# Measurement-Induced Phase Transitions in Sparse Nonlocal Scramblers

Tomohiro Hashizume  Department of Physics and SUPA, University of Strathclyde,
Glasgow G4 0NG, United Kingdom  Gregory Bentsen  Martin A. Fisher School of
Physics, Brandeis University, Waltham, Massachusetts 02465, USA  Andrew J.
Daley  Department of Physics and SUPA, University of Strathclyde, Glasgow G4
0NG, United Kingdom

###### Abstract

Measurement-induced phase transitions arise due to a competition between the
scrambling of quantum information in a many-body system and local
measurements. In this work we investigate these transitions in different
classes of fast scramblers, systems that scramble quantum information as
quickly as is conjectured to be possible – on a timescale proportional to the
logarithm of the system size. In particular, we consider sets of deterministic
sparse couplings that naturally interpolate between local circuits that slowly
scramble information and highly nonlocal circuits that achieve the fast-
scrambling limit. We find that circuits featuring sparse nonlocal interactions
are able to withstand substantially higher rates of local measurement than
circuits with only local interactions, even at comparable gate depths. We also
study the quantum error-correcting codes that support the volume-law entangled
phase and find that our maximally nonlocal circuits yield codes with nearly
extensive contiguous code distance. Use of these sparse, deterministic
circuits opens pathways towards the design of noise-resilient quantum circuits
and error correcting codes in current and future quantum devices with minimum
gate numbers.

##  I Introduction

Measurement-induced phase transitions (MIPT) [1, 2, 3, 4] are transitions
driven by a competition between scrambling dynamics, which tends to generate
many-body entanglement across all degrees of freedom of a quantum many-body
system, and local measurements, which tend to destroy this entanglement.
Measurement-driven transitions of this type have garnered an extraordinary
amount of theoretical interest recently [5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 19, 20], and have been studied in a wide range of physical
systems, including in random quantum circuits, free fermions [21, 22, 23], and
non-Hermitian Brownian models [24]. They have also been extended to systems
with long-range interactions [25, 26, 27, 28], including models with all-to-
all coupling [29, 30], and non-Hermitian Sachdev-Ye-Kitaev (SYK) models [31,
32, 33]. MIPTs have also been related to purification transitions [34], where
entanglement of the system undergoing MIPT with an ancilla can be used to
detect the transition [35], as recently demonstrated in experiments with
trapped ions [36]. Importantly, these transition have also been related to the
properties of quantum error-correcting codes (QECC) [11, 34]. These codes are
dynamically generated during the evolution and are responsible for protecting
the extensive many-body entanglement present in the system despite repeated
local measurements.

The durability of this many-body entangled phase, despite the presence of
repeated projective measurements, is particularly compelling from the
perspective of experiments with modern quantum technologies. In this context,
the goal is often to generate interesting or useful entangled quantum states,
while always contending with noise and dissipation which inevitably degrade
this entanglement [4]. In this context, it is therefore of considerable value
to identify mechanisms for building up robust patterns of entanglement as
quickly and as efficiently as possible before noise has a chance to degrade
the computation or simulation being carried out. Particularly efficient
quantum systems that build up many-body entanglement as rapidly as possible
are known as _fast scramblers_ [37, 38]. Fast scrambling was first discussed
in the context of information spreading in black holes, and refers to systems
that can scramble quantum information on timescales $t_{*}\sim\log N$ that
grow only logarithmically with the system size $N$ [39, 40, 37, 38, 41].
Random all-to-all circuits [42], and other disordered all-to-all coupled
models such as the SYK model [43, 44] are good examples of fast scramblers,
but recently, a number of deterministic and experimentally feasible systems
exhibiting fast scrambling have also been explored [45, 46, 47, 48]. These
show similar dynamics, but often on sparse nonlocal coupling graphs. In this
context, we are led to the natural questions: how do these deterministic fast
scrambling circuits perform when subjected to random measurements? Does fast
scrambling dynamics uniformly enhance the stability of the mixed phase as a
general principle, and can this lead to improved quantum error-correcting code
properties?

In this work we explore these connections by studying how the strength of
scrambling affects the properties of the MIPT mixed phase by considering
sparsely-coupled quantum circuits with tunable nonlocal interactions. By
adjusting the range of interactions in these sparse circuits, we can tune from
local spreading of information to genuine fast scrambling. Our analysis
demonstrates that sparse nonlocal interactions can significantly improve a
quantum system’s robustness to local measurements. We show that only a few
additional layers of nonlocal gates substantially increase the critical
measurement rate $p_{c}$. We further demonstrate that highly nonlocal
interactions improve the properties of the quantum error-correcting codes that
stabilize the mixed phase.

Figure 1: Sparse fast scrambling $\mathrm{PWR}2_{k}$ circuits with tunable
nonlocality $k$ subjected to random projective measurements. (a,b) Two-qubit
entangling gates $Q_{ij}$ (green, turquoise, blue, purple) are applied between
qubits $i,j$ if and only if they are separated by an integer power of 2,
$\left|i-j\right|=2^{m-1}$ for $m=1,\ldots,k$. These gates are applied in
consecutive interaction layers at times $t=1,\ldots,T$ in a nonlocal
bricklayer pattern arranged into alternating even and odd blocks. Projective
single-qubit measurements (red crosses) occur with probability $p$ between
each interaction layer. The degree $k$ of the resulting interaction graph (b)
controls the nonlocality of interactions where the longest-range interactions
occur between qubits initially separated by
$d_{\mathrm{max}}=\left|i-j\right|=2^{k-1}$. (c) The critical measurement
rates for the purification transition $p_{cp}$ (red) and entanglement
transition $p_{ce}$ (blue) increase significantly with $k$, interpolating
between the critical points found for random nearest neighbor (NN) models for
$k=1$ and random all-to-all (AA) models for larger $k$. Haar-random circuits
with identical coupling patterns show a similar improvement in robustness to
measurement as $k$ increases (green). (d) In the mixed phase for $N=256$
qubits, the maximum code distance $d_{\mathrm{code}}$ improves with $k$ at
fixed code rate $r_{\mathrm{code}}$. Error bars are shown or are smaller than
datapoints; lines are guides to the eye.

As specific examples, we choose sparse nonlocal circuits that can be realised
in the laboratory by combining nearest-neighbor Rydberg interactions with
nonlocal shuffling operations [48]. The strength of scrambling in these
circuits is governed by a nonlocality parameter $k$. In experimental
implementations with Rydberg atoms this parameter is controlled by the number
of shuffle operations applied, thereby providing a natural tunable parameter
controlling the strength of scrambling in these circuits. In particular, the
case $k=1$ corresponds to a nearest neighbour (NN) circuit with slow
scrambling, while $k\sim\log_{2}N$ yields highly nonlocal circuits which
rapidly scramble quantum information. In the following we study how these
circuits perform when subjected to random projective measurements with a view
toward understanding how nonlocal interactions can improve the stability of
the MIPT mixed phase.

The rest of this paper is organised as follows: In Section II we introduce the
deterministic $\mathrm{PWR}2_{k}$ circuit models that we study in this work.
In Section III we study measurement-induced transitions in $\mathrm{PWR}2_{k}$
circuits featuring Haar-random gates, which allows the problem to be mapped to
a classical percolation problem. In Section IV we study the entanglement
transition numerically in $\mathrm{PWR}2_{k}$ circuits featuring Clifford
gates and characterize the transition in terms of the tripartite mutual
information $I(A:B:C)$ between three consecutive regions of the output qubits.
In Section V we study the purification transition numerically for the same
Clifford circuits as characterized by the purification time $\tau$ of a single
qubit. In Section VI we numerically study the properties of the dynamically-
generated error-correcting code in the mixed phase, and demonstrate that
nonlocal interactions allow for improved code distance at fixed code rate. In
Section VII we consider the ‘complete’ $\mathrm{PWR}2_{k}$ circuit with
$k=\log_{2}N$ and compare its behavior to the random all-to-all model.
Finally, we discuss outstanding issues and prospects for future work in
Section VIII.

##  II Models

The models we study here feature sparse nonlocal interactions in which spins
residing at sites $i,j=1,\ldots,N$ of a 1d lattice are coupled if and only if
the distance between them is an integer power of 2: $\left|i-j\right|=2^{m-1}$
for $m=1,\ldots,k$ as shown in Fig. 1a. Sparse nonlocal interactions of this
type have been shown to generate fast scrambling dynamics [45], and are
inspired by the nonlocal coupling patterns that can be engineered in single-
mode cavities [49] or in Rydberg arrays with the aid of tweezer-assisted
shuffling operations [48]. We expect the fast scrambling dynamics in these
nonlocal circuits to rapidly generate many-body entanglement that is
increasingly robust to the destructive effects of local measurements relative
to circuits featuring only local interactions.

In the following we demonstrate these expectations explicitly by studying
sparse nonlocal Floquet circuits consisting of consecutive layers of pairwise
nonlocal two-qubit gates $Q_{ij}=Q_{ji}$. The gates are arranged in a nonlocal
bricklayer pattern, as illustrated in Fig. 1b, where each timestep
$t=1,\ldots,T$ represents a single interaction layer consisting of exactly
$N/2$ gates. These interaction layers are stacked into an alternating sequence
of even and odd interaction blocks. During the _even_ block $t=m=1,\ldots,k$
we place gates $Q_{ij}$ between qubits $i<j$ if and only if
$\left|i-j\right|=2^{m-1}$ and $\textrm{mod}(\lfloor i/2^{m-1}\rfloor,2)=0$.
During the subsequent _odd_ block $t=k+m=k+1,\ldots,2k$ we place gates
according to the same rules but with the odd-bricklayer condition
$\textrm{mod}(\lfloor i/2^{m-1}\rfloor,2)=1$. Finally, we apply a layer of
single-qubit phase gates $P_{i}$ to all qubits and repeat the entire sequence
until a final time $t=T$. This alternating sequence of even and odd nonlocal
interaction blocks ensures that quantum information rapidly spreads through
the system via nonlocal interactions as rapidly as possible within a single
block, but is not siloed into one of the hierarchical branches of the nonlocal
circuit that occurs if, for example, the circuit consists only of even blocks.
Unless otherwise stated, the models which appear in this paper always assume
periodic boundary conditions.

Following each interaction layer $t$ we randomly apply projective measurements
in the Pauli-$z$ basis to individual qubits with probability $p$ as
illustrated by the red crosses in Fig. 1b. These projective measurements tend
to destroy the long-range entanglement built up by the previous interaction
layers of the $\mathrm{PWR}2_{k}$ circuit. The competition between these two
effects leads to a measurement-induced phase transition in the entanglement
structure of the output state [7, 34, 29]. In this paper, we are particularly
interested in how this transition changes as we add more nonlocal
interactions.

In the $\mathrm{PWR}2_{k}$ family, this degree of nonlocality is controlled by
the parameter $k\leq\log_{2}N$, which determines the distance of the longest-
range interactions $d_{\mathrm{max}}=\left|i-j\right|_{\mathrm{max}}=2^{k-1}$.
Thus, $\mathrm{PWR}2_{k}$ circuits with $k\sim O(1)$ consist only of short-
range couplings which restrict the spread of quantum information [50, 51] and
therefore generate only slow information scrambling. By contrast, circuits
with $k\sim\log_{2}N$ consist of highly nonlocal interactions that span the
entire system and are known to rapidly scramble quantum information [52, 45,
48]. The nonlocality parameter $k$ therefore provides a convenient means by
which to interpolate between the local and nonlocal limits of the
$\mathrm{PWR}2_{k}$ family.

These two extreme cases can be compared to two prototypical scrambling
circuits known to exhibit measurement-induced transitions and that have been
studied extensively in existing literature. For local interactions $k\sim
O(1)$ we compare to a random 1+1D nearest-neighbor (NN) circuit [34] in which
the scrambling of quantum information is highly restricted by local Lieb-
Robinson bounds [50, 51, 52]. For highly nonlocal interactions
$k\sim\log_{2}N$ we compare to a random all-to-all (AA) circuit in which
qubits interact pairwise at random without regard to their spatial location
[34, 29]. These maximally nonlocal AA models were first proposed by Sekino and
Susskind [37] as prototypical examples of _fast scramblers_ , systems which
scramble quantum information at the fastest rate allowed by quantum mechanics.

In the remainder of this paper, we study the entanglement properties of these
monitored $\mathrm{PWR}2_{k}$ circuits as a function of measurement rate $p$
and nonlocality $k$ using a combination of analytical and numerical methods.
We consider two types of gates $Q_{ij}$ in this paper. In Sec. III we take
$Q_{ij}$ to be Haar-random unitary gates acting between pairs of qudits with
Hilbert space dimension $q$, which maps to a classical bond percolation
problem on a degree-4 network in the limit $q\rightarrow\infty$. In Secs. IV
\- VI we take $Q_{ij}=\mathrm{CZ}_{ij}H_{i}H_{j}$, where
$\mathrm{CZ}_{ij}=\mathrm{CZ}_{ji}$ is the controlled-$Z$ gate on qubits $i,j$
and $H_{i},P_{i}$ are the Hadamard and Phase gates on qubit $i$, respectively.
In this case the entire circuit is composed of Clifford gates and can
therefore be efficiently simulated on a classical computer [53, 54]. In each
of these cases we find that the critical measurement rate $p_{c}$
significantly improves with $k$, indicating that nonlocal interactions
substantially improve a deterministic quantum circuit’s ability to withstand
local measurements by leveraging nonlocal scrambling interactions.

##  III Percolation Transition in Haar-Random $\mathrm{PWR}2_{k}$ Circuits

To demonstrate that nonlocal sparse interactions can improve a system’s
robustness to local measurements, we first study $\mathrm{PWR}2_{k}$ circuits
consisting of Haar-random nonlocal gates $Q_{ij}$ acting between pairs of
qudits with local Hilbert space dimension $q$. In the limit
$q\rightarrow\infty$, the circuit and the randomly applied projective
measurements can be mapped to a bond percolation problem [2, 34, 29]. We
illustrate this mapping in Fig. 2a, where each two-qubit gate $Q_{ij}$ (green)
corresponds to a vertex in the percolation network while the qubit worldlines
(black) correspond to four bonds connected to this vertex. Projective
measurements (red) correspond to cutting these bonds. We discuss this mapping
more explicitly in Appendix A [55]. Applying this mapping to the entire
Floquet circuit illustrated in Fig. 1b yields a nonlocal bond percolation
network whose bonds are cut with probability $p$.

For sufficiently small $p$ this nonlocal network percolates, in the sense that
the network contains at least one connected component that extends from the
input qubit bonds at $t=0$ to the output bonds at $t=T$. In the thermodynamic
limit $N\rightarrow\infty$ and for extensive time $T=\mathcal{O}(N)$, this
corresponds to a network with at least one connected component of infinite
extent. The percolation critical point, $p_{c,\mathrm{Haar}}$, is defined as
the maximum measurement rate $p$ for which the lattice percolates in the
thermodynamic limit. For $p<p_{c,\mathrm{Haar}}$, successful percolation
implies that the output qubits at $t=T$ retain partial information about the
input state at $t=0$. By contrast, for $p>p_{c,\mathrm{Haar}}$, failure to
percolate indicates that any information embedded in the initial state at
$t=0$ is lost due to the proliferation of projective measurements in the
circuit.

Classical bond-percolation transitions of this type are well-studied in the
literature [56], including on a wide variety of local and nonlocal networks.
In the extreme local case of nearest-neighbor interactions (i.e. $k=1$ in the
$\mathrm{PWR}2_{k}$ family), the percolation mapping yields a bond percolation
problem on a two dimensional square lattice, whose critical properties are
known analytically [57, 56]. In this limit, one can naturally define a
correlation length $\xi$ which captures the typical size of connected
components in the network. Near the critical point
$p_{c,\mathrm{Haar}}^{\mathrm{NN}}=1/2$ the correlation length diverges
algebraically like $\xi\sim|p-p_{c,\mathrm{Haar}}|^{-\nu}$, where $\nu=4/3$ is
the critical exponent [57, 56, 2].

The classical bond-percolation transition has also been studied in the
opposite nonlocal limit where interactions become arbitrarily long-range. The
percolation critical point of the random AA model was studied early on by
Gullans and Huse [34] and more extensively by Nahum et al. [29]. The critical
point of the AA network is known to be at
$p_{c,\mathrm{Haar}}^{\mathrm{AA}}=2/3$. Such a high critical point comes from
the system’s locally tree-like structure in the limit of large system size,
where the probability of the network having a local loop vanishes as $1/N$
[29].

Here we interpolate between these two extreme limits using the nonlocality
parameter $k$ in the Haar-random $\mathrm{PWR}2_{k}$ circuit family, where
$k=1$ corresponds to the NN case and $k\sim\log_{2}N$ approaches the AA case.
In order to find the critical point $p_{c,\mathrm{Haar}}$ for each value of
$k$, we numerically simulate bond percolation on the corresponding network
using the Newman-Ziff Algorithm [58]. The critical point $p_{c,\mathrm{Haar}}$
and the critical exponent $\nu$ of the correlation length can be identified by
calculating the binder cumulant [59, 60],

$\displaystyle b(p)=\frac{1}{2}\left(3-\frac{\left\langle
C_{\max}^{4}(p)\right\rangle}{\left\langle
C_{\max}^{2}(p)\right\rangle^{2}}\right)$ (1)

where $C_{\max}$ is the maximum cluster size in the nonlocal percolation
network and $\langle\dots\rangle$ denotes the averaging over different
configurations of randomly-cut bonds. Near the critical point, we expect the
Binder cumulant to obey the scaling law [59, 60, 61]

$\displaystyle b(p)=f\left((p-p_{c,\mathrm{Haar}})N^{1/\nu}\right),$ (2)

which is governed by the same critical exponent $\nu$ that controls the
divergence of the correlation length $\xi$ near the critical point.

Figure 2:  Measurement-induced transitions in the $\mathrm{PWR}2_{k}$ circuit
family with Haar-random gates. (a) Nonlocal circuits featuring Haar-random
gates can be mapped to a classical percolation network, where gates in the
original circuit (green) correspond to sites in the percolation network, and
projective measurements in the Haar-random circuit (red crosses) correspond to
cut bonds in the percolation network. (b) The Binder cumulant $b(p)$ of the
maximum cluster size in the classical percolation network for $k=1,3,5$ (top
to bottom) and system sizes $N=256,\ldots,2048$ (light to dark blue). The
critical point $p_{c,\mathrm{Haar}}$ is extracted from the crossing point of
$b(p)$ across finite-size systems (insets). We observe good scaling collapse
with critical exponent $\nu=4/3$ expected for a 1+1D NN circuit even for large
nonlocality $k$. (c) The percolation critical point $p_{c,\mathrm{Haar}}$
(green) increases with $k$; the critical exponent $\nu$ (black) is nearly
constant as $k$ varies, indicating that these nonlocal circuits near
criticality likely belong to the same universality class for all
$k<\log_{2}N$. Error bars are shown or are smaller than markers; lines are
guides to the eye.

We plot the results of these numerical simulations in Fig. 2b for $k=1,3,5$
across system sizes $N=64,\ldots,1024$ (light to dark blue). To extract the
critical point $p_{c,\mathrm{Haar}}$ we plot the Binder cumulant $b(p)$
(insets of Fig. 2b) and locate the crossing point as a function of system size
$N$. At each fixed $k$ in this analysis, we only consider the crossing point
for sufficiently large system sizes $N>2^{k}$, such that the longest-range
interactions $d_{\mathrm{max}}=2^{k-1}$ are never extensive. We then fit the
scaling form (2) to the data near the critical point and use this to extract
an estimate of the critical exponent $\nu$ for each value of $k$. Our
resulting estimates for the critical point and critical exponent can be used
to collapse the Binder cumulant $b(p)$ near the critical point to a single
universal curve (2) as shown in the main panels of Fig. 2b.

We plot the resulting critical points $p_{c,\mathrm{Haar}}$ and critical
exponents $\nu$ as a function of $k$ in Fig. 2c. The critical points clearly
increase with nonlocality $k$; for $k\geq 5$, the critical point is closer to
the AA limit than the NN limit. The critical exponent $\nu$, on the other
hand, appears to be largely independent of $k$ and is consistent with the
critical exponent $\nu=4/3$ expected for a 1+1 NN model (Fig. 2c, dotted black
line). These results suggest that all Haar-random $\mathrm{PWR}2_{k}$ models
with $k<\log_{2}N$ fall into the same universality class as the local 1+1D
model in the thermodynamic limit. Nevertheless, these results also demonstrate
that the critical point $p_{c,\mathrm{Haar}}$, a non-universal parameter,
significantly increase with nonlocality $k$.

##  IV Entanglement Transition in Clifford $\mathrm{PWR}2_{k}$ circuits

The nonlocal classical percolation network we studied in the previous section
demonstrated that just a few additional layers of nonlocal interactions can
significantly increase the critical point of the Haar-random
$\mathrm{PWR}2_{k}$ circuit. In the following sections we turn our attention
to deterministic nonlocal circuits composed entirely of Clifford gates. We
choose interaction gates $Q_{ij}=Q_{ij}=\mathrm{CZ}_{ij}H_{i}H_{j}$ and show
that the circuit’s ability to withstand the destructive effects of local
measurement is improved as a function of the nonlocality parameter $k$.

Figure 3:  Measurement-induced entanglement transition in the
$\mathrm{PWR}2_{k}$ circuit family with Clifford gates. (a) The circuit is
initialized with a separable pure input state, while the output is divided
into four equal regions $A,B,C,D$. (b) Tripartite mutual information
$I(A:B:C)$ for consecutive regions $A,B,C$ in the $\mathrm{PWR}2_{k}$ circuit
for $k=1,3,5$ (top to bottom) and system sizes $N=64,\ldots,2048$ (light to
dark blue). The critical point $p_{ce}$ is extracted from finite-size scaling
for $N=64,\ldots,2048$ (insets); only sufficiently large system sizes
$N>2^{k}$ are used to estimate the critical point. Near $p_{ce}$ we observe a
scaling collapse with the critical exponent $\nu\approx 1.28$. (c) The
critical measurement rate $p_{ce}$ (blue) increases significantly with $k$
while the critical exponent $\nu$ (black) is consistent with the 1+1D critical
exponent (dotted black) to within statistical fluctuations for all values of
$k$. Error bars are shown or are smaller than datapoints; lines are guides to
the eye.

We begin our study of these nonlocal Clifford circuits by characterizing the
entanglement at the output of the circuit as a function of the measurement
rate $p$. We initialize the monitored $\mathrm{PWR}2_{k}$ circuit with a pure,
separable $z$-polarized state, and find a phase transition in the tripartite
mutual information $I(A:B:C)$ between three equal-size consecutive regions
$A,B,C$ of the output state as shown in Fig 3a. The tripartite mutual
information

$\displaystyle I(A:B:C)=I(A,B)+I(A,C)-I(A,BC)$ (3)

is defined in terms of the mutual information
$I(A,B)=S^{(2)}_{A}+S^{(2)}_{B}-S^{(2)}_{AB}$, where
$S^{(2)}_{A}=-\ln\mathrm{Tr}\left[\rho_{A}^{2}\right]$ is the Renyi entropy of
the subsystem $A$. Because the circuit consists entirely of Clifford gates,
the Renyi entropy $S^{(2)}_{A}$ is always equal to the conventional von
Neumann entropy $S_{A}=-\mathrm{Tr}\left[\rho_{A}\ln\rho_{A}\right]$ and we
may therefore characterize the entanglement of the system entirely in terms of
the Renyi entropies $S^{(2)}_{A}$.

To extract the entanglement critical point $p_{ce}$, we perform a finite-size
scaling analysis similar to the previous section. In Fig. 3b, we plot the
tripartite mutual information as a function of measurement rate $p$ for
various system sizes $N=64,\ldots,1024$ (light to dark blue). The critical
point $p_{ce}$ is determined by the crossing point of the tripartite mutual
information across system sizes as shown in the insets of Fig. 3b. Only
sufficiently large system sizes $N>2^{k}$ are used to extract the critical
point. Near the critical point $p_{ce}$ we expect the tripartite mutual
information to obey the universal scaling law

$\displaystyle I(A:B:C)=f\left((p-p_{ce})N^{1/\nu}\right)$ (4)

where $f$ is a universal function and $\nu$ is the critical exponent of the
correlation length $\xi$. After determining the critical point $p_{ce}$ we fit
this scaling form to each curve in Fig. 3b and use this to extract an estimate
for the critical exponent $\nu$. The resulting estimates for $p_{ce},\nu$
allow us to collapse the original data down to a universal curve as shown in
the main panels of Fig. 3b [34].

We plot the resulting estimates for the critical point $p_{ce}$ and critical
exponent $\nu$ in Fig. 3c for $k=1,\ldots,6$. Similar to our findings in the
previous section, the non-universal critical point $p_{ce}$ increases
significantly with $k$. The critical exponent, however, is nearly constant
across $k$, and is consistent with the critical exponent $\nu\approx 1.28(2)$
found numerically for 1+1D NN models [34] to within statistical fluctuations
(Fig. 3c). This indicates that the measurement-induced transitions in these
models likely fall into the same universality class as the completely local NN
models. Nevertheless, the fact that the critical point $p_{ce}$ increases with
the nonlocality parameter $k$ demonstrates that just a few additional layers
of nonlocal interactions can substantially improve a circuit’s ability to
retain complex many-body entanglement even in the presence of local
measurements, similar to our observations in previous sections.

##  V Single-Qubit Purification in Clifford $\mathrm{PWR}2_{k}$ Circuits

Figure 4:  Single-qubit purification transition in the $\mathrm{PWR}2_{k}$
circuit family with Clifford gates. (a) Schematic diagram for single-qubit
purification. (b) Finite-size scaling for single-qubit purification time
$\tau$ in the $\mathrm{PWR}2_{k}$ circuit for $k=1,3,5$ (top to bottom) and
system sizes $N=64,\ldots,2048$ (light to dark blue). Main figures show
scaling collapse with critical exponent $\nu\sim 1.30$ and dynamical critical
exponent $z=1$ for $k=1,3,5$. The critical point $p_{cp}$ is determined by the
crossing point of $\tau/L^{z=1}$ across system sizes $N>2^{k}$ (insets). (c)
The purification critical point $p_{cp}$ increases as a function of $k$,
closely mirroring the increase in the entanglement critical point $p_{ce}$
found in the previous section. The critical exponent, on the other hand,
agrees with the $\nu$ for $k=1$ found in the entanglement criticality in the
previous section within the error (green solid line, with 1-sigma fluctuations
indicated by the green dashed line). Error bars are shown or are smaller than
the data points; lines are guides to the eye.

Another measure of a circuit’s robustness to measurements is the timescale
$\tau$ required to purify a single qubit that has been entangled with the
system [35, 26]. Absent any measurements such a qubit will remain entangled
with the system forever, but carefully-placed projective measurements can
destroy the entanglement between the system and qubit, causing the qubit to
collapse into a pure state. The typical timescale $\tau$ required for this
purification process to occur can be used to characterize our circuit’s
ability to retain quantum information encoded in the initial state when
subjected to noise. In particular, below a critical measurement rate
$p<p_{cp}$ we expect the purification time to become extensive $\tau\sim
N^{z}$, indicating that quantum information can be robustly stored in the
circuit despite the presence of repeated projective measurements at a rate
$p$.

Although for nearest-neighbor circuits the purification critical point
$p_{cp}$ coincides with the entanglement critical point $p_{ce}$, we emphasize
that these transitions can generically be different, as has been pointed out
in the literature [34]. We are especially interested whether highly nonlocal
circuits support an intermediate phase in between the two critical points
$p_{cp},p_{ce}$ where the reference qubit has been purified, but where there
is nevertheless still volume-law entanglement in the final state.

Here we study the possibility of such an intermediate phase in our nonlocal
$\mathrm{PWR}2_{k}$ circuits by determining the single-qubit purification time
$\tau$. To determine the purification critical point $p_{cp}$, we maximally
entangle a single reference qubit $R$ with the system $Q$ and apply $t=4N$
layers of a unitary NN circuit to the system such that the qubit of
information is scrambled deeply within the system [26] (Fig. 4a,
$U_{\mathrm{thermalizer}}$). Then the monitored $\mathrm{PWR}2_{k}$ circuit is
applied as illustrated in Fig. 4a. At the end of each timestep $t$ we compute
the Renyi entropy $S_{Q}^{(2)}$ of the system. The single qubit purification
time $\tau$ is the number of timesteps required for the Renyi entropy to
vanish from its initial value of $S_{Q}^{(2)}(0)=\ln 2$.

To extract the critical point $p_{cp}$ for each $k<\log_{2}N$ we perform a
finite-size scaling analysis as shown in Fig. 4b for $k=1,3,5$. Similar to the
analysis of previous sections, the critical point is determined by the
location of the crossing point of $\tau(p)$ as the system size $N$ is varied,
as shown in the insets of Fig. 4b. The critical points $p_{cp}$ obtained from
this analysis are plotted in Fig. 4c and grow significantly with $k=1,\dots,6$
in agreement with earlier analysis. In each of these cases the critical point
$p_{cp}$ of the purification transition agrees with the entanglement critical
point $p_{ce}$ within error bars. This strongly suggests that for
$k<\log_{2}N$ the critical points $p_{cp}$, $p_{ce}$ are in fact identical and
that there is no intermediate phase between the purification and entanglement
phases [34].

Near the critical point, we expect the purification time to obey the universal
scaling law

$\displaystyle\tau(p)=N^{z}f\left((p-p_{cp})N^{1/\nu}\right)$ (5)

where $z$ is the dynamical exponent and $\nu$ is the critical exponent of the
correlation length $\xi$. Based on our findings in previous sections, we
expect the purification transition studied here for $k<\log_{2}N$ to be in the
same universality class as the purification transition in 1+1D NN models,
which are believed to be governed by a conformal field theory with dynamical
exponent $z=1$ [2, 34, 26]. We therefore assume $z=1$ and use the scaling form
(5) to fit the data plotted in Fig. 4b. These fits yield estimates for the
critical exponent $\nu$, which we plot in Fig. 4c. The resulting critical
exponents $\nu=1.30\pm 0.02$ are largely independent of $k$ and agree with the
critical exponent of the entanglement transition of the 1+1D NN model. This
consistent with previous results showing that the critical behaviour of the
$\mathrm{PWR}2_{k}$ with fixed $k<\log_{2}N$ is in the same universality class
as the conventional 1+1D MIPT transition.

##  VI Quantum Error-Correcting Code Properties

Below the critical measurement rate $p<p_{c}$, the mixed phase is underpinned
by a dynamically-generated quantum error-correcting code [11, 8, 9, 17]. The
improvement in the critical measurement rates
$p_{c,\mathrm{Haar}},p_{cp},p_{ce}$ as a function of $k$ observed in the
previous sections suggests that circuits with highly nonlocal interactions
generate improved quantum error-correcting codes in the mixed phase. An
important characteristic of any QECC is its _code distance_ , which is the
smallest number of single-qubit errors required to transform any code state
into any other – equivalently, the code distance is the minimal weight of all
nontrivial logical operators. Here we estimate the contiguous code distance
$d_{\mathrm{code}}$ for our nonlocal Clifford circuits, and show that the
improved robustness to local measurements observed in the previous sections
also generates QECCs with improved code distance.

To characterize the QECC in the mixed phase, we maximally entangle the system
$Q$ with a reference system $R$ and study the Renyi entropy $S^{(2)}_{R}$ of
the reference as well as the mutual information $I(A,R)$ between the reference
and a subset $A\subset Q$ of the output qubits as shown in Fig. 5a. In the
language of quantum error-correction, the Renyi entropy determines the _code
rate_ $r_{\mathrm{code}}=S^{(2)}_{R}/N\ln 2$ of the $\mathrm{PWR}2_{k}$
Clifford circuit, or the number of logical qubits that are encoded within the
$N$-qubit system $Q$. We plot $r_{\mathrm{code}}$ as a function of $p$ in Fig.
5b for $N=256$ and find that the code rate substantially improves with
nonlocality $k=1,\ldots,8$ at any fixed measurement rate $p$, consistent with
previous results. From Fig. 5b it is clear that for any fixed $k$, our choice
of measurement rate $p<p_{c}$ below the critical point determines the code
rate $r_{\mathrm{code}}$ of the underlying QECC.

We can also study the contiguous code distance $d_{\mathrm{code}}$ of the QECC
in the mixed phase by analyzing the mutual information $I(A,R)$ between the
reference $R$ and subregions $A\subset Q$ of the system as shown in Fig. 5a
[17]. For sufficiently small subregions $A$ in the mixed phase $p<p_{c}$, one
generically finds vanishing mutual information $I(A,R)=0$, indicating that the
region $A$ contains no information about the reference $R$. In this
circumstance we may safely discard any of the qubits in $A$ and still reliably
recover the quantum information shared between the reference and the system.
In the language of error correction, we can view the subregion $A$ as a set of
qubits that has possibly been corrupted by errors. So long as $I(A,R)=0$ we
may simply discard these corrupted qubits and still recover the information
contained in $R$. On the other hand, sufficiently large subregions $A$ will
ultimately yield $I(A,R)>0$, implying that sufficiently large errors can
degrade the correlations between the system and reference. The crossover point
$\left|A\right|^{*}$ at which the mutual information becomes nonzero provides
an estimate of the system’s code distance $d_{\mathrm{code}}$ [17]. We review
these arguments in more technical detail in Appendix E [55].

Figure 5:  Code rate and code distance for the $\mathrm{PWR}2_{k}$ Clifford
circuit family at time $t=8N$. (a) To determine
$r_{\mathrm{code}},d_{\mathrm{code}}$, we examine the entropy $S_{R}^{(2)}$ of
a maximally-entangled reference $R$ and the mutual information $I(A,R)$
between the reference and a subregion $A\subset Q$ of the output qubits. (b)
The code rate in the $\mathrm{PWR}2_{k}$ circuit for $N=256$ as a function of
measurement rate $p$ improves significantly with $k$ (green through red). (c)
Normalized mutual information as a function of subregion size for $p=0.12$,
deep in the mixed phase. The effective contiguous code distance
$d_{\mathrm{code}}$ is determined by finding the minimum size of the linear
bipartition $|A|$ to have the mutual information $I(A,\mathrm{Ref.})$ of $\ln
2$. (d) Code distance versus $k$ at fixed code rates
$r_{\mathrm{code}}=0.1,0.2,0.3$ (black, red, orange) for the system size
$N=256$. Error bars are shown or are smaller than data points; lines are
guides to the eye.

To extract an estimate of the contiguous code distance $d_{\mathrm{code}}$, we
plot the mutual information $I(A,R)$ as a function of the subregion size
$\left|A\right|/N$ and look for the crossover point $\left|A\right|^{*}$ where
the mutual information increase by one bit $\Delta I(A^{*},R)=\ln 2$ (Fig.
5c). The effective code distance is given by
$d_{\mathrm{code}}=\langle\left|A\right|^{*}\rangle$ where
$\langle\dots\rangle$ is an average over different realizations of projective
measurements in the circuit. We plot the resulting effective contiguous code
distance as a function of $k$ in Fig. 5d, which shows a striking improvement
of $d_{\mathrm{code}}$ with increasingly nonlocal interactions $k$. Moreover,
by tuning the measurement rate $p<p_{c}$ and the nonlocality parameter $k$ we
can obtain quantum error-correcting codes with a variety of code rates and
code distances. For fixed $k$ we find the expected tradeoff between code rate
and code distance that is typical in quantum error-correcting codes. By
increasing the nonlocality parameter $k$ we generate codes with significantly
improved code distance $d_{\mathrm{code}}$ for any fixed code rate
$r_{\mathrm{code}}$. In this sense, the nonlocal interactions for
$k\sim\log_{2}N$ substantially improve the quantum error-correcting code
properties in the mixed phase.

##  VII Fully Nonlocal and All-to-All Models

So far, we have considered $\mathrm{PWR}2_{k}$ circuits at fixed $k<\log_{2}N$
where the longest-range interactions $d_{\mathrm{max}}=2^{k-1}$ have been
strictly smaller than the system size $N$. In these cases we found that
nonlocal interactions could substantially improve both the critical
measurement rate $p_{c}$ and the code distance $d_{\mathrm{code}}$ in the
mixed phase. We now consider what happens in the ‘complete’
$\mathrm{PWR}2_{k}$ circuit with $k=\log_{2}N$, where the longest-range
interactions $d_{\mathrm{max}}=N/2$ are extensive, and show that the behavior
radically changes relative to the cases previously studied. In particular, the
complete $\mathrm{PWR}2_{k}$ circuit without measurements is known to be a
fast scrambler capable of generating system-wide volume-law entanglement after
only $t_{*}\propto\log N$ interaction layers [48]. In this section we
demonstrate that the fast scrambling dynamics in this circuit leads to
markedly improved code properties, including a nearly extensive contiguous
code distance $d_{\mathrm{code}}$. We also observe many similarities in this
section between the complete $\mathrm{PWR}2_{k}$ circuit and a maximally
nonlocal random AA model, another known fast scrambler which has no spatial
geometry whatsoever. These similarities highlight the central role played by
fast scrambling in determining the physics of the mixed phase in these
circuits, and suggest that fast scrambling circuits may be governed by the
same universal physics near the measurement-induced critical point.

Figure 6:  Maximally nonlocal ‘complete’ $\mathrm{PWR}2_{k}$ circuits with
$k=\log_{2}N$ and random AA circuits. (a) Crossover points
$p_{\mathrm{peak,Haar}}$ as a function of system size $N$ for the complete
Haar-random $\mathrm{PWR}2_{k}$ circuit (purple) and for the random AA model
(light blue). (b) The maximum cluster size $C_{\max}$ at the peak
$p_{\mathrm{peak,Haar}}$ grows as a power law $N^{\mu}$ with system size for
the complete $\mathrm{PWR}2_{k}$ circuit (purple) and the AA circuit (light
blue), with exponents $d_{f}=1.809\pm 0.005$ and $d_{f}=1.786\pm 0.005$
respectively. Black dashed line shows the fractal dimension, $d_{f}=91/48$, of
the cluster at the critical point of two-dimensional system. (c) The code
distance for the complete $\mathrm{PWR}2_{k}$ Clifford circuit (top) and the
random AA Clifford circuit (bottom) for code rates
$r_{\mathrm{code}}=0.1,0.2,0.3$ (black, red, orange). In both cases, the code
distance is nearly extensive, scaling like $d_{\mathrm{code}}\propto
N^{\beta}$ with $\beta=0.96\pm 0.04$ ($\mathrm{PWR}2$) and $\beta=0.97\pm
0.02$ (AA). (d) Scaling collapse of the normalized single-qubit purification
time $\tilde{\tau}(p)$ in the complete $\mathrm{PWR}2_{k}$ Clifford circuit
(top) and the random AA Clifford circuit (bottom). In both cases the data
exhibits strong collapse according to the nonstandard scaling form in Eq. (7).

We first study the complete $\mathrm{PWR}2_{k}$ circuit with Haar-random
gates, and numerically simulate the corresponding classical percolation
network similar to section III. In that prior analysis we computed the Binder
cumulant $b(p)$ of the network and estimated the critical point by fixing the
parameter $k$ and performing a finite-size scaling analysis in the system size
$N$. In the present case, finite-size scaling is difficult to define
consistently because the connectivity $k=\log_{2}N$ of the graph is coupled to
the system size $N$. Instead, we estimate a finite-size crossover point by
computing the susceptibility

$\chi(p)=\langle C^{2}_{\max}(p)\rangle-\langle C_{\max}(p)\rangle^{2}$ (6)

of the resulting classical percolation network. In the thermodynamic limit
$N\rightarrow\infty$ the susceptibility diverges at the critical point; here
we estimate the finite-size transition point by locating the peak
$p_{\mathrm{peak,Haar}}$ of the susceptibility $\chi(p)$ for each $N$.

The resulting finite-size crossover points are plotted in Fig. 6a, where we
compare to the crossover points for a random AA model analyzed using the same
methods. The crossover points noticeably increase with $N$ in both cases, and
should therefore be considered as finite-size crossover points and not bona
fide critical points. We also plot the value of the maximum cluster size at
the peak $C_{\max}(p_{\mathrm{peak,Haar}})$ in Fig. 6b, and find that it
increases as a power law with system size
$C_{\max}(p_{\mathrm{peak,Haar}})\propto N^{\mu}$ with fractal dimension
$d_{f}=1.809\pm 0.005$ for the complete $\mathrm{PWR}2_{k}$ circuit and
$1.786\pm 0.005$ for the random AA circuit. The similarity in the fractal
dimension $d_{f}$ between these two circuits suggests that they may be
governed by the same universal physics near the critical point. Moreover, the
fractal dimension in both cases differs from the analytical prediction
$d_{f}=91/48\approx 1.896$ obtained from the two-dimensional percolation
universality class. This suggests that there is an abrupt change in the
universality class from the $k<\log_{2}N$ circuits to the complete
$\mathrm{PWR}2_{k}$ circuit.

Next, we turn our attention to circuits composed of Clifford circuits with
two-qubit gates $Q_{ij}=\mathrm{CZ}_{ij}H_{i}H_{j}$ and study the quantum
error-correcting codes that support the mixed phase. Again, we find striking
similarities with the random AA circuit, suggesting that these models may be
governed by the same universal physics. Using the methods discussed in Sec.
VI, we extract the code distance $d_{\mathrm{code}}$ of the QECC in the mixed
phase at fixed code rate $r_{\mathrm{code}}$. We plot the results in Fig. 6c
for code rates $r_{\mathrm{code}}=0.1,0.2,0.3$. Linear fits indicate a nearly
extensive code distance, where $d_{\mathrm{code}}\propto N^{\beta}$ with
$\beta=0.98\pm 0.05$ for the $\mathrm{PWR}2$ circuit and $0.96\pm 0.01$ for
the random AA circuit. The fast scrambling dynamics common to both of these
models apparently generates similar quantum error-correcting codes with
excellent properties deep in the mixed phase.

Finally, we study the single-qubit purification time $\tau$ in the complete
$\mathrm{PWR}2_{k}$ circuit with $k=\log_{2}N$. Instead of the conventional
scaling law (5) near the critical point, here we empirically find strong
scaling collapse of $\tau$ only after normalizing by the number of interaction
layers $k=\log_{2}N$ within each even (or odd) block. This leads to a
nonstandard scaling law

$\tau/\log_{2}N=\tilde{\tau}(p)=N^{\tilde{z}}\tilde{f}\left((p-\tilde{p}_{c})N^{1/\tilde{\nu}}\right),$
(7)

which yields strong data collapse for both the complete $\mathrm{PWR}2$ model
and the random AA model as shown in Fig. 6d. This suggests that the critical
point in the fast-scrambling limit may be governed by a logarithmic scaling
law (7) as opposed to the conventional scaling (5) that governs the phase
transition in general short- and long-range models [26, 62]. The origin of
this scaling behaviour is an interesting topic for further investigation.

##  VIII Discussion and Outlook

In this paper we studied the role of sparse nonlocal interactions in
protecting quantum information against the destructive effects of local
measurements. We characterized the protection afforded by these nonlocal
interactions by numerically studying measurement-induced phase transitions in
a class of sparse, nonlocal $\mathrm{PWR}2_{k}$ circuits with tunable
nonlocality $k$. These studies demonstrated that non-universal properties of
the transition, such as the critical points
$p_{c,\mathrm{Haar}},p_{ce},p_{cp}$, can be significantly improved by the
nonlocality of the interaction graph. At the same time, we found that the
universal properties of the transition such as the critical exponent $\nu$ are
largely unaffected by the presence of these nonlocal interactions so long as
$k<\log_{2}N$. In the language of the renormalization group, it appears that
sparse nonlocal interactions are an irrelevant perturbation to the 1+1D fixed
point. It would be interesting to make this notion of irrelevancy more precise
within a renormalization group framework.

While the phase transition for fixed $k<\log_{2}N$ appears to be governed by
the same universality class as the 1+1D model, we find that the properties of
the system deep in the mixed phase can change considerably in the presence of
nonlocal interactions. In particular, we have demonstrated that the code
distance $d_{\mathrm{code}}$ (at fixed code rate $r_{\mathrm{code}}$) can be
significantly improved by the presence of sparse, nonlocal interactions. By
varying the parameter $k$, we can thus tune between the code properties of a
NN circuit and the improved code properties of an AA circuit.

We also studied ‘complete’ $\mathrm{PWR}2_{k}$ circuits with $k=\log_{2}N$ and
observed that these circuits are closely comparable to known fast scramblers
such as the random AA circuit. Although a proper finite-size scaling analysis
is not available for these circuits, we numerically studied finite-size
crossover points and showed that the critical properties of the complete
$\mathrm{PWR}2_{k}$ circuit are markedly similar to those of a random AA
circuit. We also studied the code properties of the complete
$\mathrm{PWR}2_{k}$ and AA Clifford circuits, and found codes with nearly
extensive code distance $d_{\mathrm{code}}\propto N^{\beta}$ with
$\beta\approx 1$ in both cases. Finally, we observed a non-standard scaling
collapse in the single-qubit purification time $\tau$ for both models, another
signal that the critical properties of these models may be described by the
same universality class. In future work we hope to make these connections more
explicit.

One of the most exciting prospects for engineering fast scrambling circuits in
the lab is the ability to generate good quantum error-correcting codes using
the fewest gates possible. Here we have demonstrated that deterministic fast
scrambling circuits with local projective measurements are capable of
generating good quantum error-correcting codes, and that the properties of
these codes can be easily tuned. It is natural to ask whether one could
reasonably use these monitored circuits to generate useful codes in near-term
experiments. One outstanding problem from an experimental perspective,
however, is the post-selection problem. Because quantum projective
measurements are inherently probabilistic, one can only force a particular
outcome by repeating the protocol ad infinitum until the correct measurement
record is obtained. This necessarily requires the experiment typically be
repeated an exponentially large number of times, which is impractical. In the
case of Clifford circuits it is known in principle how to apply feedback to
effectively correct for undesired measurement outcomes [36]. It would be
interesting to apply these ideas to our fast scrambling Clifford circuits to
understand whether single-shot experiments with suitable feedback can be used
to generated useful quantum error-correcting codes in near-term experiments
with cold neutral atoms or trapped ions. The data for this manuscript is
available in open access at [63].

###### Acknowledgements.

We thank Hans Peter Büchler and Nicolai Lang for helpful discussions. GSB is
supported by the DOE GeoFlow program (DE-SC0019380). Work at the University of
Strathclyde was supported by the EPSRC Programme Grant DesOEQ (EP/P009565/1),
the EPSRC Quantum Technologies Hub for Quantum Computing and simulation
(EP/T001062/1), the European Union’s Horizon 2020 research and innovation
program under grant agreement No. 817482 PASQuanS, and AFOSR grant number
FA9550-18-1-0064.

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##  Appendix A Haar-Random Circuits and Bond Percolation

###  A.1 Percolation Mapping

Figure 7:  (a) Haar-random two-site gates typically generate nearly maximal
entanglement. In this case the zeroth Renyi entropy $S_{0}$ of site $i=0$ is
proportional to the minimum number of cuts to required to isolate it,
$S_{0}/\ln 2=1$. (b) Haar-random gates applied between sites $i,j=1,2$ and
sites $i,j=0,3$. In this case $S_{0}$ takes a value of $S_{0}/\ln 2=2$ for the
subsystem consisting of sites $i=0,1$.

The zeroth order Renyi entropy is $S_{0}=\log M$, where $M$ is the rank of the
reduced density matrix $\rho_{A}$ of the subsystem $A$. When an independent
qubit is entangled with $A$, the rank of $\rho_{A}$ increases by a factor of
$2$. A projective measurement, on the other hand, isolates the measured qubit
from the rest of the system, typically undoing the effect of the entangling
gates applied previously to the measurement. In the tensor network picture,
the calculation of $S_{0}/\ln 2$ maps to a classical minimal cut problem. In
this picture the zeroth Renyi entropy is given by the minimum number of legs
one must cut in order to isolate the qubits in $A$.

By considering the two extreme limits of the probability of the projective
measurement $p$, one finds that in the limit of $p=0$ the initial state and
final state is guaranteed to be connected, and in the limit of $p=1$, there is
no connection between them. Between these two limits there must be a critical
probability $p=p_{c,\mathrm{Haar}}$ which separates the two phases. For
$p<p_{c,\mathrm{Haar}}$, there always exist a path between qubits in the
initial state and the final state, and for $p_{c,\mathrm{Haar}}<p$, they are
independent.

If two qubit gates are taken from the Haar-random distribution and applied to
a pair of independent qubits, then two qubit gates are almost always
guaranteed to be entangled. In this case, the circuit can be drawn as a tensor
network diagram consists of entangling tensors (Haar-random gates) which act
on the pairs of qubits accordingly to the interaction graph, with the legs
being cut with probably $p$ due to the projective measurements. This is
equivalent to the bond-percolation with the bond occupation probability
$q=1-p$. The critical point, $p_{c,\mathrm{Haar}}$, therefore, can be
determined by solving the classical bond percolation problem of the mapped
network.

###  A.2 Numerical Method for Simulating Bond Percolation

The Newman-Ziff algorithm [58] is used for simulating the critical behaviour
of bond percolation. This algorithm calculates the various observables at
different values of the bond occupation probability $q=1-p$ in a computational
time which scales only linearly with the number of bonds $M$ in the network.

The algorithm proceeds by generating random configurations of a given network
with $m=1,2,\dots,M$ bonds being occupied; where this is done by adding one
random bond at each step starting from an empty network. At each step of this
process, one can keep track of the sizes of the connected components
(clusters) in the network with the Union-Find algorithm [64, 65]. Observables
such as cluster sizes and their moments can then be stored at the end of each
step of the algorithm. The observable $Q$ as a function of the bond occupation
probability $q$ can then be calculated by taking the microcanonical ensemble
of the different configurations:

$\displaystyle Q(p)=\sum_{m}B(m,M,q)Q_{m}=\sum_{m}\begin{pmatrix}M\\\
m\end{pmatrix}q^{m}q^{M-m}Q_{m}$ (8)

where $Q_{m}$ is the value of observable $Q$ with $m$ occupied bonds and
$\begin{pmatrix}M\\\ m\end{pmatrix}=\frac{M!}{m!(M-m)!}$ is the usual binomial
coefficient.

The calculation of the binomial distribution $B(m,N,p)$ is numerically
unstable because one requires evaluations of the factorials of large numbers
and addition of numbers which differ largely in the order of the magnitudes.
Here we adopt a more stable method for evaluating the microcanonical ensemble
introduced by Newman and Ziff [58]. If we normalize the binomial distribution
by its peak value $m_{\max}=pM$, this normalized binomial distribution
$\tilde{B}(m,M,p)$ is defined recursively as

$\displaystyle\tilde{B}(m,M,q)=\begin{cases}\tilde{B}(m-1,M,q)\frac{M-m+1}{m}\frac{q}{1-q}&(m>m_{\max})\\\
\tilde{B}(m+1,M,q)\frac{m+1}{M-m}\frac{1-q}{q}&(m<m_{\max})\end{cases}$ (9)

Therefore the observable of interest can be calculating the following

$\displaystyle
Q(q)=\frac{\sum_{m^{\prime}}\tilde{B}\left(m^{\prime},M,q\right)Q_{m}}{\sum_{m^{\prime}}\tilde{B}\left(m^{\prime},M,q\right)}.$
(10)

###  A.3 Renormalization Group Solution for full PWR2

Figure 8:  Block-decimation procedure for the $k=2$ $\mathrm{PWR}2_{k}$
circuit. The entire structure on the left is reduced down to a single ribbon-
like structure on the right via the renormalization transformation.

The PWR2 circuit with size $N_{0}$ can be constructed from two sub-systems of
size $N_{0}/2$. This can be done by interleaving the degrees of freedom of the
two sub-systems and coupling them together with nearest neighbour
interactions.

This self-similar structure allows us to define a renormalization
transformation. This transformation consists of two steps: First, a large
block as depicted in Fig. 8 (left), is renormalized to a ribbon-like structure
in Fig. 8 (left). Here the renormalized is done by collapsing the bonds with
different colors in Fig. 8 (left) to the bonds of the corresponding colors in
Fig. 8 (right). This step reduces the size of the vertical dimension by half.
In order to reduce the size of the horizontal direction, two of the ribbon-
like structure is merged into one. After these steps, the size of the network
of $N/2$ by $T$ is reduced to $N/4$ by $T/2$, where $N$ is the number of
qubits and $T$ is the number of layers in the original circuit.

Let $q=1-p$ be the probability of a bond being present in the network. The
renormalized probability $R(q)$ in terms of the probability which a ribbon
structure (Fig. 8, right) is spanned ($Q_{\mathrm{ribbon}}$) is

$\displaystyle R(q)$
$\displaystyle=Q_{\mathrm{ribbon}}^{2}+2Q_{\mathrm{ribbon}}(1-Q_{\mathrm{ribbon}}).$
(11)

Now we define the condition of the ribbon to be spanned when there exists at
least one bond coming out of each the nodes. In terms of the probability of at
least a bond being spanned $q_{\mathrm{bond}}$, $Q_{\mathrm{ribbon}}$ is

$\displaystyle Q_{\mathrm{ribbon}}$
$\displaystyle=q_{\mathrm{bond}}^{4}+4q_{\mathrm{bond}}^{3}(1-q_{\mathrm{bond}})+2q_{\mathrm{bond}}^{2}(1-q_{\mathrm{bond}})^{2}.$
(12)

Finally, the probability of the bond being spanned, $q_{\mathrm{bond}}$, as a
function of $q$ is

$\displaystyle q^{\prime}$ $\displaystyle=q^{4}+4q^{3}(1-q)+4q^{2}(1-q)^{2}.$
(13)

The equation $R(q^{*})=q^{*}$ has a non-trivial fixed point solution in the
real domain of $0<q^{*}<1$, which is $q^{*}\approx 0.335\dots$ or equivalently
$p^{*}\approx 0.665\dots$.

##  Appendix B Clifford Circuits

###  B.1 Stabilizer States

The stabilizer formalism is a powerful tool for understanding a special class
of many-body quantum states and quantum error-correcting codes whose dynamics
can be efficiently simulated on a classical computer [53, 54]. Consider the
Pauli group $\mathcal{P}(Q)$ of all Pauli strings acting on a set of qubits
$Q$. We define an Abelian _stabilizer_ subgroup
$\mathcal{S}\leq\mathcal{P}(Q)$ generated by a set
$\mathcal{G}=\\{g_{1},g_{2},\ldots,g_{m}\\}$ of linearly-independent,
mutually-commuting Pauli strings $[g_{\ell},g_{\ell^{\prime}}]=0\
\forall\ell,\ell^{\prime}$. A stabilizer group with $m\leq N=\left|Q\right|$
independent generators has order $\left|\mathcal{S}\right|=2^{m}$. Given a
stabilizer group $\mathcal{S}$ we define a _stabilizer state_ (also called a
_code state_)

$\rho_{Q}(\mathcal{S})=2^{-\left|Q\right|}\sum_{g\in\mathcal{S}}g$ (14)

which is the unique density matrix stabilized by all elements
$g\in\mathcal{S}$. If we specify a ‘complete’ set of $m=N$ stabilizers, then
(14) is a rank-1 projector onto the unique pure state
$\left|\Sigma\right\rangle$ stabilized by the group $\mathcal{S}$ (i.e.
$g\left|\Sigma\right\rangle=+\left|\Sigma\right\rangle$ for all
$g\in\mathcal{S}$).

Quantum error correction is particularly easy to understand in the stabilizer
formalism. Suppose we prepare a code state (14) defined by a stabilizer group
$\mathcal{S}$, and consider disturbing it with an error operator
$e\in\mathcal{P}(Q)$. There are three possibilities: first, if
$e\in\mathcal{S}$ then the error is _trivial_ because it leaves the state (14)
unchanged. Second, if $e\notin\mathcal{S}$ fails to commute with one or more
stabilizers $g$ then this is a _detectable_ error because we can detect (and
subsequently correct) the error by measuring the stabilizer $g$. The final
possibility is if $e\notin\mathcal{S}$ but $e\in\mathcal{C}(\mathcal{S})$ is
in the centralizer of $\mathcal{S}$. These errors are _undetectable_ because
they modify the quantum state but cannot be detected by measuring any of the
stabilizers $g$. The collection of all undetectable errors gives the logical
operator group $e\in\mathcal{L}\equiv\mathcal{C}(\mathcal{S})/\mathcal{S}$.

###  B.2 Classical Simulation of Clifford Circuits

Clifford group on $N$ qubits, $\mathcal{C}^{N}$, is a group formed by unitary
operators which transforms the elements of $N$ qubit Pauli group to other
elements in the same group. The action of those unitaries on a stabilizer of
$\mathcal{O}_{\left|\Sigma\right\rangle}$ therefore transforms the stabilizer
to other stabilizer $\mathcal{O}_{\left|\Sigma^{\prime}\right\rangle}$. This
transformation is equivalent to the transformation of state
$\left|\Sigma\right\rangle$ to $\left|\Sigma^{\prime}\right\rangle$.
Therefore, the evolution of a stabilizer state under the unitaries from the
Clifford group can be tracked by keeping track of how the initial stabilizer
transforms.

The evolution with unitaries from the Clifford group is proven to be able to
be computed classically in polynomial time by Gottesman and Knill [53, 54].
This is done with a binary matrix $M$ of size $N$ by $2N$. In this
representation, Pauli strings in the stabilizer are mapped to each rows of $M$
and the actions of Clifford gates are mapped as series of logical
operations[54].

In this simulation a Pauli strings are mapped to a row in the following
fashion. A Pauli operators which corresponds to the $l$th qubit is encoded by
mapping the number of Pauli-$X$ to $l$th column and the number of Pauli-$Z$ to
$N+l$th column. If $l$th operator is Pauli $Y$, then because of the identity,
$Y=iXZ$, the values of the $l$th and $N+l$th columns is registered as $1,1$.
If $l$th operator is an identity, both of the values are registered as $0$.

To keep track of the evolution by the unitaries in the Clifford group, it is
useful to identify the transformation rule of the binary matrix for the
generators of the Clifford group. The generator of $N$-qubit Clifford group
are Hadamard ($H$), Phase ($P$) and Controlled-NOT (C-NOT) gates [53, 54, 66].
Hadamard gate acting on a site, $l$, maps operators $Z_{l}$ to $X_{l}$ and
$X_{l}$ to $Z_{l}$ while keeping $Y_{l}$ to be $Y_{l}$. This is equivalent to
swapping the columns $l$ and $N+l$. Phase gate acting on a site, $l$, on the
other hand, maps $X_{l}$ to $Y_{l}$ and $Y_{l}$ to $X_{l}$ while keeping
$Z_{l}$ unchanged. This is equivalent to setting the column $N+l$ as the
modulo $2$ sum of columns $l$ and $N+l$. Let qubit $l$ be a control and qubit
$m$ be the target, then the Controlled-NOT gate acting on those qubits flips
the target basis whenever the state of the $l$ is $\left|1\right\rangle$. By
transforming all the elements in the two-qubit Pauli group, one finds that
there are only four non-trivial rules, which are : $X_{l}I_{m}$
to$X_{l}X_{m}$, $I_{l}X_{m}$ to $I_{l}X_{m}$, $Z_{l}I_{m}$ to $Z_{l}I_{m}$,
and $I_{l}Z_{m}$ to $Z_{l}Z_{m}$. Since $H$, $P$, and C-NOT gates are the
generators of the Clifford group, one may systematically generate all the
operators in $n$-qubit group by following the algorithms such as the one
proposed by Selinger [66].

###  B.3 Entanglement Entropy

The stabilizer is equivalently the density matrix of the stabilizer state
because the stabilizer is a projector which projects the state onto the
stabilizer state. With the density matrix $\rho_{Q}(\mathcal{S})$,
entanglement entropy of a subregion $A$ can be calculated [67, 34].

The reduced density matrix on the subregion $A$, $\rho_{A}$ ($A\in Q$) is
obtained by tracing out $\bar{A}$. This is equivalent to tracing out the Pauli
operators which belong to $\bar{A}$. However, Pauli operators are traceless,
therefore

$\displaystyle\rho_{A}=\mathrm{Tr}_{\bar{A}}\\{\rho_{Q}\\}=\frac{2^{|\bar{A}|}}{2^{Q}}\sum_{g_{A}\in\mathcal{G}_{A}}g_{A}=\frac{1}{2^{|A|}}\sum_{g_{A}\in\mathcal{G}_{A}}g_{A},$
(15)

where $\mathcal{G}_{A}\in\mathcal{G}$ is a set of all $g$ where trace over
$\bar{A}$ is non-zero. [67]. Let $N_{A}$ be the number of linearly independent
Pauli strings that generate $g_{A}$, then
$\sum_{g_{A}\in\mathcal{G}_{A}}g_{A}$ is also a projector of rank
$2^{|A|-N_{A}}$ because this projects out the $-1$ eigenstates of its
generators. The von Neumann or Renyi entropy $S_{A}$ is

$\displaystyle S_{A}=\left(|A|-N_{A}\right)\ln 2$ (16)

From the identities $2^{N_{A}+N_{\bar{A}}}=2^{N}$ and
$N_{\bar{A}}=\mathrm{rank}_{\mathrm{binary}}M_{\bar{A}}$, where $M$ is the
binary matrix of the stabilizer state, $\mathrm{rank}_{\mathrm{binary}}$ is
the binary rank, and $M_{\bar{A}}$ is the matrix which its columns corresponds
to the Pauli operators of region $\bar{A}$ in $M$,

$\displaystyle
S_{A}=\left(\mathrm{rank}_{\mathrm{binary}}(M_{\bar{A}})-|\bar{A}|\right)\ln
2.$ (17)

Using $S_{A}=S_{\bar{A}}$, we obtain

$\displaystyle
S_{A}=\left(\mathrm{rank}_{\mathrm{binary}}(M_{A})-|A|\right)\ln 2.$ (18)

##  Appendix C Finite-Size Scaling

###  C.1 Finite Size Scaling of the Percolation Critical Point for
$k=1,\dots,6$

Figure 9:  (a) The binder cumulant $b(p)$ of the percolation network of
$\mathrm{PWR}2_{k}$ for $k=1,2,\d{t}s6$ for the system sizes
$N=2^{8},2^{9},2^{10},2^{11}$ (light to dark blue). The solid vertical lines
are the estimated values of the $p_{c,\mathrm{Haar}}$ and the shaded regions
are the corresponding $1$-sigma errors. (b) Collapsed Binder cumulant for
$k=1,2,\dots 6$ for the system sizes $N=2^{8},2^{9},2^{10},2^{11}$ (light to
dark blue).

Finite-size scaling analysis for the percolation transition of the
$\mathrm{PWR}2_{k}$ circuit for $k=1,3,5$ is presented in Fig. 2 b. Here we
show the finite-size scaling and the crossings of the binder cumulants for all
values of $k=1,2,\dots,6$ for the system sizes $N=2^{8},2^{9},2^{10},2^{11}$.

The error bars are estimated by taking the standard error of the 4000
realizations of the Newman-Ziff algorithm (Appendix A). The critical points
and their errors are estimated by computing the average intersection points of
the curves with 5000 different realizations of the fluctuations added to each
of the points. Fluctuations are drawn from the appropriate distribution with
the standard error as the standard deviation at each point (Fig. 9 a). The
critical exponents and their errors are also estimated by collapsing the 5000
different realizations of the fluctuations added from the same distribution
(Fig. 9 b).

###  C.2 Finite Size Scaling of the Entanglement Critical Point for
$k=1,\dots,6$

Figure 10:  (a) The tripartite mutual information $I(A;B;C)$ of initial
$z$-polarized state evolved under a deterministic Clifford $\mathrm{PWR}2_{k}$
circuit until $t=8N$ for $k=1,2,\dots 6$ for the system sizes
$N=2^{6},\dots,2^{11}$ (light to dark blue). The solid vertical lines are the
estimated value of the $p_{ce}$ and the shaded region is the corresponding
$1$-sigma errors. (b) Collapsed tripartite mutual information for $k=1,2,\dots
6$ for the system sizes $N=2^{6},\dots,2^{11}$ (light to dark blue).

The finite-size scaling analysis of the entanglement criticality of
$\mathrm{PWR}2_{k}$ circuit for $k=1,3,5$ are presented in Fig. 3 b. Here we
show the finite-size scaling and the crossings of the tripartite mutual
information $I(A;B;C)$ for all the values of $k=1,2,\dots,6$ for system sizes
$N=2^{6},\dots,2^{11}$ (the system sizes that are smaller than $2^{k+2}$ are
not used in the analysis).

The error bars are estimated by taking the standard error of up to 1000
realizations of the random projective measurement of the evolution to $t=8N$.
The critical points and their errors are calculated by computing the average
intersection points of the curves with 5000 different realizations of the
fluctuations added to each of the points that are drawn from the appropriate
distribution with the standard error as the standard deviation at the point
(Fig. 10 a). The critical exponents and their errors are also estimated by
collapsing the 5000 different realizations of the fluctuations added to the
data points from the same distribution (Fig. 10 b).

###  C.3 Finite Size Scaling of the Purification Critical Point for
$k=1,\dots,6$

Figure 11:  (a) The single-qubit purification time for $k=1,2,\dots 6$ for the
system sizes $N=2^{6},\dots,2^{11}$ (light to dark blue). The initial stat is
prepared such that there is a qubit entangled to one of the $N$ qubits
(system), then the system qubits are scrambled with nearest neighbour random
Clifford circuit up to $t=4N$. The solid vertical lines are the estimated
value of the $p_{cp}$ and the shaded region is the corresponding $1$-sigma
errors. (b) Collapsed tripartite mutual information for $k=1,2,\dots 6$ for
the system sizes $N=2^{6},2^{11}$ (light to dark blue).

The finite-size scaling analysis of the purification criticality of
$\mathrm{PWR}2_{k}$ circuit for $k=1,3,5$ are presented in Fig. 4 b. Here we
show the finite-size scaling and the crossings of the single-qubit
purification time for all the values of $k=1,2,\dots,6$ for system sizes
$N=2^{6},\dots,2^{11}$ (the system sizes that are smaller than $2^{k+2}$ are
not used in the analysis).

The error bars are estimated by taking the standard error of up to 1000
realizations of the random projective measurement of the evolution until the
state is purified. The critical points and their errors are estimated by
computing the average intersection points of the curves with 5000 different
realization of the fluctuations added to each of the points that are drawn
from the appropriate distribution with standard error as its standard
deviation at each points (Fig. 11 a). The critical exponents and their errors
are also estimated by collapsing the 5000 different realizations of the
fluctuations added to the data points from the same distribution (Fig. 11 b).

Figure 12:  (a) Susceptibility, $\chi(p)$, of the complete $\mathrm{PWR}2_{k}$
(left) and AA (right) percolation networks for the system sizes
$N=2^{6},2^{7},\dots,2^{10}$. Although there exists sharp peaks, the positions
of the peaks depends significantly on the system size. The size of the
circuits in this calculations is taken to be $N\times T$. The observables of
the bond percolation of the networks are calculated with Newman-Ziff algorithm
by taking the microcanonical ensemble of over 4000 trajectories. (b)
$\tilde{\tau}/N^{\tilde{z}}$ as a function of $p$ without the scaling collapse
for complete $\mathrm{PWR}2_{k}$ (left) and AA (right) circuits for the system
sizes $N=2^{6},2^{7},\dots,2^{10}$ with $\tilde{z}=0.21\pm 0.01$
($\mathrm{PWR}2$) and $\tilde{z}=0.170\pm 0.006$ (AA). The critical point
determined from the scaling collapse is marked by the black line and the
1-sigma error is marked by the grey shade. The critical points and their error
are $\tilde{p}_{c}=0.267\pm 0.001$ and $\tilde{p}_{c}=0.312\pm 0.001$
respectively.

###  C.4 Finite Size Scaling of the Full $\mathrm{PWR}2$ and AA circuits

The finite-size scaling of the MIPTs of the complete $\mathrm{PWR}2_{k}$
circuit is difficult due to the long-range interactions which give rise to
strong boundary effects and the loss of locality. In the percolation
transition, this appears as the strong system size dependence of the
$p_{\mathrm{peak},\mathrm{Haar}}$ as shown in the figures Fig. 6 a. The
positions of the peaks are determined from the $\chi(p)$ plotted in Fig. 12 a.

The loss of locality makes the determination of the entanglement transition
impossible. Due to the coupling of distance $N/2$, even in the regime near
$p=1$, the entanglement entropy of the region of size $A<N/2$ is almost
guaranteed to be $\sim A(1-p)$ at the end of the application of $2\log_{2}N-1$
layers of the gates. Therefore we do not argue on the existence of the
entanglement transition in the complete $\mathrm{PWR}2_{k}$ and AA circuits.

The percolation transition suffers from the similar boundary problem. However,
the transition does not ask the geometry of the circuit as it only asks the
entropy of the reference qubits that are entangled to the system at $t=0$. The
problem in the determination of the critical properties of the purification
transition is the number of layers of the even (or odd) blocks which depends
on the system size. In this paper, we proposed the potential workaround, which
is normalizing the time $t$ by the number of layers in the even (or odd)
blocks by defining $\tilde{t}=t/\log_{2}N$. This lead to the empirical scaling
law of the form in Eq. (7).

As shown in Fig. 6 d, the single-qubit purification time $\tau(p)$ collapsed
surprisingly well with Eq. (7) with the $\tilde{p}_{c}=0.27\pm
0.01$,$\tilde{\nu}_{c}=2.08\pm 0.08$, and $\tilde{z}_{c}=0.21\pm 0.01$
($\mathrm{PWR}2$) and $\tilde{p}_{c}=0.31\pm 0.01$,$\tilde{\nu}_{c}=2.29\pm
0.05$, and $\tilde{z}_{c}=0.170\pm 0.006$ (AA). For this simulation,
$\tilde{\tau}(p)$ is estimated by taking the average over up to 500
realizations of the random projective measurements. The errors in the critical
exponents and critical points are estimated by performing the scaling collapse
on the 5000 different realizations of the data set with random fluctuations.
The random fluctuations are taken from the appropriate distribution with
standard deviation of the standard error at each point.

##  Appendix D The Gap Between Entanglement ($p_{ce}$) and Purification
($p_{cp}$) Critical Points of $\mathrm{PWR}2_{k}$

Figure 13:  The difference between entanglement and purification critical
points of $\mathrm{PWR}2_{k}$ circuit are plotted (red) for $k=1,\dots,6$. The
blue line shows the error of entanglement critical points centered at the
expected value. There is no statistically significant deviation observed
between the entanglement and purification critical points.

Whether there exists a gap between the entanglement and purification critical
points is a topic of an ongoing debate. Gullans and Huse[34] show that in
1+1D, they occur at the same point in general. Here we show that such a gap is
not observed in the $\mathrm{PWR}2_{k}$ circuits for the values of
$k=1,\dots,6$. No statistically significant deviations between the
entanglement and the purification critical points are observed (Fig. 13).

##  Appendix E Quantum Error-Correcting Code Properties

###  E.1 Code Distance for Stabilizer Circuits

Here we review an argument of Li and Fisher that connects the code distance
$d_{\mathrm{code}}$ of a stabilizer circuit to the mutual information $I(A:R)$
between a subregion $A\subset Q$ of the output qubits and a maximally-
entangled reference $R$ [17]. In this work we are primarily interested in
_contiguous_ regions $A$, but the arguments in this Appendix hold equally well
for arbitrary, possibly disconnected regions. For any bipartition
$A\cup\overline{A}=Q$ of the system we define the quotient group

$\mathcal{L}_{A}\equiv\frac{\\{g\in\mathcal{C}(\mathcal{S})|\mathrm{proj}_{\overline{A}}(g)\in\mathrm{proj}_{\overline{A}}(\mathcal{S})\\}}{\mathcal{S}}.$
(19)

Here $\mathrm{proj}_{\overline{A}}$ is the group homomorphism from
$\mathcal{P}(Q)$ to $\mathcal{P}(\overline{A})$ defined by

$\mathrm{proj}_{\overline{A}}:g_{A}\otimes g_{\overline{A}}\rightarrow
g_{\overline{A}}$ (20)

which projects any Pauli string $g=g_{A}\otimes g_{\overline{A}}$ down to its
support on the region $\overline{A}$. The group $\mathcal{L}_{A}$ is the group
of undetectable errors that are localizable on $A$. In other words, any $g$ in
this group can be localized to $A$ simply by multiplying by a suitable choice
of $g^{\prime}\in\mathcal{S}$. The order
$\left|\mathcal{L}_{A}\right|=2^{\ell_{A}}$ of this group gives the number of
independent undetectable errors that can be localized to the region $A$. We
must have $\ell_{A}=0$ in order for a stabilizer code to successfully protect
quantum information from errors acting on $A$.

In the following we compute the order $\left|\mathcal{L}_{A}\right|$ of the
group $\mathcal{L}_{A}$ and show that
$\ell_{A}\equiv\log_{2}\left|\mathcal{L}_{A}\right|$ is proportional to the
mutual information $I(A:R)$. Starting from the definition (19), we find that
the order of the group can be written

$\displaystyle\left|\mathcal{L}_{A}\right|=\frac{\left|\mathcal{C}(\mathcal{S})\right|\cdot\left|\mathrm{proj}_{\overline{A}}(\mathcal{S})\right|}{\left|\mathcal{S}\right|\cdot\left|\mathrm{proj}_{\overline{A}}(\mathcal{C}(\mathcal{S}))\right|}$
(21)

(see Appendix A of [17] for a rigorous group-theoretic derivation). Our goal
is to convert the quantities on the RHS into Renyi entropies, allowing us to
relate $\ell_{A}$ to $I(A:R)$.

As a preliminary step, we first show how to compute the Renyi entropy
$S^{(2)}(\rho_{A}(\mathcal{S}))$ of a reduced density matrix
$\rho_{A}(\mathcal{S})$ for a subregion $A\subset Q$ of the full system. The
Renyi entropy of such a region $A$ is given by:

$\displaystyle\rho_{A}(\mathcal{S})$
$\displaystyle=\mathrm{Tr}_{\overline{A}}\left[\rho_{Q}(\mathcal{S})\right]$
$\displaystyle=2^{-\left|Q\right|}\sum_{g\in\mathcal{S}}\mathrm{Tr}_{\overline{A}}\left[g\right]$
$\displaystyle=2^{-\left|A\right|}\sum_{g\in\mathcal{S}\cap\mathrm{Ker}\
\mathrm{proj}_{\overline{A}}}g$ (22)

where in the third line we used the fact that
$\mathrm{Tr}_{\overline{A}}\left[g\right]=0$ except when
$\mathrm{proj}_{\overline{A}}(g)=\mathbb{I}_{\overline{A}}$ (i.e.
$g\in\mathrm{Ker}\ \mathrm{proj}_{\overline{A}}$). Introducing the group
$\mathcal{S}_{A}\equiv\mathcal{S}\cap\mathrm{Ker}\
\mathrm{proj}_{\overline{A}}\cong\mathcal{S}/\mathrm{proj}_{\overline{A}}(\mathcal{S})$
consisting of all stabilizer operators that act trivially outside the
subregion $A$, we can simply write the reduced density matrix as

$\rho_{A}(\mathcal{S})=2^{-\left|A\right|}\sum_{g\in\mathcal{S}_{A}}g$ (23)

which is a natural generalization of (14). Finally, it is easy to show that
the Renyi entropy of the reduced state $\rho_{A}(\mathcal{S})$ is given by

$\displaystyle(\ln 2)^{-1}S^{(2)}(\rho_{A}(\mathcal{S}))$
$\displaystyle=\left|A\right|-\log_{2}\left|\mathcal{S}_{A}\right|$
$\displaystyle=\left|A\right|-\log_{2}\left|\mathcal{S}\right|+\log_{2}\left|\mathrm{proj}_{\overline{A}}(\mathcal{S})\right|$
(24)

where we have used
$\left|\mathcal{S}_{A}\right|=\left|\mathcal{S}\right|/\left|\mathrm{proj}_{\overline{A}}(\mathcal{S})\right|$.

Finally, combining (21) and (24) we find:

$\displaystyle(\ln 2)^{-1}I(A:R)$ $\displaystyle=(\ln
2)^{-1}\left(S_{R}+S_{A}-S_{AR}\right)$ $\displaystyle=\left|R\right|+(\ln
2)^{-1}S_{A}-(\ln
2)^{-1}S_{AR}+(\log_{2}\left|\mathcal{C}(\mathcal{S})\right|-\log_{2}\left|\mathcal{C}(\mathcal{S})\right|)+(\log_{2}\left|\mathcal{S}\right|-\log_{2}\left|\mathcal{S}\right|)$
$\displaystyle=\log_{2}\left|\mathcal{C}(\mathcal{S})\right|-\log_{2}\left|\mathcal{S}\right|+\left((\ln
2)^{-1}S_{A}-\left|A\right|+\log_{2}\left|\mathcal{S}\right|\right)-\left((\ln
2)^{-1}S_{AR}-\left|AR\right|+\log_{2}\left|\mathcal{C}(\mathcal{S})\right|\right)$
$\displaystyle=\log_{2}\left|\mathcal{C}(\mathcal{S})\right|-\log_{2}\left|\mathcal{S}\right|+\log_{2}\left|\mathrm{proj}_{\overline{A}}(\mathcal{S})\right|-\log_{2}\left|\mathrm{proj}_{\overline{A}}(\mathcal{C}(\mathcal{S}))\right|$
$\displaystyle=\log_{2}\left|\mathcal{L}_{A}\right|=\ell_{A}$

where we have used the shorthand $S_{A}\equiv S^{(2)}(\rho_{A}(\mathcal{S}))$
to shorten notation, and in going from the third to fourth line we have used
the identity (24) and its corresponding generalization for the Renyi entropy
$S_{AR}$ of the combined regions $A,R$ [17].

###  E.2 Quantum Hamming Bound

Here we review the quantum Hamming bound, which places a fundamental bound on
achievable code rates and code distances for correctable nondegenerate QECCs.
Suppose we have a QECC with rate $r_{\mathrm{code}}=k/N$ that can correct all
errors of weight less than or equal to $w$. There are a total of
$\sum_{i=0}^{w}3^{i}\binom{N}{i}$ such errors (Pauli strings). For
nondegenerate codes, we must have
$\left\langle\psi\right|EE^{\prime}\left|\psi^{\prime}\right\rangle=0$ for all
errors $E,E^{\prime}$ and all code states
$\left|\psi\right\rangle,\left|\psi^{\prime}\right\rangle$. That is, every
code state $\left|\psi\right\rangle$ and all of its erroneous versions must be
orthogonal to every other code state and all of its erroneous versions [68].
The only way all of these orthogonal states fit into the same
$2^{N}$-dimensional Hilbert space is when

$2^{k}\sum_{i=0}^{w}3^{i}\binom{N}{i}\leq 2^{N}$ (25)

For large $N$ and fixed $k/N,w/N$ this is equivalent to

$\frac{k}{N}\leq 1-\frac{w}{N}\log_{2}3-H(w/N)$ (26)

where $H(x)=-x\log x-(1-x)\log(1-x)$ is the classical Shannon entropy.

One can derive a similar bound for the case where errors only occur
contiguously. In this case, there are a total of $\sum_{i=0}^{w}3^{i}(N+1-i)$
contiguous errors of weight less than or equal to $w$. For large $N$ and fixed
$k/N,w/N$ this yields a bound

$\frac{k}{N}\leq 1-\frac{w}{N}\log_{2}3-\frac{1}{N}\log_{2}(N-w)$ (27)

where the final term vanishes in the strict limit $N\rightarrow\infty$, but is
still relevant for large but finite size systems.