# Quantum Coding Transitions in the Presence of Boundary Dissipation

Izabella Lovas  Utkarsh Agrawal  Kavli Institute for Theoretical Physics,
University of California, Santa Barbara, CA, 93106  Sagar Vijay  Department of
Physics, University of California, Santa Barbara, CA 93106

###### Abstract

We investigate phase transitions in the encoding of quantum information in a
quantum many-body system due to the competing effects of unitary scrambling
and boundary dissipation. Specifically, we study the fate of quantum
information in a one-dimensional qudit chain, subject to local unitary quantum
circuit evolution in the presence of depolarizating noise at the boundary. If
the qudit chain initially contains a finite amount of locally-accessible
quantum information, unitary evolution in the presence of boundary dissipation
allows this information to remain partially protected when the dissipation is
sufficiently weak, and up to time-scales growing linearly in system size $L$.
In contrast, for strong enough dissipation, this information is completely
lost to the dissipative environment. We analytically investigate this “quantum
coding transition” by considering dynamics involving Haar-random, local
unitary gates, and confirm our predictions in numerical simulations of
Clifford quantum circuits. We demonstrate that scrambling the quantum
information in the qudit chain with a unitary circuit of depth
$\mathcal{O}(\log L)$ before the onset of dissipation can perfectly protect
the information until late times. The nature of the coding transition changes
when the dynamics extend for times much longer than $L$. We further show that
at weak dissipation, it is possible to code at a finite rate, i.e. a fraction
of the many-body Hilbert space of the qudit chain can be used to encode
quantum information.

##  I Introduction

The chaotic unitary evolution of an isolated quantum systems will spread
initially localized quantum information over non-local degrees of freedom, a
process known as quantum information scrambling [1, 2, 3, 4]. This
delocalization of information aids in protecting quantum information against
external interference from local noise, which is present in any real physical
system. Studying the robustness of quantum information in the presence of both
unitary scrambling and dissipation is important both to understand new
dynamical regimes of quantum many-body dynamics, and from a practical
standpoint, to design quantum codes and to appropriately interpret studies of
quantum many-body evolution in near-term quantum simulators. While dissipative
dynamical phases of matter have been the subject of intense research for
decades [5, 6, 7, 8, 9], addressing the dynamics of quantum information in
this context opens a new perspective. Similarly to how understanding the
spreading of information has led to a deeper understanding of quantum chaos
and thermalization [2, 10, 11, 12, 13, 14, 15, 16], studying quantum
information in dissipative systems can shed light on the structure of
(possibly new) dynamical regimes of quantum matter.

Besides its fundamental relevance for the dissipative dynamics of generic
quantum systems, the fate of quantum information in the presence of unitary
scrambling and destructive local noise or measurements has been explored in
the context of quantum information theory, leading to the development of the
theory of quantum error correcting codes [17, 18, 19, 20]. A key result in the
theory of quantum error correction (QEC) is the threshold theorem, stating
that for error rates below some threshold, one can reverse the effects of the
errors by applying additional quantum gates [21, 22, 23]. In other words, it
is possible to correct errors faster than they are created.

The threshold theorem is essential in designing fault-tolerant quantum
computers. Applying additional gates, trying to preserve the code-space
against the noise, allows one to perform logical operations for long times
with high precision. Such an active error correction is feasible in artificial
quantum systems with a “digital” architecture, in which real-time measurements
and unitary evolution can be executed over targeted degrees of freedom.
However, in analog quantum simulators realized, e.g., with ultracold atoms,
the options for active error correction are more restricted and costly due to
the limited control over the dynamics. This provides a strong motivation for
exploring whether the system’s intrinsic dynamics alone can protect
information, by hiding it from destructive local noise. Despite this
fundamental relevance, the conditions for obtaining such a robust, self-
generated coding dynamics in a generic quantum system without any degree of
external control, are still not fully explored.

Recently, the robustness of a self-generated code space against a special
class of local perturbations has been investigated, taking the form of local
projective measurements. These studies revealed a phase transition driven by
the measurement rate, such that the code space can store an extensive amount
of information, as long as the rate of measurements remains below a finite
threshold [24, 25, 26, 27, 28]. However, this result cannot be generalized to
more generic noise channels. For example, a quantum many-body system evolving
in the presence of random erasures occurring in the bulk with finite rate
destroys all quantum information in constant time [29, 30], and active error-
correction during the dynamics is required to protect the information beyond
this time scale. Understanding the conditions (if any) that unitary evolution
and local errors have to satisfy to guarantee the emergence of a robust, self-
generated code space, without the need for an active error correction during
the dynamics, is an open question of utmost relevance.

$\begin{array}[]{cc}\includegraphics[width=195.12767pt]{Circuit_a.png}&\includegraphics[width=173.44534pt]{phase_diagram.pdf}\\\
(a)&(b)\end{array}$

Figure 1: (a) Quantum information is encoded in a qudit chain which
subsequently evolves with a “brickwork” array of Haar-random, two-site unitary
gates and dissipation at the boundary. One timestep of these dynamics
corresponds to two layers of unitary gates along with depolarizing noise at
the boundary, as shown schematically in (a). A phase diagram for the coding
transition is shown in (b). The blue critical line is the coding transition
when the total number of timesteps $T\lesssim L/p$, see Section III. This
transition also corresponds to the de-pinning transition of an Ising domain
wall in a statistical mechanical description of quantum information in these
dynamics, as derived in the main text (Section II). This transition occurs
when $R$ is localized near the boundary and is not scrambled across the
system. The red critical line is the coding transition as the system approches
thermalization (see Section IV), across which the system becomes maximally
entangled with the environment resulting in information loss.

###  I.1 Summary of Results

With these motivations, we take a step towards understanding the dynamics of
quantum information under generic scrambling and local noise, by exploring the
fate of quantum information, subjected to the competing effects of boundary
dissipation and unitary spreading in a one-dimensional chaotic quantum system.
For concreteness and simplicity, we focus on the setup sketched in Fig. 1a,
which shows a single timestep of a random quantum circuit with a
depolarization channel acting at the left boundary. We note that it is known
both in classical coding theory [31, 32, 33, 34, 35] and in the quantum case
[36, 37, 38] that random unitary dynamics provides an optimal encoding of
information. We entangle one external reference qubit $R$ near the boundary
into a Bell pair, thereby encoding one qubit of quantum information initially
localized near the dissipative boundary. We then ask what happens to this
information as the system is subject to noisy dynamics, up to time scales $T$
scaling linearly with the system size $L$, such that $T/L$ is fixed.
Importantly, by taking the thermodynamic limit $L\to\infty$ and the long time
limit $T\to\infty$ simultaneously, with $T/L$ constant, we probe the system on
time scales where it is expected to thermalize.

Interestingly, we find that this quantum information can remain robust even at
these long times, giving rise to a rich dynamical phase diagram as a function
of dissipation strength $p$ and the ratio $T/L$, as displayed in Fig. 1b. The
left panel shows the case where the noisy dynamics starts immediately after
the encoding of the quantum information locally, near the leftmost boundary.
We find a dissipation-induced quantum coding phase transition, separating a
region where the coherent information remains partially protected and gets
delocalized within the system, and a phase where all of this information
leaked to the environment. The nature of the coding transition, however,
depends on the ratio $T/L$. For $T/L\lesssim 1$ the right boundary is
effectively decoupled from the dynamics of information and we observe a
continuous second-order phase transition (blue line). For even larger ratios
$T/L$, the right boundary plays a crucial role and gives rise to a first order
phase transition (red). We also demonstrate that adding a unitary “pre-
scrambling” step after the local encoding, before the onset of the dissipative
dynamics, can efficiently increase the robustness of the encoded information.
In particular, as shown in the right panel of Fig. 1b, a pre-scrambling time
$t_{scr}$ scaling logarithmically with system size, $t_{scr}\sim\log L$,
ensures that quantum information remains perfectly protected for small enough
dissipation strengths $p$, up to time scales $T\sim L/p$.

We gain a detailed understanding of these different types of coding
transitions, by mapping the dynamics of quantum information in a circuit with
Haar-random unitary gates and boundary dissipation to the statistical
mechanics of a two-dimensional lattice magnet. This mapping, which has been
extensively employed to understand unitary circuit quantum dynamics as well as
dynamics with projective measurements (see Ref. [39, 40] for a review), allows
us to obtain analytical predictions, as well as instructive numerical results.
While the entanglement measures of interest which diagnose the quantum coding
transition require taking a formal replica limit of this lattice magnet (akin
to a limit arising when considering “quenched” disorder), we focus our
attention on understanding this lattice magnet away from the replica limit
(akin to studying an “annealed” disorder-average). Specifically, we focus on
the “annealed” disorder average of the second Rényi mutual information between
the output of the circuit $A$, and the reference qubit $R$. In this limit, the
circuit with the boundary depolarization can be mapped to the statistical
mechanis of an Ising magnet, in which a single Ising domain wall experiences
an attractive/repulsive potential at one boundary of the two-dimensional
system, whose strength is tuned by the dissipation strength. In this language,
the coding transition at times $T/L\lesssim 1$ can be understood as a second
order pinning/depinning transition of the Ising domain wall at the noisy
boundary; we provide conjectures as to the true nature of this transition in
the replica limit. At later times $T/L>1/p$, the right boundary gives rise to
a different, first order transition by “absorbing” the Ising domain wall.
Insights gained from this classical statistical picture are confirmed by large
scale numerical simulations performed on Clifford quantum random circuits.

Finally, we show that the coding transition for $T/L>1/p$ can also be
understood as a transition arising from the monogamy of entanglement. In this
case, as the system of $L$ qubits becomes entangled with a growing number of
environmental degrees of freedom, scaling as $pT$, eventually it can no longer
stay simultaneously entangled with the reference qubit, and all information
leaks to the environment. We conclude with the interesting scenario of
encoding an extensive amount of information in the system. Specifically, we
show that a similar coding transition persists when we entangle an extensive
number of reference qubits into Bell pairs with the qubits of the system. In
particular, we identify two threshold values for the dissipation strength $p$,
$p_{th,1}$ and $p_{th,2}$, separating three regions according to the behavior
of the information density. The information density is perfectly protected in
the system for $p<p_{th,1}$, while it starts to leak into the environment
above this threshold. A finite density of information still survives in the
region $p_{th,1}<p<p_{th,2}$, until eventually reaching zero at the upper
threshold $p_{th,2}$.

The rest of the paper is organized as follows. In Sec. II, we introduce the
mapping between the coherent quantum information in random circuits and the
properties of an Ising domain wall experiencing a repulsive/attractive
boundary on the left and an absorbing boundary on the right, by considering
the “annealed” second Rényi mutual information between the circuit output and
the encoded information. We derive the random walk model in Sec. II.1. We then
show in Sec. II.2 that different phases on either side of the coding
transition can be understood by inspecting the weighted trajectories of the
Ising domain wall in this statistical mechanical model.

We turn to the detailed discussion of the second order coding transition in
the regime $T\lesssim L/p$, induced by the dissipative boundary alone without
the interference of the clean boundary, in Sec. III. We first rely on the
random walk model to gain a qualitative understanding of the phase transition,
and discuss the classical pinning/depinning transition of the Ising domain
wall in Sec. III.1. Building on these insights, we verify the presence of the
quantum coding transition and study its properties numerically in Sec III.2,
by performing large scale numerical simulations on Clifford quantum circuits,
before discussing the nature of this transition in more detail in Sec. III.3.
To end the section, in Sec. III.4 we comment on increasing the robustness of
the encoded information by applying a unitary pre-scambling before the onset
of dissipative dynamics. We show that a pre-scrambling time $t_{\mathrm{scr}}$
scaling logarithmically with system size provides perfect protection for the
coherent information for weak enough dissipation $p$, up to time scales
$T/L\sim O(1)$.

We turn to the first order coding transition, induced by the interplay of the
dissipative left boundary and the clean right boundary at times $T\gtrsim
L/p$, in Sec. IV. First, we discuss that this phase transition can be
understood in the statistical mechanical framework as the absorption of the
entanglement domain wall by the right boundary and is driven by the monogamy
of entanglement as the system becomes entangled with a growing number of
environmental qubits. We present and analyze the numerical results obtained
from Clifford circuit simulations in Sec. IV.1, and find good agreement with
the predictions of the statistical mechanics of the Ising lattice magnet. We
argue that this coding transition is of first order, and discuss its scaling
properties in Sec. IV.2. Finally, Sec V serves as an outlook to the case of
encoding an extensive amount of information into the system. Here we consider
entangling a finite density of reference qubits with the system, and find a
monogamy induced coding transition at late times $T\gtrsim L/p$, similar to
the one observed for a single bit of quantum information. Here we find three
phases, with the information perfectly protected for $p<p_{th,1}$, a finite
density of information surviving for $p_{th,1}<p<p_{th,2}$, and the density
reaching zero above $p_{th,2}$. We conclude by summarizing our results, and
discussing open questions in Sec. VI.

###### Contents

  1. I Introduction
    1. I.1 Summary of Results
  2. II Dissipation in Quantum Circuit Evolution
    1. II.1 Statistical Mechanics of Random Unitary Evolution and Dissipation
    2. II.2 Boundary Dissipation and the Encoding of Quantum Information
  3. III Quantum Coding Transition
    1. III.1 Annealed Mutual Information, and the Pinning of an Ising Domain Wall
    2. III.2 Numerical Study
    3. III.3 The Replica Limit and the Nature of the Phase Transition
    4. III.4 Perfect information protection using scrambling
  4. IV Coding transition on the approach to thermalization
    1. IV.1 Numerical Study
    2. IV.2 Nature of the Phase Transition
  5. V Encoding at a Finite Rate
  6. VI Summary and Discussion
  7. A Lattice Partition Function and the Annealed Phase Transition
  8. B Alternative random circuit protocols

##  II Dissipation in Quantum Circuit Evolution

###  II.1 Statistical Mechanics of Random Unitary Evolution and Dissipation

Past studies of random local unitary evolution [39, 40], evolution with
projective measurements [24, 25, 26] and with dissipation [30, 29, 41, 42, 43,
44] have uncovered a wealth of universal structures governing the dynamics of
information-theoretic quantities such as the Rényi entanglement entropy.
Averaging over an ensemble of unitary gates in this setting gives rise to an
emergent classical statistical mechanics of quantum entanglement, which must
be understood in an appropriate “replica limit” in order to recover the
behavior of the information-theoretic quantities of interest. A qualitatively-
accurate understanding of the behavior of quantum entanglement in chaotic
unitary dynamics, and in dynamics with projective measurements can still be
obtained even without taking the replica limit [45, 46, 13, 47], though these
approaches often fail to capture quantitative, universal properties
characterizing distinct regimes of quantum many-body evolution (e.g. of the
volume-law-entangled phase of infrequently monitored quantum many-body
evolution [48]) or of critical points (e.g. separating different phases of
monitored quantum dynamics).

Here, we consider the evolution of qudits under random, local unitary gates
and boundary dissipation. Averaging over the ensemble of unitary gates, in the
calculation of the evolving _purity_ of subsystem, leads to an emergent
statistical mechanics of an Ising magnet. We present the various ingredients
that the unitary evolution and dissipation correspond to in this setting,
before using these ingredients extensively in subsequent sections to
understand the stability of encoded quantum information under this evolution.

$\begin{array}[]{c}\includegraphics[width=164.77771pt]{Circuit_b}\\\
\includegraphics[width=117.07924pt]{Circuit_c_2}\\\ \end{array}$

Figure 2: Top. Performing a Haar-average over the unitary gates in the
calculation of the purity of the evolving state gives rise to an Ising magnet,
whose partition function may be written as the product of transfer matrices,
given in Eq. (5), (6) and (7). Bottom. A coarse-grained description of this
Ising magnet involves a single Ising domain wall (green)in the presence of a
boundary magnetic field (shaded red). The boundary conditions at the bottom of
the Ising magnet, which are fixed by the initial state of the quantum system,
are not shown.

We focus our attention on a one-dimensional chain of qudits, with Hilbert
space dimension $q$ at each lattice site. The dissipation acts on the boundary
qudit, and is described by the depolarizing channel $\Phi$ acting on the
density matrix $\rho$ of this qudit as

$\displaystyle\Phi(\rho)=(1-p)\,\rho+p\cdot\frac{\mathds{1}_{q\times q}}{q}$
(1)

with $p\in[0,1]$ parametrizing the “strength” of the dissipation. For future
convenience, we choose to rewrite the depolarizing channel as an _operator_
$\hat{\Phi}$ which acts within a Hilbert space of dimension $q^{2}$. The
operator $\hat{\Phi}$ takes the form

$\displaystyle\hat{\Phi}=\sum_{i,j=1}^{q}\left[(1-p)\ket{i,j}\bra{i,j}+\frac{p}{q}\ket{i,i}\bra{j,j}\right]$
(2)

where $\ket{i}$ for $i\in\\{1,\ldots,q\\}$ denotes an orthonormal basis of
states of a single qudit111The qudit density matrix
$\rho\equiv\sum_{i,j}\rho_{ij}\ket{i}\bra{j}$ is a _state_
$\ket{\rho}\equiv\sum_{i,j}\rho_{ij}\ket{i,j}$ in the doubled Hilbert space on
which the operator $\hat{\Phi}$ acts as
$\hat{\Phi}\ket{\rho}=(1-p)\ket{\rho}+(p/q)\sum_{i}\ket{i,i}$. .

Apart from the dissipation, the remaining qudits will be chosen to evolve
according to two-site unitary gates, chosen from the uniform (Haar) measure
for the unitary group U$(q^{2})$. Given such a two-qudit unitary gate $U$, we
note that the average over the Haar measure of $U\otimes U^{*}\otimes U\otimes
U^{*}$ – a quantity which will naturally appear in subsequent sections – is
given by

$\displaystyle V\equiv$ $\displaystyle\langle U\otimes U^{*}\otimes U\otimes
U^{*}\rangle$
$\displaystyle=\sum_{\sigma,\tau\in\\{\uparrow,\downarrow\\}}\mathrm{wg}_{2}(\sigma\tau)\ket{\tau,\tau}\bra{\sigma,\sigma}$
(3)

where $\langle\cdots\rangle$ denotes the Haar average, the Weingarten function
is given as $\mathrm{wg}_{2}(+)=\frac{q^{2}}{q^{4}-1}$ and
$\mathrm{wg}_{2}(-)=\frac{-1}{q^{4}-1}$, and the states $\ket{\uparrow}$ and
$\ket{\downarrow}$ are defined as
$\ket{\uparrow}\equiv\sum_{i,j=1}^{q}\ket{i,i,j,j}$ and
$\ket{\downarrow}\equiv\sum_{i,j=1}^{q}\ket{i,j,j,i}$ so that

$\displaystyle\braket{\sigma}{\tau}=(q^{2}-q)\delta_{\sigma,\tau}+q.$ (4)

From these expressions, it is clear that

$\displaystyle V\ket{\uparrow\uparrow}=\ket{\uparrow\uparrow}\hskip
50.58878ptV\ket{\downarrow\downarrow}=\ket{\downarrow\downarrow}$ (5)
$\displaystyle
V\ket{\uparrow\downarrow}=V\ket{\downarrow\uparrow}=\frac{q}{q^{2}+1}\left[\ket{\downarrow\downarrow}+\ket{\uparrow\uparrow}\right]$
(6)

From Eq. (2), the operator $D\equiv\hat{\Phi}\otimes\hat{\Phi}$ acts on these
states as

$\displaystyle D\ket{\uparrow}=\ket{\uparrow}\hskip
21.68121ptD\ket{\downarrow}=(1-p)^{2}\ket{\downarrow}+\frac{p(2-p)}{q}\ket{\uparrow}$
(7)

###  II.2 Boundary Dissipation and the Encoding of Quantum Information

We now consider a qudit chain consisting of $L$ qudits, into which quantum
information has been encoded. We may imagine that this quantum information is
represented by physical reference qudits which are maximally-entangled with
the one-dimensional system. This system subsequently evolves according to a
unitary circuit composed of Haar-random unitary gates in a “brickwork” array,
together with dissipation which acts near the boundary. We first focus on the
case where only a single qudit is encoded in the one-dimensional system, and
with dissipation acting periodically in time on the boundary qudit, as shown
schematically in Fig. 2a. A single timestep of this evolution corresponds to
the application of two layers of two-site unitary gates, followed by the
depolarizing channel (1) on the boundary qudit.

To diagnose whether this qudit of encoded information can remain in the
system, even as the boundary dissipation continues to act, we study the
behavior of the bipartite mutual information between the reference qudit
($R$), and the system ($A$) at a time $t$; this mutual information is defined
as

$\displaystyle I_{A,R}(t)=S_{A}(t)+S_{R}(t)-S_{A\cup R}(t)$ (8)

where $S_{A}\equiv-\Tr\left[\rho_{A}(t)\log_{q}\,\rho_{A}(t)\right]$ is the
von Neumann entanglement entropy of subsystem $A$ at a time $t$. We note that
$I_{A,R}(t)$ is related to the coherent information present in the system. If
$I_{A,R}=2$ the entangled qudit can be perfectly recovered by applying a
recovery operation to the system alone whereas for $I_{A,R}=0$ the information
has leaked to the environment, that is, $I_{E,R}=2$ [49, 50].

The mutual information (8) averaged over realizations of the random unitary
evolution, thus diagnoses whether quantum information remains in the system,
even in the presence of boundary dissipation. Instead of considering the Haar-
average of the mutual information, we turn our attention on the “annealed”
average of the second Rényi mutual information between $A$ and $R$, defined as

$\displaystyle I_{A,R}^{(\mathrm{ann})}(t)\equiv\log_{q}\langle
q^{\,I_{A,R}^{(2)}(t)}\rangle$ (9)

where $I_{A,R}^{(2)}(t)=S_{A}^{(2)}(t)+S_{R}^{(2)}(t)-S_{A\cup R}^{(2)}(t)$,
with the second Rényi entropy defined as
$S_{A}^{(2)}\equiv-\log_{q}\mathrm{Tr}\rho_{A}(t)^{2}$, and
$\langle\cdots\rangle$ denotes the Haar average over the unitary gates in the
circuit. The behavior of the annealed mutual information (9) can provide a
qualitative understanding of the quantity of interest (8), as discussed at the
beginning of this section, though quantitative details may differ, as we will
later clarify.

We proceed to calculate the annealed mutual information (9). We initialize the
qudits in a product state, except for the qudit at a site $x_{0}$ away from
the boundary which is maximally entangled with the reference qudit. As the
system evolves in the presence of unitary gates and dissipation, it is evident
that the purity of the reference qudit remains unchanged,
$\Tr\rho_{R}(t)^{2}=q^{-1}$ for all times $t$. Furthermore, calculation of
$\langle\Tr\rho_{A}(t)^{2}\rangle$ and $\langle\Tr\rho_{A\cup
R}(t)^{2}\rangle$ involves performing a Haar average of four copies of the
quantum circuit. Following the discussion in the previous section, it is thus
clear that these Haar-averaged purities may be written as partition functions
for an Ising magnet of finite extent in the vertical direction – corresponding
to the time direction in the quantum circuit – and with horizontal extent
fixed by the number of qudits in the system. The Ising spins live on the links
of a square lattice, and are acted upon by the transfer matrices matrices $V$
and $D$, as given in Eq. (5), (6) and (7), depending on whether a Haar-random
unitary gate or dissipation is applied at a particular point in spacetime in
the quantum circuit, respectively. The full transfer matrix is shown
schematically in Fig. 2b.

The boundary conditions for the Ising partition sum, at the ($i$) bottom and
($ii$) top boundaries are determined by ($i$) the initial state of the qudit
chain along with the location of the reference qudit, and ($ii$) the subsystem
over which the purity is being calculated, respectively. First, fixing Ising
spins at the top boundary to be in the $\downarrow$ state corresponds to
keeping the corresponding qudit within the region for which the purity is
being calculated. As a result, the spins at the top boundary are all fixed in
the $\downarrow$ state for both the calculation of
$\langle\Tr\,\rho_{A}(t)^{2}\rangle$ and $\langle\Tr\,\rho_{A\cup
R}(t)^{2}\rangle$, as shown in Fig. 2b. These two purities thus only differ in
their bottom boundary conditions. Here, the boundary spins are allowed to
freely fluctuate, with the exception of the spin corresponding to the qudit at
a distance $x$ away from the boundary; the state of this Ising spin determines
whether the reference qudit is included in the subsystem whose purity is being
computed. More precisely, this spin is fixed in the $\uparrow$ or $\downarrow$
state in the calculation of the quantities,
$\langle\Tr\,\rho_{A}(t)^{2}\rangle$ and $\langle\Tr\,\rho_{A\cup
R}(t)^{2}\rangle$, respectively.

It is convenient to evaluate these partition functions by contracting the
transfer matrix from the top boundary condition, i.e. “backwards” in time with
respect to the arrow of time in the quantum circuit. Let $Z(t)$ denote the
partition sum obtained by evolving the all-down state of the Ising spins for
$t$ timesteps by repeatedly applying the row transfer matrix corresponding to
a single timestep of the dynamics. The partition sum $Z(t)$ describes a
single, directed Ising domain wall, which can only be created/annihilated at
the boundary of the system. This can be seen as follows. First, starting with
the all-down state, the dissipation (7) can flip the boundary Ising spin from
$\ket{\downarrow}$ to $\ket{\uparrow}$, thus creating an Ising domain wall
near the boundary. The effect of the Haar-random unitary gates (5), (6) in the
bulk of the quantum circuit is to simply move the domain wall. Notably, Eq.
(5) implies that the Haar-random gates cannot create or annihilate Ising
domain walls in the bulk of the system, though gates acting near the boundary
can annihilate the Ising domain wall. Once the state of the boundary spin is
$\ket{\uparrow}$, the dissipation cannot alter this state since
$D\ket{\uparrow}=\ket{\uparrow}$; this is simply a consequence of the fact
that the depolarizing channel (1) leaves the maximally-mixed density matrix
$\rho=\mathds{1}_{q\times q}/q$ unchanged.

The partition sum $Z(t)$ is thus performed over histories of the entanglement
domain wall trajectories, which can propagate in the bulk of the system, or be
created/annihilated at the boundary. Formally, we write

$\displaystyle Z(t)=\sum_{x\geq 0}z(x,t)$ (10)

where $z(x,t)$ is a restricted sum over trajectories of the entanglement
domain wall where the domain wall ends up between sites $x-1$ and $x$ at time
$t$. In this convention, $z(0,t)$ corresponds to trajectories where the
entanglement domain wall no longer exists at time $t$, as it has been
annihilated at the left interface.

We may now write the Haar-averaged purities as

$\displaystyle\langle\Tr\,\rho_{A}(t)^{2}\rangle=q^{2}\sum_{y>x_{0}}z(y,t)+q\sum_{y\leq
x_{0}}z(y,t)$ (11) $\displaystyle\langle\Tr\,\rho_{A\cup
R}(t)^{2}\rangle=q^{2}\sum_{y\leq x_{0}}z(y,t)+q\sum_{y>x_{0}}z(y,t)$ (12)

This is due to the fact that $\langle\Tr\,\rho_{A\cup R}(t)^{2}\rangle$
involves a sum over trajectories of the entanglement domain wall, with an
additional weight $q^{2}$ given to trajectories which end at a position
$y>x_{0}$ and a weight $q$ given to trajectories ending at $y\leq x_{0}$,
where $x_{0}$ is the location of the entangled reference qudit. The opposite
weighting scheme is true for $\langle\Tr\,\rho_{A}(t)^{2}\rangle$. These
additional weights arise due to the fact that depending on the final position
of the entanglement domain wall, the boundary spin at $x$ is contracted with
the state $\ket{\uparrow}$ or $\ket{\downarrow}$. These overlaps are given in
Eq. (4). With these expressions, it is straightforward to see that

$\displaystyle
I_{A,R}^{(\mathrm{ann})}(t)=\log_{q}\left[\frac{q^{2}-q(q-1)P(x_{0},t)}{1+(q-1)P(x_{0},t)}\right]$
(13)

where

$\displaystyle P(x_{0},t)\equiv\frac{1}{Z(t)}\sum_{y\geq x_{0}}z(y,t)$ (14)

is the probability that the domain wall ends at a position $y\geq x_{0}$ at
time $t$.

##  III Quantum Coding Transition

In this section, we study the behavior of the encoding of quantum information
in the system, after evolving the system by the quantum circuit for $T$
timesteps, for a fixed dissipation strength $p$. The number of timesteps of
the evolution $T$ can be large so that $T/L\sim O(1)$ but is taken to be small
enough throughout the entirety of this section, so that the left and right
ends of the one-dimensional qudit chain are causally disconnected. As $p$ is
increased from zero, we will find an “quantum coding” transition, where
information initially encoded in the system is lost to the environment above a
threshold $p=p_{c}$.

###  III.1 Annealed Mutual Information, and the Pinning of an Ising Domain
Wall

First, we investigate the behavior of $I_{A,R}^{(\mathrm{ann})}$ as the
dissipation strength $p$ is tuned, by studying the Ising lattice magnet that
emerges after performing a Haar-average over the unitary gates in the quantum
circuit.

As discussed in Sec. II.2, the partition sum $Z(T)$ describes a single Ising
domain wall which can propagate through the bulk of the two-dimensional
system, and be created/annihilated at the left boundary of the system. Tuning
the dissipation strength, which alters the Ising symmetry-breaking field
applied at the boundary, modulates an effective “pinning potential” for the
Ising domain wall. This can be clearly seen in the limiting cases when $p=0$
or $1$. In the former case, the dissipation is completely absent, and Eq. (5)
implies that the all-down state is left invariant by the transfer matrix for
the Haar-averaged circuit. Thus, in this limit, there is no Ising domain wall.
In contrast, when $p=1$, the boundary spin is fixed in the $\ket{\uparrow}$
state, and the domain wall is effectively repelled from the left boundary.

Increasing the dissipation strength can then drive a pinning/de-pinning phase
transition for the entanglement domain wall. Similar phase transitions due to
the presence of a boundary magnetic field in an Ising magnet have been studied
in the literature (see, e.g. Ref. [51, 52, 53]). Equivalently, the temporally-
directed nature of the Ising domain wall also suggests these paths may be
thought of as the imaginary-time trajectories of a single quantum-mechanical
particle on the half-line, which experiences a potential near the boundary,
which is tuned by the dissipation strength. $Z(T)$ is thus an amplitude for
this particle to propagate under imaginary time-evolution by this Hamiltonian.
In this setting, the particle can undergo a localization transition when the
potential is _sufficiently_ attractive [52]. This result is to be contrasted
with the well-studied problem of a particle on the full line, with a delta-
function potential near the origin, which always forms a bound-state in the
potential well as long as the potential is attractive.

The annealed mutual information precisely measures the localization of the
Ising domain wall, as is evident from Eq. (13). Deep within a localized phase,
where the transverse wandering of the domain wall is governed by a length-
scale $\ell_{\perp}$, the probability $P(x_{0},T)\sim e^{-x_{0}/\ell_{\perp}}$
($\ell_{\perp}\ll x_{0})$, so that $I^{(\mathrm{ann})}_{A,R}$ is a constant,
deviating from its maximal value of $2$ by a constant correction which changes
within the localized phase. In contrast, in the delocalized phase, the
probability $P(x_{0},T)\overset{T\rightarrow\infty}{=}1$, where the limit is
taken, keeping the ratio $T/L=\mathrm{const.}$ fixed.

Properties of this coding transition, as seen by annealed-averaged
observables, such as the annealed mutual information, may be obtained by
studying the lattice partition function for the Ising domain wall, which we
present in Appendix A, due to the technical nature of the calculations
involved. From this study, we find that

  1. 1.

The phase transition occurs at a probability $p_{c}$ which varies as a
function of the on-site Hilbert space dimension $q$. The behavior of $p_{c}$
as $q$ is tuned may be determined by studying the lattice partition function.
In the limit $q\rightarrow\infty$, the coding transition is absent.
Specifically, we find that

$\displaystyle p_{c}=1-O(q^{-2})$ (15)

so that information is always preserved in the system in the limit that the
on-site Hilbert space dimension is strictly infinite.

  2. 2.

Near the phase transition, the annealed mutual information takes the universal
scaling form

$\displaystyle
I^{(\mathrm{ann})}_{A,R}(T)=T^{-\beta/\nu}F(T^{1/\nu}(p-p_{c}))$ (16)

where $\beta=1/2$ and $\nu=2$. The function $F(x)\sim x^{\beta}$ as
$x\rightarrow-\infty$. This relation is obtained by determining that in the
thermodynamic limit, the annealed mutual information should vanish on
approaching the transition as $I^{(\mathrm{ann})}_{A,R}\sim\ell_{\perp}^{-1}$,
where $\ell_{\perp}$ is the distance of a transverse excursion of the Ising
domain wall in the pinned phase. This length scale is shown to diverge as
$\ell_{\perp}\overset{p\rightarrow p_{c}^{-}}{\sim}(p_{c}-p)^{-\beta}$ upon
approaching the phase transition.

The above scaling form for the annealed mutual information is in good
quantitative agreement with numerical studies, which we perform by directly
studying the transfer matrix for the Ising magnet. A numerically-obtained
scaling collapse for the annealed mutual information is shown in Fig. 3, which
is consistent with Eq. (16).

Figure 3: Scaling collapse of the annealed mutual information, consistent with
the scaling form in Eq. (16). The inset shows the behavior of the annealed
mutual information as a function of dissipation strength $p$, indicating the
presence of an coding transition. The exponents $\beta=1/2$, $\nu=2$ are
determined from properties of the pinning transition of the Ising domain wall.
The system size is taken to be large enough that the left and right ends of
the qudit chain are causally disconnected.

We expect that the qualitative behaviors presented here hold for the
“quenched-averaged” quantities of interest, such as the averaged von Neumann
mutual information $\langle I_{A,R}(t)\rangle$, which truly diagnose the loss
of quantum information from the system, as the dynamics proceed. The true
nature of the phase transition, however, will be different, as we discuss in
Sec. III.3.

###  III.2 Numerical Study

Having obtained a qualitative understanding of the coding transition by
considering the “annealed” Haar average of the Rényi mutual information, we
now demonstrate the presence of this transition in numerical studies of
quantum circuit evolution in a qubit chain ($q=2$ on-site Hilbert space
dimension). Here, the unitary time evolution of the bulk is governed by
Clifford random unitary gates, arranged in a brickwork structure. This setup
allows us to simulate the dynamics of large systems for sufficiently long
times to study the phase transition introduced above, by relying on the
stabilizer formalism. The boundary dissipation is realized as a random erasure
channel, acting on the leftmost qubit with probability $p$ in each time step,
by deleting the information stored in the qubit. In the stabilizer formalism,
this boundary erasure channel is implemented by deleting all stabilizers
acting non-trivially (as a non-identity operator) on the leftmost qubit.

We note that besides the protocol described above, we also considered other
forms of boundary dissipation and Clifford scrambling, all giving rise to
similar results for the behavior of the mutual information. Specifically, we
implemented an alternative dissipation channel, by applying a CNOT gate
entangling the boundary qubit with an environmental ancilla qubit that was
subsequently traced out from the density matrix. Moreover, we considered
protocols with sparse bulk scrambling, where each unitary gate in the
brickwork structure is a random Clifford unitary with probability $p_{U}<1$,
but the trivial identity operator with probability $1-p_{U}$. This scenario
allowed us to tune the efficiency of the scrambling through the parameter
$p_{U}$, while keeping the boundary noise fixed, leading to a phase transition
similar to the one discussed in the main text. We discuss these alternative
protocols in more detail, and present supplementary numerical results in
Appendix B.

The Bell pair is encoded in the initial state at the leftmost site, by
entangling the boundary qubit with a reference qubit, while the remaining
qubits are initialized in a random product state. We run the dissipative
dynamics for time $T$, with system size $L$ chosen to keep $T/L<1$ fixed, such
that the right boundary of the system is not causally connected to the Bell
pair. This setting allows us to detect the coding transition induced by a
single boundary, by increasing the evolution time $T$. Importantly, due to the
fixed ratio $T/L$, the long time limit $T\to\infty$ and the thermodynamics
limit $L\to\infty$ are performed simultaneously, therefore, we are probing the
mutual information on time scales where the system is expected to become
thermalized.

Figure 4: Coding transition induced by a single boundary. The mutual
information between the reference qubit and the output of the circuit shown as
a function of dissipation strength $p$, for $T/L<1$ fixed, with boundary
dissipation realized as a random erasure channel. The scaling with circuit
depths $T$ points to a phase transition between a phase with partially
protected information, and a phase with all information lost.

The mutual information $I_{A,R}$ between the output of the dissipative quantum
circuit $A$ and the reference qubit $R$ is shown in Fig. 4, for different
dissipation strengths $p$ and circuit depths $T$. These results are consistent
with a coding transition tuned by the dissipation strength $p$, between a
phase where the system retains part of the encoded information, and a strongly
dissipative phase with all information lost. We note that determining the
critical exponents and critical point of this transition from finite time data
is numerically challenging. Nevertheless, we attempt to estimate these
parameters by noting that the mutual information obeys the finite size scaling
$I_{A,R}\sim T^{-\beta/\nu}$ at the critical dissipation strength $p_{c}$,
while it saturates to a finite value as $T\rightarrow\infty$ for $p<p_{c}$.
Relying on this observation, we identify $p_{c}$ with the smallest $p$ where
the numerical data are consistent with $I_{A,R}$ approaching zero
algebraically as $T\rightarrow\infty$, yielding the estimate $p_{c}\approx
0.5$. We then use the critical scaling $\left.I_{A,R}\right|_{p=p_{c}}\sim
T^{-\beta/\nu}$ to fit the ratio $\beta/\nu$, see Fig. 5a. Finally, we fit
estimate $\nu$ by requiring a good scaling collapse for the full set of data
from Fig. 4. We obtain the critical parameters $p_{c}=0.5$, $\beta/\nu=0.34$
and $\nu=2$, yielding the scaling collapse shown in Fig. 5b. We note, however,
that due to the large number of fitting parameters, the critical exponents
extracted this way carry a considerable uncertainty. We leave the more
thorough investigation of critical properties for future work.

Figure 5: Critical properties of the coding transition for a single boundary.
(a) Critical power law scaling of the mutual information with respect to
circuit depth $T$ at the estimated transition point, $p_{c}=0.5$. The scaling
relation $I_{A,R}\sim T^{-\beta/\nu}$ is used to extract $\beta/\nu=0.34$
(dashed line). (b) Full scaling collapse of rescaled mutual information
$T^{\beta/\nu}I_{A,R}$ as a function of $T^{1/\nu}\left(p-p_{c}\right)$, using
$\nu=2$.

###  III.3 The Replica Limit and the Nature of the Phase Transition

The behavior of quenched-averaged quantities, e.g. the Haar-averaged Rényi
mutual information $\langle I_{A,R}^{(2)}(t)\rangle$, close to the coding
phase transition are quantitatively distinct from the annealed-averaged mutual
information studied in Sec. III.1. This is suggested by the numerical studies
in the previous section, which present strong evidence that the coding phase
transition is in a different universality class from a de-pinning phase
transition for a single Ising domain wall. Here, we will provide some
conjectures on the nature of this phase transition, based on analytic
arguments.

We will focus our attention on the averaged second Rényi mutual information
$\langle I_{A,R}^{(2)}(t)\rangle$ whose behavior may be obtained via a
“replica trick”; the second Rényi entropy may be obtained in the limit
$S_{A}^{(2)}(t)=\displaystyle\lim_{k\rightarrow
0}\left(1-\left[\Tr\rho_{A}(t)^{2}\right]^{k}\right)/k$, so that the
calculation of the Haar-averaged mutual information reduces to evaluating
quantities such as $\langle\left[\Tr\rho_{A}(t)^{2}\right]^{k}\rangle$ in a
replica limit $k\rightarrow 0$. After the Haar average, these quantities may
be regarded as partition functions for lattice magnets with “spins” taking
values in the permutation group on $2k$ elements $S_{2k}$ [39]. A drastic
simplification in the limit of large, but finite, on-site Hilbert space
dimension $q$ occurs [54], whereby
$\langle\left[\Tr\rho_{A}(t)^{2}\right]^{k}\rangle$ may be regarded as $k$
copies of an Ising magnet, with weak inter-replica interactions at each
spacetime point where a Haar-random unitary gate has been applied. The intra-
replica interactions for each Ising magnet are described by the statistical
mechanical rules presented in Sec. II.1. The inter-replica interactions are
known to be attractive, and vanish in the limit that $q$ is strictly infinite
[54]. As already derived in II.1, the boundary dissipation acts as an Ising
symmetry-breaking field, giving rise to a boundary potential for the Ising
domain wall within each replica.

Figure 6: The Haar-averaged Rényi mutual information between the reference
qudit(s) and the system, $\langle I^{(2)}_{A,R}(t)\rangle$ is described in the
large-$q$ limit, by the $k$ Ising domain walls in the presence of attractive,
inter-replica interactions, and an attractive interface within each replica,
in the limit $k\rightarrow 0$. This is described by the path integral in Eq.
(18).

The replica limit of the resulting theory may thus be regarded as the
description of a directed path in a random environment [55, 56], restricted to
the half-line $x\geq 0$, and in the presence of a potential near this
boundary, due to the dissipation. The path integral for this problem for a
given realization of the disorder is formally given by

$\displaystyle Z[V]=\int\,Dx(\tau)\,e^{-S[x,V]}$ (17)

where

$\displaystyle S[x,V]\equiv\int
d\tau\left[\frac{1}{2}\left(\frac{dx}{d\tau}\right)^{2}+V[x,\tau]-u\,\delta[x]\right].$
(18)

Here $x(\tau)$ is the coordinate of the path at time $\tau$. The random
potential in the bulk $V[x,\tau]$ is taken to have zero mean, and is short-
range-correlated in spacetime, e.g. we may take the potential to be delta-
function-correlated as
$\overline{V[x,\tau]V[x^{\prime},\tau^{\prime}]}=\sigma^{2}\delta(x-x^{\prime})\delta(\tau-\tau^{\prime})$,
where $\overline{\cdots}$ denotes an average over the probability distribution
for the disorder. The statistical mechanics of the replicated theory
$\overline{Z^{k}}$ thus describes $k$ interacting paths in the presence of a
boundary potential, and thus resembles that of the Haar-averaged quantities
$\langle\left[\Tr\rho_{A}(t)^{2}\right]^{k}\rangle$,
$\langle\left[\Tr\rho_{A\cup R}(t)^{2}\right]^{k}\rangle$ in the limit of
large, but finite, $q$. A schematic depiction of this replicated theory is
shown in Fig. 6.

The weak inter-replica interactions are known to be a relevant perturbation at
the critical point describing the pinning of a single Ising domain wall [57].
Remarkably, the new critical point describing the pinning/de-pinning of a
directed polymer to an interface, has been understood exactly [57] by Bethe
ansatz techniques. The characteristic wandering length of the polymer
transverse to the interface diverges with an exponent $\nu_{\perp}=2$ on
approaching the phase transition from the localized phase, while the
divergence of the specific heat is characterized by the exponent $\alpha=0$.
For time-independent dissipation (e.g. the depolarizing channel is applied
identically at the boundary at each time of the quantum circuit evolution), we
thus expect the coding transition to be in the universality class of this de-
pinning phase transition for a directed polymer.

In contrast, if the boundary dissipation varies randomly in time - as was
studied in Sec. III.2 \- then the nature of the phase transition is not
completely understood. This problem corresponds to having an imaginary-time-
dependent boundary potential $u(\tau)=u_{0}+v(\tau)$ in (18), where $v(\tau)$
has zero mean and is short-range-correlated in spacetime; for simplicity, we
take
$\overline{\overline{v(\tau_{1})v(\tau_{2})}}=\mu^{2}\delta(\tau_{1}-\tau_{2})$,
with $\overline{\overline{\cdots}}$ denoting the average over the distribution
for $v(\tau)$.

We may study the relevance of randomness in this boundary potential at the de-
pinning transition. Here, the action is invariant under coarse-graining and
re-scaling $\tau^{\prime}=\tau/b^{z}$, and $x^{\prime}\equiv x/b$ where $z$ is
the dynamical critical exponent at the phase transition. Under this
transformation, the random boundary potential becomes $\int
d\tau\,v(\tau)\delta[x]\longrightarrow
b^{z-1}\int\,d\tau^{\prime}\,v(b^{z}\tau^{\prime})\delta[x^{\prime}]$, so that
we identify $v^{\prime}(\tau^{\prime})\equiv b^{z-1}v(b^{z}\tau^{\prime})$ as
the renormalized potential in the coarse-grained theory. The correlations of
the renormalized potential are thus

$\displaystyle\overline{\overline{v^{\prime}(\tau^{\prime}_{1})v^{\prime}(\tau^{\prime}_{2})}}=\mu^{2}b^{z-2}\delta(\tau^{\prime}_{1}-\tau^{\prime}_{2})$
(19)

Therefore, the strength of the disorder decreases under renormalization when
$z<2$. It has been conjectured [58] that $z=3/2$ at the pinning transition for
the directed polymer, so that the randomness in the boundary potential should
be irrelevant by Eq. (19), so that the same fixed-point describing the de-
pinning of a directed polymer studied in Ref. [57] should describe the
resulting transition in the presence of randomness.

We are, however, unaware of the correctness of this result in Ref. [58] for
the dynamical exponent. The numerical studies presented in Sec. III.2 further
suggest that $\nu_{\parallel}=2$ (as opposed to
$\nu_{\parallel}=z\nu_{\perp}=3$, which is what would be predicted on the
basis of $z=3/2$ and $\nu_{\perp}=2$), though more extensive numerical studies
are required to pin down the nature of this transition. We note, for
completeness, that Eq. (19) suggests that the random boundary potential is a
marginal perturbation exactly at the de-pinning phase transition for the Ising
domain wall (which has $z=2$ [53]). A Wilsonian renormalization-group
calculation to higher order further suggests that the disorder is marginally
_relevant_ [59]. The nature of the resulting critical point is not understood,
and deserves further investigation.

###  III.4 Perfect information protection using scrambling

In the low-dissipation phase of the coding transition, quantum information is
only partially protected. One would expect that the information protection can
be improved by first scrambling the information with unitary gates, which can
effectively act like a random encoding, before the dissipation is turned on;
we refer to this as a “pre-scrambling” step. Here we argue that for fixed
system size $L$ and dissipation strength $p$, scrambling the initially local
quantum information via a random unitary circuit of logarithmic depth
$t_{\mathrm{scr}}=k\log L$ for some sufficiently large $k$, can lead to
perfect protection of quantum information within the system, up to times of
order $T\sim L/p$. For a pre-scrambling step with a fixed depth
$t_{\mathrm{scr}}=k\log L$ and for low $k$, we can observe the coding
transition by tuning the dissipation strength $p$. The coding transition will
now be manifest in a step-function-like behavior of the mutual information
$I_{A,R}$ across the transition due to the perfect preservation of information
for sufficiently low dissipation.

Figure 7: The behavior of the Ising domain wall in the presence of a pre-
scrambling step, whereby the initially local quantum information is evolved by
a quantum circuit of depth $t_{\mathrm{scr}}$. We consider propagation of the
domain wall backwards in time, with respect to the arrow of time in the
quantum circuit. In this picture, trajectories of the domain wall which are
survive in the bulk into the pre-scrambling step (right) are exponentially
suppressed relative to trajectories which are annihilated at the boundary
beforehand (left).

To gain some intuition for this result, we again consider the statistical
mechanics of the Ising domain wall. As before, the domain wall is naturally
thought of as propagating in a direction which is opposite to the arrow of
time in the quantum circuit evolution. The domain wall thus propagates through
$T$ timesteps of the circuit involving boundary dissipation, and then
encounters the pre-scrambling step where the dissipation is absent. This
corresponds to free evolution of the domain wall without the symmetry-breaking
field at the boundary. When this field at the boundary is turned off,
trajectories of the domain wall which have already been annihilated at the
boundary – such as the one shown in the left panel of Fig. 7 – do not cost
additional weights in the partition sum. On the other hand, “surviving” domain
wall trajectories in the bulk – such as the one shown in the right panel of
Fig. 7 – incur a weight of $q/(q^{2}+1)$ at each time step. Thus the weights
of the bulk trajectories of the domain wall are exponentially suppressed in
time relative to trajectories terminating at the boundary.

Let $Z_{a}(t,T)$ be the partition function for the Ising domain wall, after
the $T$ timesteps of the dynamics with dissipation have taken place, followed
by an additional $t$ timesteps of pre-scrambling, and so that the domain wall
has been annihilated at the boundary of the system. In contrast, let
$Z_{b}(t,T)$ be the partition function for the Ising domain wall to “survive”
in the bulk of the system after the same evolution. To determine the behavior
of the annealed mutual information, we wish to determine the probability that
the domain wall ends at position $x\geq x_{0}$ after another $t$ steps of the
dissipation-free evolution, as per Eq. (13), where $x_{0}$ is the location of
the entangled reference qubit of quantum information. For simplicity of
presentation, we take $x_{0}$ to be at the boundary of the qubit chain, so
that this probability $P(t,T)$ is

$\displaystyle P(t,T)$
$\displaystyle=\frac{Z_{b}(t,T)}{Z_{a}(t,T)+Z_{b}(t,T)}$ (20)

To make progress, we note that since the “surviving” trajectories contributing
to $Z_{b}(t,T)$ are exponentially suppressed in time, we may write that
$Z_{b}(t,T)=Z_{b}(0,T)e^{-\gamma t}$, where $\gamma$ is a phenomenological
decay rate which will be a function of the local Hilbert space dimension, and
the dissipation strength. We further approximate the partition sum
$Z_{a}(t,T)$ by its value before the pre-scrambling step, so that
$Z_{a}(t,T)=Z_{a}(0,T)$. With these approximations, we may write

$\displaystyle P(t,T)$ $\displaystyle=\frac{P(0,T)}{P(0,T)+[1-P(0,T)]e^{\gamma
t}}$ (21)

The annealed mutual information is now obtained from Eq. (13). At sufficiently
long times, so that $P(t,T)\ll 1$, we thus find that the mutual information
deviates from its maximal value by

$\displaystyle
2-I^{(\mathrm{ann})}_{A,R}(t)=\frac{q^{2}-1}{q}\cdot\frac{P(0,T)}{P(0,T)+[1-P(0,T)]e^{\gamma
t}}$ (22)

In the pinned phase of the domain wall, we expect $P(0,T)$ is exponentially
small in the number of timesteps $T$. In contrast, in the de-pinned phase, the
probability that the domain wall has been annihilated at the interface decays
as a power-law in time due to the diffusive nature of the Ising domain wall,
so that $P(0,T)=1-O(T^{-a})$, with $a$ a constant. For fixed $T$, we thus find
that for a sufficiently long pre-scrambling time $t$, the mutual information
deviates from its maximal value as

$\displaystyle 2-I^{(\mathrm{ann})}_{A,R}(t)\sim\begin{cases}e^{-\gamma
t}&p<p_{c}\\\ T^{a}e^{-\gamma t}&p>p_{c}\end{cases}.$ (23)

Evaluating this expression at the scrambling time $t_{\mathrm{scr}}=k\log L$
yields

$\displaystyle 2-I_{A,R}^{\mathrm{(ann)}}(t)\sim\begin{cases}L^{-\gamma
k}&p<p_{c}\\\ L^{a-\gamma k}&p>p_{c}\end{cases}.$ (24)

$\begin{array}[]{c}\includegraphics[width=433.62pt]{log_scram_final.pdf}\\\
\text{(a)}\\\ \includegraphics[width=433.62pt]{Imutual_logscrambling.pdf}\\\
\text{(b)}\end{array}$

Figure 8: Coding transition with logarithmic-depth pre-scrambling. In (a),
$I_{A,R}^{\mathrm{(ann)}}$ vs $p$ is plotted with a pre-scrambling circuit of
depth $t_{\mathrm{scr}}\sim\log L$. The subsequent evolution with dissipation
proceeds for a total number of timesteps $T=L$. The main plot is for
$t_{\mathrm{scr}}=4\log_{2}(L)$. The annealed mutual information approaches
the maximum value as $L$ is increased indicating that logarithmic-depth
encoding is enough to protect the information against boundary dissipation.
Inset shows the plot for $t_{\mathrm{scr}}=\log_{2}(L)$ with
$I_{A,R}^{\mathrm{(ann)}}$ going through a transition with respect to $p$. The
results agree with eq. (24) derived in the main text. In (b), the mutual
information, as calculated in Clifford dynamics, for dynamics with pre-
scrambling of depth $t_{\mathrm{scr}}=\log_{2}(L)$, plotted as a function of
dissipation strength $p$. Boundary dissipation is realized as a random erasure
channel, and $T/L=1/2$ is kept fixed for different system sizes. The mutual
information reveals a phase transition, with the critical point appearing as a
crossing point of the data for different system sizes.

The above calculation implies that for $t_{\mathrm{scr}}=k\log L$, with $k$
large enough, quantum information is perfectly preserved. Logarithmic
scrambling is enough to protect the information against noise. For low values
of $k$, the mutual information can exhibit different behavior depending on
whether $a-\gamma k$ is positive or negative. We show the results obtained
from studying the annealed MI numerically in Fig. 8a, and find good agreements
with the considerations above.

We now turn to the simulation of Clifford quantum circuit dynamics. To explore
how logarithmic pre-scrambling affects the coding transition induced by a
single boundary, we modify the circuit protocol to include a unitary, non-
dissipative pre-scrambling step, with pre-scrambling time scaling
logarithmically with system size, $t_{\mathrm{scr}}=k\log L$, before applying
the dissipative dynamics for time $T$. We then approach the thermodynamic
limit by increasing $T$ and $L$, while keeping the aspect ratio $T/L<1$ fixed.
In accordance with the insights gained above from the annealed Haar average,
we find a phase transition for $k=1$ as a function of $p$ between a phase
retaining information between the input and output of the circuit, and a phase
with all information destroyed by dissipation, as shown in Fig. 8b. The
critical properties are different from the case without pre-scrambling
discussed in the previous subsection, and, as predicted by the annealed model,
the critical point is signaled by a crossing point in the mutual information
obtained for different system sizes. We find a similar coding transition for
$k\leq k_{\rm max}$, with $k_{\rm max}\sim O(1)$. For even larger values of
$k$, the mutual information remains maximal for all values of $p$.

##  IV Coding transition on the approach to thermalization

In the previous section, we studied systems of size $L$ with dissipation
acting near the left boundary in the regime $T\lesssim L$ so that the right
boundary did not play a role in the dynamics. More precisely, as long as $L/T$
remains larger than the velocity of the entanglement domain wall, which is
less than the lightcone velocity in the quantum circuit, the coding transition
can be understood as a depinning transition of the domain wall, such that for
noise rate $p$ below the critical value $p_{c}$ some amount of information
survives.

In this section, we study what happens when the dynamics in the coding phase
extend for even longer periods of time, and show that the surviving
information will eventually be lost to the environment as the system
completely thermalizes. We may understand this result by considering the
dynamics of the Ising domain wall, which describes the behavior of the
annealed mutual information. For sufficiently large $T/L$ the domain wall will
escape and get annihilated at the right boundary. Thus using eq. (13)
$I_{A,R}^{\mathrm{(ann)}}$ becomes zero and the information gets leaked to the
environment. Intuitively speaking, the system gets entangled with $pT$ number
of environment qubits, and when $pT\gtrsim L$ the system gets maximally
entangled with the environment and become thermalized. By the monogamy of
entanglement, the reference qudits can no longer be entangled with the system
but are lost to the environment. Therefore for large $T/L$ there is a
transition with respect to the dissipation strength $p$, and the location of
the critical point scales as $p_{d}\sim T/L$; for $p>p_{d}$ the information
gets completely entangled with the environment. This transition is also
visible with respect to $T$ and fixed dissipation strength $p$.

We study this coding transition by performing $t_{\mathrm{scr}}=L$ steps of
pre-scrambling before turning on the noise. As explained in the previous
section, linear pre-scrambling perfectly protects the information for all
strengths of dissipation, and when $T/L$ is sufficiently small. This pre-
scrambling step has the effect of making the transition appear as a “step
function” in the mutual information $I_{A,R}$ as a function of dissipation
strength. Indeed, $I_{A,R}^{(\mathrm{ann})}(T)$ vs $p$ for $T/L=4$ in Haar
random circuit in Fig. 9 shows such a behavior, and appears to be a scaling
function of $(p-p_{d})L$ (see inset).

Figure 9: Plot of $I_{A,R}^{(\mathrm{ann})}$ in Haar random circuits. $T/L=4$
and $t_{\mathrm{scr}}=L$. Inset. The data collapse to a single curve as a
function of $(p-p_{d})L$.

###  IV.1 Numerical Study

We also verify the above transition in the Clifford circuit setting introduced
in the previous section. Here, after initializing the Bell pair at the left
boundary of the chain, we run a pre-scrambling step linear in system size,
$t_{\mathrm{scr}}=L$, followed by the dissipative dynamics applied for time
$T$. As before, we examine the finite size scaling by increasing $T$ and $L$,
while keeping $T/L>1$ fixed. As already discussed in the annealed framework,
we find a phase transition for large enough aspect ratio $T/L>1$. In Fig. 10a,
we plot the mutual information between the reference qubit and the output of
the circuit as a function of $p$ for different system sizes $L$, using a pre-
scrambling time $t_{\mathrm{scr}}=L$ and aspect ratio $T/L=4$. In perfect
agreement with the annealed picture, the mutual information curve approaches a
step function in the thermodynamic limit, confirming a phase transition
between a phase with all the information protected, and a phase with all
information destroyed.

We find a good scaling collapse with the scaling function depending on
$(p-p_{d})L^{1/2}$, see Fig. 10b. The form of the scaling function differs
from the annealed result. This deviation can be understood by noting that for
the annealed case we applied a deterministic boundary depolarization channel,
Eq. (1) whereas the dissipation in the Clifford circuit is applied at random
time steps, and this disorder may change the properties of the transition.
Indeed, the effect of randomness in the dissipation channel can be studied by
introducing disorder into the annealed model and applying channel (1) at
random times which leads to scaling function depending on $(p-p_{d})L^{1/2}$
(data not shown), in perfect agreement with the Clifford circuit results. The
discrepancy between the factor of $L$ and $L^{1/2}$ can be understood as
follows. With randomness, the number of environment qubits entangled with the
system increase linearly with $T$ but has fluctuations of order $\sqrt{T}$.
This results in the critical point fluctuating as $\delta p/\sqrt{T}$ leading
to $(p-p_{d})L^{1/2}$ dependence of the mutual information.

Figure 10: Coding transition upon approaching thermalization. (a) Mutual
information between the input and the output of the circuit shown as a
function of dissipation strength $p$, converging towards a step function in
the thermodynamic limit. Pre-scrambling time is set to $t_{\mathrm{scr}}=L$,
followed by dissipative dynamics for time $T$, with $T/L=4$ fixed. (b) Data
collapse as a function of $(p-p_{d})L^{1/2}$, with the critical point
$p_{d}=0.136$ corresponding to the crossing point of finite size data.

###  IV.2 Nature of the Phase Transition

We end this section by discussing the nature of the transition explored above.
We argue below that coding transition in this regime is a first-order phase
transition.

To begin with, let us consider the large qudit limit such that
$1/q\ll(1-p)^{2}$. The partition function in the annealed picture contains
contributions coming from all possible trajectories of the domain wall. The
contribution at time $t$ from trajectories having the domain wall at $n_{DW}$
number of time steps is of order $(1/q)^{n_{DW}}((1-p)^{2})^{t-n_{DW}}$. The
entropic factor, due to there being more configurations with the domain wall
as opposed to without it, can only renormalize the $1/q$ factor. Thus the
partition function is dominated by the term having no domain wall at any point
of time, $(1-p)^{2t}$. However, for $(1-p)^{2t}>(1/q)^{L}$, it is preferable
for the domain wall to go all the way to the right boundary and get
annihilated there. Thus at $t_{c}\sim\frac{\log 1/q}{\log(1-p)}L$ the nature
of the domain wall changes discontinuously from being stuck at the noisy
boundary to getting annihilated at the un-noisy boundary indicating a first-
order transition. The finite $q$ corrections to the above picture only act as
thermal fluctuations which causes the domain wall to have some excursions
inside the bulk. The contributions from these excursions will be sub-leading
and we expect the transition to remain first-order. Note that similar time
scales were also identified in [30] for the system to become perfectly
thermalized in the presence of noise.

As in the standard theory of first-order phase transition, the two boundaries
correspond to the two local minima for the domain wall and the system
discontinuously jumps from one to another. The mutual information then is a
function of the probability that the system is in one of the two minima (see
eq. (13)). Since the free energy is extensive, the probability of being in a
particular minimum scales as a function of $\delta gV$ where $\delta g$ is the
tuning parameter for the transition and $V$ is the total volume of the system.
In our case, the volume is equal to $T$. This explains the observed finite-
size collapse as a function of $(p-p_{d})T$ in Fig. 9.

##  V Encoding at a Finite Rate

So far we looked into the dynamics of a single bell pair localized near the
noisy boundary. But it is equally interesting to understand the effects of the
noise when we have an extensive number of Bell pairs in the initial state. We
denote the code rate, defined as the fraction of the system’s qubits entangled
in Bell pairs, by $C=N_{R}/L$ where $N_{R}$ is the total number of Bell pairs.
For the purpose of this section, we will consider code density $C=1/2$ but we
believe that the qualitative results should not change for different values of
$C$ as long as $C$ is not close to $1$. To make the final results independent
of the distribution of the Bell pairs at the initial time we will perform
random encoding by performing unitary scrambling for time
$t_{\mathrm{scr}}=L$.

We plot the annealed mutual information between the input and output,
$I_{A,R}^{(\mathrm{ann})}$, in Fig. 11 as a function of the dissipation
strength for $T=7L$. We find two threshold values for the noise rate,
$p_{th,1},p_{th,2}$. For $p<p_{th,1}$, the information is perfectly protected
and $I_{A,R}^{\mathrm{(ann)}}$ is equal to the maximal value $2CL$. For
$p_{th,1}<p<p_{th,2}$, the information starts leaking to the environment but
still a finite density of it remains in the system. Finally when $p>p_{th,2}$
the information is completely leaked to the environment. Note that the values
of $p_{th}$ change with the ratio $T/L$.

Similarly to the strategy followed in the previous sections, we verify these
predictions by performing numerical simulations in Clifford quantum random
circuits. We show the density of the mutual information between the output of
the circuit $A$ and the reference qubits, $I_{A,R}/N_{R}$, with $N_{R}=L/2$
denoting the number of input Bell pairs, as a function of dissipation strength
$p$ in Fig. 12, for different system sizes $L$ with $T/L=4$ fixed. As noted
above, here we applied a linear unitary pre-scrambling step for time
$t_{\mathrm{scr}}=L$, before the onset of the noisy dynamics, such that the
results do not depend on the spatial distribution of the Bell pairs in the
initial state. We find a phase with perfectly protected information for small
enough dissipation strength $p$, followed by a crossover region with a finite
density of preserved coherent information decreasing continuously with $p$,
eventually decaying to zero for large $p$.

$\begin{array}[]{c}\includegraphics[width=433.62pt]{Extensive_Coding_Rate.pdf}\\\
\\\ \includegraphics[width=359.90538pt]{finite_rate_figure.pdf}\end{array}$

Figure 11: Top. Schematic representation of the statistical mechanics of the
Ising domain wall in the calculation of the annealed mutual information, when
coding at a finite rate. Typical domain wall trajectories when
$p_{th,1}<p<p_{th,2}$ are shown. In $Z_{\Downarrow}$ the domain wall remains
localized whereas it is delocalized for $Z_{\Uparrow}$, as explained in the
text. Bottom. Plot of the annealed mutual information between Bell pairs
entangled with the system’s qubits at alternate sites ($C=1/2$) and the
system. The Bell pairs are scrambled by a unitary circuit for time
$t_{\mathrm{scr}}=L$. The system is evolved in presence of the boundary
dissipation for time $T=7L$. We find that for $p<p_{th}^{1}\approx 0.06$, full
information is preserved, while for $p_{th}^{1}<p<p_{th}^{2}\approx 0.2$, a
finite density of information is protected. The threshold values decrease as
$T$ is increased. Inset. For low $p<p_{th}^{1}$ there is no information loss
even for $T=7L$, that is, the difference between $I_{A,R}^{\mathrm{(ann)}}$
and the maximum value $L$ goes to zero with system size. Thus all Bell pairs
can be perfectly recovered by a recovery operation acting on the system.
Figure 12: Coding transition for finite code rate. Density of mutual
information between the output of the circuit and the reference qubits shown
as a function of dissipation strength $p$, for fixed evolution time $T/L=4$
and number of initial Bell pairs $N_{R}=L/2$. Pre-scrambling time is
$t_{\mathrm{scr}}=L$, followed by noisy dynamics with a random boundary
erasure channel. The information density is perfectly protected for weak
enough dissipation $p$, then decays continuously towards zero with $p$ in a
crossover region, with all information leaked to the environment for $p$ large
enough.

To understand this behavior we again resort to the statistical mechanics of
the Ising domain wall. This model for a finite code rate differs importantly
from that of the model when an $O(1)$ amount of quantum information is
encoded. In the case of finite coding rate there are an extensive number of
Ising spins at the top boundary whose state is fixed by the boundary
conditions, though the bulk dynamics of the domain wall remain the same. This
leads to an exponential amplification of the trajectories that minimize the
number of domain walls at the top boundary (note that these domain walls at
the boundary are different than the Ising domain wall performing random walk
in the bulk). As shown at top of Fig. 11, the annealed $I$ is given by

$\displaystyle
I_{A,R}^{\mathrm{(ann)}}=CL+\log\left(\frac{Z_{\Downarrow}}{Z_{\Uparrow}}\right)$
(25)

where $Z_{\Downarrow},Z_{\Uparrow}$, are the partition function of the
statistical mechanics model with down and up spins respectively at the
locations of the encoded Bell pairs; the log is in the base of $q$. As
discussed in Sec. IV the domain wall discontinuously changes from being at the
left boundary to being at the right boundary. To a good approximation, we can
thus only keep these two trajectories in the partition function. For clarity
of the expressions we also introduce $\tilde{p}\equiv 1-p$. The partition
functions $Z_{\Downarrow},Z_{\Uparrow}$ can thus be written as

$\displaystyle
Z_{\Downarrow}\approx\tilde{p}^{2T}q^{2CL}+\left(\frac{1}{q}\right)^{L}q^{CL}$
(26) $\displaystyle
Z_{\Uparrow}\approx\tilde{p}^{2T}q^{CL}+\left(\frac{1}{q}\right)^{L}q^{2CL}$
(27)

Putting the above expression in eq. (25) and identifying the threshold values
to be $1-p_{th,1}\sim q^{-(1-C)L/(2T)},1-p_{th,2}\sim q^{-(1+C)L/(2T)}$, we
get

$\displaystyle I_{A,R}^{\mathrm{(ann)}}\approx\begin{cases}2CL&p<p_{th,1}\\\
2CL-2T\log\left(\frac{1-p_{th,1}}{1-p}\right)&p_{th,1}<p<p_{th,2}\\\
0&p>p_{th,2}.\end{cases}$ (28)

Intuitively, for low $p$ the domain wall remains localized near the noisy
boundary and mutual information is maximal. As $p$ is increased, it is easier
for the DW in $Z_{\Uparrow}$ to delocalize compared to $Z_{\Downarrow}$ as in
the former delocalization results in an exponential reduction in the cost
associated with having domain walls at the boundary. Thus the critical point
at which the DW delocalizes is different for the two boundary conditions
resulting in the two thresholds discussed above.

##  VI Summary and Discussion

In this work, we studied one-dimensional quantum many-body systems with a
noisy boundary. We focused on the dynamics of the information of an initially
localized Bell pair near the (noisy) boundary by studying the mutual
information $I_{A,R}(t)$ between the inert spin of the Bell pair with the
system at later times where $A$ is the system and $R$ is the inert spin. This
is also related to the coherent information about the Bell pair remaining in
the system [49, 50]. We find that the chaotic scrambling due to the unitary
dynamics is sufficient to protect a part of this information against getting
leaked to the environment for noise rate $p<p_{c}$ and long times $T\lesssim
L/p$ by allowing the information to escape away from the boundary. We further
show that a random encoding of the Bell pair via noise-less scrambling
dynamics of depth $\mathcal{O}(\log L)$, is sufficient to perfectly protect
the information for all strengths of the noise upto time $T\lesssim L/p$. See
Fig. 1.b for a schematic representation of the phase diagram.

In the regime when the total time of evolution $T\gtrsim L/p$, any remaining
information in the system is revealed to the environment and the system go
through a first-order coding transition. This transition can also be seen as a
result of the system approaching thermalization to infinite temperature. We
expect this form of coding transition to be present in all noisy channels
though in the case of the boundary noise considered here, the timescales
associated with the transition increase parametrically with the system size
[30].

We also look at the coding dynamics for finite code rate, that is, when an
extensive number $N_{R}=CL$, with $C<1$, of the system’s qubits are entangled
in Bell pairs. We find that the code space can be perfectly preserved for
noise strength below some threshold $p_{th,1}$ and for strength above
$p_{th,2}$ the code space is completely destroyed, see Fig. 11, 12. We can
also look at the time for which the information stays in the system for a
fixed noise rate $p$ and equivalently define two threshold times
$T_{th,1}<T_{th,2}$ both of which scales linearly with system size.

This work provides new insights into the competition between scrambling and
decoherence. Normally, active feedback in the form of error correction is
needed to counter the decoherence effects of the noise. However, we present
the case of boundary noise where it is possible to have stable quantum error
codes (QEC) in presence of generic noise, with the code space dynamically
protected by scrambling. Previously such dynamical protection of information
was also observed for the special case of dephasing noise which can be
unraveled into quantum trajectories corresponding to projective measurements,
but there an extensive number of ancilla qubits that act as register for the
measurement outcomes are made part of the system [27]. It would be of interest
to generalize our results and techniques in the presence of ancilla qubits for
cases different from the boundary noise. We leave this for future work.

Other interesting directions to explore are the presence of similar coding
transitions in purely unitary evolution. It seems possible for quantum
information to remain confined in part of a system evolving under chaotic
unitary dynamics for a long time, and before the system thermalizes. We leave
a detailed discussion of this direction to future work [60].

The competition between chaos and decoherence has also been studied in the
context of open quantum systems. Previous studies have mostly focussed on
level statistics and quantities like spectral form factor, purity, and
Loschmidt echo to study the effect of decoherence in chaotic dynamics [61, 62,
63, 64, 65, 66, 67, 68]. It is an open question to study such probes in our
context and whether the coding transitions can also be seen in these
quantities. There is also a close relationship between the input-output mutual
information and operator spreading (measured via out-of-time-correlators
(OTOCs)) in noise-free unitary dynamics [4]. It is interesting to understand
how OTOCs in noisy systems are related to the emergent QEC property of the
noisy dynamics [69, 70, 71]. Or more generally, how is the dynamics of
information related to the above-mentioned quantities for open quantum
systems?

The coding transitions imply protection of the code-space against noise and
the potential existence of a decoding protocol that brings back the code-space
to its initial state. Such a protocol is notoriously hard for random dynamics
having little structure, except in a few special cases like the Preskill-
Hayden black hole protocol [1, 72] or for special types of noises like erasure
channel. For Clifford circuits with boundary dissipation considered here, an
efficient decoder can probably be constructed for the erasure channel. Another
interesting direction in further understanding the error-correcting properties
of the coding transitions is to look into the code distance of the resulting
code. We leave a detailed study of the decoding protocols and code distance
for future studies.

We also find similar coding transitions for bulk defects where noise acts on
the same site in the bulk. Protection of quantum information against bulk
defects is important for the design of modular quantum computers in which
smaller modules of quantum memory/computer are connected together to form a
bigger block. In this case, one expects the noise in the gates connecting the
two modules to be far greater than the noise in the bulk of the individual
modules. Thus the existence of an error-threshold against a bulk defect and
the availability of the decoding protocol discussed above gives a fault-
tolerant way of building a modular quantum computer.

A possible extension of our work is to study information dynamics in noisy
symmetric systems. The behavior of information in symmetric systems with local
charge density in presence of measurements has been shown to be qualitatively
different than without symmetry [73, 74, 75, 76]. It is also known that
systems with local charge conservation can have charge transport and long-time
operator entanglement growth even in the presence of strong dephasing noise
[77, 78]. This may potentially lead to a more robust encoding of the
information for when the code-space is spread across different charge sectors
as opposed to being confined to one sector. We leave this for future studies.

###### Acknowledgements.

The authors thank the Kavli Institute for Theoretical Physics (KITP), where
this research was initiated and partly performed. The KITP is supported, in
part, by the National Science Foundation under Grant No. NSF PHY-1748958. S.V.
thanks Matthew Fisher for helpful discussions. U.A. thanks Ali Lavasani for
helpful discussions. I.L. acknowledges support from the Gordon and Betty Moore
Foundation through Grant GBMF8690 to UCSB. This work was supported by the
Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons
Foundation (651440, U.A.).

## References

  * Hayden and Preskill [2007] P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP 09, 120.
  * Sekino and Susskind [2008] Y. Sekino and L. Susskind, Fast scramblers, JHEP 10, 065.
  * Lashkari  _et al._ [2013] N. Lashkari, D. Stanford, M. Hastings, T. Osborne, and P. Hayden, Towards the fast scrambling conjecture, Journal of High Energy Physics 2013, 22 (2013), arXiv:1111.6580 [hep-th] .
  * Hos [2016] Chaos in quantum channels, Journal of High Energy Physics 2016, 1 (2016).
  * Carr [2010] L. Carr,  _Understanding Quantum Phase Transitions_ (CRC Press, 2010).
  * Breuer and Petruccione [2002] H. P. Breuer and F. Petruccione,  _The theory of open quantum systems_ (Oxford University Press, Great Clarendon Street, 2002).
  * Müller  _et al._ [2012] M. Müller, S. Diehl, G. Pupillo, and P. Zoller, Engineered open systems and quantum simulations with atoms and ions, in  _Advances in Atomic, Molecular, and Optical Physics_, Advances In Atomic, Molecular, and Optical Physics, Vol. 61, edited by P. Berman, E. Arimondo, and C. Lin (Academic Press, 2012) pp. 1–80.
  * Carusotto and Ciuti [2013] I. Carusotto and C. Ciuti, Quantum fluids of light, Rev. Mod. Phys. 85, 299 (2013).
  * Daley [2014] A. J. Daley, Quantum trajectories and open many-body quantum systems, Advances in Physics 63, 77 (2014), https://doi.org/10.1080/00018732.2014.933502 .
  * Maldacena  _et al._ [2016] J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, Journal of High Energy Physics 2016, 106 (2016), arXiv:1503.01409 [hep-th] .
  * Xu  _et al._ [2020] T. Xu, T. Scaffidi, and X. Cao, Does scrambling equal chaos?, Phys. Rev. Lett. 124, 140602 (2020).
  * Nahum  _et al._ [2017] A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum Entanglement Growth under Random Unitary Dynamics, Physical Review X 7, 031016 (2017), arXiv:1608.06950 [cond-mat.stat-mech] .
  * Nahum  _et al._ [2018] A. Nahum, S. Vijay, and J. Haah, Operator Spreading in Random Unitary Circuits, Physical Review X 8, 021014 (2018), arXiv:1705.08975 [cond-mat.str-el] .
  * von Keyserlingk  _et al._ [2018] C. W. von Keyserlingk, T. Rakovszky, F. Pollmann, and S. L. Sondhi, Operator Hydrodynamics, OTOCs, and Entanglement Growth in Systems without Conservation Laws, Physical Review X 8, 021013 (2018), arXiv:1705.08910 [cond-mat.str-el] .
  * Mezei and Stanford [2017] M. Mezei and D. Stanford, On entanglement spreading in chaotic systems, Journal of High Energy Physics 2017, 65 (2017), arXiv:1608.05101 [hep-th] .
  * Luitz and Bar Lev [2017] D. J. Luitz and Y. Bar Lev, Information propagation in isolated quantum systems, Phys. Rev. B 96, 020406 (2017).
  * Shor [1995] P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, R2493 (1995).
  * Steane [1996] A. M. Steane, Error correcting codes in quantum theory, Phys. Rev. Lett. 77, 793 (1996).
  * Bennett  _et al._ [1996] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Mixed-state entanglement and quantum error correction, Phys. Rev. A 54, 3824 (1996).
  * Knill and Laflamme [1997] E. Knill and R. Laflamme, Theory of quantum error-correcting codes, Phys. Rev. A 55, 900 (1997).
  * Knill  _et al._ [1998] E. Knill, R. Laflamme, and W. H. Zurek, Resilient quantum computation: error models and thresholds, Proceedings of the Royal Society of London Series A 454, 365 (1998), arXiv:quant-ph/9702058 [quant-ph] .
  * Aharonov and Ben-Or [1999] D. Aharonov and M. Ben-Or, Fault-Tolerant Quantum Computation With Constant Error Rate, arXiv e-prints , quant-ph/9906129 (1999), arXiv:quant-ph/9906129 [quant-ph] .
  * Shor [1996] P. Shor, Fault-tolerant quantum computation, in  _Proceedings of 37th Conference on Foundations of Computer Science_ (1996) pp. 56–65.
  * Li  _et al._ [2018] Y. Li, X. Chen, and M. P. A. Fisher, Quantum Zeno effect and the many-body entanglement transition, Phys. Rev. B 98, 205136 (2018), arXiv:1808.06134 [quant-ph] .
  * Skinner  _et al._ [2019] B. Skinner, J. Ruhman, and A. Nahum, Measurement-Induced Phase Transitions in the Dynamics of Entanglement, Physical Review X 9, 031009 (2019), arXiv:1808.05953 [cond-mat.stat-mech] .
  * Chan  _et al._ [2019] A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, Unitary-projective entanglement dynamics, Phys. Rev. B 99, 224307 (2019), arXiv:1808.05949 [cond-mat.stat-mech] .
  * Gullans and Huse [2020] M. J. Gullans and D. A. Huse, Dynamical Purification Phase Transition Induced by Quantum Measurements, Physical Review X 10, 041020 (2020), arXiv:1905.05195 [quant-ph] .
  * Choi  _et al._ [2020] S. Choi, Y. Bao, X.-L. Qi, and E. Altman, Quantum Error Correction in Scrambling Dynamics and Measurement-Induced Phase Transition, Phys. Rev. Lett. 125, 030505 (2020), arXiv:1903.05124 [quant-ph] .
  * Noh  _et al._ [2020] K. Noh, L. Jiang, and B. Fefferman, Efficient classical simulation of noisy random quantum circuits in one dimension, Quantum 4, 318 (2020).
  * Li  _et al._ [2023a] Z. Li, S. Sang, and T. H. Hsieh, Entanglement dynamics of noisy random circuits, Phys. Rev. B 107, 014307 (2023a).
  * Shannon [1948] C. E. Shannon, A mathematical theory of communication, The Bell System Technical Journal 27, 379 (1948).
  * Gallager [1962] R. Gallager, Low-density parity-check codes, IRE Transactions on Information Theory 8, 21 (1962).
  * Gallager [1973] R. Gallager, The random coding bound is tight for the average code (corresp.), IEEE Transactions on Information Theory 19, 244 (1973).
  * CodesDavid  _et al._ [1996] CodesDavid, C. J., and MacKayCavendish, Near shannon limit performance of low density parity check codes (1996).
  * Richardson and Urbanke [2001] T. Richardson and R. Urbanke, Efficient encoding of low-density parity-check codes, IEEE Transactions on Information Theory 47, 638 (2001).
  * Brown and Fawzi [2013] W. Brown and O. Fawzi, Short random circuits define good quantum error correcting codes, in  _2013 IEEE International Symposium on Information Theory_ (IEEE, 2013).
  * Brown and Fawzi [2015] W. Brown and O. Fawzi, Decoupling with random quantum circuits, Communications in Mathematical Physics 340, 867 (2015).
  * Gullans  _et al._ [2021] M. J. Gullans, S. Krastanov, D. A. Huse, L. Jiang, and S. T. Flammia, Quantum coding with low-depth random circuits, Phys. Rev. X 11, 031066 (2021).
  * Fisher  _et al._ [2022] M. P. A. Fisher, V. Khemani, A. Nahum, and S. Vijay, Random quantum circuits, Annual Review of Condensed Matter Physics 14 (2022).
  * Potter and Vasseur [2022] A. C. Potter and R. Vasseur, Entanglement dynamics in hybrid quantum circuits, in  _Entanglement in Spin Chains: From Theory to Quantum Technology Applications_ (Springer, 2022) pp. 211–249.
  * Jian  _et al._ [2021] S.-K. Jian, C. Liu, X. Chen, B. Swingle, and P. Zhang, Quantum error as an emergent magnetic field, arXiv e-prints , arXiv:2106.09635 (2021), arXiv:2106.09635 [quant-ph] .
  * Li and Fisher [2021] Y. Li and M. P. A. Fisher, Robust decoding in monitored dynamics of open quantum systems with Z_2 symmetry, arXiv e-prints , arXiv:2108.04274 (2021), arXiv:2108.04274 [quant-ph] .
  * Sá  _et al._ [2020] L. Sá, P. Ribeiro, T. Can, and T. c. v. Prosen, Spectral transitions and universal steady states in random kraus maps and circuits, Phys. Rev. B 102, 134310 (2020).
  * Weinstein  _et al._ [2022] Z. Weinstein, Y. Bao, and E. Altman, Measurement-induced power-law negativity in an open monitored quantum circuit, Physical Review Letters 129, 10.1103/physrevlett.129.080501 (2022).
  * Fan  _et al._ [2021] R. Fan, S. Vijay, A. Vishwanath, and Y.-Z. You, Self-organized error correction in random unitary circuits with measurement, Physical Review B 103, 174309 (2021).
  * Li and Fisher [2021] Y. Li and M. P. A. Fisher, Statistical mechanics of quantum error correcting codes, Physical Review B 103, 104306 (2021).
  * Bao  _et al._ [2020] Y. Bao, S. Choi, and E. Altman, Theory of the phase transition in random unitary circuits with measurements, Phys. Rev. B 101, 104301 (2020), arXiv:1908.04305 [cond-mat.stat-mech] .
  * Li  _et al._ [2023b] Y. Li, S. Vijay, and M. P. A. Fisher, Entanglement domain walls in monitored quantum circuits and the directed polymer in a random environment, PRX Quantum 4, 010331 (2023b).
  * Schumacher and Nielsen [1996] B. Schumacher and M. A. Nielsen, Quantum data processing and error correction, Phys. Rev. A 54, 2629 (1996).
  * Schumacher and Westmoreland [2001] B. Schumacher and M. D. Westmoreland, Approximate quantum error correction, arXiv e-prints , quant-ph/0112106 (2001), arXiv:quant-ph/0112106 [quant-ph] .
  * Abraham [1980] D. Abraham, Solvable model with a roughening transition for a planar ising ferromagnet, Physical Review Letters 44, 1165 (1980).
  * Abraham [1981] D. Abraham, Binding of a domain wall in the planar ising ferromagnet, Journal of Physics A: Mathematical and General 14, L369 (1981).
  * Chalker [1981] J. Chalker, The pinning of a domain wall by weakened bonds in two dimensions, Journal of Physics A: Mathematical and General 14, 2431 (1981).
  * Zhou and Nahum [2019] T. Zhou and A. Nahum, Emergent statistical mechanics of entanglement in random unitary circuits, Physical Review B 99, 174205 (2019).
  * Huse and Henley [1985] D. A. Huse and C. L. Henley, Pinning and roughening of domain walls in ising systems due to random impurities, Phys. Rev. Lett. 54, 2708 (1985).
  * Kardar and Zhang [1987] M. Kardar and Y.-C. Zhang, Scaling of directed polymers in random media, Phys. Rev. Lett. 58, 2087 (1987).
  * Kardar [1985] M. Kardar, Depinning by quenched randomness, Phys. Rev. Lett. 55, 2235 (1985).
  * Lipowsky and Fisher [1986] R. Lipowsky and M. E. Fisher, Wetting in random systems, Physical review letters 56, 472 (1986).
  * Vijay and Fisher [2022] S. Vijay and M. P. A. Fisher, Unpublished (2022).
  * Lovas  _et al._ [2023] I. Lovas, U. Agrawal, and S. Vijay, Unpublished (2023).
  * Kawabata  _et al._ [2022] K. Kawabata, A. Kulkarni, J. Li, T. Numasawa, and S. Ryu, Dynamical quantum phase transitions in syk lindbladians (2022), arXiv:2210.04093 [cond-mat.stat-mech] .
  * Yan  _et al._ [2020] B. Yan, L. Cincio, and W. H. Zurek, Information scrambling and loschmidt echo, Phys. Rev. Lett. 124, 160603 (2020).
  * Xu  _et al._ [2019] Z. Xu, L. P. García-Pintos, A. Chenu, and A. del Campo, Extreme decoherence and quantum chaos, Phys. Rev. Lett. 122, 014103 (2019).
  * Can [2019] T. Can, Random Lindblad dynamics, Journal of Physics A Mathematical General 52, 485302 (2019), arXiv:1902.01442 [quant-ph] .
  * Jalabert and Pastawski [2001] R. A. Jalabert and H. M. Pastawski, Environment-independent decoherence rate in classically chaotic systems, Phys. Rev. Lett. 86, 2490 (2001).
  * Karkuszewski  _et al._ [2002] Z. P. Karkuszewski, C. Jarzynski, and W. H. Zurek, Quantum chaotic environments, the butterfly effect, and decoherence, Phys. Rev. Lett. 89, 170405 (2002).
  * Cucchietti  _et al._ [2003] F. M. Cucchietti, D. A. R. Dalvit, J. P. Paz, and W. H. Zurek, Decoherence and the loschmidt echo, Phys. Rev. Lett. 91, 210403 (2003).
  * Peres [1984] A. Peres, Stability of quantum motion in chaotic and regular systems, Phys. Rev. A 30, 1610 (1984).
  * Schuster and Yao [2022] T. Schuster and N. Y. Yao, Operator growth in open quantum systems (2022), arXiv:2208.12272 [quant-ph] .
  * Zanardi and Anand [2021] P. Zanardi and N. Anand, Information scrambling and chaos in open quantum systems, Phys. Rev. A 103, 062214 (2021), arXiv:2012.13172 [quant-ph] .
  * Yoshida and Yao [2019] B. Yoshida and N. Y. Yao, Disentangling scrambling and decoherence via quantum teleportation, Phys. Rev. X 9, 011006 (2019).
  * Yoshida and Kitaev [2017] B. Yoshida and A. Kitaev, Efficient decoding for the hayden-preskill protocol (2017), arXiv:1710.03363 [hep-th] .
  * Agrawal  _et al._ [2022] U. Agrawal, A. Zabalo, K. Chen, J. H. Wilson, A. C. Potter, J. Pixley, S. Gopalakrishnan, and R. Vasseur, Entanglement and charge-sharpening transitions in u(1) symmetric monitored quantum circuits, Physical Review X 12, 10.1103/physrevx.12.041002 (2022).
  * Barratt  _et al._ [2022a] F. Barratt, U. Agrawal, A. C. Potter, S. Gopalakrishnan, and R. Vasseur, Transitions in the learnability of global charges from local measurements, Physical Review Letters 129, 10.1103/physrevlett.129.200602 (2022a).
  * Barratt  _et al._ [2022b] F. Barratt, U. Agrawal, S. Gopalakrishnan, D. A. Huse, R. Vasseur, and A. C. Potter, Field theory of charge sharpening in symmetric monitored quantum circuits, Physical Review Letters 129, 10.1103/physrevlett.129.120604 (2022b).
  * Oshima and Fuji [2023] H. Oshima and Y. Fuji, Charge fluctuation and charge-resolved entanglement in a monitored quantum circuit with U (1 ) symmetry, Phys. Rev. B 107, 014308 (2023), arXiv:2210.16009 [cond-mat.dis-nn] .
  * Wellnitz  _et al._ [2022] D. Wellnitz, G. Preisser, V. Alba, J. Dubail, and J. Schachenmayer, Rise and fall, and slow rise again, of operator entanglement under dephasing, Phys. Rev. Lett. 129, 170401 (2022).
  * Cai and Barthel [2013] Z. Cai and T. Barthel, Algebraic versus exponential decoherence in dissipative many-particle systems, Phys. Rev. Lett. 111, 150403 (2013).

##  Appendix A Lattice Partition Function and the Annealed Phase Transition

The annihilation of the Ising domain wall at the boundary, or the free
propagation of the domain wall through the bulk describe two distinct phases
which may be accessed by tuning the dissipation strength, as described in
detail in Sec. II. Here, we make this connection precise by studying the
lattice partition function for the domain wall using the weights derived in
Sec. II.1. We consider the quantum circuit evolution shown schematically in
Fig. 13a, where each site (in blue) denotes the action of a two-site unitary
gate on a qudit chain, while dissipation (in orange) acts periodically on the
boundary qudit. Let $Z(T)$ denote the partition function for the domain wall
propagating for a time $T$, defined so that at the initial and final times,
the domain wall is absent (i.e. has been annihilated at the $x=0$ interface).
This partition sum may be calculated as follows. First, we define $Z_{a}(t)$
to be the partition functions when there is no domain wall for a time interval
$t$ (it has been annihilated), while $Z_{f}(t)$ is the partition function when
the domain wall is created at the $x=0$ interface, wanders and first returns
back to the interface after a time $t$, after which it is then annihilated
(the domain wall is free). With these definitions, we observe that
$Z_{0}({T})$ is given by summing over all possible domain wall histories as

$\displaystyle Z(T)=Z_{a}(T)+Z_{f}(T)+\sum_{t<T}Z_{a}(t)Z_{f}(T-t)+\cdots$
(29)

where the ellipsis denotes all possible domain wall configurations in which,
at intermediate timesteps, the domain wall wanders away or is annihilated at
the interface.

It is convenient to consider the discrete Laplace transform of the partition
function

$\displaystyle z(w)\equiv\sum_{T\geq 0}w^{T}Z(T).$ (30)

The inverse of this transformation is given by

$\displaystyle Z(T)=\frac{1}{2\pi i}\oint_{\Gamma}dw\frac{z(w)}{w^{T+1}}$ (31)

where the contour $\Gamma$ encloses the origin in the complex $w$ plane. This
relation is easily verified by substituting Eq. (30). As a result, the
smallest real singularity of $z(w)$ – denoted $w_{*}$ – determines the
behavior of the partition function at long times. Equivalently, the free
energy density $f=-T^{-1}\log Z$ is given by

$\displaystyle f\overset{T\rightarrow\infty}{\sim}\log w_{*}$ (32)

The Laplace transform of $Z(T)$ is straightforward to evaluate, since each
term in the expansion (29) is a discrete convolution of products of $Z_{a}$
and $Z_{f}$. As a result, the Laplace transform of each term in this sum is
simply the product of the Laplace transformations of appropriate products of
$Z_{a}$ and $Z_{f}$. We thus find that

$\displaystyle z(w)$
$\displaystyle=\frac{z_{a}(w)+z_{f}(w)+2\,z_{a}(w)z_{f}(w)}{1-z_{a}(w)z_{f}(w)}$
(33)

with $z_{a}(w)$ and $z_{f}(w)$ defined as the Laplace transforms of $Z_{a}(t)$
and $Z_{f}(t)$, respectively.

Observe that $Z_{a}(t)=(1-p)^{2t}$ so that

$\displaystyle z_{a}(w)=\frac{w(1-p)^{2}}{1-w(1-p)^{2}}$ (34)

Similarly, we note that when $t\geq 2$

$\displaystyle
Z_{f}(t)=\frac{p(2-p)}{q}\left(\frac{q}{q^{2}+1}\right)^{2t-3}N_{2t-4}$ (35)

Here, $p(2-p)/q$ is the weight to create the Ising domain wall, as indicated
in Eq. (7). The domain wall is acted upon by $2t-3$ two-site unitary gates,
incurring a weight $q/(q^{2}+1)$ for the action of each gate. Finally,
$N_{2k}$ is the number of walks on the rotated square lattice – such as the
one shown in Fig. 13b – which start at a site closest to the boundary, and
which and return to the same point after $2k$ steps, without touching the
boundary. This counting of paths is easily determined to be

$\displaystyle N_{2k}=\left(\begin{array}[]{c}2k\\\
k\end{array}\right)-\left(\begin{array}[]{c}2k\\\ k+1\end{array}\right).$ (40)

Performing the Laplace transform thus yields

$\displaystyle
z_{f}(w)=\frac{p(2-p)}{2q}\frac{w(q^{2}+1)}{q}\left[1-\sqrt{1-\frac{w}{w_{1}(q)}}\right]$
(41)

which has a singularity when the argument of the square root vanishes at

$\displaystyle w_{1}(q)\equiv(q^{2}+1)^{2}/4q^{2}.$ (42)

We note that $z(w)$ is also singular at $w=w_{2}$ such that
$z_{a}(w_{2})z_{f}(w_{2})=1$. Finally, we note that while $z_{a}(w)$ contains
a pole at $w=1/(1-p)^{2}$, it is clear from (33) that this does not give rise
to a singularity in $z(w)$.

$\begin{array}[]{cc}\includegraphics[width=99.73074pt]{Ising_DW_1}&\includegraphics[width=95.39693pt]{Ising_DW_2}\\\
\text{(a)}&\text{(b)}\end{array}$

Figure 13: A depiction of the quantum circuit which is applied to the qudit
chain is shown in (a). Here, each blue vertex indicates the application of a
two-site unitary gate while the orange sites indicate the periodic application
of a single-qudit depolarizing channel. The calculation of the corresponding
Ising partition sum can be performed, with spin configurations living on bonds
of the square lattice, as in (b), and which are naturally thought of as
propagating in the indicated “time” direction by the transfer matrix for the
Ising magnet. Shown is a contribution to $Z_{f}(t=5)$, where the domain wall
is created by the dissipation at the initial time, and is annihilated four
timesteps later. The trajectory of the Ising domain wall can be thought of as
a path on the lattice, which starts from the first unitary gate which acts on
a pair of anti-aligned spins, and ends when the domain wall is annihilated.

When $p>p_{c}$, the smallest real singularity of $z(w)$ occurs at
$w=w_{1}(q)$, so that the free energy

$\displaystyle f=2\log\left(\frac{q^{2}+1}{2q}\right)\hskip
14.45377pt(p>p_{c})$ (43)

A phase transition occurs at $p=p_{c}$ when the two singularities merge
$w_{1}=w_{2}$, and for $p<p_{c}$ the singularity at $w_{*}=w_{2}$ determines
the free energy density. The phase transition therefore occurs when

$\displaystyle z_{f}(w_{1})z_{a}(w_{1})=1$ (44)

This equation may be solved numerically to obtain $p_{c}$ for any finite $q$.
The critical probability increases with increasing Hilbert space dimension
$q$. In the limit $q\rightarrow\infty$, we may analytically solve this
equation to find that $p_{c}$ approaches one as

$\displaystyle p_{c}=1-O(q^{-2})$ (45)

so that the phase transition is absent when the on-site Hilbert space
dimension is strictly infinite.

Finally, we may study the singular part of the free energy near the transition
at $p=p_{c}$. Expanding the equation $z_{f}(w_{2})z_{a}(w_{2})=1$ for
$p=p_{c}-\delta p$ with $\delta p\ll p_{c}$ yields the result that the
singularity $w_{*}=w_{2}=w_{1}-\delta w$ where $\delta w\sim(\delta p)^{2}$.
As a result, the free energy difference vanishes when approaching the critical
point as

$\displaystyle\Delta f(p)\equiv f(p_{c})-f(p)\overset{p\rightarrow
p_{c}^{-}}{\sim}(p-p_{c})^{2}$ (46)

On general grounds, the singular part of the free energy density should vanish
as $\Delta f\sim 1/\xi_{\parallel}$ where $\xi_{\parallel}$ is the correlation
length along the time direction. This correlation length thus diverges as
$\xi_{\parallel}\sim(p-p_{c})^{-\nu_{\parallel}}$ with $\nu_{\parallel}=2$.

Finally, we may determine the typical length of an excursion $\ell_{\perp}$
that the domain wall will make into the bulk of the quantum circuit, and how
this distance diverges as we approach the phase transition from the pinned
phase $p\leq p_{c}$. First, observe that the weight for the Ising domain wall
to make an excursion for a time $t$ is $Z_{f}(t)/Z(t)$. Then the typical
duration of an excursion is

$\displaystyle\tau=\frac{\displaystyle\sum_{t}t\,Z_{f}(t)/Z(t)}{\displaystyle\sum_{t}Z_{f}(t)/Z(t)}\sim\frac{\displaystyle\sum_{t}t\,w_{*}^{t}Z_{f}(t)}{\displaystyle\sum_{t}w_{*}^{t}Z_{f}(t)}=\frac{\partial\ln
Z_{f}(w)}{\partial\ln w}\Big{|}_{w=w_{*}}$

where in the second expression, we have used the fact that
$Z(t)\overset{t\rightarrow\infty}{\sim}w_{*}^{-t}$. On approaching the
transition from the localized phase $p=p_{c}-\delta p$, the singularity
$w_{*}=w_{2}=w_{1}-\delta w$ with $\delta w\sim\delta p^{2}$, as derived
previously, which yields the result that $\tau\sim(p_{c}-p)^{-1}$ as
$p\rightarrow p_{c}^{-}$. Assuming a diffusive wandering of the domain wall,
the transverse distance covered by the domain wall diverges on approaching the
depinned phase as

$\displaystyle\ell_{\perp}\overset{p\rightarrow
p_{c}^{-}}{\sim}(p_{c}-p)^{-1/2}$ (47)

Approaching the phase transition, when $\ell_{\perp}\gg x_{0}$, the
probability that the domain wall has reached a point $y\geq x_{0}$ is
approximately $P(x_{0},t)=1-O(x_{0}/\ell_{\perp})$. Substituting this into Eq.
(13) yields the result that the annealed mutual information vanishes as
$I^{(\mathrm{ann})}_{A,R}\sim\ell_{\perp}^{-1}\sim(p_{c}-p)^{\beta}$ (with
$\beta\equiv 1/2$) when approaching the phase transition. This behavior, along
with the knowledge of $\nu_{\parallel}=2$ motivates the finite-size scaling
form for the annealed mutual information, which we use in the main text
$I^{(\mathrm{ann})}_{A,R}(T)=T^{-\beta/\nu}F(T^{1/\nu}(p-p_{c}))$.

##  Appendix B Alternative random circuit protocols

To show that the phase transition in the mutual information persists
irrespective of the precise form of the boundary dissipation and scrambling
dynamics, here we introduce and examine four different protocols for the
random circuit. We consider the following two types of time evolution, each of
them with two different realizations of the boundary dissipation.

$\bullet$ Random boundary dissipation + maximal Clifford scrambling. In each
time step, the dissipation acts on the leftmost qubit with probability $p$.
Scrambling is provided by random Clifford gates arranged in a brickwork
structure. Therefore, the relative strength of the dissipation compared to the
efficiency of scrambling is tuned through the parameter $p$.

$\bullet$ Periodic boundary dissipation + sparse Clifford scrambling. The
dissipation acts on the leftmost qubit periodically, with periodicity $T_{\rm
period}$. The unitary gates providing the scrambling of information are
applied in a sparse brickwork structure, where each gate in the brickwork is a
random Clifford unitary with probability $p_{U}$, and the identity with
probability $1-p_{U}$. In this scenario, the relative strength of the
dissipation compared to the efficiency of scrambling is determined by two
parameters, $T_{\rm period}$ and $p_{U}$.

Figure 14: Coding transition induced by a single boundary for different
circuit protocols. Mutual information between the input and the output of the
circuit (a) as a function of dissipation strength $p$ for boundary dissipation
realized as a CNOT coupling to an ancilla qubit with maximal bulk Clifford
scrambling, (b)-(c) varying the strength of sparse bulk Clifford scrambling
$p_{U}$ with periodic boundary erasure channel (b), or periodic boundary CNOT
gate to an ancilla (c). All data are consistent with a coding transition
between a phase with partially protected information for weak dissipation /
strong enough scrambling, and a dissipative phase with all encoded information
lost. No pre-scrambling step was used for these plots.

As described in the main text, the Bell pair is encoded in the initial state
at the left boundary, optionally followed by a pre-scrambling step logarithmic
or linear in system size, depending on the type of phase transition that we
consider. We note that the pre-scrambling is realized by a full or sparse
brickwork of Clifford unitary gates, in the first and second types of
dynamics, respectively.

As mentioned above, we consider two different realizations of the boundary
dissipation.

$\bullet$ Boundary erasure channel. The dissipation acts by deleting the
information stored in the leftmost qubit.

$\bullet$ Coupling to an ancilla qubit. Here, we first couple the leftmost
qubit of the system to an ancilla qubit through a CNOT gate, and then trace
out the ancilla. In the stabilizer formalism, this operation results in
deleting all stabilizers containing a $Y$ or $Z$ Pauli operator at the left
end of the chain. To restore rotational invariance and obtain a smooth limit
$p_{U}\rightarrow 0$, for sparse Clifford scrambling we also act with a random
single site Clifford gate on the leftmost qubit before applying the CNOT gate.

In the main text we mainly focused on the case of random boundary dissipation
and maximal Clifford scrambling, with the dissipation realized as a boundary
erasure channel. We also briefly commented on the effect of a periodic
boundary noise, modifying the critical properties for linear pre-scrambling
compared to the random case. Below we provide supplementary numerical results
for the other protcols, showing a similar phase transition in the mutual
information.

We show the coding transition in the mutual information without pre-
scrambling, induced by a single boundary with aspect ratio $T/L<1$, in Fig. 14
for three different protocols. We cross the phase transition by tuning the
strength of dissipation $p$ in Fig. 14a, realized with a random CNOT coupling
between the boundary spin and an ancilla qubit. In contrast, in Fig. 14b and c
the tuning parameter is the strength of sparse bulk scrambling $p_{U}$, while
we apply a fixed strength periodic boundary dissipation, realized as an
erasure channel in Fig. 14b, and as a CNOT gate with an ancilla qubit in Fig.
14c. We recover the coding transition between a phase with partially protected
coherent information and a phase where all information is destroyed for all
protocols. Due to the difficulties in fitting critical exponents from finite
size data mentioned in the main text, we leave the detailed study of critical
properties for future work. In the cases with periodic boundary dissipation we
used $T_{\rm period}=5$ (b), and $T_{\rm period}=3$ (c).