# Slow measurement-only dynamics of entanglement in Pauli subsystem codes

Benedikt Placke<EMAIL_ADDRESS>S. A. Parameswaran
<EMAIL_ADDRESS>Rudolf Peierls Centre for Theoretical
Physics, University of Oxford, Oxford OX1 3PU, United Kingdom

###### Abstract

We study the non-unitary dynamics of a class of quantum circuits based on
stochastically measuring check operators of subsystem quantum error-correcting
codes, such as the Bacon-Shor code and its various generalizations. Our focus
is on how properties of the underlying code are imprinted onto the
measurement-only dynamics. We find that in a large class of codes with
nonlocal stabilizer generators, at late times there is generically a nonlocal
contribution to the subsystem entanglement entropy which _scales_ with the
subsystem size. The nonlocal stabilizer generators can also induce slow
dynamics, since depending on the rate of competing measurements the associated
degrees of freedom can take exponentially long (in system size) to purify
(disentangle from the environment when starting from a mixed state) _and_ to
scramble (become entangled with the rest of the system when starting from a
product state). Concretely, we consider circuits for which the nonlocal
stabilizer generators of the underlying subsystem code take the form of
subsystem symmetries. We present a systematic study of the phase diagrams and
relevant time scales in two and three spatial dimensions for both Calderbank-
Shor-Steane (CSS) and non-CSS codes, focusing in particular on the link
between slow measurement-only dynamics and the geometry of the subsystem
symmetry. A key finding of our work is that slowly purifying or scrambling
degrees of freedom appear to emerge only in codes whose subsystem symmetries
are nonlocally generated, a strict subset of those whose symmetries are simply
nonlocal. We comment on the link between our results on subsystem codes and
the phenomenon of Hilbert-space fragmentation in light of their shared
algebraic structure.

##  I Introduction

Quantum error correction (QEC) is essential in order to enable the robust
processing of quantum information. It played a key role in establishing that
scalable fault-tolerant quantum computation is possible in principle, and
remains of central practical importance in the quest to move beyond the
“noisy, intermediate-scale quantum” (NISQ) regime [1] achieved by present-day
platforms. Despite these pragmatic origins, QEC is deeply linked to many
fundamental problems in quantum many-body physics. For example, the concept of
topological order in equilibrium quantum systems is intimately related to the
non-recoverability of local information in a quantum code [2, 3], while the
search for a self-correcting quantum memory led to the identification of
various types of fracton topological order [4, 5, 6], with implications both
for equilibrium phase structure and glassy dynamics out of equilibrium. Error
correction itself is a concrete instance of what has sometimes been dubbed
“quantum interactive dynamics” [7, 8, 9, 10, 11, 12, 13], where the future
time evolution of a system is influenced by the knowledge acquired, via
measurements, about its present state: in this setting, quantum error
correcting codes can be viewed as undergoing dynamical phase transitions as
the error rate is varied [14, 15, 16, 17, 18, 19].

Recent attention has focused on the _dynamical generation_ of quantum error
correcting codes. This builds on the idea that characterizing the (late-time)
state of a system can be fruitfully viewed as a problem in error correction.
This task is particularly simple when the local dynamics take a restricted
“Clifford” form, since in this case at any time the dynamical state of a
system of $N$ qubits initialized in an appropriately chosen computational
basis state is the simultaneous eigenstate of $N$ commuting operators from the
Pauli group $\operatorname{\mathcal{P}_{N}}$ (though the set of operators can
itself change). Any such “stabilizer group” naturally defines a quantum code,
whose properties can be used to characterise the state itself. This paves the
way to generating previously unknown codes through dynamics. These can be
produced either by unitary circuits built from local few-qubit gates arranged
in a random [20] or deterministic [21] pattern in space-time, or through
sequences of projective measurements. In the latter case, again, operators can
be measured either randomly in space and time [22], or specifically-chosen
ones can be measured according to a fixed time-periodic schedule — as in the
celebrated “Floquet codes” [23, 24, 25]. This framework also offers a useful
alternative perspective on circuits which include both unitary gates and
measurements. For example, the measurement-induced phase transition observed
originally in such a hybrid unitary-projective setting [26, 27, 28] can be
understood as an error-correction transition where information scrambling by
unitary evolution in a dynamically generated code either succeeds or fails in
concealing information from projective measurements, leading to differing
scaling behaviour of entanglement in the time-evolved state [29].

Figure 1:  (a) Measurement-only circuits based on subsystem codes. A subsystem
code is defined by a check group $\mathcal{G}$, which is generated by
geometrically local check operators $g_{j}$ acting on $N$ qubits. In the
corresponding circuit, we measure $N$ randomly chosen check operators per time
step. We sketch the purification and scrambling dynamics of such circuits in
panels (b) and (c), respectively. We observe a separation of typical time
scales for both the decay of the von Neumann entropy $S_{N}$ and for the
growth of the half-system entanglement entropy $S_{\rm E}$. The larger time
scale $t_{\infty}$, which is exponentially large in a subextensive power of
the linear system size, arises if the stabilizer group (see Fig. 2) of the
underlying subsystem code has $N_{S}>0$ non-local generators (which may exist
even if all measured operators are geometrically local). In panel (d), we
tabulate all codes analysed in this work. All the codes we study are defined
on a hypercubic lattice of dimension $D$ (so $N=L^{D}$) and the $N_{S}$
nonlocal stabilizer generators take the form of subsystem symmetries acting on
subsystems of dimension $\Delta$ (so $N_{S}\sim L^{D-\Delta}$). The number of
logical qubits is denoted by $K$.

Here, we draw inspiration from these developments but study the converse
question: rather than generating codes from dynamics, we instead study the
dynamics generated by codes. In other words, we build a specified code
structure into a measurement-only quantum circuit, and examine how this
structure is imprinted onto the properties of its steady state and late-time
dynamics.

To do so, we consider codes defined by a specified set of “check operators”
and measure a randomly chosen check operator at each step in time. For the
best-known class of error-correcting codes — stabilizer codes, all of whose
check operators commute — this results in trivial dynamics. In order to
generate richer structure, measurements of stabilizers can be interspersed
with those of single qubits, leading to complex phase diagrams [30, 31, 32];
the link to error correction is then via the resilience of the original code
to single-qubit noise [33, 34]. This is a rich subject in its own right;
however, the addition of single-qubit measurements necessarily breaks the
algebraic (symmetry) structure inherited by the circuit from its underlying
quantum code. In order to generate nontrivial dynamics while preserving code
structure, we instead study measurement-only circuits based on _subsystem
codes_. Such codes are defined in terms of a group of check operators
$\mathcal{G}$ which do not, in general, commute. As is illustrated in Fig. 1
(a), choosing a generating set $\\{g_{j}\\}$ for $\mathcal{G}$, at each time
step we randomly measure one of these generators (for convenience we choose a
unit of time in which $N$ measurements are possible). Since the $g_{j}$ do not
generally commute, such circuits can have nontrivial measurement-only dynamics
in their own right but in addition they inherit from the code nontrivial
symmetries in the form of its stabilizer and logical operators (which together
form the centralizer
$\mathcal{C}_{\operatorname{\mathcal{P}_{N}}}(\mathcal{G})$ of $\mathcal{G}$).
These symmetries strongly constrain many aspects of the dynamics;
understanding precisely how is our central focus here.

Although individual examples of this kind have been studied previously [33,
35, 36], a comprehensive analysis of how spatial dimension and stabilizer
geometry — among other properties of the underlying codes — manifest
themselves in the approach towards and time evolution within the steady state
of such dynamics has been lacking. It is this overarching understanding that
we supply in the present work. More generally, our work complements earlier
studies on the role of on-site symmetries in monitored circuits [37, 38, 39].

Our main findings are that for a large class of subsystem codes, the resulting
measurement-only circuit dynamics (i) exhibit nontrivial steady states whose
bipartite entanglement entropy has _leading-order_ nonlocal contributions,
even in cases where it satisfies area-law scaling with subsystem size; (ii)
can show very slow approaches to steady-state scaling of the entanglement
entropy, only saturating it at a time scale that can be typically
exponentially large in (a subextensive power of) the system size. More
precisely, both these features hold if the stabilizer group of the subsystem
code (the center of $\mathcal{G}$) underlying the circuit has _nonlocal
generators_. This results in two distinct dynamical regimes, perhaps most
clearly visible in ‘purification dynamics’: initializing the system in a
maximally mixed initial state $\rho(t=0)=0$, and tracking the von Neumann
entropy $S_{\rm N}$ as a function of time. In this case, one can view the
nonlocal stabilizer generators as encoding information about the state $\rho$
that can be learned by local check measurements (since stabilizers are, by
definition, products of checks), but for which the typical time on which is
learned by the dynamics can be long, depending on the rate of competing (i.e.,
non-commuting) measurements. As sketched in Fig. 1 (b), the von Neumann
entropy initially decreases exponentially with rate $\tau^{-1}_{*}$
independent of system size and plateaus, after a typical time of
$t_{*}\sim\tau_{*}\log(N/(N_{S}+K))$, at a value of $N_{S}+K$, where $N_{S}$
is the number of nonlocal stabilizer generators that take long to be measured
effectively and $K$ is the number of logical qubits of the code. This fast
initial drop in entropy corresponds to the dynamics learning all information
that is efficiently obtained by local measurements. After time $t_{*}$, the
entropy continues decreasing as all nonlocal stabilizer generators are
measured, but with a rate $\tau_{\infty}^{-1}$ that is exponentially small in
system size. The infinite-time value, given by $S_{\rm N}(t\to\infty)=K$
(since the logical degrees of freedom cannot purify) is then reached after a
typical time $t_{\infty}=\tau_{\infty}\log(1+N_{S}/K)$. These two distinct
time scales are also probed by the subsystem entanglement entropy $S_{\rm E}$,
as sketched in Fig. 1 (c). When starting an initial random product state, the
system, after an initial transient, scrambles its locally available
information at time $t^{*}$, but picks up an additional contribution scaling
with $N_{S}$ at exponentially long times $t_{\infty}$. This contribution is
generally subleading but can contribute at leading order in area law phases.

We demonstrate the above picture in detail by giving physically intuitive
arguments in the stabilizer formalism and by deploying extensive numerical
simulations on an array of examples summarized in Fig. 1 (d). In all cases,
the codes are defined on a hypercubic lattice of dimension $D$ (i.e. $N\sim
L^{D}$), and _all_ stabilizer generators are nonlocal and take the form of
subsystem symmetries acting on subsystems of dimension $\Delta$, with
$N_{S}\sim L^{D-\Delta}$. We also study the phase diagrams of these codes as a
function of relative measurement rates of different check operators and
characterise how both the topology of the phase diagram as well as the scaling
of $\tau_{\infty}$ in these phases depends on the geometry of the subsystem
symmetry. We find that while in circuits with line-like stabilisers
($\Delta=1$) dynamical transitions can separate distinct area-law phases where
different subsets of the full stabilizer group purify slowly, sheet-like
stabilizers ($\Delta=2$) purify slowly for any finite rate of competing
measurements. In both cases, the time scale on which slow stabilizers are
measured scales as $\tau_{\infty}\sim\exp(\gamma L^{\Delta})$. We also compare
circuits based on CSS and non-CSS codes and find that only the latter generate
steady states with a volume-law scaling of the entanglement entropy as part of
their phase diagram, in consonance with a recent conjecture. Intriguingly, our
work also reveals a connection between measurement-only subsystem code
dynamics and the phenomenon of Hilbert space fragmentation observed for
_unitary_ dynamics, and hence our results may be regarded as the first step
towards a more general understanding of how symmetries can also have non-
trivial consequences for non-unitary quantum dynamics.

The rest of the paper proceeds as follows. In Sec. II we introduce subsystem
quantum error correction codes, explain how we construct corresponding
measurement-only circuits, derive the relation between code properties and
circuit dynamics and discuss the relation to Hilbert space fragmentation. We
begin Sec. III by giving an overview of our results. We then present several
concrete examples in two and three spacial dimensions that help to both
motivate and establish these results. We consider circuits with string- and
membrane-like subsystem symmetries, as well as examples based on CSS and non-
CSS codes, and explain the consequences of these choices on the phase diagrams
of our models, and on purification and scrambling dynamics. We conclude in
Sec. IV with a summary and a discussion of possible future directions.

##  II Background and methods

In this paper, we focus on stochastic measurement-only dynamics [22]: a
discrete time evolution where at each time step a (geometrically) local
operator is drawn from a specified distribution and projectively measured.
Throughout, we will consider systems of $N$ qubits, and will choose the
operators uniformly from a set of “check” operators of a given subsystem code
(these terms are defined below). Unless specified otherwise, we fix the units
of time $t$ such that $N$ operators have a chance to be measured in an
interval $\Delta t=1$. This setup is also sketched in Fig. 1 (a).

In the remainder of this section, we first introduce the notion of subsystem
codes and discuss their algebraic structure and its link to the commutant-
algebra formulation of the phenomenon of Hilbert-Space fragmentation in
unitary dynamics [40]. We then introduce the measurement-only dynamics and
entanglement measures deployed in the rest of this paper. Readers familiar
with these concepts could either omit this section or simply skim it in order
to familiarize themselves with our notation, and skip ahead to our results in
Section Sec. III

###  II.1 Stabilizer Subsystem Codes: Algebraic Structure

Figure 2: Algebraic structure of stabilizer subsystem codes. The check
operators generate the group $\mathcal{G}$. Its center,
$\mathcal{Z}(\mathcal{G})$, is the stabilizer group of the code. The bare
logical operators of the code are given by
$\mathcal{C}_{\operatorname{\mathcal{P}_{N}}}(\mathcal{G})-\mathcal{Z}(\mathcal{G})$,
where $C_{\operatorname{\mathcal{P}_{N}}}(\mathcal{G})$ is the centralizer of
$\mathcal{G}$ in the $N$-qubit Pauli group $\operatorname{\mathcal{P}_{N}}$.

Subsystem codes are generalizations of the perhaps better-known stabilizer
codes, whose properties we first summarize to serve as a point of comparison.
A stabilizer code is defined by fixing an abelian subgroup of the Pauli group
$\operatorname{\mathcal{P}_{N}}$ on $N$ qubits, termed the stabilizer group
$\mathcal{S}$ of the code. This is usually done by specifying a set of
mutually commuting stabilizer generators $S_{j}$ such that
$\mathcal{S}=\expectationvalue{S_{j},\,j=1\dots N_{S}}$. This induces a
decomposition of the full Hilbert space of the form

$\mathscr{H}=\bigoplus_{\bm{\sigma}\in\mathbb{Z}_{2}^{N_{S}}}\mathscr{C}(\bm{\sigma}),$
(1)

where the sum is over all $2^{N_{S}}$ possible bitstrings $\bm{\sigma}$ of
eigenvalues of the stabilizer generators $S_{j}$. (In other words, the Hilbert
space decomposes into a direct sum of subspaces with fixed values of the
$N_{S}$ stabilizers.) Logical information is encoded into one of these
subspaces, termed the ‘code space’. This is conventionally taken to be the
$+1$ eigenspace of all the stabilizers, although practical error correction
routines typically infers the set of stabilizer eigenvalues (not necessarily
all $+1$) and performs appropriate classical post-processing whenever
possible.

Stabilizers by definition act trivially on the code space and hence represent
the logical identity. Operators which preserve the code space but act
nontrivially on it are called logical operators. For stabilizer codes, logical
operators are exactly those operators that commute with all stabilizers but
cannot themselves be written as a product of stabilizers. In other words, the
group of logical Pauli operators is the centralizer111Recall that the
centralizer or commutant of a subset $\mathcal{S}$ of a group $G$ is the set
of $C_{G}(\mathcal{S})$ of elements in $G$ that commute with every element in
$S$; equivalently, $gsg^{-1}=s$ for every $s\in\mathcal{S}$ and $g\in
C_{G}(\mathcal{S})$. of $\mathcal{S}$ in $\operatorname{\mathcal{P}_{N}}$,
denoted $C_{\operatorname{\mathcal{P}_{N}}}(\mathcal{S})$. An automatic
corollary is that multiplication by stabilizers induces an equivalence
relation between logicals: two logicals that only differ by a product of
stabilizers are identified. When linked with an appropriate notion of locality
(e.g., on a lattice), this equivalence relation endows the code with an
appealing topological interpretation, as in the case of the toric or surface
codes or even more exotic fracton models. The dimension of the logical
subspace is $\rm dim\mathscr{C}=2^{K}$, where $K$ is the number of logical
qubits, and the support of the smallest nontrivial logical operator [i.e.,
element of $C_{\operatorname{\mathcal{P}_{N}}}(\mathcal{S})$] is called the
code distance. A stabilizer code with $N$ physical qubits, $K$ logical qubits,
and distance $D$ is denoted an $[[N,K,D]]$ code.

In an extension of this structure, subsystem codes [41] encode logicals not in
a subspace of the total Hilbert space, but instead in a logical subsystem
(hence the name). Focusing specifically on the case of stabilizer222More
general subsystem codes only demand the decomposition of the Hilbert space as
$\mathscr{H}=\mathscr{R}\oplus(\mathscr{G}\otimes\mathscr{L})$. subsystem
codes, this corresponds to a further decomposition of the Hilbert space beyond
the direct-sum structure of a stabilizer code, viz.

$\mathscr{H}=\bigoplus_{\bm{\sigma}\in\mathbb{Z}_{2}^{N_{S}}}\mathscr{L}(\bm{\sigma})\otimes\mathscr{G}(\bm{\sigma}),$
(2)

where $\mathscr{L}$ and $\mathscr{G}$ are respectively termed the logical and
gauge subsystems (The latter terminology is historical and should not be
confused with the ‘gauging’ that arises when codes are studied in physical
settings; sometimes $\mathscr{G}$ is termed the ‘junk’ or ‘garbage’ subsystem
for this reason.) The decomposition in Eq. (missing) 2 is induced by
specifying a set of check operators $g_{j}\in\operatorname{\mathcal{P}_{N}}$
which, in contrast to stabilizer generators, need not mutually commute. The
checks generate the gauge group $\mathcal{G}$, and the center
$\mathcal{Z}(\mathcal{G})$ of the gauge group —– i.e., its maximal abelian
subgroup — is identified with the stabilizer group of the subsystem code. The
subspaces
$\mathscr{C}(\bm{\sigma})=\mathscr{L}(\bm{\sigma})\otimes\mathscr{G}(\bm{\sigma})$
are labeled the eigenvalues $\bm{\sigma}$ of all stabilizer generators.
Conventionally, the subspace used for the code is the one with
$\bm{\sigma}=(1\dots 1)$, that is the common +1 eigenspace of all stabilizers.
Evidently, when $\mathcal{G}$ is abelian so that
$\mathcal{G}=\mathcal{Z}(\mathcal{G})$, we recover the stabilizer group
structure of Eq. (missing) 1. We can thus view a stabilizer subsystem code as
arising from decomposing each stabilizer subspace of a stabilizer code into
gauge and logical subsystems.

The logical operators of the subsystem code are those operators that leave
each logical subsystem $\mathscr{L}(\bm{\sigma})$ invariant. However, in
contrast to stabilizer codes, logical operators of a subsystem code are
equivalent up to multiplication with gauge operators in $\mathcal{G}$, rather
than just up to multiplication with stabilizers. A particular set of
representatives of the logical operators of the code are the so-called bare
logical operators. These act trivially on the gauge subsystem $\mathscr{G}$,
that is for any bare logical $L$, we can write

$L=\bigoplus_{\bm{\sigma}\in\mathbb{Z}_{2}^{N_{S}}}L_{\mathscr{L}(\bm{\sigma})}\otimes
I_{\mathscr{G}(\bm{\sigma})}$ (3a) where $I_{\mathscr{V}}$ is the identity
operator on vector space $\mathscr{V}$. Stabilizers are trivial bare logical
operators, while in general the bare logical operators are the elements of the
centralizer of the gauge group
$C_{\operatorname{\mathcal{P}_{N}}}(\mathscr{G})$. Any logical operator can
then be written as the product of a bare logical operator and a gauge
operator. For completeness, we mention that by definition the gauge group
preserves the logical subspace so
$g=\bigoplus_{\bm{\sigma}\in\mathbb{Z}_{2}^{N_{S}}}I_{\mathscr{L}(\bm{\sigma})}\otimes
g_{\mathscr{G}(\bm{\sigma})}$ (3b) for any check operator
$g\in\mathcal{G}$.333In fact, as will become important in the next section,
the decompositions in Eq. (missing) 3b are possible not only for the elements
of $C_{\operatorname{\mathcal{P}_{N}}}(\mathscr{G})$ and $\mathcal{G}$, but
also for elements of their respective linear spans. The algebraic (operator)
structure of subsystem codes is sketched in Fig. 2.

As before, $\dim\mathscr{C}=2^{K}$, where $K$ is the number of logical qubits
of the code and its distance is the support of the smallest nontrivial logical
operator. Importantly, for subsystem codes this includes dressed (that is non-
bare) logicals. The dimension of the gauge subspace is usually written as
$\dim[\mathscr{G}(\bm{\sigma})]=\dim(\mathscr{G})=2^{R}$, where $R=N-K-N_{S}$
is the number of gauge qubits and $N_{S}$ as before denotes the number of
independent stabilizer generators, while that of the logical subspace is given
by $\dim[\mathscr{L}(\bm{\sigma})]=\dim(\mathscr{L})=2^{K}$. Note that the
dimensions of both the gauge and logical subspaces are independent of the
stabilizer eigenvalues. A subsystem code with $N$ physics qubits, $K$ logical
qubits, $R$ gauge qubits, and distance $D$ is called an $[[N,K,R,D]]$ code.

A Calderbank-Shor-Steane (CSS) code [42, 43] is a special case where all the
check operators $g_{j}$ of the code are either products of only Pauli-$X$ or
Pauli-$Z$ operators. A well-known example of a CSS subsystem code is the
Bacon-Shor code[44], while a well-known example of a non-CSS subsystem code is
provided by Kitaev honeycomb model; however when viewed as a subsystem code
the latter has a trivial (i.e. zero-dimensional) logical subspace [45].

####  II.1.1 Connection to Hibert-Space Fragmentation

We also comment on a link between the algebraic structure of subsystem codes
and the bond-algebraic perspective on the phenomenon of Hilbert space
fragmentation [40]. To make the connection precise, consider a Hamiltonian
$H=\sum_{j}\alpha_{j}g_{j}$, where $\alpha\in\operatorname{\mathbb{R}}$ and
the $g_{j}\in\mathcal{G}$ are the check operators of a given stabilizer
subsystem code. The time evolution operator444Note that a similar structure
can also be defined for general unitary time evolution operators that lack a
well-defined local Hamiltonian, e.g. in a Floquet system, but we discuss the
Hamiltonian version for concreteness. $U(t)=\exp(-itH)$ is then in the linear
span $\operatorname{\mathrm{Span}}\mathcal{G}$ of elements in the gauge group,
which for Hamiltonian systems is termed the bond algebra, and hence preserves
both the full set of stabilizer eigenvalues as well as the state of the
logical subsystem of the code.

Figure 3: Block-diagonal structure of the matrix representation of an element
$\hat{G}$ of the gauge algebra $\operatorname{\mathrm{Span}}{\mathcal{G}}$ in
an appropriately chosen tensor-product basis
$\ket{\bm{\sigma},\bm{z}_{b}}\otimes\ket{\bm{\sigma},\bm{z}_{l}}$ [see main
text for details]

To draw the parallel in full, first note that the decompositions shown in Eq.
(missing) 3b are in fact not only possible for the bare logical operators
$C_{\operatorname{\mathcal{P}_{N}}}(\mathscr{G})$ and the check operators
$\mathcal{G}$, but also for any element in their respective linear span
$\operatorname{\mathrm{Span}}C_{\operatorname{\mathcal{P}_{N}}}(\mathscr{G})$
and $\operatorname{\mathrm{Span}}\mathcal{G}$. Returning to the Hamiltonian
example, this means in particular that any element of the bond algebra
$\operatorname{\mathrm{Span}}\mathcal{G}$, and hence also $U(t)$, is block
diagonal as sketched in Fig. 3, with $2^{N-R}$ blocks of dimension $2^{R}$
when written in an appropriate product basis
$\ket{\bm{\sigma},\bm{z}_{b}}\otimes\ket{\bm{\sigma},\bm{z}_{l}}$. The
bitstrings $\bm{z}_{b}$ and $\bm{z}_{l}$ respectively specify bases of the
gauge and logical subsystems for a given stabilizer sector labelled by
$\bm{\sigma}$. For the case of the time evolution operator $U(t)$, the block-
diagonal structure implies the existence of $2^{N-R}$ invariant subspaces,
which are called Krylov subspaces in the context of Hilbert space
fragmentation [46, 47, 48]. From the preceding discusson, we see that this
corresponds precisely to the decomposition in Eq. (missing) 2: in other words,
the Hilbert space of a subsystem code is ‘fragmented’ according to this bond-
algebraic definition [40]. [We observe that the fact that code Hamiltonians
can naturally _evade_ of thermalization provides an elegant counterpoint to
the fact that eigenstates of _chaotic_ Hamiltonians obeying the Eigenstate
Thermalization Hypothesis (ETH) form approximate quantum error-correcting
codes [*[Tothebestofourknowledge,
thisobservationfirstappearedinprintinaclearformin][.However,
S.A.P.recallsbeingeducatedontheparallelsbetweenETHandtheKnill-
LaflammeconditionsforQECbySteveFlammiaandthelateDavidPoulinduringanilluminating2015discussioninMoorea,
FrenchPolynesia.]brandao2019qec_eth].]

The fact that such a structure in the bond algebra of a family of Hamiltonians
naturally leads to the emergence of multiple Krylov subspaces even within a
single global symmetry sector, and the consequences for unitary dynamics
(which include the possibility of ergodicity breaking) was first noted in Ref.
[40] (which also mentioned but did not systematically explore the connection
to subsystem codes). This is believed to underpin various exotic dynamical
phenomena in a range of isolated quantum systems, and has been experimentally
explored in the setting of one-dimensional Rydberg arrays. Below, we show that
with an additional condition on the stabilizer geometry, one can define a
measurement-only analogue of the ‘anomalous’ dynamics often associated with
Hilbert-space fragmentation in the unitary setting.

###  II.2 Stochastic measurement-only dynamics

We now define a class of stochastic measurement-only dynamics [22] that can be
associated with a specified subsystem code. In each time step, a random check
operator of the subsystem code is measured. Different check operators will in
general be measured with different rates, while preserving translational
invariance of the ensemble. Such dynamics has been considered previously, in
particular in Ref. 50, which studies measurement-only circuits based on the
Bacon-Shor code (see subsection III.3 for a definition) and uncovers an
interesting spin-glass like structure of correlation functions in the ensemble
of steady states. However Ref. 50 did not explore the purification dynamics
and slow entanglement growth that will be the main topics of this paper.
Recent studies of the measurement-only Kitaev model [35, 36, 51] also
highlight its interpretation as a subsystem code. This is useful for gaining
intuition about the dynamics, despite the fact that the Kitaev model (when
viewed as a subsystem code) does not encode a logical qubit and is hence
trivial.

We focus specifically on Pauli stabilizer subsystem codes since their dynamics
can be efficiently computed numerically using the stabilizer formalism [52,
53], by exploiting the fact that evolution under a restricted set of
operations can be efficiently computed in the Heisenberg picture, as we now
describe. First, consider a state described by the density matrix $\rho$ that
is the $+1$ eigenstate of a set of independent and mutually commuting Pauli
operators $s_{i}\in\operatorname{\mathcal{P}_{N}}$, $i=1\dots m$. The
operators $s_{i}$ form the minimal generating set of the stabilizer group
$\mathcal{S}(\rho)$ of the state $\rho$. A pure state on $N$ qubits is
uniquely fixed by a stabilizer group with $N$ independent generators. More
generally, if the stabilizer group $\mathcal{S}(\rho)$ has $m<N$ generators,
the state is the maximally mixed state on the common $+1$ eigenspace of
$\mathcal{S}(\rho)$, so that one can identify

$\rho=\frac{1}{2^{N}}\sum_{s\in\mathcal{S}(\rho)}s.$ (4)

The von Neumann entropy of the state is then $S=-\tr\rho\log_{2}\rho=N-m$.
Evidently, we can specify a state by simply listing a (non-unique) set of
generators $\\{s_{1},s_{2}\ldots,s_{m}\\}$ for its stabilizer group, and
$\mathcal{S}(\rho)=\expectationvalue{s_{1},s_{2},\ldots,s_{m}}$, where
$\expectationvalue{\ldots}$ denotes the group generated by the listed
stabilizers. [Note that this introduces a second notion of stabilizer group,
namely that of the stabilizer group of a state, $\mathcal{S}(\rho)$, which is
an efficient description of its density matrix in the sense of Eq. (missing)
4. This is distinct from the previously-encountered notion of the stabilizer
group of a code, $\mathcal{S}=\mathcal{Z}(\mathcal{G})$ ( Fig. 2), which
describes the (abelian) symmetries of the circuit. We use both notions in this
paper; where there is potential for ambiguity, we will explicitly refer to
code stabilizers and state stabilizers, but we may omit these qualifiers when
the meaning is clear from context.]

We can compute the dynamics of a state $\rho$ specified in this manner under
projective measurements of operators from the Pauli group555More generally,
one can efficiently classically simulate the Heisenberg evolution under any
set of operations that preserves the Pauli group, but in this work we restrict
ourselves to measurement-only dynamics. as follows. Given a state $\rho$
specified by $\mathcal{S}(\rho)=\expectationvalue{s_{1},s_{2},\ldots,s_{m}}$
and a measured Pauli operator $g\in\operatorname{\mathcal{P}_{N}}$, our task
is to give a prescription for the post-measurement state $\rho^{\prime}$ [or
equivalently $\mathcal{S}(\rho^{\prime})$]. To that end, we distinguish three
cases:

  1. 1.

If $\pm g\in\mathcal{S}(\rho)$ so that $g$ is an element of the stabilizer
group of $\rho$, then the state is unchanged by measurement, so that
$\rho^{\prime}=\rho$ and hence $\mathcal{S}(\rho)=\mathcal{S}(\rho^{\prime})$,
and the measurement outcome is the eigenvalue of $\rho$ with respect to $g$.

  2. 2.

If $\pm g\notin\mathcal{S}(\rho)$, but if $g$ commutes with every generator
$s_{i}\in S(\rho)$, then we simply add $g$ to $\mathcal{S}$, i.e. the post-
measurement state is generated by the stabilizer set
$S(\rho^{\prime})=\expectationvalue{\pm g,s_{1},s_{2},\dots s_{m}}$.

  3. 3.

Finally, if $\pm g\notin\mathcal{S}(\rho)$ and $g$ anticommutes with some
$s_{i}\in\mathcal{S}(\rho)$, we can without loss of generality assume that $g$
anticommutes with exactly one generator $s_{1}$, since we can simply redefine
$s_{j}\to s_{1}s_{j}$ for each $s_{j}\neq s_{1}$ that also anticommutes with
$g$. In this case, we remove $s_{1}$ from the stabilizer group and replace it
with $g$, so that $S(\rho^{\prime})=\expectationvalue{\pm g,s_{2},\dots
s_{m}}$.

In both cases (2) and (3), the sign of $g$ in $\mathcal{S}(\rho)$ is given by
the measurement outcome.

It is worth briefly commenting on the nature of such dynamics for stabilizer
and subsystem codes. In the stabilizer case, since the measured operators
always commute, Rule 3 above is never invoked and the state stabilizer group
can grow but is otherwise fully static. In contrast, in a subsystem code where
the measured check operators need not commute, adding new state stabilizers
can remove existing ones. A special case occurs when a state stabilizer
coincides with one of the code stabilizers, whereupon — as it then by
definition commutes with all the check operators — it is never removed from
the state stabilizer group. Thus the measurement only dynamics of subsystem
codes can be viewed in terms of a gradual accumulation of code stabilizers
into the state stabilizer group as time evolves; this will be central to our
discussion of the slow dynamics below.

###  II.3 Entanglement-based characterization of dynamical states

As noted by earlier studies [32, 35, 36, 51, 50, 54], the ultimate fate of the
measurement-only dynamics for $t\to\infty$ and any finite $N$ is for the
system to reach a density matrix that is invariant under the stabilizer group
of the code. In most models of interest, any stabilizer generator can be
measured effectively by measuring a certain sequence of check operators. This
is also the case for all codes considered here, although we also present a
simple example where this is not possible in App. A. Logical operators cannot
be measured by measuring products of check operators; hence, the entanglement
between the logical subsystem and the environment is an invariant of the
dynamics.666The easiest way to see this is to note that any state $\rho$ with
nontrivial entanglement between the logical subsystem and the environment can
be purified into a state on system and environment whose stabilizer group
involves some number of operators that are products of logicals and operators
on the environment. Since all measurements commute with the logicals (and
trivially with environment operators) and logicals are themselves never
measured, these operators always remain part of the state stabilizer group and
hence the logical-environment entanglement is preserved.

However, the approach to this late-time state can involve several distinct
dynamical regimes whose timescales can diverge in the thermodynamic limit, as
we discuss in detail the next section. As we will demonstrate, these regimes
can be understood by characterizing the entanglement properties of the system.
When considering how the measurements purify initially mixed states, the
relevant notion of entanglement is that of the system with the environment, as
captured by the von Neumann entropy $S(t)=-\tr\rho(t)\log_{2}\rho(t)$. On the
other hand, when characterizing how measurements generate entanglement in
initially unentangled pure states, we instead focus on the bipartite
entanglement entropy; given a bipartition of the system into a subsystem $A$
and its complement $\bar{A}$, this is given by
$S_{A}(t)=-\tr\rho_{A}(t)\log_{2}\rho_{A}(t)$, with
$\rho_{A}(t)\equiv\tr_{\bar{A}}\rho(t)$. In both cases we will be interested
in the scaling of the respective entropies with (sub)system size.

In the case of stabilizer states [Eq. (missing) 4], the von Neumann entropy is
given by the difference between the number of qubits and the number of
independent generators $\mathcal{S}(\rho)$ of the stabilizer group, while the
entanglement entropy $S_{A}(t)$ is given by the minimal number of generators
that cannot be supported on only either $A$ or $\bar{A}$ [55]. The latter
computation involves some linear algebra in order to perform the requisite
minimization procedure on the state stabilizer generators.

We study purification dynamics by considering near-maximally mixed initial
states with $S(0)\propto N$ that are highly entangled with the environment; at
late times we expect conversely that $S(t)$ is lower-bounded by $K$, the
number of logical qubits, with the exact value as $t\to\infty$ controlled by
the details of the initial state. Scalings intermediate between these arise
can arise during the evolution owing to the interplay of the code structure
with the locality of measurements, and require additional analysis.

Turning to the scaling of the entanglement entropy of a subsystem, we employ
the usual practice of distinguishing between volume law ($S_{A}\propto|A|$)
and area law ($S_{A}\propto|\partial A|$) scaling of the dominant
contribution, and between topological and trivial subleading contributions in
area-law phases. Since when exploring scrambling we typically begin with a
product state, $S_{A}=0$ initially and approaches either a volume or area law
at late times.

Area law states can in turn be distinguished by subleading contributions to
$S_{A}$. For instance, in a two-dimensional topologically ordered phase we
have

$S_{A}=c\absolutevalue{\partial A}-\gamma,$ (5)

where $\absolutevalue{\partial A}$ is the size of the perimeter of the region
$A$, $c$ is some non-universal constant, and $\gamma$ is called the
topological entanglement entropy (TEE) [56, 57]. In our analysis of
intermediate-time behaviour of measurement-only dynamics of subsystem codes,
we will find additional, non-local area-law contributions to the entanglement
entropy that are leading (subleading) in spacial dimension $D=2$ ($D=3$) that
arise as a consequence of their special symmetry properties.

Note that in probing “generic” scrambling behaviour it is important to choose
a random initial state. This is because special initial states, such as
product states in the local $Z$ basis, can be eigenstates of non-local
stabilizer generators of the code and hence evade slow dynamics linked to the
formation of these stabilizers.

##  III Results

###  III.1 Overview

We begin by giving an overview of our results, that we obtain from different
examples of subsystem-code circuits with non-locally generated stabilizer
groups. Each of these circuits illustrates a distinct aspect of the
measurement-only dynamics, and as we discuss in detail in subsequent sections.
A summary of the models that we study are shown in Fig. 1 (d).

####  III.1.1 Slow Purification and Scrambling: “Learning” an Unknown State
and the Role of Stabilizer Geometry

Figure 4:  Purification dynamics as learning of the state in Hilbert space.
Assume for simplicity that all stabiliser generators of the code purify slowly
(in the figure, $N_{S}=3$) and that there are no locical qubits ($K=0$). The
Hilbert space $\mathscr{H}$ is the direct sum of subspaces $\mathscr{S}$, each
labelled by a list of stabilizer eigenvalues
$\bm{\sigma}\in\mathbb{Z}_{2}^{N_{S}}$. Purification dynamics then proceeds as
follows: at time $t=0$, the state is maximally mixed
$\rho=\operatorname{\mathds{1}}$, i.e. we have no information about the state
[panel(1)]. As check operators of the code are measured, we gradually learn
about the state [panel (2)] and by the time $t_{*}$ we have typically learned
everything about the state _except_ the stabilizer subspace it is in [panel
(3)]. Note that since the check operators do not commute, the state has
dynamics within each stabilizer subspace on $\order{1}$ time scales [arrows in
panel (3-5)]. On a much longer time scale $t_{\infty}$, we also learn about
the subspace that the state is in, by effectively measuring code stabilizer
generators. In panel (4) for example, only the second generator has been
measured with outcome $\sigma_{2}=0$. Finally after a typical time
$t_{\infty}$, all stabilizer eigenvalues have been measured and the state is
fully purified. The state continues to have dynamics within the subspace, but
cannot leave the subspace, i.e. the value of $\bm{\sigma}$ is a constant of
motion.

As we have noted, the study of measurement-only dynamics motivates the
consideration of two quite different processes: purification and scrambling.
In exploring purification dynamics, we initialize the system in a maximally
mixed state (so that the state stabilizer group is initially empty) and
examine the time evolution of the von Neumann entropy $S(t)$. For an
$[[N,K,D]]$ code and an initially maximally mixed state recall that we expect
that $S(0)=N$, the number of qubits, and that $S(t\to\infty)=K$, the number of
logicals, since the associated degrees of freedom can not be purified by
measuring check operators alone. For scrambling dynamics in contrast the
initial state is taken to be a trivial product density matrix, usually with
$N$ single-qubit stabilizers. When not further specified, we randomly choose a
single qubit Pauli $X_{j}$, $Y_{j}$, $Z_{j}$ independently for each site $j$
with probability $1/3$.

At late times we expect that the density matrix will be invariant under the
full stabilizer group of the code, since such density matrices are fixed
points of the dynamics. Starting from any initial state, code stabilizers will
be added to the state stabilizer group when a specific sequence of check
operators is measured. Once a code stabilizer is added to the state stabilizer
group, it will not be subsequently removed since by definition it commutes
with all measurements performed on the state.

Given that the late-time state is well-understood, the natural open question
about the dynamics is to characterize the approach to this state, which is
closely linked to the time scale on which the stabilizer generators are
measured. As was observed in Refs. 35, 36, a local stabilizer generator will
typically be measured in constant time $\tau_{*}\sim\order{1}$, whereas to
measure a general stabilizer generator with support $Q$ can take takes a time
exponentially long in the size of this support
$\tau_{\infty}\sim\order{\exp\absolutevalue{Q}}$. Such potentially long time
scales make these natural candidates for generating “slow” dynamics and
motivates our interest in models all of whose stabilizer generators are non-
local in the sense of the preceding subsection, namely that they have support
on some $\Delta$-dimensional subsystem.

We emphasize that in order to have slow dynamics at intermediate times it is
important to have non-local stabilizer _generators_ rather than either non-
local logical operators or local stabilizer generators, since the former
cannot be measured by dynamics that preserve the code structure while a
stabilizer that is a local product of check operators is quickly measured in
$\order{1}$ steps. In contrast to the measurement-only Kitaev model considered
in Refs. 35, 36 which has a finite number of non-local stabilizer generators,
here we consider circuits where the number of non-local stabilizer generators
$N_{S}$ grows with system size ($N_{S}\sim\order{L^{D-\Delta}}$). We study
cases with both string-like $\Delta=1$ and sheet-like $\Delta=2$ generators.

As the relative rates of non-commuting check measurements are varied, whether
a given non-local stabilizer generator is in fact a slow degree of freedom is
dictated by its geometry. We find that while string-like stabilizer generators
of a code can be measured quickly if the rate of competing check measurements
is sufficiently low, sheet-like stabilizer generators take an exponentially
long time to be measured for any non-zero measurement rate of competing check
operators. Here, by competing check operators we mean those that anti-commute
with some intermediate state of the sequence of check measurements necessary
to measure the stabilizer generator in question. The difference in stability
may be intuitively understood by estimating the number of competing check
operators, as we detail in subsection III.4. While the above result might seem
technical, one can gain an intuitive understanding by considering the dynamics
of purification, within a fixed trajectory of measurement outcomes, as the
process of learning information about an initially unknown state of the system
by means of repeated measurements. This is illustrated in Fig. 4, where we
show how the uncertainty about the state in the Hilbert space (indicated by
orange shading) evolves in time during purification. For simplicity, in the
figure we assume that all $N_{S}=3$ stabilizer generators are slow and that
there are no logical degrees of freedom, so that $K=0$. Starting from zero
knowledge, $\rho=\operatorname{\mathds{1}}$ at $t=0$ in panel (1), measuring
the outcomes of local measurements quickly fixes the state within each
stabilizer subspace $\mathscr{C}(\bm{\sigma})$ (indicated by gray lines in the
figure), but does notm, in general, determine which subspace the state is in.
The typical time for this ‘local purification’, i.e. for the state to learn
everything except the code stabilizer eigenvalues is $t_{*}$, as shown in
panel (3). Note that even after $t_{*}$, the state has dynamics as indicated
by the arrows in panel (3), but remains pure within each subspace
$\mathscr{C}(\bm{\sigma})$. Local measurements learn code stabilizer
generators at a much lower rate ($\tau_{\infty}^{-1}$), which gradually fixes
the state [panel (4)] until it is fully purified at a typical time
$t_{\infty}$ shown in panel (5). Even at the longest times the system has
dynamics within its final code stabilizer subspace, but the subspace remains
fixed.

####  III.1.2 Nonlocal Area Laws

For circuits based on codes, the steady-state entanglement entropy can receive
nonlocal contributions either from logical operators, observed e.g. in the
toric code [32], or from non-local stabilizers as was observed in the
measurement-only Kitaev model [35, 36].

While in both of these cases, the nonlocal contribution is topological (in the
sense of Eq. (missing) 5) and does not scale with system size, a key
innovation in our work is to consider circuits where _all_ generators of the
stabilizer group are necessarily non-local and take the form of subsystem
symmetries. Specificially, they act nontrivially on a $\Delta<D$ dimensional
subsystem, where $D$ is the full spatial dimension of the circuit.

The number of independent generators then grows subextensively with the
codimension of the stabilizers, i.e. as $\order{L^{D-\Delta}}$. As we show in
Sec. III, if $\Delta=1$ so that the stabilizer generators each act on a line-
like subsystem, they give a finite area-law contribution to the entanglement
entropy of the steady state in any dimension $D\geq 2$. This contribution is
non-local and hence at odds with the common wisdom that area law terms in the
entanglement entropy stem exclusively from local contributions (similar to
what is observed in Ref. 58 for the ground states of a class of local
Hamiltonians). Note that $D=3$ codes admit the possibility of sheet-like
stabilizers, but the corresponding non-local contribution is subleading to the
area law as it scales as $\order{L}$. Interestingly, only in the case where
the code admits string-like stabilizer generators do we see a sharp phase
transition between distinct area-law states as a function of measurement rate,
suggesting that a leading non-local contribution may be important to the phase
distinction associated to this transition.

Non-local stabilizers linked to subsystem symmetries are also known to give
rise to spurious contributions to the ground state entanglement entropy in
certain stabilizer codes [59]. However, in the dynamical setting there is an
important distinction between codes with stabilizer groups that cannot be
locally _generated_ and those with stabilizer groups that contain nonlocal
elements but nevertheless admit a local generating set. While both can lead to
unusual entanglement scaling, for the reasons discussed above, we do not in
general expect slow measurement-only dynamics in cases where the stabilizer
group can be generated by local operators.

####  III.1.3 Frustration Graphs and Volume-Law Entanglement

We also comment on a link between the late-time entanglement scaling and the
characterization of a circuit in terms of its frustration graph, originally
conjectured by Ref. 22. Each node $j$ of the frustration graph corresponds to
a check operator $g_{j}$; two nodes $i$, $j$ are connected by and edge if and
only if the two corresponding checks do not commute, $[g_{i},g_{j}]\neq 0$.
The authors conjectured that a volume-law phase can be generated by
measurement-only dynamics only if the frustration graph of the circuit is non-
bipartite. For the circuits based on subsystem codes considered in this work,
the frustration graph is bipartite if the underlying code is of the
Calderbank-Shor-Steane (CSS) form. We find that of the circuits we study only
those based on non-CSS codes support volume law steady states, consistent with
the conjecture of Ref. 22. [Note, however, that while CSS code circuits always
have bipartite frustration graphs, they are not the unique circuits with this
property. Examples of non-CSS code circuits with bipartite frustration graphs
include the measurement-only toric code with single-qubit $Y$ measurements
[32], and the one-dimensional chain with $\\{XX,YY,ZZ\\}$ checks on nearest-
neighbours, such that checks on even sites only anticommute with those on odd
sites [22].]

####  III.1.4 Measurement-only Analogue of Hilbert Space Fragmentation

Given the equivalence of the algebraic structure of subsystem codes and the
bond-algebraic formulation of Hilbert-space fragmentation, it is natural to
ask how fragmented Hilbert-space structure impacts dynamics that lack
unitarity. While one can endow non-unitary time evolution (e.g. by imposing a
similar block structure on Lindblad superoperators) with similar structure to
the unitary case, extending the notion of fragmentation to the measurement-
only setting has not to date been explored.

First note that as long as we monitor the measurement outcomes, the
measurement-only dynamics maps pure states to pure states. The resulting pure-
state trajectories are then also restricted to only to exploring a fixed
stabilizer subspace, and hence would resemble Panel (5) of Fig. 4. This is
analogous to the fragmented dynamics one would expect for unitary dynamics in
a subsystem code, and hence might lead to similar signatures such as
persistent oscillations in certain (spin-glass) observables. While systematic
exploration of such aspects is likely to be interesting in its own regard,
here we focus on the slow dynamics of _entanglement_. In this context our
results show that, on imposing the additional caveat that we consider
subsystem codes with non-local stabilizer generators, we can indeed define a
type of ‘fragmented measurement-only dynamics’ of scrambling or purification,
where the system persists in a sub-maximally entangled or sub-maximally-
purified state for a time scale that diverges (albeit sub-extensively) in
system size. We conjecture that imposing this kind of additional structure
beyond just the Hilbert-space decomposition of Eq. (missing) 2 is in fact
necessary for such slow dynamics of entanglement to arise in measurement-only
models. Furthermore, to the best of our knowledge, no special role has been
identified for nonlocally-generated symmetries in the context of fragmented
unitary dynamics; it would be interesting to consider this in future work.

####  III.1.5 Stability to Perturbations

We also comment on a related but distinct problem: namely, the probability of
(and hence the time scale on which) logical operators of error correcting
codes are measured by single-qubit measurements. This been studied in a series
in recent works [33, 34], which suggest a remarkable robustness of the logical
subspace against local measurements.

A natural question is then to ask how robust the phenomenology discovered in
this work (consisting of slow entanglement dynamics and the non-local late-
time area laws) is to adding perturbations that break the symmetry of the
circuit. We find that, as in most cases of Hilbert space fragmentation,
neither the slow dynamics nor the non-local area law contribution persists
upon introducing a finite probability of measuring single qubit operators — at
least in the examples considered here. As we now discuss, this is a direct
consequence of the nonlocal nature of _all_ stabilizer generators in our
examples.

Consider an environment initially entangled with the degree of freedom
associated with the code stabilizer $S_{j}$. This means that the stabilizer
group describing the state of the composite system contains generators of the
form

$E_{j}=O^{(\rm env)}_{j}\otimes S_{j}$ (6)

but neither $O_{\rm env}$ nor $S_{j}$ alone. When all measurements done on the
system commute with $S_{j}$, then the only way to disentangle the environment
is to measure $S_{j}$ itself, which takes time $\order{\exp L^{\Delta}}$. If
we now perturb the dynamics by introducing a small density of measurements
$M_{k}$ which do not commute with the $S_{j}$ however, then measuring the
$M_{k}$ can also disentangle the system.

If there are both nonlocal and local code stabilizers, then the local
stabilizers might be able to “protect” the nonlocal ones from local non-
commuting measurements $M_{k}$. This is because in order to purify a nonlocal
stabilizer generator, the operator $M_{k}$ has to commute with all local
stabilizer generators of the code [33]. Since the local stabilizer generators
are measured with some finite rate (as is the case for example in the
measurement-only Kitaev model [35, 36]), there is then competition between
adding operators to the state stabilizer group by measuring stabilizers and
removing them by measuring the $M_{k}$.

In contrast, the circuits considered in this work are based on codes where all
stabilizer generators are nonlocal and hence are measured with a rate that is
exponentially small in system size. In this case, since single-qubit
measurements remove stabilizers with finite probability $p_{s}$, any $p_{s}>0$
leads to purification in logarithmic time. We have verified this numerically
for the examples considered in this paper.

Note that the above argument is strongly reminiscent of the reasoning used to
argue that the Bacon-Shor code does not have a error-correction threshold [60]
and in particular no measurement threshold [34]. It hence serves as another
striking examples of the parallels between entanglement dynamics in circuits
and error correction (specifically, in this case, the error correction
properties of the underlying code).

###  III.2 Warmup: The Ising Chain

A particularly simple example of the above formalism is furnished by the
projective 1D transverse-field Ising model777We thank Matteo Ippoliti for
suggesting this illustrative example. [61]. This can be viewed as the
measurement-only dynamics of a trivial subsystem code whose competing check
operators are $g_{i}^{(X)}=X_{i}$, $g_{i}^{(Z)}=Z_{i}Z_{i+1}$, where
$i=1,\ldots,N$ denote sites (and we identify the sites at $i=1$ and $i=N$),
and $X/Z$ are Pauli operators. There is a single stabilizer
$S_{i}^{(X)}=\prod_{i}g_{i}^{(X)}$ with $\Delta=1$, and no logical qubits so
that $K=0$. If we consider measurement-only dynamics in the ‘paramagnetic’
which we measure the $g_{i}^{(X)}$ more frequently, we quickly measure the
single stabilizer and the system purifies quickly. However if instead we
consider the ‘spin glass’ phase where we measure the $g_{i}^{(Z)}$ more
frequently, then the system quickly learns all local information, but needs to
perform a system-spanning sequence of $X$ checks in order to diagnose the
global parity bit (encoded by the eigenvalue of $S_{i}^{(X)}$); since this is
frustrated by measuring the $Z$ checks, there is an exponentially slow
timescale for learning this single stabilizer and hence slow purification.

###  III.3 The 2D Bacon-Shor Code

Figure 5:  Entanglement Dynamics of the two-dimensional measurement-only
Bacon-Shor circuit. (a) Summary of the circuit geometry. At each time step, we
measure a random check operator $g_{ij}^{(X)}$, $g_{ij}^{(Z)}$ with
probability $p_{x}$, $p_{z}=1-p_{x}$, respectively. The (Pauli-)symmetry group
of the circuit is generated by the stabilizers $S_{i/j}^{X/Z}$ and logical
operators $\bar{X}/\bar{Z}$, which can and cannot be written as a product of
check operators, respectively. (b) Entanglement entropy across a cut in the
$y$-direction (see inset), as a function of time for $p_{x}=0.49$ when
starting from a trivial state. After an initial transient, the entanglement
entropy settles to a value given by the local contribution to the area law,
while the non-local contribution to the steady-state value $S_{\infty}$ builds
only at exponentially long times. (c) Purification dynamics when starting from
a fully mixed state. The inset shows a fit of the system-size dependence long
time scale $t_{\infty}$.

We begin our suite of examples by considering the best-known example of a code
with only non-local stabilizer generators: the two-dimensional Bacon-Shor
subsystem code [44]. The geometry of the code is sketched in Fig. 5 (a).
Qubits reside on the sites of a square lattice and a single two-body check
operator is defined on each bond, depending on its orientation:

$\displaystyle g_{(i,j)}^{(X)}=X_{(i,j)}X_{(i+1,j)},\quad
g_{(i,j)}^{(Z)}=Z_{(i,j)}Z_{(i+1,j)},$ (7)

where $(i,j)$, $i,j=1\dots L$ denote the sites and $X/Z$ are Pauli operators.
The stabilizer group of the Bacon-Shor code is generated by the so-called
“Bacon strips”888Pun intended, but not original to us., given by the product
of $Z$ and $X$ checks along a given row and column, respectively

$\displaystyle S_{i}^{(X)}=\prod_{j}g_{(i,j)}^{(X)},\qquad
S_{j}^{(Z)}=\prod_{i}g_{(i,j)}^{(Z)}.$ (8)

Bare logical operators in turn are given by products of $Z$ and $X$ operators
along a single row and column, respectively:

$\displaystyle\bar{X}=\prod_{j}X_{(0,j)},\qquad\bar{Z}=\prod_{i}Z_{(i,0)}.$
(9)

Note that the choice of row/column of the logical operator is arbitrary, since
different choices are related by multiplication with a code stabilizer.

In Fig. 5 (b), we show the half-system entanglement entropy $S(t)$ as a
function of time for stochastic measurement-only dynamics (see subsection II.2
for details) starting from a product state. The data clearly reveals the
existence of two dynamical regimes at long times: after an initial transient,
the system settles to a local area law, while the non-local area law state
that we expect for $t\to\infty$ is reached only after a time exponentially
long in the linear system size $L$.

Figure 6:  Scrambling in the 2D Bacon-Shor Code for $p_{z}>p_{x}$. We sketch a
typical set of state stabilizer generators for three characteristic times. (1)
At $t=0$, the system is in a product state and the state stabilizer group is
generated by randomly-choosen single-qubit Pauli operators. (2) At $t\sim
t_{*}$, the state stabilizer group can still typically be generated by only
local operators, but will no longer be an exact product state (and hence will
generically have area-law entanglement). (3) We now draw $X$ and $Z-$type
state stabilizer generators separately for better visibility. The state
stabilizer group at late times $t\sim t_{\infty}$ typically does not admit a
local generating set of operators.

We can rationalize this behaviour and our characterization of the entanglement
at $t\sim t_{*}$ and $t\sim t_{\infty}$ as local and nonlocal respectively by
considering the behaviour of the typical state stabilizer generators at these
these characteristic times. A representative sketch of these generators is
shown in Fig. 4, for the case $p_{z}>p_{x}$, which we now discuss. The system
is initialized in a product state at $t=0$, whose state stabilizer group is
generated by randomly chosen single-qubit operators. As time proceeds and we
measure check operators, the state stabilizers are no longer single qubits.
Since the check operators in different directions do not commute, however,
$Z$\- and $X-$type state stabilizers grow along different axes, but their
growth can be truncated by competing measurements; however, once a state
stabilizer has grown to span the system it coincides with a code stabilizer
and is thereafter invariant under the dynamics.

To understand the consequences for entanglement, recall that the computation
of the bipartite entanglement entropy $S_{E}$ follows by choosing state
stabilizer generators such that a minimal number of them straddle the
entanglement cut [55]. Evidently, if a code stabilizer is also part of this
minimal generating set, this leads to a nonlocal contribution to $S_{E}$.
However, whether this is the case depends on the nature of the typical local
stabilizers. (Since these are all state stabilizers, for conciseness we omit
that qualifier below.) In the steady state of our measurement-only dynamics,
this is controlled by the relative probability of $Z$\- and $X-$type
measurements. For $p_{z}>p_{x}$, $\mathcal{S}(\rho)$ typically contains more
local $Z-$type stabilizers than $X$-type ones. In this case, although a
nonlocal $Z$-type code stabilizer $S_{j}^{(Z)}$ will also be in
$\mathcal{S}(\rho)$ with high probability, there are typically many local
$Z$-type stabilizers that can be used to generate it; so any minimal
generating set of $\mathcal{S}(\rho)$ is unlikely to require nonlocal
generators in order to generate $S_{j}^{(Z)}$. Conversely, a nonlocal $X$-type
code stabilizer $S_{i}^{(X)}$ is unlikely to be in $\mathcal{S}(\rho)$ at
early times since it requires a rare event (the probability is exponentially
small in linear system size) but once created it remains in
$\mathcal{S}(\rho)$ for all later times. Since at any given time there are
typically very few local $X$-type stabilizers in $\mathcal{S}(\rho)$ that
could generate such an $S_{i}^{(X)}$, it is highly likely that any generating
set of $\mathcal{S}(\rho)$ will require nonlocal elements999Note that we
cannot guarantee that any specific nonlocal code stabilizer is in the
generating set, since it is nonunique.. Thus the $S_{i}^{(X)}$s, once
measured, will with high probability yield a nonlocal contribution to $S_{E}$.

This results in the following picture of stabilizer dynamics for
$p_{z}>p_{x}$. We begin in a product state, and as noted we start building
local entanglement by measuring check operators. After an initial transient,
at a time $t\sim t_{*}$ the the state stabilizer group can still typically be
generated by local (but generically not _single-qubit_) operators, and hence
the subsystem entanglement entropy follows the usual area-law scaling with
subsystem size. Furthermore the $S_{j}^{(Z)}$ have with high probability been
effectively measured already. However, as noted above, they do not typically
yield a nonlocal contribution to $S_{E}$. As the system evolves, an occasional
rare event will create an $S_{i}^{(X)}$; as noted above this will with high
probability require another nonlocal generator for $\mathcal{S}(\rho)$ and
hence yields a nonlocal contribution to $S_{E}$ (for a cut along
$\bm{\hat{x}}$-direction). Each such rare event increases the nonlocal piece
of $S_{E}$ by one bit, until a time $t\sim t_{\infty}$ — corresponding to when
all $N_{S}\sim L$ of the $S_{i}^{(X)}$ have been measured — the entanglement
reaches its saturation value $S_{\infty}$, the nonlocal contribution to which
scales with the extent of the subsystem along the $\bm{\hat{x}}$ direction.

The behavior for the case $p_{x}>p_{z}$ is equivalent, but with the roles of
the stabilizers $S_{j}^{(Z)}$ and $S_{i}^{(X)}$ reversed, and the nonlocal
contribution to $S_{E}$ given by the extent of the subsystem in the
$\bm{\hat{y}}$-direction. Evidently, in both cases, there will be no slow
buildup of entanglement for cuts that are not straddled by the slow
stabilizers, since this direction does not have a nonlocal area-law
contribution.

The dynamics can also be probed by studying the process of purification, i.e.
by tracking the von Neumann entropy of the mixed state $\rho$ as a function of
time when starting from a fully mixed initial state. We show this in Fig. 5
(c), where again two dynamical regimes are clearly visible: the system quickly
[in a time of $\order{\log N}$] purifies down to an entropy $S=L-1$, and then
decays slowly towards the infinite-time limit $S\to 1$ (which is set by the
fact that the logical qubit cannot be disentangled from the environment by
this choice of dynamics). To verify this picture quantitatively, we fit the
von Neumann entropy $S(t)$ at times after the initial transient to an
exponential decay, $S-1\propto e^{-t/\tau}$, which defines a purification time
scale $\tau$. We show the system-size dependence of $\tau$ in the inset of
Fig. 5 (c), and find that it is well-described by an exponential, $\tau\propto
e^{0.18L}$.

The purification dynamics also shows two distinct dynamical phases as a
function of the relative measurement rates $p_{x}$ and $p_{z}$. For
$p_{z}>p_{x}$ (the case discussed above), the $Z$ stabilizers, despite their
large support, are measured quickly whereas the the $X$ stabilizers take an
exponentially long time (in $L$) to be measured. For $p_{z}<p_{x}$ the roles
of $X$ and $Z$ stabilizers are reversed. Finally, for $p_{x}=p_{z}$ the system
shows critical dynamics: there is no longer a separation of purification
timescales between $X$\- and $Z$-type stabilizers, and instead the system
shows algebraic scaling of the purification time, $S(t)\sim\exp(-t/L^{z})$
with $z=3/2$ [50, 32].

###  III.4 Going 3D: Membranes vs. strings in the 3D Bacon-Shor and CSS-
Plaquette Code

In the preceding section, we demonstrated the unusual scrambling and
purification dynamics en route to a non-local late-time area law state in a
two-dimensional model model with string-like non-local stabilizer generators.
A natural next step is to clarify the role played by the geometry of the
stabilizer generators in this problem:- for instance, if the non-local
generators were sheet-like rather than string-like. In order to investigate
the effect of such changes to the geometry, we are led to consider three-
dimensional subsystem codes.

Figure 7:  Purification dynamics of the 3D Bacon-Shor code. (a) geometry of
the check operators in the 3D Bacon-Shor code. (b) von Neumann entropy $S$ as
a function of time $t$ for linear different system sizes $L$. (c) purification
time scale of the more frequently measured stabilizer (here $S^{(Z)}$) as a
function of system size for different measurement rates $p_{x}$. (d) growth
rate of the fast stabilizer purification time scale, $\alpha$, as a function
of measurement rate $p_{x}$.

Concretely, we study two such codes: (1) the three-dimensional Bacon-Shor code
[44] with sheet-like stabilizer generators ($\Delta=2$) and (2) the CSS-
Plaquette code which has string-like stabilizer generators ($\Delta=1$). We
begin by discussing the former, which is a generalization of the two-
dimensional case and sketched in Fig. 7 (a). Qubits reside on the sites of a
simple cubic lattice and checks are defined for bonds as follows: we define
one $Z$($X$)-type check for each bond in the $\hat{z}$($\hat{x}$)-direction
and _both types_ of checks for bonds in the $\hat{y}$ direction

$\mathcal{G}=\expectationvalue{X_{\bm{r}}X_{\bm{r}+\hat{\bm{x}}},X_{\bm{r}}X_{\bm{r}+\hat{\bm{y}}},Z_{\bm{r}}Z_{\bm{r}+\hat{\bm{y}}},Z_{\bm{r}}Z_{\bm{r}+\hat{\bm{z}}};\bm{r}}.$
(10)

Restricting ourselves to odd $L$101010For even $L$, the three-dimensional
Bacon-Shor code still has non-local stabilizer generators, but has no logical
qubit (see e.g. the discussion in Sec. IX B of Ref. 62)., the stabilizer
generators of this code are given by generalizing “Bacon strips” to
“nailbeds”: products of checks along planes perpendicular to the $\hat{z}$ and
$\hat{x}$ direction, respectively. The logical operators are products of
single-qubit Pauli operators along the same set of planes [see $\overline{X}$
and $\overline{Z}$ indicated in Fig. 7 (a)]. Similar to the two-dimensional
case, the choices of plane here is arbitrary, since products of single-qubit
paulis along any give plane are related by multiplication with stabilizers.
The three-dimensional Bacon-Shor code hence encodes a single logical qubit.

The presence of nonlocal, sheet-like stabilizer generators again leads to slow
scrambling and purification dynamics, yet with the important difference that
both sets of sheet-like stabilizers purify slowly _independent_ of the
measurement rate of their corresponding check operators. To demonstrate this,
we show in Fig. 7 (b) the measurement-only purification dynamics of the three-
dimensional Bacon-Shor code. The von Neumann entropy clearly exhibits two
plateaus whose width grows with system size, and its late-time behavior is
indeed fit by a functional dependence of the form

$S(t)=1+A\,e^{-t/\tau_{Z}}+B\,e^{-t/\tau_{X}},$ (11)

where both $\tau_{Z}$ and $\tau_{X}$ grow exponentially with the square of the
linear system size. This is shown explicitly in Fig. 7 (c), where we plot the
faster of the two time scales ($\tau_{Z}$ since $p_{z}>p_{x}$) as function of
system size $L^{2}$ for a range of measurement rates $p_{x}$. The solid lines
are fits to

$\tau_{Z}\sim\exp(\alpha L^{2})$ (12)

Remarkably, the growth rate $\alpha$ itself depends exponentially on the
measurement rate $p_{x}$, as shown in Fig. 7 (c). In particular, for
$p_{x}\lesssim 0.47$ the $p_{x}$-dependence of $\alpha$ is well described by
the functional form

$\alpha\sim\exp[-\gamma(\tfrac{1}{2}-p_{x})],$ (13)

where $\gamma\approx 75.8\pm 3.1$ as extracted from a linear fit.

In summary, the numerical results suggest that for any $p_{x}>0$ ($p_{z}>0$)
the purification of $Z$ stabilizers ($X$ stabilizers) is exponentially slow in
the system size $L^{2}$, even though the prefactor of this exponential grow is
parametrically small for small $p_{x}$ ($p_{z}$). In particular this means
that for all $p_{x},p_{z}>0$ the model is in the same dynamical phase, in
contrast to the case of the two-dimensional Bacon-Shor code. The critical
point is replaced by a crossover in the purification timescales $\tau_{Z}$ and
$\tau_{X}$, which coincide at the point $p_{x}=p_{z}$.

We attribute the differing measurement-only dynamics of the two- and three-
dimensional Bacon-Shor codes to the contrasting geometry of the stabilizer
generators. To see this, consider how stabilizer operators are measured in
stochastic measurement-only dynamics. Viewing the dynamics in terms of the
state stabilizer group, to measure a stabilizer with support on a
$\Delta$-dimensional subsystem, one has to measure a sequence of $L^{\Delta}$
check operators. At any intermediate step, the state stabilizer group will
contain ‘incomplete’ code stabilizers: these are either open strings that
would eventually grow into a string-like ($\Delta=1$) stabilizers, or open
membranes that would eventually grow into sheet-like ($\Delta=2$) stabilizers.
Such a code stabilizer is measured (‘completed’) only when this open string or
open membrane spans the full system. Apart from the trivial outcome of no
change, there are two possible fates for each incomplete code stabilizer in
the subsequent time step: it can either be entirely removed from the state
stabilizer group by measuring a non-commuting check operator; or it can grow
if a commuting check is measured at its perimeter. Now, the checks of the
subsystem code that fail to commute with a given incomplete code stabilizer
$S_{C}$ and are hence capable of eliminating it from the state stabilizer
group are precisely those at the perimeter of $S_{C}$. Thus, both growth and
removal of $S_{C}$ occurs by measuring check operators at its perimeter. For a
string-like code stabilizer, it is now as likely that the incomplete operator
is grown by a single step as it is that the entire operator is removed. The
processes hence do not compete fairly, yet if the measurement probabilities
are sufficiently biased, string-like code stabilizers can be measured quickly.
For sheet-like stabilizers, the competition is even more unfair: to grow the
sheet by one step, one has to measure the correct check along the entire
perimeter, but measurement of any non-commuting check along its perimeter
still removes the incomplete operator in its entirety. Because of this, even
biasing the check measurement rates in favor of the stabilizer may be
insufficient to measure it in subexponential time.

To test the above hypothesis, we also consider a three-dimensional model in
which all the stabilizer generators are string-like, and confirm that in this
case there are again two distinct dynamical phases separated by a critical
point at $p_{x}=p_{z}$ (akin to the two-dimensional Bacon-Shor code).
Naturally, this model also features a leading-order non-local contribution to
the entanglement entropy in the area law phase, owing to the $\order{L^{2}}$
string-like code stabilizers.

This three-dimensional model is based on a code that we term the “CSS
plaquette code”. This is again defined on qubits situated on the sites of a
simple cubic lattice, but the check operators now involve four qubits defined
on the elementary plaquettes of the cubic lattice. As is shown in Fig. 8 (a),
four-body Pauli-$X$ checks are defined on every plaquette perpendicular to the
$\hat{x}$ and $\hat{y}$ directions, and four-body Pauli-$Z$ checks are defined
on every plaquette perpendicular to the $\hat{z}$ and $\hat{y}$ directions.
The geometry of the full stabilizer group is quite involved and is different
for odd and even system sizes (see App. B for details). Most importantly for
the present purposes, the code has $\order{L^{2}}$ string-like stabilizer
generators, sketched in Fig. 8 (b), which suffices to induce a phase
transition as well as a non-local contribution to the area law. This is
demonstrated in Fig. 8 (c), where we show purification dynamics in the phase
where the $Z$ stabilizers purify fast (top) as well as for the critical point
(bottom). For $p_{x}\neq p_{z}$, the von Neumann entropy shows a clear plateau
at $S\sim L^{2}$ (indicated by a dotted line for the largest system size),
whose width grows exponentially with the linear system size $L$. There is also
a second plateau at $S\sim L$ (indicated by a dash-dotted line, again only for
the largest system size), whose width grows exponentially with the square of
the linear system size $L^{2}$; this is only present for even system sizes for
which the system also has sheet-like stabilizer generators. For odd system
sizes, these sheet-like operators are logicals and hence they simply
contribute to the infinite-time limit $S(t\to\infty)$. Finally, for
$t\to\infty$ the entropy approaches the limit $S\to k$ (indicated by a dashed
line), where $k=L$ is the number of logical qubits of the CSS-plaquette code
for even $L$. For $p_{x}=p_{z}$, the system shows critical scaling of the
purification time: at late times the data is compatible with

$S(t)\sim\exp(-\gamma\,t/L^{z}),$ (14)

with, remarkably, a subdiffisuive scaling $z=4$ [see bottom panel of Fig. 8
(c)]. This dynamical exponent is different both from that seen at the critical
point of the two-dimensional Bacon-Shor circuit [50, 32] ($z=3/2$) as well as
from those seen in the critical phase of the measurement-only Kitaev honeycomb
model ($z=1/2$ at early times [36] and $z=2$ at late times [35, 36]). In both
those two-dimensional cases the $z\leq 2$ scaling is superdiffusive in
contrast to the subdiffusive scaling seen here. As was suggested in Ref. 32,
such unusual exponents might be connected to the presence of subsystem
symmetries. Drawing a precise connection between symmetry properties and
dynamical exponents remains an important topic for future work. Finally, we
note from the bottom panel of Fig. 8 (c) that there is no additional plateau
associated with the sheet-like stabilizers at the critical point, suggesting
that these purify in polynomial time.

Figure 8: Purification dynamics of the CSS plaquette code. (a) Geometry of
four-body checks of the code. (b) geometry of string-like stabilizer
generators. (c) von Neumann entropy as a function of time for the phase where
Pauli-$Z$ stabilizers purify fast $p_{x}<p_{z}$ (top) and at the critical
point $p_{x}=0.5$ (bottom). The horizontal lines in the top panel are a guide
to the eye (see main text for details). For the critical point, we plot the
entropy density and fit the purification time $\tau$ at late times. The
scaling of $\tau$ with system size is compatible with a power law with
exponent $z=4$ (see inset).

###  III.5 Adding Frustration: 3D Compass Code and non-CSS Plaquette Code

We now turn to measurement-only circuits with non-bipartite frustration
graphs. Consistent with the connection originally conjectured in Ref. 22, we
find that such circuits support a volume-law phase as part of their phase
diagram. Specifically, we consider circuits based on two different three
dimensional non-CSS codes, each of which is obtained by modifying one of the
CSS codes studied in the preceding section.

The first of these is a modification of three-dimensional Bacon-Shor code,
where as before check operators are defined as before on bonds of simple cubic
lattice. However, we now impose Pauli-$X$, $Y$ and $Z$ checks on bonds in the
$\hat{x}$, $\hat{y}$ and $\hat{z}$ directions, respectively:

$\displaystyle
g_{\bm{r}}^{(\alpha)}=\alpha_{\bm{r}}\alpha_{\bm{r}+\hat{\bm{\alpha}}},,$ (15)

where $\alpha\in\\{X,Y,Z\\}$. It is easy to verify that this subsystem code
still encodes a single qubit, and has $3L-3$ stabilizers, which are products
of $X$, $Y$ and $Z$ checks across planes perpendicular to their respective
bond direction. In the following, we use the term “three-dimensional compass
circuits” to refer to measurement-only circuits based on this code, since when
considering the Hamiltonian $H=-J\sum_{g\in\mathcal{G}}g$, we obtain the so-
called “three-dimensional 90∘ compass model” [63].

The phase diagram that we conjecture based on studying purification dynamics
along two cuts in parameter space is shown in Fig. 9. A volume law phase
occupies the center of the phase diagram, around $p_{x}=p_{y}=p_{z}$, and
biasing the measurements towards one of these choices induces a transition
into an area-law phase, as shown in panel Fig. 9 (b). This shows the von
Neumann entropy after polynomial time $t=L^{3}$, when starting from a
maximally mixed initial state. We expect that this number grows with the
volume of the system $\order{L^{3}}$ in the volume-law phase [64] while it
grows with the number of slowly-purifying stabilizers [here $\order{L}$] in
the area-law phase. Indeed, plotting $S(t=L^{3})/L$ against $L^{2}$, we see
that the data shows linear growth for $p_{z}\lesssim 0.52$, it saturates to an
number independent of system size for $0.52\lesssim p_{z}$. We note that for
the system sizes studied here, the fastest stabilizers $S^{(Z)}$ have already
been measured at time $t=L^{3}$, so we observe $S(t=L^{3})=2L-1$ for
$0.52\lesssim p_{z}$. However, based on the arguments in the previous section,
namely that sheet-like stablizer generators always purify on a time scale
exponentially large in system size, we actually expect $S(t=L^{3})=3L-2$. We
therefore deduce that the “missing” entropy is in fact a finite-size effect
stemming from the highly biased measurement rate, that can strongly
parametrically suppress the growth rate of the $S^{(Z)}$ stabilizers [cf. Eq.
(missing) 12 and Eq. (missing) 13].

We also numerically studied purification dynamics along the line
$p_{z}=p_{y}$, shown as gray-dashed arrow in Fig. 9 (a), finding that the
volume law phase does extend at least to $p_{x}=0.01$. We therefore conjecture
that the system is in fact in a volume law phase for any $p_{x}>0$. Note that
this is not guaranteed by simple arguments since we do not expect the three
area law phases separated by the volume law phase to have distinct
entanglement dynamics based on the arguments in the previous section. Along
the boundaries of the phase diagram (i.e., where one of the three $p_{i}=0$),
the system decouples (up to application of a local unitary) into $L$
independent copies of two-dimensional $L\times L$ Bacon-Shor code circuits.

Figure 9:  Conjectured entanglement phase diagram of the 3D compass circuit
(see main text for details). (a) Check geometry and topology of the phase
diagram. We numerically scanned the phase diagram along the lines indicated in
gray. (b) von Neumann entropy after polynomial time as a function of
measurement rate $p_{z}$ along the line $p_{x}=p_{y}$ indicated [solid gray
line in panel (a)]. The data shows a clear transition from volume to area-law
behavior at a critical rate of about $p_{c}\approx 0.54\pm 0.02$.

Finally, we can similarly define a non-CSS version of the plaquette code that
we introduced in the previous section (also see App. B for details). The check
operators of this “non-CSS plaquette code” are four-body Pauli-$X$, $Y$ and
$Z$-type checks, all defined on the plaquettes on a simple-cubic lattice:
instead of measuring both $X$ and $Z$ operators on plaquettes perpendicular to
the $\hat{y}$ direction as in the CSS version, here we measure a four-body
Pauli-$Y$ check with probability $p_{y}$. We have numerically computed
purification dynamics along the same paths in parameter space as for the
three-dimensional compass circuits [gray solid and dashed lines in Fig. 9
(a)]. Again, our numerical results suggest that the system undergoes a
transition from a volume to area law phase along the line $p_{z}>p_{x}=p_{y}$
($p_{c}\approx 0.51\pm 0.02)$ and that it is in a volume law phase for
$p_{z}=p_{y}>p_{x}>0$. In contrast to the three-dimensional compass circuit
case, the area law phases now have a non-local contribution to the
entanglement entropy at leading order as well as distinct entanglement
dynamics. Potentially, this could lead to an even richer phase diagram than
the one conjectured in Fig. 9 (a) for the compass model circuit, but a
detailed investigation of this is beyond the scope of the present work.

##  IV Concluding Remarks

We have presented the first systematic study of measurement-only dynamics in
subsystem quantum error-correcting codes; for reasons described in the
introduction such codes offer a particularly natural route to building code
properties into circuit dynamics. In this context, our work singles out a
special subset of codes whose stabilizer groups cannot be generated by purely
local operators. Such codes are most conveniently constructed by considering
models with nonlocally-generated subsystem symmetries: the symmetry generators
are the nonlocal stabilizers, and their number diverges (subextensively) in
the thermodynamic limit.

The presence of such nonlocal stabilizers has striking consequences:

  * •

the entanglement entropy along steady-state trajectories of the measurement-
only dynamics of such codes acquires an unusual contribution linked to the
nonlocal stabilizer generators. Strikingly, where the entanglement entropy of
these states satisfies an area law, such nonlocal terms can contribute at
leading order, in contrast to the conventional wisdom that area law
contributions stem from short-range entanglement.

Furthermore, the approach to the late-time state is itself characterized by
two distinct dynamical regimes, manifest in both scrambling and purification:

  * •

in the scrambling dynamics of an initial product state, the local contribution
to the entanglement saturates quickly, in $\order{\log L}$ time steps, whereas
the non-local piece is only saturated in a time that is exponentially long in
system size.

  * •

in the purification dynamics of an initially maximally mixed state, the system
initially purifies relatively quickly up to a time of $\order{\log L}$ when
all the local degrees of freedom have been disentangled from the environment
or equivalently, all but the nonlocally encoded information about the state
has been extracted from the system; thereafter, the rate of purification slows
down and as the nonlocal stabilizers are purified, which again takes a time
exponentially long in system size, leaving only the logicals (which cannot be
measured) still entangled with the environment, or equivalently, their value
remains unknown even to an observer that has access to the full history of
measurement outcomes.

To establish and rationalize these results, we numerically simulated
measurement-only dynamics in several different codes in spatial dimensions
$D=2$ and $3$. In each of these models _all_ stabilizer generators of the code
are nonlocal and involve symmetries that act on a subsystem of dimension
$0<\Delta<D$, leading to a non-local contribution to the entanglement entropy
that scales with the number of stabilizer generators, $N_{S}\sim
L^{D-\Delta}$. We also showed that whether a given stabilizer generator is
“slow”, [meaning that the time scale on which it is measured diverges as
$\tau_{\infty}\sim\exp(\gamma L^{\Delta})$], in general depends on its
geometry as well as the relative measurement rate of different check
operators. To wit, we found that sheet-like stabilizer generators with
$\Delta=2$ are always measured slowly, with a rate that vanishes exponentially
in the limit $L\to\infty$ for generic choices of parameters that exclude
critical points. In contrast, string-like ($\Delta=1$) stabilizer generators
can be measured quickly, i.e. on a $\order{\log L}$ timescale, but only if the
measurement rates of the check operators operators are appropriately biased.
Translating this behaviour to the phase diagrams of our circuits as a function
of measurement rates for different kinds of check operators, this implies that
codes with string-like stabilizer generators can host critical points between
area-law phases with distinct dynamical behavior (as found e.g. in the 2D
Bacon-Shor code), whereas no such critical points are possible for codes with
sheet-like stabilizer generators (such as the 3D Bacon-Shor code). We also
found that in accord with a conjecture made in Ref. 22, circuits based on CSS
codes do not support volume-law phases in their phase diagram, while those
based on non-CSS codes do.

Our results present a further step in the exploration of the dynamics of
quantum many-body systems out of equilibrium. Perhaps most importantly, they
are a clear demonstration that the algebraic structure of operators underlying
a given dynamics (here, the operators measured in the measurement-only
dynamics) can have striking consequences for both the steady-state
entanglement properties as well as the approaching to this state, even beyond
the setting of unitary dynamics. We also drew a concrete parallel between the
latter observation and the phenomenon of Hilbert space fragmentation in
unitary dynamics, and its explication based on commutant algebras [40].

Multiple aspects of our work suggest directions worthy of future study. It
would be fascinating to see to what extent our results can be generalized
beyond stabilizer dynamics and the measurement-only setting. Staying within
the context of measurement-only dynamics based on subsystem codes, first an
important question for future work remains to understand the universality
classes of the dynamical transitions, given that the critical behaviour is
neither captured by existing mappings to percolation [32] nor by those to loop
models [65, 66]. Second, there are other examples whose study could yield
entirely novel phenomenology, such as circuits based on two-dimensional
compass codes [67], where the size of stabilizer generators follows a broad
distribution. The direct parallel of our results to Hilbert space
fragmentation suggests that all the codes studied here may be relevant in that
context as well. Also, while the thermalization of subsystem codes under
unitary dynamics has been studied previously [68], there is no systematic
exploration of the difference between the unitary dynamics of systems with
(subsystem) symmetry groups that can be generated by local operators (such as
the parent Hamiltonian of the cluster state [59]) and those that cannot (e.g.
the 2D Bacon-Shor code). Finally, the three-dimensional plaquette codes
studied in this work have to the best of our knowledge not been previously
studied, yet have favorable code parameters (see App. B). It may be of
independent interest to explore their error-correcting properties even away
from the dynamical setting considered here.

###### Acknowledgements.

We thank Matteo Ippoliti, Steven Simon, Victor Albert, Sarang Gopalakrishnan,
and Vedika Khemani for valuable discussions and correspondence on related
topics, and are especially grateful to Matteo Ippoliti for detailed comments
on an earlier manuscript. B.P. acknowledges funding through a Leverhulme-
Peierls Fellowship at the University of Oxford and the Alexander von Humboldt
foundation through a Feodor-Lynen fellowship. S.A.P. acknowledges support from
EPSRC Grants EP/S020527/1 and EP/X030881/1.

##  Appendix A Measurement-only dynamics of the Yao-Kivelson Model

In this section we give an example of a case where not all _stabilizer_
generators of the code can be measured by a sequence of check measurements.
The example is based on a Kitaev Hamiltonian on a decorated honeycomb lattice,
first considered by Yao and Kivelson in Ref. 69. The geometry of the lattice
is shown in the bottom left of the top panel in Fig. 10: it is obtained from
the honeycomb lattice by replacing each site with a triangle, which preserves
its three-colorability. Here, we consider stochastic measurement-only dynamics
analogous to that studied for the Kitaev model on the honeycomb lattice in
Ref. 35, 36. That means given the three-coloring indicated in Fig. 10, at each
time step we measure a two-body Pauli-$X$/$Y$/$Z$ operator on a randomly
chosen red/green/blue bond with probability $p_{x}$/$p_{y}$/$p_{z}$
respectively.

Before discussing the numerical results in more detail, we flag perhaps most
striking difference between the Yao-Kivelson circuit and the standard Kitaev
circuit: namely, that the former does not purify the system even at infinite
times. We find $S(t\to\infty)=1$, whereas $S(t\to\infty)=0$ for the Kitaev
circuit on the honeycomb lattice. It turns out that this is a direct
consequence of the presence of odd-length loops (triangles) in the model, as
we now discuss. First note that when viewed as a subsystem code, the Yao-
Kivelson model has no logical qubit and the stabilizer group is given by
products of checks along closed loops on the lattice. Hence, when we consider
the system on a torus, the stabilizer group is generated by products of checks
around the elementary plaquettes of the lattice (triangles and octagons), as
well as two non-local loops that wrap around the two inequivalent handles of
the torus. While even-length loops of check operators can be measured
effectively by a sequence of check measurements (in the sense that after this
sequence of measurements, the respective product will be part of the state-
stabilizer group), this is not true for odd-length loops. In fact, it is
straightforward to verify that, given three sites $i,j,k$, no sequence of
measurements of the operators $X_{i}X_{j}$, $Y_{j}Y_{k}$ and $Z_{i}Z_{k}$ will
effectively measure their product, i.e. the operator $Y_{i}Z_{j}X_{k}$. For
the Yao-Kivelson circuits this means that exactly one stabilizer degree of
freedom, associated with a single triangle, remains unpurified even for
$t\to\infty$ when measuring only nearest-neighbor check operators. Note that
one can effectively measure products around two triangles, so that only a
single degree of freedom will remain mixed globally.

Figure 10: Purification dynamics of the measurement-only Yao-Kivelson circuit.
The geometry of the measured two-body operators is sketched in the top panel.
The bottom panel shows von Neuman Entropy density $(S-1)/L^{2}$ as a function
of time at the isotropic point $p_{x}=p_{y}=p_{z}$, which is compatible with
critical scaling. The top panel shows the same quantity for strongly
anisotropic measurement rates, compatible with a topological ordered phase.

For completeness, in Fig. 10 we show the purification dynamics for both the
isotropic point $p_{x}=p_{y}=p_{z}$ (bottom panel) and for strongly
anisotropic measurement rates $p_{x}=p_{y}=0.15$, $p_{z}=0.7$ (top panel). The
data is consistent with the two points being in distinct phases. At the
isotropic point, the system in the observed time purifies all acessible
degrees of freedom, i.e. $S\to 1$. For strong anisotropy, the nonlocal
stabilizer generators purify slowly, leading to $S\to 3$ in the observed time.
These two behaviours are consistent with the system being in a critical and
topologically ordered phase for $p_{x}=p_{y}=p_{z}$ and $p_{x}=p_{y}=0.15$,
$p_{z}=0.7$, respectively.

A detailed study of the two phases and the transition is beyond the scope of
this appendix, but remains an important task for future work. In particular it
would be interesting to study the consequences of the fact that the lattice
here is not bipartite, which means that the Yao-Kivelson model presumably maps
onto a classical loop model _with crossings_ [66]. This in particular is
different to the measurement-only honeycomb model, which can be mapped onto a
classical loop model without crossings [65, 70],

##  Appendix B Detailed description of the Plaquette codes

In this appendix, we provide some details about the CSS and non-CSS plaquette
codes studied in the main text.

Qubits for both codes reside on the sites of a simple cubic lattice and the
check operators are four-body Pauli operators situated on plaquettes of the
lattice. In the following, we give a detailed description of all check
operators, stabilizers, and logical operators.

For the CSS codes, there are four check operators defined per lattice site
$\bm{r}$

$\displaystyle g_{\bm{r},\hat{\bm{x}}}^{(X)}$
$\displaystyle=X_{\bm{r}}\,X_{\bm{r}+\hat{\bm{y}}}\,X_{\bm{r}+\hat{\bm{z}}}\,X_{\bm{r}+\hat{\bm{y}}+\hat{\bm{z}}},$
(16a) $\displaystyle g_{\bm{r},\hat{\bm{y}}}^{(X)}$
$\displaystyle=X_{\bm{r}}\,X_{\bm{r}+\hat{\bm{x}}}\,X_{\bm{r}+\hat{\bm{z}}}\,X_{\bm{r}+\hat{\bm{x}}+\hat{\bm{z}}},$
(16b) $\displaystyle g_{\bm{r},\hat{\bm{z}}}^{(Z)}$
$\displaystyle=Z_{\bm{r}}\,Z_{\bm{r}+\hat{\bm{x}}}\,Z_{\bm{r}+\hat{\bm{y}}}\,Z_{\bm{r}+\hat{\bm{x}}+\hat{\bm{y}}},$
(16c) $\displaystyle g_{\bm{r},\hat{\bm{y}}}^{(Z)}$
$\displaystyle=Z_{\bm{r}}\,Z_{\bm{r}+\hat{\bm{x}}}\,Z_{\bm{r}+\hat{\bm{z}}}\,Z_{\bm{r}+\hat{\bm{x}}+\hat{\bm{z}}}.$
(16d)

We sketch the geometry of checks, stabilizers and logical operators for both
even and odd linear system sizes in Fig. 11. On the top left of the figure we
sketch the check operators also written in Eq. (missing) 16. Next to it, we
show string-like products of checks, which can be written as

$S^{(\alpha,\mathrm{str})}_{\bm{r}}=\prod_{j=1}^{L}g^{(\alpha)}_{\bm{r}+j\hat{\bm{\alpha}},\hat{\bm{\alpha}}}$
(17)

and which commute with all checks and are hence in the stabilizer group of the
code. On the bottom left of the figure, we show string-like products of $X$
and $Z$ operators, which also commute with all check operators and hence are
logical operators of the code

$\displaystyle\bar{X}^{(\mathrm{str})}_{\bm{r}}=\prod_{j=1}^{L}X_{\bm{r}+j\hat{\bm{x}}}\qquad\bar{Z}^{(\mathrm{str})}_{\bm{r}}=\prod_{j=1}^{L}Z_{\bm{r}+j\hat{\bm{z}}}.$
(18)

Finally, on the bottom right of the figure we show sheet-like products of
Pauli operators, which always commute with all checks, but can be written as a
product of checks only if the linear system size $L$ is even. This means,
these operators are bare logicals for odd $L$ but stabilizer generators for
even $L$.

Figure 11: Geometry of the CSS plaquette code. Checks are defined on the
plaquettes of a simple cubic lattice of linear size $L$. For even $L$ (solid
lines), there two kinds of stabilizers: one takes the form of string-like
products of checks (top right) and another one takes the form of sheets of
Pauli operators (bottom right). The bare logical operators (bottom left) take
the form of strings of single-qubit Pauli operators. For odd $L$, the sheet-
like stabilizer operators become logical operators.

Using the above, it is straightforward to see that the CSS plaquette codes on
a $L\times L\times L$ simple cubic lattice with periodic boundary conditions
are a family of codes with

$[[N,K,D]]=\begin{cases}[[L^{3},3L-2,L]]&\text{if $L$ even}\\\
[[L^{3},L,L]]&\text{if $L$ odd}\end{cases}.$ (19)

There are $2(L-1)^{2}$ independent string-like stabilizer generators if the
system size is odd and an additional $4(L-1)$ independent sheet-like
stabilizers if the system size is even.

For the non-CSS plaquette code we define three check operators per site
$\bm{r}$ as

$\displaystyle g_{\bm{r}}^{(X)}$
$\displaystyle=X_{\bm{r}}\,X_{\bm{r}+\hat{\bm{y}}}\,X_{\bm{r}+\hat{\bm{z}}}\,X_{\bm{r}+\hat{\bm{y}}+\hat{\bm{z}}},$
(20a) $\displaystyle g_{\bm{r}}^{(Y)}$
$\displaystyle=Y_{\bm{r}}\,Y_{\bm{r}+\hat{\bm{x}}}\,Y_{\bm{r}+\hat{\bm{z}}}\,Z_{\bm{r}+\hat{\bm{x}}+\hat{\bm{z}}},$
(20b) $\displaystyle g_{\bm{r}}^{(Z)}$
$\displaystyle=Z_{\bm{r}}\,Z_{\bm{r}+\hat{\bm{x}}}\,Z_{\bm{r}+\hat{\bm{y}}}\,Z_{\bm{r}+\hat{\bm{x}}+\hat{\bm{y}}}.$
(20c)

The geometry of stabilizers and logical operators is also very similar to that
sketched for the CSS version in Fig. 11. We obtain one more set of string-like
Pauli-$Y$ stabilizer generators, so that now we have $3(L-1)^{2}$ independent
string-like stabilizer generators for both both odd and even system sizes. As
in the CSS case, the stabilizer group of the code has sheet-like elements for
even $L$, however these sheets are not independent for different planes
anymore since one can (for even $L$) use products of e.g. string-like $X$ and
$Y$ stabilizers to obtain the plane-like $Z$ stabilizers.

In summary, the non-CSS plaquette codes are a family of codes with

$[[N,K,D]]=\begin{cases}[[L^{3},3L-2,L]]&\text{if $L$ even}\\\
[[L^{3},3L-1,L]]&\text{if $L$ odd}\end{cases}.$ (21)

and $3(L-1)^{2}$ independent stabilizer generators for odd system sizes and
$3(L-1)^{2}+1$ independent stabilizer generators for even system sizes.

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