# Subsystem symmetry, spin glass order, and criticality from random
measurements in a two-dimensional Bacon-Shor circuit

Vaibhav Sharma<EMAIL_ADDRESS>Chao-Ming Jian<EMAIL_ADDRESS>Erich J Mueller<EMAIL_ADDRESS>Laboratory of Atomic and Solid State
Physics, Cornell University, Ithaca, New York

###### Abstract

We study a 2D measurement-only random circuit motivated by the Bacon-Shor
error correcting code. We find a rich phase diagram as one varies the relative
probabilities of measuring nearest neighbor Pauli XX and ZZ check operators.
In the Bacon-Shor code, these checks commute with a group of stabilizer and
logical operators, which therefore represent conserved quantities. Described
as a subsystem symmetry, these conservation laws lead to a continuous phase
transition between an $X$-basis and $Z$-basis spin glass order. The two phases
are separated by a critical point where the entanglement entropy between two
halves of an $L\times L$ system scales as $L\ln L$, a logarithmic violation of
the area law. We generalize to a model where the check operators break the
subsystem symmetries (and the Bacon-Shor code structure). In tension with
established heuristics, we find that the phase transition is replaced by a
smooth crossover, and the $X$\- and $Z$-basis spin glass orders spatially
coexist. Additionally, if we approach the line of subsystem symmetries away
from the critical point in the phase diagram, some spin glass order parameters
jump discontinuously.

##  I Introduction

Random circuits provide a platform for exploring quantum dynamics and quantum
algorithms, revealing how information propagates and how entanglement evolves
[1, 2, 3, 4, 5]. They provide simple examples of quantum dynamical systems [6,
7] and also act as models of noisy hardware [8]. A central insight has been
the establishment of analogies between thermodynamic phase transitions and the
behavior of quantum circuits consisting of both random unitary operations and
measurements [9, 10, 11, 12, 13, 7, 14, 15, 16, 17, 18, 19, 20, 7, 21, 22,
23]. As one varies model parameters, such as the probabilities of various
gates or measurements, these quantum circuits can exhibit different phases
characterized by distinct entanglement scalings, which are separated from each
other by continuous phase transitions. As exemplified in Refs. 24, 25, 26,
quantum circuits consisting of only measurements can also exhibit rich phase
diagrams when the measurements performed at different time steps do not
commute with each other. Apart from certain examples [24, 25, 19, 20, 27], the
dynamical systems evolving under measurement-only quantum circuits have not
been systematically explored beyond one dimension. In this paper, we focus on
a two-dimensional (2D) measurement-only circuit on a square lattice where we
are able to explore the role of both global symmetries and “subsystem”
symmetries in quantum dynamics. We analyze the phase transitions in this
circuit using a detailed analysis of order parameters and correlation
functions. We find that our 2D model displays intriguing behaviors that are
not seen in 1D measurement-only circuits.

Our model is motivated by the Bacon-Shor code [28], which is a quantum error
correcting code on a 2D square lattice that relies upon repeated measurement
of check operators. Also known as gauge operators, these are a set of non-
commuting local operators which can be used to detect errors without
scrambling the stored quantum information. In the Bacon-Shor code, the check
operators consist of the product of any two nearest neighbor Pauli $X$
operators on a horizontal bond, and the product of any two nearest neighbor
Pauli $Z$ operators on a vertical bond (Fig. 1 (a)). We will denote these
check operators as $\begin{smallmatrix}X\,X\end{smallmatrix}$ and
$\begin{smallmatrix}Z\\\ Z\end{smallmatrix}$. While the check operators do not
commute with one another, they do commute with the Bacon-Shor stabilizer
group, generated by the products of $Z$ operators along any two horizontal
rows, and the products of $X$ operators along any two vertical columns (Fig. 1
(b),(c)). Both the Bacon-Shor stabilizers and the check operators commute with
logical operators: A product of $Z$’s along a single row or a product of $X$’s
along a single column (Fig. 1 (b),(c)). We will consider a model where the
check operators are randomly measured. The fact that these checks commute with
the stabilizers and logic operators can be considered as a symmetry. The
logical operators and the Bacon-Shor stabilizers are symmetry generators.
After reorganizing these operators, there is effectively an independent
symmetry generator for every row and every column. Since the action of each
generator is concentrated only on a lower dimensional subsystem (i.e. the
spins in a single row or a single column), these are referred to as subsystem
symmetries. This designation distinguishes them from global symmetries which
act on all degrees of freedom or local symmetries which involve only a single
site.

In quantum mechanics, a state is changed by the act of measuring it. This
feature is extremely powerful: One can build a general-purpose quantum
computer that solely uses a sequence of measurements to process information
[29, 30]. In this manner, random measurements introduce stochastic dynamics.
In our first numerical experiment, we randomly measure the Bacon-Shor
$\begin{smallmatrix}Z\\\ Z\end{smallmatrix}$ checks with probability $p_{1}$,
and the $\begin{smallmatrix}X\,X\end{smallmatrix}$ checks with probability
$\bar{p}_{1}=1-p_{1}$. We study the properties of the steady state ensemble
produced by this quantum Markov chain [31].

We characterize the steady state ensemble using spin glass order. In the limit
of applying only $\begin{smallmatrix}X\,X\end{smallmatrix}$ checks
($p_{1}=0$), each row develops a perfect X-spin glass order and similarly,
only applying $\begin{smallmatrix}Z\\\ Z\end{smallmatrix}$ checks ($p_{1}=1$)
leads to a perfect Z-spin glass order along columns. As we vary $p_{1}$, the
ensemble displays a phase transition at $p_{1}=1/2$ from an independent
$X$-spin glass order along each horizontal row for $p_{1}<1/2$ and an
independent $Z$-spin glass order along each vertical column when $p_{1}>1/2$.
This form of order was used by Sang and Hseih to characterize the behavior of
a 1D system [19], and it will be explained in detail in Sec. IV.1. We
characterize the ordered phases and the phase transition at $p_{1}=1/2$.

In addition to looking at spin-glass order, we calculate the entanglement
entropy between two halves of the system. Such entanglement based measures are
analogous to thermodynamic quantities like specific heat: They show
divergences/discontinuities at phase transitions with critical exponents that
encode universal features of the model and are independent of the microscopic
details. Away from $p_{1}=1/2$, we find that the entanglement entropy has an
area law scaling $S\sim L$, where $L$ is the linear size of our square array.
At $p_{1}=1/2$, we instead see $S\sim L\ln L$, which is referred to as a
logarithmic violation of the area law. We explain the structure of the spin-
glass order and the behavior of the entanglement entropy in terms of the
subsystem symmetries of the model.

In a recent paper [20], Lavasani et al. studied a model which (in certain
limits) maps onto our random Bacon-Shor code. They discovered the entanglement
scaling described above, but did not explore the possibility of spin-glass
order (which would be a non-local order in their model, and would not have
been natural). Another related model was studied by Lavasani et al. [24] and
Sriram et al. [25]. They considered a 2D model on a honeycomb lattice where
one randomly measures non-commuting two-site checks, chosen so that each
hexagonal plaquette has a conserved quantity. Hence their model contains a
subsystem symmetry, but of a very different form than the Bacon-Shor code.
Their model contains a topologically ordered spin-liquid phase (with area-law
entanglement entropy, $S\sim L$) and a critical phase (with entanglement
entropy scaling $S\sim L\ln L$). Neither of those phases display any spin-
glass order.

After characterizing the random Bacon-Shor code, we break the subsystem
symmetry by allowing $\begin{smallmatrix}X\\\ X\end{smallmatrix}$ and
$\begin{smallmatrix}Z\,Z\end{smallmatrix}$ checks, with probability
proportional to $p_{2}$ (Fig. 1 (d)). In this case, the only symmetries are
global symmetries, generated by the product of $X$ operators of all spins in
the system and its $Z$-operator counterpart. With these extra
$\begin{smallmatrix}X\\\ X\end{smallmatrix}$ and
$\begin{smallmatrix}Z\,Z\end{smallmatrix}$ check operators, we find that the
spin-glass order is no longer confined to rows or columns, and that the $X$\-
and $Z$-spin-glass orders can coexist. At the extreme limits of applying only
$\begin{smallmatrix}X\,X\end{smallmatrix},\begin{smallmatrix}X\\\
X\end{smallmatrix}$ checks, we get a perfect global X spin glass order whereas
in the limit of applying only
$\begin{smallmatrix}Z\,Z\end{smallmatrix},\begin{smallmatrix}Z\\\
Z\end{smallmatrix}$ checks, we get a perfect global Z spin glass order. One
can continuously pass from the region of the phase diagram dominated by the
$X$-spin-glass order to the region dominated by the $Z$-spin-glass order
without encountering a phase transition. There is some qualitative similarity
with the critical point separating the liquid and gas phases of water where
one can circumvent the thermodynamic singularity by taking an appropriate path
through the parameter space.

One important caveat with this analogy is that we find discontinuities as
$p_{2}\to 0$. An infinitesimal breaking of the subsystem symmetry leads to a
discontinuous change in the spin-glass order parameters which describe
$X$-correlations between rows or $Z$-correlations between columns. The
$X$-spin glass order in a single row (or $Z$-spin glass order in a single
column) changes continuously. In other words, an infinitesimal $p_{2}$ leads
causes the independent spin-glass orders in individual columns and rows to
merge into a global 2-dimensional spin-glass.

Through our study, we learn that some prior heuristics have limitations. Our
model (even with the broken subsystem symmetry) features a bipartite
frustration graph. That means, the check operators divide into two sets:
$\\{\begin{smallmatrix}X\,X\end{smallmatrix},\begin{smallmatrix}X\\\
X\end{smallmatrix}\\}$ and
$\\{\begin{smallmatrix}Z\,Z\end{smallmatrix},\begin{smallmatrix}Z\\\
Z\end{smallmatrix}\\}$. Within each set, the operators commute with each
other, but any operator from one set anti-commutes with at least one operator
from the other set. In the context of 1D models, Ippoliti et al. [26]
conjectured that this feature would ensure a phase transition as one varied
the probabilities of measuring the various operators. Our model instead shows
a crossover, unless we impose the subsystem symmetry.

##  II Model and notation

Figure 1:  The Bacon-Shor error correction model on a square lattice with
qubits on the vertices (a,b,c), and the phase diagram of our measurement-only
random circuit (d). (a) The operators
$\begin{smallmatrix}X\,X\end{smallmatrix}$ on horizontal bonds and
$\begin{smallmatrix}Z\\\ Z\end{smallmatrix}$ on vertical bonds are check
operators. (b,c) A product of $X$ operators along two adjacent columns and a
product of $Z$’s along two adjacent rows are the Bacon-Shor code stabilizers,
which can be formed from products of check operators, but commute with all
checks. The check operators also commute with the logical operators, products
of $X$/$Z$ operators along a single column/row. These logical operators
generate the subsystem symmetries. (d) Parameter space of our model. We
measure 2-qubit operators:
$\begin{smallmatrix}X\,X\end{smallmatrix},\begin{smallmatrix}Z\\\
Z\end{smallmatrix},\begin{smallmatrix}Z\,Z\end{smallmatrix},\begin{smallmatrix}X\\\
X\end{smallmatrix}$ with probabilities
$\bar{p}_{1}\bar{p}_{2},\bar{p}_{1}p_{2},p_{1}p_{2},p_{1}\bar{p}_{2}$, where
$p_{1}$ and $p_{2}$ run from 0 to 1, and $\bar{p}_{j}=1-p_{j}$. The solid red
lines at $p_{2}=0,1$ illustrate the regions of parameter space having the
Bacon-Shor code structure with subsystem symmetries, and the blue dots
correspond to critical points.

We consider a square $L\times L$ grid of qubits. The qubits do not evolve
under any Hamiltonian, and instead the only dynamics comes from measurements.
At each time step, we measure a rantom 2-qubit check operator:
$\begin{smallmatrix}X\,X\end{smallmatrix},\begin{smallmatrix}Z\\\
Z\end{smallmatrix},\begin{smallmatrix}Z\,Z\end{smallmatrix},\begin{smallmatrix}X\\\
X\end{smallmatrix}$ with probabilites
$\bar{p}_{1}\bar{p}_{2},\bar{p}_{1}p_{2},p_{1}p_{2},p_{1}\bar{p}_{2}$, where
$p_{1}$ and $p_{2}$ run from 0 to 1, and $\bar{p}_{j}=1-p_{j}$. The resulting
parameter space is conveniently visualized by the square in Fig. 1(d). At each
corner, only one type of operator is measured.

We start with a random state in the Bacon-Shor code space and use the
stabilizer formalism in Sec. III to calculate how the state evolves under each
measurement. We will use $\langle O\rangle$ or $\langle\psi|O|\psi\rangle$ to
denote an expectation value in a single state, and $\overline{O}$ or
$\overline{\langle O\rangle}$ to denote the ensemble average. In our numerics,
each ensemble average corresponds to time averages over 32 independent runs,
each starting with a different initial state. Each run contains $100L^{2}$
measurement ‘time’ steps.

Each qubit is labeled by indices $(i,j)$, where $i$ and $j$ are integers
between 1 and $L$. We will use Greek letters to denote the 2D coordinate, so a
Pauli operator on such a qubit will be designated $X_{i,j}$ or $X_{\alpha}$.
Following the notation of matrices, we will consider the first index, $i$ to
specify a row, while $j$ specifies a column. This is opposite to the
convention of Cartesian coordinates.

Note that we can map $p_{2}\to 1-p_{2}$ by rotating the lattice by 90 degrees.
We can also map $p_{1}\to 1-p_{1}$ by simultaneously rotating the lattice and
swapping $X\leftrightarrow Z$. Thus the entire phase diagram can be
constructed from one quadrant.

###  II.1 Symmetries

The system always possesses two global symmetries: The products over all
sites, $\prod_{i=1}^{L}\prod_{j=1}^{L}X_{i,j}$ and
$\prod_{i=1}^{L}\prod_{j=1}^{L}Z_{i,j}$ commute with all check operators. On
the red lines shown in Fig. 1(d), corresponding to $p_{2}=0,1$, the system has
additional subsystem symmetries. When $p_{2}=0$, all of the check operators
commute with the Bacon-Shor stabilizer generators: $\prod_{i}X_{i,j}X_{i,k}$
and $\prod_{i}Z_{j,i}Z_{k,i}$ where $j\neq k$. On this high symmetry line, the
checks also commute with the logic operators: $\prod_{i}X_{ij}$ and
$\prod_{j}Z_{ij}$. These operators are illustrated in Fig. 1(b),(c). Note that
the Bacon-Shor stabilizers are products of the logical operators.

The same structure is found when $p_{2}=1$, but with $X$ and $Z$ reversed. For
simplicity, in our numerics we always choose our initial state to be an
eigenstate of the relevant symmetry operators.

##  III Calculation Technique

###  III.1 Stabilizer formalism

We use the quantum trajectories approach in the stabilizer formalism to
simulate the dynamics [21]. In the stabilizer formalism, the many-body quantum
state of $L^{2}$ qubits is specified by $L^{2}$ linearly independent and
mutually commuting operators called stabilizer generators[9].

Each of these stabilizer generators is a Pauli string and the many-body
quantum state is a simultaneous eigenstate of all the generators. For our
calculation, we do not need to keep track of the eigenvalues, which are all
$\pm 1$. One can interpret our results as an average over all such
possibilities. The instantaneous stabilizer group formed by taking all
possible products of the stabilizer generators should not be confused with the
Bacon-Shor stabilizer group which describes the symmetries when $p_{2}=0,1$.

At each time step we randomly measure a check operator
($\begin{smallmatrix}X\,X\end{smallmatrix},\begin{smallmatrix}Z\\\
Z\end{smallmatrix},\begin{smallmatrix}Z\,Z\end{smallmatrix}$, or
$\begin{smallmatrix}X\\\ X\end{smallmatrix}$) on a random bond on the lattice
according to probabilities $p_{1}$ and $p_{2}$ defined in Sec. II. Based on
the measurement, we update the generators.

Each generator has the form
$g=\prod_{\alpha\beta}X_{\alpha}^{n_{\alpha}}Z_{\beta}^{m_{\beta}}$ where the
$n$’s and $m$’s can be 0 or 1. Thus the stabilizer can be stored as a length
$2L^{2}$ binary string, $V$, containing the $n$’s and $m$’s. In this
representation two operators commute if they share an even number of non-zero
entries, after the $X$ and $Z$ entries of one of them are swapped. The
generators are linearly independent if the $L^{2}\times 2L^{2}$ matrix of the
$V$’s is of full rank, modulo 2. This property can readily be checked by row
reduction.

The expectation value of a Pauli operator/string $O$ can be easily calculated:
If $O$ commutes with all of the instantaneous stabilizer generators, then it
must be a product of them, and $|\psi\rangle$ is an eigenstate. Thus
$\langle\psi|O|\psi\rangle=\pm 1$. Conversely, suppose $O$ anticommutes with
at least one generator, $g$. (Pauli operators which do not commute instead
anticommute.) Then $\langle\psi|O|\psi\rangle=\langle\psi|gOg|\psi\rangle$
because $g|\psi\rangle=\pm|\psi\rangle$, Since $g$ and $O$ anticommute, we
conclude $\langle O\rangle=0$.

When we randomly choose to measure a check operator $S$ during our dynamics,
one of two scenarios are possible: (1) If $S$ commutes with all of the
generators of the instantaneous stabilizer group, then $S$ belongs to this
stablizer group and the measurement does not change the state. (2) If $S$
anticommutes with generators $g_{1},g_{2},\cdots g_{n}$, the effect of the
measurement can be captured by replacing the generators $g_{1}\to S$ and
$g_{j}\to g_{1}g_{j}$ for $j>1$ in the instantaneous stablizer group. The
generators that initially commute with $S$ will be unchanged.

###  III.2 Order Parameter

Our model always has global symmetries generated by $\prod_{\alpha}X_{\alpha}$
and $\prod_{\alpha}Z_{\alpha}$. The single spin operators $X_{\alpha}$ and
$Z_{\beta}$ anticommute with these global symmetry generators and thus the
expectation values $\langle X_{\alpha}\rangle$ and $\langle Z_{\beta}\rangle$
vanish in the long time limit. Consequently, the lowest order correlation
function which can be used to characterize the states in our ensemble are the
two-point functions $\langle X_{\alpha}X_{\beta}\rangle$ and $\langle
Z_{\alpha}Z_{\beta}\rangle$. As argued in Sec. III.1, for a given state, these
expectation values are either $0,1$, or $-1$ – corresponding to the two spins
being uncorrelated, aligned, or anti-aligned in the appropriate basis. We
calculate the ensemble average ${\cal X}_{\alpha\beta}=\overline{\langle
X_{\alpha}X_{\beta}\rangle^{2}}$ which measures the degree of correlation
between the two spins, irrespective of the sign.
${\mathcal{Z}}_{\alpha\beta}=\overline{\langle
Z_{\alpha}Z_{\beta}\rangle^{2}}$ plays the same role in the $Z$-basis. We use
superscripts of $c$ or $r$ when the correlation functions are between spins on
the same column or row.

A non-vanishing correlation function ${\cal X}_{\alpha\beta}$ (or ${\cal
Z}_{\alpha\beta}$) at large distances indicates a spin-glass order. We refer
to the spin glass order detected by ${\cal X}_{\alpha\beta}$ (${\cal
Z}_{\alpha\beta}$) as the $X$-spin-glass ($Z$-spin-glass) order.

Equivalently, to detect the spin-glass orders, one can use the classic
Edwards-Anderson order parameter [32] which is the average of ${\cal
X}_{\alpha\beta}$ or ${\cal Z}_{\alpha\beta}$ for all pairs of sites $\alpha$
and $\beta$ in the entire system (for a spin glass order with respect to a
global symmetry) or within a subsystem (for a spin glass order with respect to
a subsystem symmetry). Sang and Hsieh used this approach to describe the spin
glass order in a 1D random circuit [19].

In principle, one could consider six different correlation functions:
$XX,XY,XZ,YY,YZ,ZZ$, which can be written as matrix elements of a $3\times 3$
symmetric tensor. Of these, the $XX$ and $ZZ$ correlators are most relevant:
The operators $X_{\alpha}Z_{\beta}$, $X_{\alpha}Y_{\beta}$ and
$Z_{\alpha}Y_{\beta}$ all anticommute with one of the global stablilizers
$\prod_{\alpha}X_{\alpha}$ or $\prod_{\alpha}Z_{\alpha}$, and hence have
vanishing expectation values. For a single realization, the $Y$-correlator
$\langle Y_{\alpha}Y_{\beta}\rangle$ can be expressed as $\langle
Y_{\alpha}Y_{\beta}\rangle=-\langle X_{\alpha}X_{\beta}\rangle\langle
Z_{\alpha}Z_{\beta}\rangle$ due to the properties of the instantaneous
stabilizer group in this model. By the Cauchy-Schwartz inequality,
$\left(\overline{\langle
Y_{\alpha}Y_{\beta}\rangle^{2}}\right)^{2}\leq\overline{\langle
X_{\alpha}X_{\beta}\rangle^{2}}\langle\overline{Z_{\alpha}Z_{\beta}\rangle^{2}}={\cal
X}_{\alpha\beta}{\cal Z}_{\alpha\beta}$. Thus we cannot have $Y$ spin-glass
order without having both $X$ and $Z$ order. We will therefore not report on
the $Y$ correlators.

We can always choose our instantaneous stabilizers to separately involve
products of Pauli $X$ operators or products of Pauli $Z$ operators: We call
these $X$-stabilizers and $Z$-stabilizers respectively. A useful observation
is that $\langle X_{\alpha}X_{\beta}\rangle^{2}=1$ is equivalent to the
statement that the only $Z$ stabilizers which pass through site $\alpha$ also
pass through site $\beta$, and vice-versa. This observation is a corollary of
the fact that $\langle X_{\alpha}X_{\beta}\rangle^{2}=1$ if and only if
$X_{\alpha}X_{\beta}$ commutes with every stabilizer generator. The
stabilizers which anticommute with $X_{\alpha}X_{\beta}$ are the
$Z$-stabilizers which pass through only $\alpha$ or $\beta$, but not both.

When $p_{2}=0$, the Bacon-Shor code stabilizers (either a product of two
columns of $X$ operators or two rows of $Z$ operators) are part of the
stabilizer group. Thus $\langle X_{\alpha}X_{\beta}\rangle^{2}=0$ unless
$\alpha$ and $\beta$ lie on the same row. Similarly, $\langle
Z_{\alpha}Z_{\beta}\rangle^{2}=0$ unless the spins lie on the same column.
(For $p_{2}\neq 0,1$, no such constraints are present.) The natural
interpretation of this row-only $X$ spin-glass order, or column-only $Z$ spin-
glass order, is that each row or column forms an independent spin glass.

We find that in the presence of the subsystem symmetry ($p_{2}=0,1$), the $X$
and $Z$ spin-glass orders are mutually exclusive, When the subsystem symmetry
is explicitly broken (by taking $p_{2}\neq 0,1$), the two-types of spin glass
order coexist as we will show later.

###  III.3 Entanglement Entropy and Mutual Information

In addition to the spin-glass order parameter, we characterize the states in
our ensemble via their entanglement properties. In particular, we split the
system in half by a vertical cut, and calculate the resulting entanglement
entropy. We then explore how this entanglement scales with system size.

For computational simplicity, we use the second Renyi entropy,
$S_{EE}=-\ln\operatorname{Tr}{\rho_{A}^{2}}$, where $\rho_{A}$ is the density
matrix formed by tracing over half the system. We calculate this entropy in
the stabilizer formalism by truncating the $L^{2}\times 2L^{2}$ matrix of the
stabilizers generators to an $L^{2}\times L^{2}$ matrix, tracing over half of
the sites in the system. We calculate the rank $R$ of this truncated matrix.
The entropy is given by, $S_{EE}=R-L^{2}/2$. [1, 10]

As already argued, the entanglement entropy acts like a thermodynamic
variable, and will display discontinuities or cusps at phase transitions. It
can also distinguish between certain types of states. For example, a great
deal of work has been devoted to studying phase transitions between “area law”
and “volume law” phases in systems of dimension $d$ [17, 33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45, 46, 9, 10, 11, 26, 12, 20, 21]. In the area-
law phases, the entanglement entropy between regions of size $L$ scales as
$L^{d-1}$, corresponding to the area of the boundary. The entanglement entropy
in volume-law phases instead scales as $L^{d}$. An intuitive heuristic is that
local measurements favor area-law phases, while longer-range measurements and
unitary gates favor volume law phases. Most of the previous work focuses on
$d=1$. In our case, we focus on $d=2$. As we will see later, our model
exhibits an interesting phase diagram without a volume-law phase.

In addition to entanglement entropy, we can also consider the mutual
information between two qubits, ${\cal
I}_{\alpha\beta}=S_{\alpha}+S_{\beta}-S_{\alpha\cup\beta}$. Here $S_{\alpha}$
is the entanglement entropy between qubit $\alpha$ and the remainder of the
system. The quantity $S_{\alpha\cup\beta}$ is the entanglement entropy between
the two spins and the rest of the system. The mutual information captures the
correlations between the two qubits. In our system, it carries the same
information as the spin correlation functions, and ${\cal I}$ can be taken as
a basis independent measure of the spin-glass correlations. More precisely,
our system has the global symmetries, generated by $\prod_{i,j}X_{i,j}$ and
$\prod_{i,j}Z_{i,j}$, that forbid single-site operators to appear as
instantaneous stabilizers. Together with the fact that the instantaneous
stabilizers group is generated by $X$\- and $Z$-stabilizers, the mutual
information ${\cal I}_{\alpha\beta}$ between the two sites $\alpha$ and
$\beta$ can be written as ${\cal I}_{\alpha\beta}=\ln 2\left(\langle
X_{\alpha}X_{\beta}\rangle^{2}+\langle Z_{\alpha}Z_{\beta}\rangle^{2}\right)$.

##  IV Results

###  IV.1 Behavior on high symmetry line, $p_{2}=0$

We first consider the bottom red line in Fig. 1(d) where $p_{2}=0$. Along this
line, we measure horizontal $\begin{smallmatrix}X\,X\end{smallmatrix}$ checks
with probability $1-p_{1}$ and vertical $\begin{smallmatrix}Z\\\
Z\end{smallmatrix}$ checks with probability $p_{1}$. As discussed in Sec. II.1
these operators commute with the Bacon-Shor stabilizers and logical operators.
The resulting subsystem symmetries exclude all two-point correlation functions
$\langle X_{\alpha}X_{\beta}\rangle$ along a column or $\langle
Z_{\alpha}Z_{\beta}\rangle$ along a row. This implies that in the $X$-basis,
the rows appear independent while in the $Z$-basis, the columns appear
independent. These symmetry constraints are properties of each individual
element of the ensemble and do not require ensemble averaging to observe.

Figure 2: Long-range spin-spin correlators in the $X$ basis along rows, ${\cal
X}^{r}=L^{-2}\sum_{ij}\overline{\langle X_{i,j}X_{i,j+L/2}\rangle^{2}}$ and in
the $Z$ basis along columns, ${\cal Z}^{c}=L^{-2}\sum_{ij}\overline{\langle
Z_{i,j}Z_{i+L/2,j}\rangle^{2}}$, as a function of probability $p_{1}$, along
the high symmetry line $p_{2}=0$ for a $L\times L$ system of size $L=36$. The
system has $X$ correlations along rows for $p_{1}<0.5$ and $Z$ correlations
along columns for $p_{1}>0.5$. The point $p_{1}=0.5$ is critical and there the
value of the correlator is set by finite size effects. Inset: Spatial
dependence of the $X$-correlator ${\cal
X}^{r}_{\delta}=L^{-2}\sum_{ij}\overline{\langle
X_{i,j}X_{i,j+\delta}\rangle^{2}}$ at the critical point as a function of
distance $\delta$ for a system of size $L=60$. The red solid line is a power
law fit to the curve, $0.49\delta^{-1.81}$.

We calculate the ensemble averaged long-range spin correlators in the $X$
basis along rows, ${\cal X}^{r}=L^{-2}\sum_{ij}\overline{\langle
X_{i,j}X_{i,j+L/2}\rangle^{2}}$ and in the $Z$ basis along columns, ${\cal
Z}^{c}=L^{-2}\sum_{ij}\overline{\langle Z_{i,j}Z_{i+L/2,j}\rangle^{2}}$ as we
vary $p_{1}$ from 0 to 1 in a $36\times 36$ lattice. The superscripts r and c
indicate correlations along a row and column respectively. The result is shown
in Fig. 2. For $p_{1}<0.5$, we find ${\cal X}^{r}\neq 0$, indicating a spin-
glass order in the $X$ basis along rows. For $p_{1}>0.5$, we instead find
${\cal Z}^{c}\neq 0$, indicating a spin glass order in the $Z$ basis along
columns. As argued below, the apparent non-zero value of ${\cal X}^{r},{\cal
Z}^{c}$ at $p_{1}=0.5$ is a finite size effect, and the two orders are never
simultaneously present. Due to the subsystem symmetries, the $X$-spin-glass
order ($Z$-spin-glass order), if present, is separately developed on each row
(column).

To elucidate the behavior at $p_{1}=0.5$, the inset of Fig. 2 shows the
spatial dependence of the $X$-correlator,
${\mathcal{X}}^{r}_{\delta}=L^{-2}\sum_{i,j}\overline{\langle
X_{i,j}X_{i,j+\delta}\rangle^{2}}$ as a function of distance $\delta$ for a
larger ($L=60$) system. We see a power law decay where
${\mathcal{X}}^{r}_{\delta}\sim 0.49\delta^{\gamma_{x}}$, with
$\gamma_{x}=-1.81(5)$. If we extrapolate to long distances,
${\mathcal{X}}^{r}_{\delta\rightarrow\infty}$ vanishes indicating the absence
of $X$-spin-glass order. The $Z$-correlator
${\mathcal{Z}}^{c}_{\delta}=L^{-2}\sum_{i,j}\overline{\langle
Z_{i+\delta,j}Z_{i,j}\rangle^{2}}$ along the columns exhibits the same scaling
behavior. From this behavior, we conclude that $p_{1}=0.5$ is a critical point
between a $Z$-spin-glass-ordered and $X$-spin-glass-ordered state. To further
demonstrate the criticality, in Fig. 3, we show that the spin-glass order
parameter obeys a scaling form
${\mathcal{X}}^{r}=L^{-\bar{\gamma}_{x}}g((p_{1}-0.5)L^{1/\nu})$, where $g$ is
a scaling function. Following the scaling collapse procedure in Ref. [11], we
find critical exponents $\bar{\gamma}_{x}=1.6(3)$ and $\nu=0.77(22)$. Within
error bars, $\gamma_{x}={\bar{\gamma}}_{x}$, and our value of $\nu$ agrees
with the one found in Ref. 20 by considering entropy scalings.

Figure 3:  Scaling collapse near the critical point $p_{1}=0.5$, $p_{2}=0$,
showing that
${\mathcal{X}}^{r}=L^{{-\bar{\gamma}}_{x}}g((p_{1}-0.5)L^{1/\nu})$ for
${\bar{\gamma}}_{x}=1.6$ and $\nu=0.77$, where $g$ is a scaling function.
Figure 4: Half cut bipartite entanglement entropy divided by system size,
$S_{EE}/L$ as a function of $\ln{L}$, where $L$ is the system size of an
$L\times L$ square lattice ranging from $L=12$ to $L=66$. The solid blue
circles show $S_{EE}/L$ at the critical point $(p_{1},p_{2})=(0.5,0)$ . The
dashed red line is a straight line fit showing logarithmic growth of
$S_{EE}/L$, violating the area law at criticality. The open black circles show
$S_{EE}/L$ at $(p_{1},p_{2})=(0.4,0)$ . The entropy saturates to a constant
value, showing area law.

We also calculate the half-cut bipartite entanglement entropy, $S_{EE}$ of the
system as we vary $p_{1}$ for different system sizes. We observe that the
system obeys area-law entanglement entropy ($S_{EE}\propto L$) for all values
of $p_{1}$ except at the critical point, $p_{1}=0.5$. At $p_{1}=0.5$, we
instead find $S_{EE}\propto L\ln{L}$ as shown in Fig. 4. In 1D critical
points, a logarithmic violation of area-law entanglement ($S_{EE}\propto\ln L$
for a system size of L) is a common feature which is often associated with
conformal symmetry. In 2D, this logarithmic violation of the area law, i.e.
$S_{EE}\propto L\ln{L}$, is more unusual. This scaling previously was found in
Fermi liquids [47] and some monitored/non-unitary dynamical systems [20, 48,
21, 49]. In our model, the scaling behavior $S_{EE}\sim L\ln{L}$ results from
the subsystems symmetry, which gives our model a 1D-like character : The order
parameters ${\mathcal{X}}^{r}$ and ${\mathcal{Z}}^{c}$ correspond to
independent ordering along each row or column. These rows/columns
simultaneously become critical at $p_{1}=0.5$. Thus it is natural to interpret
the $L\ln{L}$ entanglement scaling as being due to $\sim L$ independent 1D
critical chains, each of which have $\ln L$ entanglement.

It is a remarkable feature of quantum mechanics that the rows appear
independent if one only looks at the $X$-component of the spins. Conversely,
the columns appear independent if one only looks at the $Z$-component.
Criticality appears simultaneously in each of these bases.

###  IV.2 Without subsystem symmetry, $p_{2}=0.5$

We now consider the case where $p_{2}\neq 0,1$. This breaks all the subsystem
symmetries that were present when $p_{2}=0,1$. To illustrate the generic
behavior, we fix $p_{2}=0.5$ and vary $p_{1}$ from 0 to 1. As will be
demonstrated in Sec. IV.3, other values of $p_{2}\neq 0,1$ show qualitatively
similar behavior. That section also discusses continuity (or the lack there-
of) as $p_{2}\to 0,1$.

For $p_{2}=0.5$ measurements along horizontal and vertical bonds are equally
probable. As illustrated in Fig. 1(d), we measure
$\\{\begin{smallmatrix}Z\,Z\end{smallmatrix},\begin{smallmatrix}Z\\\
Z\end{smallmatrix}\\}$ with probability $p_{1}$ and
$\\{\begin{smallmatrix}X\,X\end{smallmatrix},\begin{smallmatrix}X\\\
X\end{smallmatrix}\\}$ with probability $1-p_{1}$. There are no constraints on
the allowed form of the $X$\- or $Z$\- spin glass order. Each order can exist
along rows, columns, and diagonally.

Figure 5: Long-range spin-spin correlators in the $X$ and $Z$ basis along rows
(${\mathcal{X}}^{r},{\mathcal{Z}}^{r}$) and
columns(${\mathcal{X}}^{c},{\mathcal{Z}}^{c}$) as a function of probability
$p_{1}$, with $p_{2}=0.5$ for an $L\times L$ system of size $L=36$. Along this
line, the correlations are isotropic: Column and row orders are of equal
strength. Near $p_{1}=0.5$, the $X$ and $Z$ orders coincide. Inset shows
spatial dependence of the $X$-correlator ${\cal
X}^{r}_{\delta}=L^{-2}\sum_{ij}\overline{\langle
X_{i,j}X_{i,j+\delta}\rangle^{2}}$ and the $Z$-correlator ${\cal
Z}^{c}_{\delta}=L^{-2}\sum_{ij}\overline{\langle
Z_{i,j}Z_{i+\delta,j}\rangle^{2}}$ at the point $(p_{1},p_{2})=(0.5,0.5)$ as a
function of distance $\delta$. The red solid line is an exponential fit to the
curve, $0.6e^{-1.36\delta}+0.06$. The correlations fall off exponentially and
saturate at a system-size independent value of 0.06.

In Fig. 5 we show the ensemble averaged long-range correlators in the $X$ and
$Z$ basis along rows (${\mathcal{X}}^{r},{\mathcal{Z}}^{r}$) and columns
(${\mathcal{X}}^{c},{\mathcal{Z}}^{c}$). Without the subsystem symmetry, the
X-spin glass order and Z-spin glass order exist identically along both rows
and columns. The $X$ order is peaked at $p_{1}=0$, while the $Z$ order peaks
at $p_{1}=1$. Despite the superficial similarity with Fig. 2, the point
$p_{1}=0.5$ is not a critical point. Both order parameters are non-zero at
$p_{1}=0.5$: The $X$ order smoothly falls to zero as one increases $p_{1}$,
and the $Z$ order behaves similarly as one decreases $p_{1}$.

To demonstrate that the order parameters are finite at
$(p_{1},p_{2})=(0.5,0.5)$, in the inset of Fig. 5, we plot the spatial
dependence of the $X$ and $Z$ correlators along rows,
${\mathcal{X}}^{r}_{\delta}=L^{-2}\sum_{i,j}\overline{\langle
X_{i,j}X_{i,j+\delta}\rangle^{2}}$ and
${\mathcal{Z}}^{r}_{\delta}=L^{-2}\sum_{i,j}\overline{\langle
Z_{i,j}Z_{i,j+\delta}\rangle^{2}}$. The correlation functions fall off
exponentially ($\sim 0.6e^{-1.36\delta}$) for small $\delta$ and saturate at a
finite value of
${\mathcal{X}}^{r}_{\delta\rightarrow\infty},{\mathcal{Z}}^{r}_{\delta\rightarrow\infty}\approx
0.06$ for large $\delta$. The long-range behavior is independent of system
size as long as $L>5$. We also calculated the $Y$ correlator,
${\mathcal{Y}}^{r}_{\delta}=L^{-2}\sum_{i,j}\overline{\langle
Y_{i,j}Y_{i,j+\delta}\rangle^{2}}$. We find that there is no long-range
$Y$-spin-glass order.

There are three forms of averaging in calculating these correlation functions:
(1) For a given state, a quantum mechanical expectation value corresponds to
the average result after many measurements of identical systems. (2) We are
performing a spatial average, summing over $i$ and $j$ in calculating $\langle
X_{i.j}X_{i,j+\delta}\rangle^{2}$. (3) We are ensemble averaging over multiple
states. It is useful to identify the role of each of these in determining
which correlation functions are zero and which are non-zero. If we take a
single state, and fix two sites, $\alpha$ and $\beta$, then (as previously
explained) $\langle X_{\alpha}X_{\beta}\rangle^{2}$ will be 0 or 1. Spatial
averaging then gives a real number which represents the fraction of correlated
sites. We find that for large system sizes the quantities are self-averaging,
meaning that the spatial average of a single realization approaches the
ensemble average, as we increase system size.

This self-averaging property implies that behavior of the ensemble can be
understood by studying a single typical state, and the coexistence of $X$ and
$Z$ spin-glass order at $(p_{1},p_{2})=(0.5,0.5)$ demonstrates that these
orders can be found in each individual state. As already explained, we find no
$Y$ spin-glass order, indicating that in any given state the $X$-correlated
spins are distinct from the $Z$-correlated spins.

To further elucidate this structure, we recall that $\langle
X_{\alpha}X_{\beta}\rangle\neq 0$ if and only if all $Z$-stabilizers which
pass through $\alpha$ also pass through $\beta$. Given that we are in an area-
law phase, we can find a basis in which the only linearly independent long-
range stabilizers are the global symmetries $\prod_{\alpha}Z_{\alpha}$ and
$\prod_{\alpha}X_{\alpha}$. The remaining generators are short-ranged, and can
be classified as either $Z$-type or $X$-type, depending on which operators
they are constructed from. Thus for well separated $\alpha$ and $\beta$,
$\langle X_{\alpha}X_{\beta}\rangle\neq 0$ if and only if there are no short-
range $Z$-generators passing through either of the two sites. An analogous
argument applies to $\langle Z_{\alpha}Z_{\beta}\rangle$.

For $\langle Y_{\alpha}Y_{\beta}\rangle$ to be non-zero, there would need to
be no short-range generators of any type passing through both sites. A simple
counting argument implies there can be at most two sites with only the global
$\prod_{\alpha}Z_{\alpha}$ and $\prod_{\alpha}X_{\alpha}$ stabilizers going
through them and no short range stabilizers touching them. In the
thermodynamic limit, this results in no $Y$-order.

We refer to a site with no short-range $Z$-stabilizer generators as a
$X$-site, and one with no short-range $X$-stabilizer generators as a $Z$-site.
All other sites have both $X$ and $Z$ short-range stabilizer generators. In
Appendix A we show typical distributions of the $X$ and $Z$ sites. They are
uniformly distributed, forming clusters whose size corresponds the length-
scale over which the correlation functions decay. The $X$-clusters and $Z$
clusters are interdigitated, and there is no global phase separation. At
$p_{1}=0.5,p_{2}=0.5$, the density of $X$-sites is $1/4$, as is the density of
$Z$-sites. This implies that the long-range correlators take the value
$\langle X_{\alpha}X_{\beta}\rangle^{2}=(1/4)^{2}\approx 0.06$ for well
separated sites $\alpha$ and $\beta$, which is consistent with Fig. 5.

As mentioned in the introduction, Ippoliti et al. [26] introduced the concept
of a frustration graph to organize their thinking about random circuits. In
the frustration graph, each check operator is a node. Two nodes are connected
if they anti-commute. Our frustration graph is bipartite, with $p_{1}$
controlling the weight given to the disconnected subgraphs formed respectively
from measuring
$\\{\begin{smallmatrix}X\,X\end{smallmatrix},\begin{smallmatrix}X\\\
X\end{smallmatrix}\\}$ and
$\\{\begin{smallmatrix}Z\,Z\end{smallmatrix},\begin{smallmatrix}Z\\\
Z\end{smallmatrix}\\}$ operators. Previous examples with a bipartite
frustration graph in 1D models displayed phase transitions [26]. Our 2D model
instead shows a smooth crossover (except in the presence of subsystem
symmetries), demonstrating the lack of a direct connection between the phase
diagram and frustration graph topology in higher dimensions.

The bipartite entanglement entropy obeys an area law for all values of $p_{1}$
when $p_{2}=0.5$. In fact, the system shows area-law behavior for all values
of $(p_{1},p_{2})$ except at the two critical points ($p_{1}=0.5$ and
$p_{2}=0,1$) where we see an $L\ln{L}$ scaling.

###  IV.3 Along $p_{1}=0.25$ and $p_{1}=0.5$

So far we have been considering fixed values of $p_{2}$ and varying $p_{1}$,
making horizontal cuts in the phase diagram in Fig. 1(d). We complete our
understanding by varying $p_{2}$ and making vertical cuts along fixed values
of $p_{1}=0.25,0.5$. The parameter $p_{2}$ quantifies the distance from the
subsystem symmetry lines at $p_{2}=0,1$.

Figure 6: (a) Long-range spin correlators in the $X$ basis along rows,
${\mathcal{X}}^{r}$ and columns, ${\mathcal{X}}^{c}$ as a function of $p_{2}$
for $p_{1}=0.25$ on an $L\times L$ system with $L=36$. ${\mathcal{X}}^{r}$
remains nearly constant even as $p_{2}\to 0$. The order is roughly isotropic,
${\mathcal{X}}^{c}\sim{\mathcal{X}}^{r}$, when $p_{2}\neq 0$, but
${\mathcal{X}}^{c}$ jumps discontinuously to 0 when $p_{2}\to 0$. Inset: Blue
solid line shows the region of the phase diagram probed here (varying $p_{2}$
at $p_{1}=0.25$). (b) Long-range spin correlators in the $X$ basis along
columns, ${\mathcal{X}}^{c}$ and in $Z$ basis along rows, ${\mathcal{Z}}^{r}$
as a function of $p_{2}$ for $p_{1}=0.5$ on an $L\times L$ system with $L=36$.
Both orders continuously go to zero as $p_{2}\to 0$. Inset: Blue solid line
shows the region of the phase diagram probed here (varying $p_{2}$ at
$p_{1}=0.5$). Error bars represent the standard error in the mean.

Figure 6 shows the long-range $X$ correlators along rows (${\mathcal{X}}^{r}$)
and columns (${\mathcal{X}}^{c}$) for different values of $p_{2}$. Panel (a)
where $p_{1}=0.25$ illustrates the generic situation, away from the critical
point. Panel (b) with $p_{1}=0.5$ instead shows the approach to criticality.
In these plots, ${\mathcal{X}}^{r}$ and ${\mathcal{X}}^{c}$ are nearly
identical at non-zero $p_{2}$. By considering systems of different sizes, we
have established that for $p_{2}\neq 0$ the discrepancy between order along
rows and columns is due to finite size effects and statistical noise.

At $p_{2}=0$ the $X$-order along columns vanishes (${\mathcal{X}}^{c}=0$, see
Sec IV.1). With $p_{1}=0.5$, ${\mathcal{X}}^{c}$ smoothly approaches zero as
we approach the critical point by tuning $p_{2}\to 0$. Conversely, for
$p_{1}=0.25$ we find (in the limit $L\to\infty$) a discontinuity (Panel (a)).
This discontinuous jump can be understood as follows. With $p_{2}=0$ and
$p_{1}<0.5$, there is effectively an independent and uncorrelated $X$-spin-
glass order for every row $i$ , associated with the subsystem symmetry
generated by $\prod_{j}Z_{i,j}$. Once $p_{2}$ becomes finite (while
$p_{1}<0.5$), this subsystem symmetry is broken down to a global symmetry
generated $\prod_{i,j}Z_{i,j}$. The existing $X$-spin-glass orders in all rows
now couple to each other and merge into a 2-dimensional $X$-spin-glass.

##  V Summary and Outlook

Randomly measuring non-commuting observables causes a quantum system to
evolve. In the thermodynamic limit, discontinuities and singularities are
found in the properties of the resulting ensembles. These points of non-
analyticity are analogous to thermodynamic phase transitions. As in
thermodynamics, both dimensionality and symmetries are crucial to understand
the universal behavior of dynamical systems undergoing both unitary evolution
and measurements. So far, the bulk of prior work in this direction has focused
on the entanglement properties of 1D systems [17, 33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46, 9, 10, 11, 26, 12, 50]. Our work is part of a
lively program to enlarge this scope [20, 24, 25, 21].

Our 2D model contains a number of symmetries which grow with system size (a
subsystem symmetry). This feature has important implications for its phase
diagram and critical behavior.

We consider a measurement-only random circuit inspired by the Bacon-Shor error
correcting code [28]. This circuit is a natural starting point, as it involves
only nearest neighbor two qubit operators,
$\begin{smallmatrix}X\,X\end{smallmatrix}$,$\begin{smallmatrix}X\\\
X\end{smallmatrix}$,$\begin{smallmatrix}Z\,Z\end{smallmatrix}$ and
$\begin{smallmatrix}Z\\\ Z\end{smallmatrix}$. As we vary the probabilities of
measuring these operators we obtain a rich phase diagram of spin-glass ordered
phases – with features that are not seen in 1D models.

Symmetry plays an important role in our 2D model. When we only measure
$\begin{smallmatrix}X\,X\end{smallmatrix}$ and $\begin{smallmatrix}Z\\\
Z\end{smallmatrix}$ ($p_{2}=0$), the system contains a large number of
symmetries/conservation laws: the product of $X$ operators along any one
column is conserved. Similarly, the product of $Z$ operators along any row is
conserved. As we vary $p_{1}$, the system undergoes a continuous phase
transition from an $X$-basis spin glass along rows to a $Z$-basis spin glass
along columns. We identify a critical point at $p_{1}=0.5$ where both spin
glass orders vanish in the thermodynamic limit and the entanglement entropy
scales as $S\sim L\ln L$.

We interpret this $L\ln L$ scaling behavior, often referred to as the
logarithmic violation to the area law, as being due to a decoupling into 1D
chains. Remarkably, in the $Z$ basis the columns form these chains, while in
the $X$ basis the rows form these chains. Criticality occurs simultaneously in
these two channels.

When we supplement the $\begin{smallmatrix}X\,X\end{smallmatrix}$ and
$\begin{smallmatrix}Z\\\ Z\end{smallmatrix}$ measurements with
$\begin{smallmatrix}X\\\ X\end{smallmatrix}$ and
$\begin{smallmatrix}Z\,Z\end{smallmatrix}$ operators, the subsystem symmetries
are broken. We find that the system now shows a smooth crossover between $X$
and $Z$ basis spin glass orders. The spin glass orders can coexist along both
rows and columns. Without subsystem symmetries, even though the frustration
graph remains bipartite, there is no criticality and the system always
displays an area-law entanglement entropy.

The system displays discontinuities as one approaches the high symmetry line
at $p_{2}=0$. Away from the critical point some spin glass order parameters
discontinuously jump to zero as $p_{2}\to 0$. This is analogous to a first
order phase transition.

An important feature of our study is that, like Sang and Hsieh [19], we go
beyond studying the entanglement properties of the quantum states, and
consider the spin correlation functions. This allows us to develop a physical
understanding of the properties of the ensemble. Similar analysis can be
applied to many of the other examples in the literature [26, 12, 20, 50, 9],
though some cases show topological order, requiring the use of string
operators rather than the simple two-point functions used here.

One interesting feature of our model is that, aside from the critical points,
it always yields states with area-law entanglement. Such behavior is expected
when one measures sufficiently local operators. It would be interesting to
characterize the minimal set of operators which would lead to a volume-law
phase in 2D.

The behaviour of 2D measurement-only random circuits is richer than their 1D
counterparts. By virtue of the larger connectivity there can exist symmetries
in 2D systems without clear analogs in 1D. Our subsystem symmetry is such an
example. In our model, this symmetry has dramatic effect, leading to
discontinuities in the spin-glass order, and converting a crossover into a
phase transition. It would be interesting to understand if other forms of
subsystem symmetries lead to similar behaviour.

## Acknowledgements

This work was supported by a New Frontier Grant awarded by the College of Arts
and Sciences at Cornell University (V.S., E.J.M.) and a faculty startup grant
at Cornell University (C.-M.J.). We thank Maissam Barkeshli and Ali Lavasani
for insightful discussions.

##  Appendix A Snapshots of spin-glass order

Here we introduce a technique to visualize the spin-glass order in a given
quantum state. We draw a grid of colored cells, each one represents a single
spin. We color the cells according to the stabilizer generators which have
support at that location: White cells have no short-range $Z$-stabilizers
passing through them. Black cells have no short-range $X$-stabilizers passing
through them. Gray cells have both short-range $X$ and $Z$ stabilizers. In the
language of Sec. IV.2, the white cells correspond to $X$-sites, and the Black
cells correspond to $Z$-sites.

By construction, any two $X$-sites ($\alpha,\beta$) will be $X$-correlated,
namely $\langle X_{\alpha}X_{\beta}\rangle^{2}=1$. Similarly, $Z$ sites will
have $\langle Z_{\alpha}Z_{\beta}\rangle^{2}=1$. Because of the area-law
entanglement, these are the only long-range correlations: In the
visualizations, there are no long-range $X$ correlations involving grey cells
or black cells; there are no long-range $Z$ correlations involving grey cells
or white cells. There can, however, be short-range correlations of these
sorts.

We refer to each visualization as a “snapshot”. Fig. 7 shows snapshots of
typical states for a variety of $p_{1},p_{2}$. In all cases we consider a
$36\times 36$ system, and run our dynamics for a large number of timesteps
before producing the visualization.

As we discussed in Sec. IV.2, the spin glass order is self averaging and thus
a spatial average on a typical state is representative of the ensemble
average.

Figure 7: Snapshots of typical long time quantum states for a system size
$36\times 36$ for different values of $p_{1},p_{2}$. (a)
$p_{1}=0.25,p_{2}=0.5$, (b) $p_{1}=0.5,p_{2}=0.5$, (c) $p_{1}=0.75,p_{2}=0.5$,
(d) $p_{1}=0.25,p_{2}=0.02$, (e) $p_{1}=0.5,p_{2}=0.02$, (f)
$p_{1}=0.75,p_{2}=0.02$. Each colored cell represents a single spin. White
cells have no short-range $Z$-stabilizers passing through them, and are
therefore $X$-correlated with one-another. Black cells have no short-range
$X$-stabilizers passing through them, and are $Z$-correlated. Gray cells have
both short-range $X$ and $Z$ stabilizers. While this coloring captures all
long-range correlations, there may be additional short-range correlations.

The top row of Fig. 7 shows the snapshots when $p_{2}=0.5$. Fig. 7(a) shows
the case where $p_{1}=0.25$. Most sites are $X$-correlated (white) here and
spread throughout the entire system. There are no black sites, which shows
that there are no long-range $Z$ correlations. Fig. 7(c) similarly shows the
case when $p_{1}=0.75$ where the system is dominated by $Z$-correlated sites
(black) with no $X$-correlated sites (white) present.

Fig. 7(b) shows the snapshot when $p_{1}=0.5$. Both white and black colored
sites are randomly spread throughout the system which shows how the $X$\- and
$Z$-spin-glass orders coexist in the system. (Though in any given state, the
sites participating in the $X$-order are distinct from the sites participating
in the $Z$-order) Roughly 1/4 of the sites are white. For any two distant
sites, the probability of them being $X$-correlated would be $(1/4)^{2}\approx
0.06$. This agrees with the asymptotic behavior of the correlation function
shown in the inset of Fig. 5.

In the bottom row of Fig. 7, we show snapshots when $p_{2}=0.02$. These points
are close to the subsystem symmetry line but are still area-law entangled.
Fig. 7(d) and Fig. 7(f) show the states with dominant $X$ correlations and $Z$
correlations respectively. The number of white and black sites are nearly
identical to what is seen when $p_{2}=0.5$ (Fig. 7(a),(c)) confirming the
results of Fig. 6(a) where the spin glass orders are only weakly dependent on
$p_{2}$. However, we see that the uncorrelated gray clusters display
interesting asymmetries: They are more horizontal for $p_{1}=0.25$ and more
vertical when $p_{1}=0.75$. This elongation is an intuitive consequence of the
different probabilities of measuring horizontal or vertical check operators.
The asymmetry can also be seen in the short-range spin correlation functions –
though the long-range correlation functions are rotationaly symmetric.

Fig. 7(e) shows the case where $p_{1}=0.5$. While we see both $X$-correlated
(white) and $Z$-correlated (black) sites randomly spread throughout the
system, the image is dominated by the uncorrelated gray sites. This is a
visual indicator of the feature of Fig. 6(b) where the $X$\- and $Z$-spin-
glass orders continuously fall to zero as $p_{2}\to 0$. As we make $p_{2}$
smaller, we will continue to see more and more uncorrelated sites.

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