\nu\approx 0.87$.

We interpret these new exponents to reflect a shortfall of the previously-
known percolation mapping, wherein all unitaries becomes impasses regardless
of their original form. On one hand, the mapping is certainly justifiable for
Haar-random unitaries, as the set of products of single-qubit unitaries
constitutes a set of measure zero. For Clifford-random, and even for 1D
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-respecting Cliffords [20], this mapping
still appears empirically justified, possibly because typical unitaries in
these ensembles remain sufficiently entangling. For our final ensemble here,
however, the large number of restrictions for the subsystem symmetries we have
imposed – especially the requirement of preservation of the Pauli $Z$ on the
center qubit – may make this simplification no longer applicable.

|  Symmetry Group Types |  | Volume/Cluster Area |  | Volume/Trivial Area
---|---|---|---|---|---
Unitary Group |  | ($p_{c},\nu$) |  | ($p_{c},\nu$)
Clifford | $\mathds{I}$ |  | $0.068\pm 0.001$ | $0.90\pm 0.03$ |  | $0.193\pm 0.005$ | $0.85\pm 0.03$
SPT Clifford | $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ |  | $0.084\pm 0.002$ | $0.83\pm 0.03$ |  | $0.170\pm 0.001$ | $0.83\pm 0.05$
SSPT Clifford | $\mathbb{Z}_{2}^{2L-1}$ |  | $0.176\pm 0.001$ | $0.38\pm 0.05$ |  | $0.612\pm 0.002$ | $0.52\pm 0.06$
Table 1:  Critical points $p_{c}$ and exponents $\nu$ along two phase
boundaries. $p_{c}$ refers to the value of $p_{u}$ with $p_{M}^{z}=0$ for the
volume/cluster area-law and $p_{M}^{z}=1.0-p_{u}$ for the volume/trivial area-
law phase transitions respectively. Figure 10:  Data/analysis using the
average 7-term topological entanglement entropy $\bar{S}_{top}$ with quasi-1D
cylindrical geometry for the cluster-to-volume transition for the SSPT
Clifford ensemble. (LABEL:fig:10a): collapsed data as a function of the
probability of applying a unitary operation $p_{u}$ (unscaled data inset).
(LABEL:fig:10a): objective landscape for the collapse using a polynomial
fitting method (slice at $p_{u}=p_{c}$ inset). The minimum is found at the
critical point $p_{c}\approx 0.176$ and critical exponent $\nu\approx 0.38$.
Figure 11:  Data/analysis using the average ancilla entanglement entropy
$S_{anc}$ for the volume-to-trivial-area-law transition. (LABEL:fig:11a):
collapsed data as a function of the probability of applying a unitary
operation $p_{u}$ (unscaled data inset). (LABEL:fig:11b): objective landscape
determining collapse (slice at $p_{u}=p_{s}$ inset). The minimum is at
critical point/exponent $p_{c}\approx 0.612$/$\nu\approx 0.52$.

##  IV Discussion

We have studied the interplay between a hierarchy of increasingly restrictive
symmetries and the phases observed in $(2+1)$D random monitored quantum
circuits. We have shown that, in contrast to the SPT case in (1+1)D [20], the
cluster area-law phase of a random monitored quantum circuit for a subsystem-
symmetry respecting ensemble of Clifford unitaries features a different
universality class than unrestricted dynamics in $(2+1)$D. Our results are
summarized in Table 1. The volume-to-cluster-area-law transition has a
critical exponent of $\nu\approx 0.38$, and the volume-to-trivial-area-law
transition has $\nu\approx 0.52$. These values are quite different from
$\nu\approx 0.88$ for globally-symmetric and unrestricted 5-qubit Cliffords.
The (1+1)D SPT case is analogous to the global symmetry considered here, where
the transitions all appear to remain within the percolation universality
class. This suggests that the presence of subsystem symmetry is fundamentally
altering certain critical dynamics.

It may be useful to compare our setting with other hierarchies of symmetries
in phase transitions. Recall that our (2+1)D Clifford, SPT Clifford, and SSPT
Clifford circuits feature symmetry groups
$\mathds{I}\subset\mathbb{Z}_{2}\times\mathbb{Z}_{2}\subset\mathbb{Z}_{2}^{2L-1}$
respectively. The $O(N)$ classical spin model, whose Hamiltonian is
$H=J\sum_{\langle\alpha,\beta\rangle}\mathbf{S}_{\alpha}\cdot\mathbf{S}_{\beta}$
for the $N$-component spin, contains the Ising, XY, and Heisenberg models as
special cases, with associated symmetry groups $\mathbb{Z}_{2}\subset
O(2)\subset O(3)$ respectively. The critical phenomena of these models have
been studied extensively, and the critical exponent $\nu$ is known to increase
with $N$ in this hierarchy – approximately equal to $0.63,0.67,$ and $0.70$
respectively in the spatial dimension $d_{s}=3$ [62, 63, 64]. It is believed
that this trend is true regardless of whether the spin is treated classically
or quantumly. It is also known that the one-loop analysis of the fixed point
in more general spatial dimension $d_{s}=4-\epsilon$ theory yields the
following expression for the exponent for general $N$:

$\displaystyle\frac{1}{\nu}=2-\frac{N+2}{N+8}\epsilon.$

This suggests that $\nu$ generally increases with $N$ [65].

In contrast, for our hierarchy of lattice symmetries, $\nu$ is about the same
for $\mathds{I}$ and $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ ($0.83$ and $0.85$),
and decreases for the subsystem symmetry to $0.38$. This is the opposite
trend. It might be fruitful to investigate the relationship between
hierarchies of symmetries and critical exponents in broader contexts.

Lastly, given the well-known kinship of SPT and SSPT symmetries and
measurement-based quantum computation (MBQC), understanding if this difference
in transition characteristics relates to a differentiation of computational
power is also a matter of interest, and will be the subject of future
investigation. It is worth noting that, just as for the full Clifford group,
the critical exponents observed for the SSPT ensemble violate the Harris
criterion [66]. This criterion states that local disorder is a relevant
perturbation (in the renormalization group sense) when the correlation length
critical exponent $\nu<2/d_{s}$, where $d_{s}$ is the spatial dimension. As
recent work has investigated the effect disorder has on the measurement-
induced phase transition [67, 68], investigating parallels between the effects
of such disorder and potential counterparts in MBQC might also be a
potentially interesting direction.

##  V Acknowledgements

This work was supported by the National Science Foundation STAQ Project
(PHY-1818914, PHY-2325080) and PHY-2310567. We would like to thank the UNM
Center for Advanced Research Computing, supported in part by the National
Science Foundation, for providing the research computing resources used in
this work.

##  Appendix A Simulation Methods

###  A.1 Stabilizer Formalism

One possible pitfall of using entanglement entropies to probe the measurement-
induced phase transition is the drifting of the critical parameter $p_{c}$ for
small system sizes [69]. Coupled with other finite size effects, it is often
necessary to simulate large numbers of qubits to attain a quantitative hold on
properties of the transition, especially for higher dimensions [58].

To reach these larger system sizes we employ stabilizer machinery [54, 55].
This well-known formalism tracks the quantum state $\rho$ in terms of its
stabilizers $S(\rho)$, which are the group of operators of which the state is
a $+1$ eigenstate. $S(\rho)$ may be conveniently handled by looking at its
generators. For the standard setting of qubits it is often the case of
interest that $S(\rho)$ is an Abelian subgroup of the Pauli group of tensor
products of the single-qubit operators $\\{I,X,Y,Z\\}$, including an overall
phase. In this case the generators themselves are Paulis with phases, and are
computationally amiable. For this reason, Pauli stabilizer states are simply
referred to as the “stabilizer states” for short.

The $n$-qubit Pauli operators constituting the stabilizers of a stabilizer
state may be represented (up to phase) by a string of $2n$ binary elements. By
convention, the bits of such a string correspond to the presence or absence of
an $X$ or $Z$ for a particular qubit, with a $Y$ understood by the presence of
both. Often all $X$ bits are grouped together, followed by the $Z$ bits.
Hence, the $2$-qubit operators $X_{1}X_{2}$, $Z_{1}X_{2}$, $I_{1}X_{2}$, and
$Y_{1}Z_{2}$ may be represented by the strings $1100$, $0110$, $0100$, and
$1011$ respectively. With this technology, the $n$ generators of a Pauli
stabilizer state’s stabilizer group may thus be represented by a $n\times 2n$
binary matrix. This results in considerable savings on space complexity
compared to the naive representation of generic states.

One typically wishes not for some quantum state by itself, but rather that
state as the input or output of some circuit consisting of unitary actions and
measurements. Stabilizer states typically cease to be such under generic
operations. However, in the case of measurements of Pauli operators and a
special subset of unitaries known as Clifford unitaries, stabilizer states are
mapped to stabilizer states. Restriction of circuit elements to these
operations permits efficient simulation of a state’s evolution [54].

Clifford unitaries are the normalizer $\mathcal{N}(\mathcal{P})$ of the set of
Pauli products $\mathcal{P}$, i.e. the subset of unitaries
$U\in\mathcal{N}(\mathcal{P})\subset U(2^{n})$ such that
$U^{-1}\mathcal{P}U=\mathcal{P}$. Limiting one’s attention to Pauli products
consisting only of one, two, three, etc. non-identity factors, one speaks of
$1$-, $2$-, $3$-, etc. Clifford operators. When a Pauli group of symmetries is
enforced on the system, the set of allowed Cliffords is further reduced by the
requirement that they belong to the centralizer of that Pauli subgroup. The
action of Clifford unitaries on a stabilizer state can be understood via their
Heisenberg representation action on the state’s stabilizer group. Representing
that group’s generators via binary matrices, Cliffords correspond to
symplectic operators over $\mathbb{Z}_{2}$. Hence, not only are stabilizer
states efficiently expressible, but so too are the natural unitary operations
upon them.

We now turn to measurements. The measurement of an operator results in a
projection of the state. As Pauli operators only have the two eigenvalues $\pm
1$, and since they square to the identity, this projection takes on the form
$\Pi^{\pm}_{P}=\tfrac{1}{2}(\mathds{I}\pm P)$, where $P$ is the Pauli operator
being measured. The density operator of a stabilizer state $\rho$ itself may
itself be represented as a product of the projectors corresponding to the
generators of its stabilizer group $S(\rho)$ – this is a consequence of basic
representation theory. We write the stabilizer state operator as

$\displaystyle\rho=\prod\limits_{i=1}^{n}\Pi_{P_{i}}$

The action of $\Pi^{\pm}_{P}$ on the density matrix $\rho$ of a state can be
understood by looking at its conjugate action on another projector
$\Pi_{P^{\prime}}$:

$\displaystyle\Pi^{\pm_{1}}_{P}\Pi_{P^{\prime}}\Pi^{\pm_{1}}_{P}$
$\displaystyle=\frac{\mathds{I}\pm_{1}P}{2}\frac{\mathds{I}+P^{\prime}}{2}\frac{\mathds{I}\pm_{1}P}{2},$
$\displaystyle=\frac{2(\mathds{I}\pm_{1}P)+(P^{\prime}\pm_{1}(PP^{\prime}+P^{\prime}P)+PP^{\prime}P)}{8}.$

Two Paulis either commutate or anticommutate with one another. When they
commutate, the above product becomes $\Pi_{P^{\prime}}\Pi^{\pm_{1}}_{P}$. When
they anticommutate, the result is $\tfrac{1}{2}\Pi^{\pm_{1}}_{P}$. As the
stabilizer group is Abelian, the $\Pi_{P_{i}}$ factors of $\rho$ may be freely
ordered. One may hence group all the $P_{i}$s commutating with $P$ at the
beginning, and move all those that anticommutate to the end. Let the
generators commuting with $P$ be $\\{R_{i}\\}$ and those anticommutating be
$\\{Q_{i}\\}$. Since any two $Q_{i}$s commutate with each other, the identity
$\Pi_{Q_{i}}\Pi_{Q_{j}}=\Pi_{Q_{i}Q_{j}}\Pi_{Q_{i}}$ is satisfied. The product
of the anticommutating generators may be written

$\displaystyle\prod\limits_{i=1}^{m}\Pi_{Q_{i}}$
$\displaystyle=\prod\limits_{i=1}^{m-1}\Pi_{Q_{1}Q_{i+1}}\Pi_{Q_{1}}.$

The double products $Q_{i}Q_{j}$ commutate with $P$, and so may be absorbed
into the commutating generators $R_{i}$. There is now only a single generator
anticommutating with $P$. The action of $\Pi^{\pm}_{P}$ on $\rho$ can thus be
written

$\displaystyle\Pi^{\pm}_{P}\rho\Pi^{\pm}_{P}$
$\displaystyle=\Pi^{\pm}_{P}\left(\prod\limits_{i=1}^{n}\Pi_{P_{i}}\right)\Pi^{\pm}_{P},$
$\displaystyle=\Pi^{\pm}_{P}\left(\prod\limits_{i}R_{i}\right)\Pi_{Q_{1}}\Pi^{\pm}_{P},$
$\displaystyle=\left(\prod\limits_{i}R_{i}\right)\Pi^{\pm}_{P}\Pi_{Q_{1}}\Pi^{\pm}_{P},$
$\displaystyle=\left(\prod\limits_{i}R_{i}\right)\Pi_{\pm P},$

Our new list of stabilizers includes all the old commutating Paulis, all the
old anticommutating Paulis (except $Q_{1}$) multiplied by $Q_{1}$, and the
measured Pauli $P$ (replacing $Q_{1}$). A shorter explanation proceeds along
the following logic. One first observes that $\rho$ may be represented using
any choice of generators of $S(\rho)$. With this freedom, all anticommutating
generators except for $Q_{1}$ can be replaced by $Q_{i}Q_{1}$. Then, in the
expansion by projections one readily sees that applying $\Pi^{\pm}_{P}$ to the
state has the effect of replacing the $Q_{1}$ projector with itself. As a
whole, the algorithm can be described like so: if a measured operator $P$
anticommutates with some generators of the stabilize group of a state, replace
one of the anticommutating generators $Q_{1}$ with $P$, then replace all other
anticommutating $Q_{i}$ with $Q_{1}Q_{i}$.

We have ignored phases in this explanation, as they are not important for our
purposes, however they introduce extra complications, even moreso when $P$
commutates with $S(\rho)$. Where the anticommutating case corresponds to a
nondeterministic measurement, the commuting case represents a deterministic
measurement. In this case, extra matrix manipulations must be performed to
determine the sign of the deterministic measurement. The ensuing complications
may be ameliorated by more thoughtful implementations [55, 70].

The advantage of the stabilizer formalism is twofold: reduced space
complexity, and access to a class of highly entangled states. A general
$n$-qubit wave function naively requires $\mathcal{O}(2^{n})$ numbers, and
quickly becomes unfeasible to work with. Tensor approximations, meanwhile,
effectively model the low-entanglement area-law phase, but fail in the volume
phase and near the critical point. Qubit stabilizers require only
$\mathcal{O}(n^{2})$ binaries, and are hardly less tractable in the volume-law
phase than in the area. They therefore present an efficient probe of
characteristics of measurement-induced phase transitions, provided our
purposes are met by the limited stabilizer polytope of Hilbert space we are
constrained to.

Our code stores the binary string corresponding to a given Pauli in chunks,
with each chunk being a 256-bit integer vector. On somewhat more modern
architectures, these vectors could be increased in size. This vectorization
makes various operations, such as the calculating the product of two Paulis,
quite fast, particularly since our purposes are agnostic to phases. Further
improvements could be made, such as implementing advanced rank-calculation
algorithms for finite fields instead of employing binary Gaussian elimination
[71].

###  A.2 Unitary Generation

To simulate our random monitored circuits we must know how to implement the
gates and measurements of interest. For measurements, this is simple. We only
employ computational basis $Z$ measurements and 2D cluster state “+”-shaped
stabilizer measurements (see Fig. 3). Carrying out these measurements in the
stabilizer formalism is particularly easy, as our purposes are unconcerned
with stabilizer phase.

For our three phase diagrams our gates are drawn from three corresponding
unitary ensembles: unconstrained-, SPT-, and SSPT 5-qubit Cliffords. This
poses a predicament, as the set of all 5-qubit Cliffords poses a prohibitive
space requirement. The set of $n$-qubit Cliffords may be generated recursively
given the set of $n-1$ qubit Cliffords via a well-known scheme [9]. At each
step, one randomly selects an $n-1$ qubit Clifford and two anticommutating
Pauli strings corresponding to what the $X$ and $Z$ operators map to for the
newest qubit. Ignoring a phase, this amounts to $4^{k+1}-1$ possibilities for
the first choice, and $4^{k}$ for the second. The number of unsigned $n$-qubit
Cliffords may be obtained via induction:

$\displaystyle\text{\\# }n\text{-qubit
Cliffords}=2^{n^{2}}\prod_{i=1}^{n}\left(2^{2(i+1)}-1\right).$ (3)

This evaluates to $6,720,1451520,47377612800,$ and
$24815256521932800$ for $n=1,2,3,4,5$ qubits respectively. If each is naively
represented by $2n\times 2n$ binary numbers, this corresponds to
$24,11520,52254720,3032167219200,$ and
$2481525652193280000$ bits respectively, amounting to around
$3b,1KB,7MB,380TB,300PB$ of space. Frontier currently boasts $700$ petabytes
of storage capacity [72], so this is not beyond the scope of modern
computational prowess, but a more parsimonious usage of resources would be
preferable

A satisfactory approach we have found generates the unconstrained Clifford and
globally-constrained SPT Clifford ensembles “on the fly” using the standard
recursion algorithm. We store all 3-qubit Cliffords, which may be generated
very quickly. By randomly selecting anticommutating 4-qubit and 5-qubit
Paulis, which have also been computed and stored in advance, we then generate
random 4-qubit and then 5-qubit Cliffords. For the SPT Clifford ensemble,
additional checks are performed to ensure that the resultant unitary is
adequate – this process may be sped up by thoughtful implementation
optimizations.

The SSPT Clifford ensemble is small enough to be collected, stored, and
sampled from as needed. The set of all 5-qubit Cliffords can be searched
through by generating it in small enough chunks to be stored and have symmetry
conditions checked. This procedure is readily parallelized. We have found that
a better (also parallelizable) method starts by generating the small number of
acceptable “Z” columns of such unitaries, then complete the remainder by
generating and checking all compatible “X” columns. This last method may be
realized in a matter of hours even with Python.

##  Appendix B Diagnostics

In the conventional classification of ground states, SPT states are short-
range entangled states from the strict topological perspective [45]. This is
to say that a shallow local circuit is able to transform them into product
states with trivial entanglement. What defines symmetry-protected classes is
that any such local circuit necessarily violates some natural symmetry. The
requirement of this symmetry makes such a transition nontrivial, thus it is
said that the symmetry eponymously protects the phase thus defined. It is not
usually difficult to distinguish SPT states from volume-law states; most
diagnostics distinguishing any area-law behavior from volume-law suffice.
Distinguishing SPT area-law entanglement phases from trivial area-law phases,
however, is more challenging. Furthermore, trying to identify strong SSPT
states like the 2D cluster state – which are not genuinely global SPT states
(see [73] for a more detailed explanation) – is an even more elusive task.

This section covers the three primary diagnostics we employ: the topological-,
ancilla-, and dumbbell entanglement entropies. The first two consistently
distinguish the cluster area-law phase from the volume phase, but do not
distinguish between other area-law phases. The dumbbell entanglement entropy
(see Fig. 12) successfully discriminates them, but is numerically less robust.

###  B.1 Topological Entanglement Entropy

The ground states of a large class of gapped 2D systems generically feature
area-law behavior. A subregion $R$’s entanglement entropy with respect the
rest of the system follows $S_{R}\sim c|\partial R|-\Gamma$. Here $|\partial
R|$ is the area-law contribution, while $\Gamma$ represents subleading
corrections encoding long-range correlations. For systems whose far-infrared
behavior is well-described by topological quantum field theories
$\Gamma=\log\mathcal{D}$, where $\mathcal{D}$ is the total quantum dimension
of the effective theory. Extraction of $\Gamma$, which nicely characterizes
the quantum phase in a single number, is typically carried out through clever
combinations of entanglement entropies of varying regions; this is the
topological entanglement entropy $S_{top}$ [38, 39].

The precise form comes in several varieties, possessing different numbers of
terms and reflecting different geometries. The 2D cluster state has zero
topological entanglement entropy for most conventional geometries, a property
that is expected to be shared by a cluster area-law phase of our random
monitored circuits. Nevertheless, variants of this quantity can still be used
as an order parameter, since volume phase states will typically take on an
extensive value.

Our primary analysis uses the 7-term variant of $S_{top}$, which is related to
the tripartite mutual information (Eq. 2) [44]. Our geometry, meanwhile, is
defined in terms of cylinders embedded in a torus, as shown in Fig. 1. Other
natural geometries (inset in Fig. 13) yield results quantitatively similar to
those covered in the main text. A key result of our simulations is the
appearance of a new universality class for the transition between the SSPT
Clifford volume-law phase and the cluster area-law phase. As an example of
this agreement, Figs. 14 and 15 both illustrate, for the SSPT Clifford
ensemble along the $p_{z}=0.0$ line, the analysis and collapse of $S_{top}$
using standard “ring” and diagonally-directed quasi-1D geometries. The
critical point is, as for Fig. 10 in the main text, found to be $p_{c}\approx
0.176$ uniformly, while the critical exponent $\nu$ varies from $0.387$ to
$0.406$.

Our sampling method is as follows. We first wait for a given instance of the
circuit to reach a sort of steady state. This happens when the entanglement
growth plateaus, which happens before some fraction of $L^{4}$ (we take that
fraction to be $1/4$ for concreteness). Then, we sample $S_{top}$ every
$L^{2}$ time steps, until time $\mathcal{O}(L^{4}$), at which point we start
an entirely new circuit again from scratch with a fresh product state input.
The sampling frequency is safely uncorrelated far away from the critical
point, but as the critical point is approached the correlation time formally
diverges. Nevertheless, we have found that the resultant statistical
compromisation has an effect negligible compared to the effects of the
accompanying growth of the statistical dispersion of the order parameter
distribution of $S_{top}$. Our typical number of sample circuits depends on
the system size, but varies from thousands to tends of thousands for smaller
and larger systems respectively.

Figure 12:  Example regions defined for the dumbbell entanglement entropy. In
this work it is the tripartite mutual information of the heads conditioned on
the handle, which is oriented along the direction of subsystem symmetries of
the 2D cluster state. Figure 13:  Unscaled data for 7-term average topological
entanglement entropy $\bar{S}_{top}$ for the SSPT Clifford ensemble with
$p_{M}^{z}=0$ as a function of the probability of applying a unitary operation
$p_{u}$ for the standard “ring” geometry (LABEL:fig:unsclextraa) and
“diagonal” geometry (LABEL:fig:unsclextrab) oriented along the direction of
subsystem symmetry (geometries inset). Analysis of this data is shown in Figs.
14 and 15 respectively. Figure 14:  Analysis using the standard “ring”
geometry. Data of the average 7-term topological entanglement entropy
$\bar{S}_{top}$ for the SSPT Clifford ensemble is depicted with $p_{M}^{z}=0$
for the cluster-to-volume transition. (LABEL:fig:subregab),
(LABEL:fig:subregcd), and (LABEL:fig:subregef) all illustrate
collapse/objective landscapes (LABEL:fig:subregace/LABEL:fig:subregbdf) using
three different fitting methods. Figure 15:  Analysis using the “diagonal”
geometry. Data of the average 7-term topological entanglement entropy
$\bar{S}_{top}$ for the SSPT Clifford ensemble is depicted with $p_{M}^{z}=0$
for the cluster-to-volume transition. (LABEL:fig:subdiagab),
(LABEL:fig:subdiagcd), and (LABEL:fig:subdiagef) all depict collapse
(LABEL:fig:subdiagace) and objective landscapes (LABEL:fig:subdiagbdf) (with
inset slices) using our three fitting methods.

###  B.2 Ancilla Entanglement Entropy

One element underlying the stability of the volume phase of random monitored
circuits is the notion of scrambling. Provided that the measurement rate is
not too great, information about the initial state is, after some initial
mixing time, encoded in long-range correlations that local measurements cannot
efficiently probe. The ancilla entanglement entropy $S_{anc}$ takes advantage
of this phenomenon [40, 10]. It is a measure of how much information about a
probe entangled with the system (size $N$) remains encoded at a time $t$.
Quantitatively, $S_{anc}$ is the von Neumann entropy of a probe entangled with
the system:

$\displaystyle S_{anc}$ $\displaystyle=S_{vN}(\rho_{anc}),$

where $\rho_{anc}$ is the state operator of the ancilla system. As with most
probes, this must be evaluated for an individual trajectory, then averaged
over measurement outcomes and circuit realizations to yield the diagnostic
$\bar{S}_{anc}$.

As long as a few physical qubits have not been measured, some nonlocal
correlations will survive between the system and the ancilla, and thus the von
Neumann entanglement will be nonzero. Deep in the area-law phase, where
measurements with probability $p$ are frequent, we can estimate the
probability of survival like so. First, one assumes that at every time step
each qubit has an independent probability $p$ of being measured. Then, the
probability of any one qubit remaining unmeasured is the compliment of the
probability that all qubits have been measured at some point in the past at a
given time step. This probability is a power of the probability that any
particular qubit has been measured at least once, which is the compliment of
the probability that it’s never been measured at time $t$ – $(1-p)^{t}$. We
can therefore write the survival probability of a probe’s information as
$P\sim 1-(1-(1-p)^{t})^{\mathcal{O}(N)}$. This entails
$\tau\sim\mathcal{O}(N)$ for large $t$ and $N$. Meanwhile, volume phase
information is destroyed by the very rare event of an extensive number of
measurements. For small $p$, this happens with probability $\sim
p^{\mathcal{O}(N)}$, implying the survival probability
$(1-p^{\mathcal{O}(N)})^{t}$ and lifetime $\tau\sim\exp(N)$. Evaluating the
entanglement at sufficiently large $t$ effectively distinguishes these two
behaviors.

Similar to the topological entanglement entropy, the ancilla entanglement
entropy takes on a value tending towards $0$ in the cluster area-law phase of
our circuits. That is to say, $S_{anc}$ does not differentiate it from the
trivial area-law phase. But equally similarly, this suffices to distinguish
the cluster area-law phase from the volume-law phase. This quantity may
accordingly be used to probe transition throughout most of the phase diagram.

Our implementation is as follows. Before starting the circuit properly, we
entangle $20$ ancillae by running a scrambling circuit for time
$T=\mathcal{O}(L^{4})$, with the 5-Clifford unitaries enacted on any arbitrary
five qubits (either ancilla or system), regardless of spatial separation.
Then, we sample the entanglement between the ancilla and the system after
$\mathcal{O}(L^{4})$ time steps. Immediately after scrambling, the
entanglement is on order of $20$, and deep in the volume-law phase this value
is retained for long periods of time. In the area-law phases, meanwhile, the
entanglement drops quickly to zero. At criticality, a constant fraction of the
original entanglement is retained in the thermodynamic limit, with the
specific factor depending on when the entanglement is sampled.

Because of the initial mixing time, and since $S_{anc}$ is only really sampled
once per circuit run, analysis of $S_{anc}$ requires more simulation time. It
makes up for this defect by being formally less noisy in spite of this defect.
Typical sample sizes ranges from high hundreds for the largest systems to high
thousands for smaller ones.

###  B.3 Dumbbell Entanglement Entropy

While the topological entanglement entropy succeeds in distilling the
topological essence of many nice models down to a single number, so-called
“spurious” contributions can arise in systems exhibiting long-range string
order [56, 57]. Though this fact spoils elegant theoretical models, its
existence can be used as a diagnostic of the presence of SPT and SSPT effects.
The dumbbell entanglement entropy $S_{dumb}$ [56] is a construction taking
advantage of this fact. For this work we define it as the 7-term tripartite
mutual information as in Eq. 2, rather than the conditional mutual information
of [56]. As opposed to the more standard geometries used for $S_{top}$, the
dumbbell entanglement entropy $S_{dumb}$ is so named for using regions forming
dumbbell with the handle directed along a line of string symmetry (see Fig.
12).

We make use of this quantity in this work to distinguish the cluster area-law
phase, exhibiting subsystem symmetry, from the trivial area-law phase
generated by an excess of local computational basis measurements. While it
succeeds in discriminating between the two behaviors, in practice it is less
robust at its job than the topological- or ancilla entanglement entropies
covered above. The defects in the curve (seen in Fig. 8) become more prominent
near criticality, and capriciously depend on the size of the “head” and
“handle.”

$S_{dumb}$ is sampled similarly to $S_{top}$ as described in Sec. B.1. We
begin sampling at $t=\tfrac{1}{4}L^{2}$, and sample every $L^{2}$ timesteps,
starting another circuit from scratch at $t=L^{4}$. This sampling frequency
produces data sufficiently uncorrelated away from the critical point, but near
the critical point the correlation time diverges along with the correlation
length. Nevertheless, the statistical influence of this fact appears
negligible compared to the growth of the spread of the order parameter
distribution of $S_{dumb}$. Typical sample sizes range from thousands to tens
of thousands for larger and smaller systems respectively.

##  Appendix C Fitting Methods

Although our presented data is analyzed using a single fitting method, we
apply three different techniques as a check on our finite-size scaling
analysis. We refer to them as “Nearest,” “Multilevel,” and “Polynomial” styles
respectively. All take as input the critical parameters $(p_{c},\nu,\gamma)$,
which scale the data as $(p,\Delta)\to((p-p_{c})L^{1/\nu},\Delta
L^{-\gamma})$. Although we have included the extensive exponent $\gamma$ here,
we have found that this is within margin of error $0$ in practice. It is
therefore set to be exactly $0$ in all figures in this work.

For the “Nearest” style, we order the scaled data following
$(p-p_{c})L^{1/\nu}$. A linear regression is performed for all interior points
using their immediate neighbors, regardless of system size, and the goodness
of fit for the whole curve is the sum of the goodness of fits for these
individual regressions. This heuristically has the effect of favoring input
parameters producing a curve that is locally as smooth as possible.

The second technique, “Multilevel,” is similar to the first. However, for each
individual point, we look at each system size greater than that of the point,
and then select neighbors from those to perform a regression. The objective
function is the sum of the goodness of fit for each point for each system size
larger than the size that point belongs to. The intuition underlying this
method is that it favors input parameters specifying a curve as smooth as
possible with respect to the samples coming from larger systems, which
hypothetically suffer less from finite size effects.

The final “Polynomial” style is a simple polynomial regression on the entirety
of the scaled data. Though crude, the output of this method produces smooth
landscapes whose minima do not deviate significantly from the prior two
methods, provided that the data is well-sampled and smooth. Eighth order
polynomials have proven sufficient for our purposes; higher orders suffer from
overfitting, while lower orders fail to capture the master curve’s
contouration.

All figures in the main text employ the “polynomial” method. However, as shown
in Fig. 14 and Fig. 15, all three methods are typically in good agreement.

##  Appendix D Behavior at Pure Measurement Critical Point

Figure 16:  Plots of the scaled entanglement versus system length for linear
(LABEL:fig:subentgrowtha) and quadratic (LABEL:fig:subentgrowthb) regressions,
and half-system entanglement growth versus time (LABEL:fig:subentgrowthc).
Colors/shapes refer to the same system sizes across all figures. The linear
fit works well for small $L$, but consistently deviates for larger sizes,
while the quadratic fit continues well. Meanwhile, the regression coefficients
for the growth over time slowly vary with $L$.

For $(1+1)$D systems at criticality it is well-known that the entanglement
entropy of a contiguous section (size $l$) of the system (size $L$) often
features entanglement entropy scaling as

$\displaystyle
S_{1D}(l)=\text{Const.}+\tfrac{c}{6}\log\left(\tfrac{L}{\pi}\sin(\tfrac{\pi
l}{L})\right),$ (4)

where $c$ is the central charge of the associated conformal field theory [74,
75]. The entanglement entropy of a region of set size (say, half the system),
meanwhile, grows logarithmically in time $S_{1D}(t)=a+b\log(t)$.

In 2D translationally invariant systems it is expected that spiritually
similar expressions continue to hold, provided that quasi-1D sections of the
system (such as cylindrical sections of a torus) are chosen. Furthermore,
while the critical properties of random monitored quantum circuits do not
precisely match the phenomenology of more conventional phase transitions,
similar relationships still hold in the $(1+1)$D setting with certain
modifications. For example, the coefficient of proportionality in $S_{1D}$
above is replaced by the primary weight of a boundary changing operator [76].

Between these principles it is reasonable to check the hypothesis of this
scaling for our $(2+1)$D system. Other $(2+1)$D monitored circuits have found
reasonable agreement for certain transitions [19]. We find that the agreement
for the transition occurring in pure measurement dynamics (shown in Fig.
LABEL:fig:subentgrowtha) is respectable. However, for larger system sizes
disagreement begins to emerge. Adding a term quadratic in
$\log\left(\tfrac{L}{\pi}\sin(\tfrac{\pi x}{L})\right)$ appreciably resolves
this discrepancy for the system sizes considered, as shown in Fig.
LABEL:fig:subentgrowthb. Quantitatively, the order of the square residual is
two orders of magnitude smaller for this quadratic regression. Although we
have no theoretical justification for this addition, it suggests that some
power series in this variable captures the finer critical features observed.
We also find in Fig. LABEL:fig:subentgrowthc that the terms in the logarithmic
linear regression for $S_{1D}(t)$ vary with system size as well.

## References

  * R. Feynman [1982] R. Feynman, Simulating Physics with Computers, Int. J. Theor. Phys. 21, 467 (1982).
  * P. Shor [1997] P. Shor, Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, SIAM J. Comput. 26, 1484 (1997).
  * J.S. Bell [1964] J.S. Bell, On the Einstein Podolsky Rosen paradox, Physics Physique Fizika 1, 195 (1964).
  * A. Cabello [2001a] A. Cabello, Bell’s theorem without inequalities and without probabilities for two observers, Phys. Rev. Lett. 86, 1911 (2001a).
  * A. Cabello [2001b] A. Cabello, All versus nothing inseparability for two observers, Phys. Rev. Lett. 87, 010403 (2001b).
  * Y. Li, X. Chen, and M.P.A. Fisher [2019] Y. Li, X. Chen, and M.P.A. Fisher, Measurement-driven entanglement transition in hybrid quantum circuits, Phys. Rev. B 100, 134306 (2019).
  * Y. Li, R. Vasseur, M.P.A. Fisher, and A.W.W. Ludwig [2024] Y. Li, R. Vasseur, M.P.A. Fisher, and A.W.W. Ludwig, Statistical Mechanics Model for Clifford Random Tensor Networks and Monitored Quantum Circuits, Phys. Rev. B 109, 1174307 (2024).
  * B. Skinner, J. Ruhman, and A. Nahum [2019] B. Skinner, J. Ruhman, and A. Nahum, Measurement-Induced Phase Transitions in the Dynamics of Entanglement, Phys. Rev. X 9 (2019).
  * Y. Li, X. Chen, and M.P.A. Fisher [2018] Y. Li, X. Chen, and M.P.A. Fisher, Quantum Zeno effect and the many-body entanglement transition, Phys. Rev. B 98, 205136 (2018).
  * M.J. Gullans and D.A. Huse [2020a] M.J. Gullans and D.A. Huse, Dynamical Purification Phase Transitions Induced by Quantum Measurements, Phys. Rev. X 10 (2020a).
  * S. Choi, Y. Bao, X.-L. Qi, and E. Altman [2020a] S. Choi, Y. Bao, X.-L. Qi, and E. Altman, Theory of the phase transition in random unitary circuits with measurements, Phys. Rev. B 101, 104301 (2020a).
  * S. Choi, Y. Bao, X.-L. Qi, and E. Altman [2020b] S. Choi, Y. Bao, X.-L. Qi, and E. Altman, Quantum Error Correction in Scrambling Dynamics and Measurement-Induced Phase Transition, Phys. Rev. Lett. 125, 030505 (2020b).
  * B. Skinner [2023] B. Skinner, Lecture Notes: Introduction to random unitary circuits and the measurement-induced entanglement phase transition, arXiv:2307.02986 (2023).
  * M.P.A. Fisher, V. Khemani, A. Nahum, and S. Vijay [2023] M.P.A. Fisher, V. Khemani, A. Nahum, and S. Vijay, Random Quantum Circuits, Annual Review of Condensed Matter Physics 14, 335 (2023).
  * A.C. Potter and R. Vasseur [2022] A.C. Potter and R. Vasseur, Entanglement dynamics in hybrid quantum circuits, in  _Entanglement in Spin Chains. Quantum Science and Technology_ (Springer, 2022) pp. 211–249.
  * A. Chan, R.M. Nandkishore, M. Pretko, and G. Smith [2019] A. Chan, R.M. Nandkishore, M. Pretko, and G. Smith, Unitary-projective entanglement dynamics, Phys. Rev. B 99, 224307 (2019).
  * X. Cao, A. Tillory, and A. De Luca [2019] X. Cao, A. Tillory, and A. De Luca, Entanglement in a fermion chain under continuous monitoring, SciPost Phys. 7, 24 (2019).
  * E. Wigner [1955] E. Wigner, Characteristic vectors of bodered matrices with infinite dimensions, Annals of Mathematics 62, 548 (1955).
  * A. Lavasani, Y. Alavirad, and M. Barkeshli [2021a] A. Lavasani, Y. Alavirad, and M. Barkeshli, Topological order and criticality in (2+1)D monitored random quantum circuits, Phys. Rev. Lett. 127, 235701 (2021a).
  * A. Lavasani, Y. Alavirad, and M. Barkeshli [2021b] A. Lavasani, Y. Alavirad, and M. Barkeshli, Measurement-induced topological entanglement transitions in symmetric random quantum circuits, Nat. Phys. 17, 342 (2021b).
  * H.J. Briegel and R. Raussendorf [2001] H.J. Briegel and R. Raussendorf, Persistent Entanglement in Arrays of Interacting Particles, Phys. Rev. Lett. 86, 910 (2001).
  * R. Raussendorf and H.J. Briegel [2001] R. Raussendorf and H.J. Briegel, A One-Way Quantum Computer, Phys. Rev. Lett. 86, 5188 (2001).
  * R. Raussendorf, D.E. Browne, and H.J. Briegel [2003] R. Raussendorf, D.E. Browne, and H.J. Briegel, Measurement-based quantum computation on cluster states, Phys. Rev. A 68, 022312 (2003).
  * D.V. Else, I. Schwarz, S.D. Bartlett, and A.C. Doherty [2012] D.V. Else, I. Schwarz, S.D. Bartlett, and A.C. Doherty, Symmetry-Protected Phases for Measurement-Based Quantum Computation, Phys. Rev. Lett. 108, 240505 (2012).
  * A.C. Doherty and S.D. Bartlett [2009] A.C. Doherty and S.D. Bartlett, Identifying Phases of Quantum Many-Body Systems That Are Universal, Phys. Rev. Lett. 103, 020506 (2009).
  * D. Gross, S.T. Flammia, and J. Eisert [2009] D. Gross, S.T. Flammia, and J. Eisert, Most Quantum States Are Too Entangled To Be Useful As Computational Resources, Phys. Rev. Lett. 102, 190501 (2009).
  * M.J. Bremner, C. Mora, and A. Winter [2009] M.J. Bremner, C. Mora, and A. Winter, Are Random Pure States Useful for Quantum Computation?, Phys. Rev. Lett. 102, 190502 (2009).
  * S. Sang, and T.H. Hsieh [2021] S. Sang, and T.H. Hsieh, Measurement-protected quantum phases, Phys. Rev. Res. 3, 023200 (2021).
  * Y. Bao, S. Choi, and E. Altman [2021] Y. Bao, S. Choi, and E. Altman, Symmetry enriched phases of quantum circuits, Annals of Physics 435, 168618 (2021).
  * H. Liu, T. Zhou, and X. Chen [2022] H. Liu, T. Zhou, and X. Chen, Measurement-induced entanglement transition in a two-dimensional shallow circuit, Phys. Rev. B 106, 144311 (2022).
  * A.-R. Negari, S. Sahu, and T.H. Hsieh [2024] A.-R. Negari, S. Sahu, and T.H. Hsieh, Measurement-induced phase transitions in the toric code, Phys. Rev. B 109, 125148 (2024).
  * A. Nahum, J. Ruhman, S. Vijay, and J. Haah [2017] A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum Entanglement Growth under Random Unitary Dynamics, Phys. Rev. X 7, 031016 (2017).
  * G. Grimmett [1999] G. Grimmett,  _Percolation_ (Springer, 1999).
  * C-M. Jian, Y.-Z. You, R. Vasseur, and A.W.W. Ludwig [2020] C-M. Jian, Y.-Z. You, R. Vasseur, and A.W.W. Ludwig, Measurement-induced criticality in random quantum circuits, Phys. Rev. B 101, 104302 (2020).
  * N. Kawashima and N. Ito [1993] N. Kawashima and N. Ito, Critical Behavior of the Three-Dimensional ±J Model in a Magnetic Field, Journal of The Physical Society of Japan 62, 435 (1993).
  * J. Houdayer, A.K. Hartmann [2004] J. Houdayer, A.K. Hartmann, Low-temperature behavior of two-dimensional Gaussian Ising spin glasses, Phys Rev. B 70 (2004).
  * N. Kawashima and H. Rieger [1997] N. Kawashima and H. Rieger, Finite-Size Scaling Analysis of Exact Ground States for ±J Spin Glass Models in Two Dimensions, Europhys. Lett. 39, 85 (1997).
  * A. Kitaev and J. Preskill [2006] A. Kitaev and J. Preskill, Topological Entanglement Entropy, Phys. Rev. Lett. 96, 110404 (2006).
  * M. Levin and X.-G. Wen [2006] M. Levin and X.-G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett. 96, 110405 (2006).
  * M.J. Gullans and D.A. Huse [2020b] M.J. Gullans and D.A. Huse, Scalable probes of measurement-induced criticality, Phys. Rev. Lett. 125, 070606 (2020b).
  * Google Quantum AI and Collaborators [2023] Google Quantum AI and Collaborators, Measurement-induced entanglement and teleportation on a noisy quantum processor, Nature 622, 481 (2023).
  * J.M. Koh, S.-N. Sun, M. Mota, and A.J. Minnich [2023] J.M. Koh, S.-N. Sun, M. Mota, and A.J. Minnich, Measurement-induced entanglement phase transition on a superconducting quantum processor with mid-circuit readout, Nat. Phys. 19, 1314 (2023).
  * C. Noel, et al. [2022] C. Noel, et al., Measurement-induced quantum phases realized in a trapped-ion quantum computer, Nat. Phys. 18, 760 (2022).
  * B. Zeng, X. Chen, D.-L. Zhou, and X.-G. Wen [2019] B. Zeng, X. Chen, D.-L. Zhou, and X.-G. Wen,  _Quantum Information Meets Quantum Matter – From Quantum Entanglement to Topological Phase in Many-Body Systems_ (Springer, 2019).
  * X. Chen, Z.-C. Gu, Z-X. Lieu, X.-G. Wen [2013] X. Chen, Z.-C. Gu, Z-X. Lieu, X.-G. Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87, 155114 (2013).
  * A. Miyake [2010] A. Miyake, Quantum Computation on the Edge of a Symmetry-Protected Topological Order, Phys. Rev. Lett. 105, 040501 (2010).
  * R. Raussendorf, C. Okay, D.-S. Want, D.T. Stephen, and H.P. Nautrup [2019] R. Raussendorf, C. Okay, D.-S. Want, D.T. Stephen, and H.P. Nautrup, Computationally Universal Phase of Quantum Matter, Phys. Rev. Lett. 122, 090501 (2019).
  * J. Miller and A. Miyake [2015] J. Miller and A. Miyake, Resource Quality of a Symmetry-Protected Topologically Ordered Phase for Quantum Computation, Phys. Rev. Lett. 114, 120506 (2015).
  * D.T. Stephen, D.-S. Wang, A. Prakash, T.-C. Wei, and R. Raussendorf [2017] D.T. Stephen, D.-S. Wang, A. Prakash, T.-C. Wei, and R. Raussendorf, Computational Power of Symmetry Protected Topological Phases, Phys. Rev. Lett. 119, 010504 (2017).
  * R. Raussendorf, D.-S. Wang, A. Prakash, T.-C. Wei, and D.T. Stephen [2017] R. Raussendorf, D.-S. Wang, A. Prakash, T.-C. Wei, and D.T. Stephen, Symmetry-protected topological phases with uniform computational power in one dimension, Phys. Rev. A 96, 012302 (2017).
  * T. Devakul and D.J. Williamson [2018] T. Devakul and D.J. Williamson, Universal quantum computation using fractal symmetry-protected cluster phases, Phys. Rev. A 98, 022332 (2018).
  * D.T. Stephen, H.P. Nautrup, J. Bermejo-Vega, J.Eisert, and R. Raussendorf [2019] D.T. Stephen, H.P. Nautrup, J. Bermejo-Vega, J.Eisert, and R. Raussendorf, Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter, Quantum 3, 142 (2019).
  * A.K. Daniel, R.N. Alexander, and A. Miyake [2020] A.K. Daniel, R.N. Alexander, and A. Miyake, Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices, Quantum 4, 228 (2020).
  * D. Gottesman [1999] D. Gottesman, The Heisenberg Representation of Quantum Computers, in  _Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics_, Group22 (Interational Press, Cambridge, MA, 1999) pp. 32–43.
  * S. Aaronson and D.Gottesman [2004] S. Aaronson and D.Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A 70, 052328 (2004).
  * D.J. Williamson, A. Dua, and M. Cheng [2019] D.J. Williamson, A. Dua, and M. Cheng, Spurious Topological Entanglement Entropy from Subsystem Symmetries, Phys. Rev. Lett. 122, 140506 (2019).
  * D.T. Stephen, H. Dreyer, M. Iqbal, and N. Schuch [2019] D.T. Stephen, H. Dreyer, M. Iqbal, and N. Schuch, Detecting subsystem symmetry protected topological order via entanglement entropy, Phys. Rev. B 100, 115112 (2019).
  * P. Sierant, M. Schiro, M. Lewenstein, and X. Turkeshi [2022] P. Sierant, M. Schiro, M. Lewenstein, and X. Turkeshi, Measurement-induced phase transitions in (d+1)-dimensional stabilizer circuits, Phys. Rev. B 106, 214316 (2022).
  * O. Lunt, M. Szyniszweski, and A. Pal [2021] O. Lunt, M. Szyniszweski, and A. Pal, Measurement-induced criticality and entanglement clusters: A study of one-dimensional and two-dimensional Clifford circuits, Phys. Rev. B 104, 155111 (2021).
  * X. Turkeshi, R. Fazio, and M. Dalmonte [2020] X. Turkeshi, R. Fazio, and M. Dalmonte, Measurement-induced criticality in (2+1)-dimensional hybrid quantum circuits, Phys. Rev. B 102, 014315 (2020).
  * Y. You, T. Devakul, F.J. Burnell, and S.L. Sondhi [2018] Y. You, T. Devakul, F.J. Burnell, and S.L. Sondhi, Subsystem symmetry protected topological order, Phys. Rev. B 98, 035112 (2018).
  * S.G. Gorishny, S.A. Larin, and F.V. Tkachov [1984] S.G. Gorishny, S.A. Larin, and F.V. Tkachov, $\epsilon$-Expansion for critical exponents: The O($\epsilon^{5}$) approximation, Phys. Lett. A , 120 (1984).
  * R. Guida and J. Zinn-Justin [1998] R. Guida and J. Zinn-Justin, Critical Exponents of the N-vector model, J. Phys. A 31, 8103 (1998).
  * M. Hasenbusch [2001] M. Hasenbusch, Eliminating leading corrections to scaling in the three-dimensional O(N)-symmetric $\phi^{4}$ model: $N=3$ and $4$, J. Phys. A 40, 8221 (2001).
  * J. Cardy [1996] J. Cardy,  _Scaling and Renormalization in Statistical Physics_ (Cambridge University Press, 1996).
  * A.B. Harris [1974] A.B. Harris, Effect of random defects on the critical behavior of Ising Models, J. Phys. C: Solid State Phys 7, 1671 (1974).
  * G. Shkolnik, A. Zabalo, R. Vasseur, D.A. Huse, J.H. Pixley, and S. Gazit [2023] G. Shkolnik, A. Zabalo, R. Vasseur, D.A. Huse, J.H. Pixley, and S. Gazit, Measurement induced criticality in quasiperiodic modulated random hybrid circuits, Phys. Rev. B 108, 184204 (2023).
  * A. Zabalo, J.H. Wilson, M.J. Gullans, R. Vasseur, S. Gopalkrishnan, D.A. Huse, and J.H. Pixley [2022] A. Zabalo, J.H. Wilson, M.J. Gullans, R. Vasseur, S. Gopalkrishnan, D.A. Huse, and J.H. Pixley, Infinite-randomness criticality in monitored quantum dynamics with static disorder, Phys. Rev. B 107, L220204 (2022).
  * A. Zabalo, M.J. Gullans, J.H. Wilson, S. Gopalakrishnan, D.A. Huse, and J.H. Pixley [2020] A. Zabalo, M.J. Gullans, J.H. Wilson, S. Gopalakrishnan, D.A. Huse, and J.H. Pixley, Critical properties of the measurement-induced transition in random quantum circuits, Phys. Rev. B 101 (2020).
  * Gidney, Craig [2021] Gidney, Craig, Stim: a fast stabilizer circuit simulator, Quantum 5, 497 (2021).
  * E. Bertolazzzi and A. Rimoldi [2014] E. Bertolazzzi and A. Rimoldi, Fast matrix decomposition in $\mathbb{F}_{2}$, Journal of Computational and Applied Mathematics 260, 519 (2014).
  * Fro [2022] Frontier: Direction of Discovery (2022), Accessed: 2023-10-23.
  * J. Miller and A. Miyake [2016] J. Miller and A. Miyake, Hierarchy of universal entanglement in 2D measurement-based quantum computation, npj Quantum Information 2 (2016).
  * P. Calabrese and J. Cardy [2004] P. Calabrese and J. Cardy, Entanglement Entropy and Quantum Field Theory, J. Stat. Mech 0406, P06002 (2004).
  * P. Calabrese and J. Cardy [2009] P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42, 504005 (2009).
  * Y. Li, X. Chen, A.W.W. Ludwig, and M.P.A. Fisher [2021] Y. Li, X. Chen, A.W.W. Ludwig, and M.P.A. Fisher, Conformal invariance and quantum nonlocality in critical hybrid circuits, Phys. Rev. B 104, 104305 (2021).