††thanks: Current Address: Centre for Engineered Quantum Systems, School of
Physics, University of Sydney, Sydney, New South Wales 2006, Australia

# Symmetry-Protected Infinite-Temperature Quantum Memory from Subsystem Codes

Julia Wildeboer<EMAIL_ADDRESS>Department of Physics and Astronomy, Iowa
State University, Ames, Iowa 50011, USA  Thomas Iadecola<EMAIL_ADDRESS>Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA  Ames Laboratory, Ames, Iowa 50011, USA  Dominic J. Williamson  Stanford
Institute for Theoretical Physics, Stanford University, Stanford, California
94305, USA<EMAIL_ADDRESS>
###### Abstract

We study a mechanism whereby quantum information present in the initial state
of a quantum many-body system can be protected for arbitrary times due to a
combination of symmetry and spatial locality. Remarkably, the mechanism is
sufficiently generic that the dynamics can be fully ergodic upon resolving the
protecting symmetry and fixing the encoded quantum state, resulting in a
quantum memory that persists up to infinite temperature. After exemplifying
the mechanism in a strongly nonintegrable two dimensional (2D) spin model
inspired by the surface code, we find it has a natural interpretation in the
language of noiseless subsystems and stabilizer subsystem codes. This
interpretation yields a number of further examples, including a nonintegrable
Hamiltonian with quantum memory based on the Bacon-Shor code. The lifetime of
the encoded quantum information in these models is infinite provided the
dynamics respects the stabilizer symmetry of the underlying subsystem code. In
the presence of symmetry-violating perturbations, we make contact with
previous work leveraging the concept of prethermalization to show that the
encoded quantum information can acquire a parametrically long lifetime under
dynamics with an enlarged continuous symmetry group. The prethermalization
mechanism hinges on the application of external fields that are much larger
than the perturbations themselves. We identify conditions on the underlying
subsystem code that enable such a prethermal enhancement of the memory
lifetime.

##  I Introduction

Generic quantum many-body systems naturally “forget” their initial conditions
as they evolve. Even in isolated systems undergoing unitary dynamics—where,
strictly speaking, the evolution of observables always depends on the initial
state—quantum information in the initial state is “scrambled” as entanglement
between local degrees of freedom develops Hayden and Preskill (2007); Sekino
and Susskind (2008); Lashkari  _et al._ (2013); Landsman  _et al._ (2019).
This tendency towards scrambling is physically important, as it underlies the
emergence of statistical mechanics in closed quantum systems via the
eigenstate thermalization hypothesis (ETH) Deutsch (1991); Srednicki (1994);
Rigol  _et al._ (2008); D’Alessio  _et al._ (2016); Deutsch (2018). However,
in the context of, e.g., quantum computation, such information loss is
undesirable, making the task of protecting quantum information against chaotic
dynamics an important practical concern.

One of the most fascinating features of a quantum many-body system is the
collective formation of a quantum error-correcting code Shor (1995). Quantum
error correction is a key requirement for the feasibility of quantum
computation in a realistic environment Aharonov and Ben-Or (2008); Knill  _et
al._ (1998). There is a well established subfield of research investigating
the connections between topological phases of matter and the topological
quantum error correcting codes they define Kitaev (2003); Bravyi and Kitaev
(1998). Much of the work focuses on utilizing zero- or near-zero-temperature
topological phases to protect quantum information either under thermal
dynamics Dennis  _et al._ (2001); Bacon (2006); Brown  _et al._ (2016a), or
via feedback and active intervention Dennis  _et al._ (2001).

More recently, the study of dynamical quantum many-body systems that naturally
possess a degree of memory has gained traction. In this setting it is crucial
to make a distinction between classical and quantum memory in a quantum
system. A classical memory is defined by the preservation of one or more
classical configurations. For example, a quantum system protecting one
classical bit possesses an effective “Pauli-$Z$” operator that is conserved
under the dynamics, indicating that the encoded bit never flips. A quantum
memory is defined by the ability to form superpositions of classical
configurations that do not decohere. For example, a quantum system protecting
one qubit possesses both the aforementioned effective Pauli-$Z$ operator as
well as a dual “Pauli-$X$” operator that anticommutes with it, signaling the
presence of an encoded qubit with an associated Bloch sphere. The effective
$Z$\- and $X$-type operators are referred to as the logical operators of the
encoded qubit.

Many examples of quantum dynamical systems with various types of memory are
known. Many-body localization (MBL) Nandkishore and Huse (2015); Altman and
Vosk (2015); Abanin  _et al._ (2019) is known to approximately protect
classical memory due to the presence of slow, logarithmic dephasing Žnidarič
_et al._ (2008); Bardarson  _et al._ (2012); Serbyn  _et al._ (2013a, b); Huse
_et al._ (2014); Swingle (2013); Imbrie  _et al._ (2017). When MBL is combined
with symmetry-protected topological order the logical operators of an encoded
qubit have been shown to remain protected under symmetry-respecting dynamics
Huse  _et al._ (2013); Chandran  _et al._ (2014); Bahri  _et al._ (2015);
however, the precondition of MBL is expected to become unstable upon coupling
to an external bath Nandkishore  _et al._ (2014); Nandkishore and
Gopalakrishnan (2016); Levi  _et al._ (2016); Fischer  _et al._ (2016); De
Roeck and Huveneers (2017); Luitz  _et al._ (2017); Rubio-Abadal  _et al._
(2019). There have also been studies of MBL combined with topological order
Huse  _et al._ (2013); Bauer and Nayak (2013); Wahl and Béri (2020). However,
for spin systems this relies on the disputed existence of MBL above one
spatial dimension De Roeck and Imbrie (2017); Wahl  _et al._ (2018);
Gopalakrishnan and Huse (2019); Doggen  _et al._ (2020); Chertkov  _et al._
(2021); Decker  _et al._ (2021); Théveniaut  _et al._ (2020); Pietracaprina
and Alet (2021) and, even if this were resolved, would also be expected to
become unstable upon coupling to an external bath. Systems with gauge
constraints or fractonic conservation laws have also been shown to lead to
dynamics with classical memory related to the fragmentation of Hilbert space
into disconnected subsectors Pai and Pretko (2019); Sala  _et al._ (2020);
Khemani  _et al._ (2020); Rakovszky  _et al._ (2020); Moudgalya  _et al._
(2020); Yang  _et al._ (2020); Moudgalya and Motrunich (2021); Moudgalya  _et
al._ (2021). Systems with dynamical symmetries can also exhibit memory effects
Buča  _et al._ (2020); Buča (2021).

Alternatively, the existence of strong zero modes (SZMs) Kitaev (2001);
Fendley (2012) in a dynamical system is capable of guaranteeing a quantum
memory Fendley (2016); Kemp  _et al._ (2017); Else  _et al._ (2017). In this
setting the SZM operator plays the role of one logical operator (say, the
$Z$-type) while a global symmetry plays the role of a dual ($X$-type) logical
operator. Symmetry-respecting dynamics then commutes with the symmetry logical
operator by definition, and commutes with the SZM logical operator in the
thermodynamic limit, again by definition. However, in this case the quantum
nature of the memory is only accessible if one considers the dynamics of
states that do not respect the global symmetry. If the system is prepared in a
state with a well-defined symmetry eigenvalue, then the encoded qubit is
automatically restricted to be polarized along the “$X$” axis.

There have been other efforts to protect quantum information through a
combination of symmetry and topological order. One approach considers a 1-form
symmetry-protected topological phase in three spatial dimensions that persists
to nonzero temperature and supports a topological boundary phase that is self-
correcting, and hence has a quantum memory, under symmetry-respecting thermal
dynamics Roberts  _et al._ (2017); Roberts and Bartlett (2020); Roberts and
Williamson (2020); Stahl and Nandkishore (2021). Alternatively, if the
anomalous 1-form symmetry corresponding to all string operators in an Abelian
topological order is strictly enforced directly in two spatial dimensions in a
system with nontrivial boundary conditions (e.g., defined on a torus), any
local symmetry-respecting dynamics must automatically support a quantum
memory, as the nonlocal logical string operators around nontrivial cycles
commute with the dynamics Bombin  _et al._ (2009); Kargarian  _et al._ (2010);
Bombin (2010); Suchara  _et al._ (2011); Bravyi  _et al._ (2013). Another
approach that has experienced an explosion of interest is quantum dynamics
involving hybrid “unitary-projective” quantum circuits in which unitary gates
are combined with projective measurements Li  _et al._ (2018); Skinner  _et
al._ (2019); Li  _et al._ (2019); Chan  _et al._ (2019). This has lead to the
discovery of systems with both random Gullans and Huse (2020); Choi  _et al._
(2020); Fan  _et al._ (2021); Fidkowski  _et al._ (2021) and topologically
structured Sang and Hsieh (2021); Lavasani  _et al._ (2021); Ippoliti  _et
al._ (2021); Lavasani  _et al._ (2020) quantum memory.

Here we take inspiration from several directions of previous work and consider
quantum dynamics that is defined by a combination of symmetry and locality.
The main example we consider is a family of Hamiltonians with finite gauge
symmetry in two spatial dimensions whose dynamics are shown to act as an exact
quantum memory for infinite time. In previous work, approximate quantum
memories have been obtained by mechanisms that tend to suppress ergodicity of
the quantum dynamics. For example, classical and quantum memories relying on
MBL require the complete breakdown of ergodicity in order to operate. Other
studies of approximate quantum memories rely on perturbing around an exactly
solvable point with simple dynamics Fendley (2016); Else  _et al._ (2017);
Kemp  _et al._ (2017) and/or considering a solvable system at finite
temperature Else  _et al._ (2017). The main example considered in this work
differs substantially in that it provides an exact quantum memory even in a
system which otherwise exhibits fully ergodic dynamics—that is, the family of
Hamiltonians we consider supports a quantum memory even at infinite
temperature. This infinite-temperature memory arises due to an exact twofold
degeneracy of every eigenstate in the spectrum of the Hamiltonian, which
implies that the memory persists under dynamics from any initial state.
Remarkably, only a global $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry is
necessary to guarantee the existence of the quantum memory under locally
generated dynamics. To study the effect of small perturbations that break
these symmetries, we leverage previous work on prethermalization Abanin  _et
al._ (2017); Else  _et al._ (2017, 2020) showing that the protecting symmetry
can in principle persist as an approximate emergent symmetry for (stretched)
exponentially long times in the presence of appropriate external fields. To do
so, we embed the protecting $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry into
a larger U(1)$\times$U(1) symmetry group, yielding a closely related class of
Hamiltonians to which the prethermalization arguments can be applied.

We find that our construction is naturally understood within the formalism of
stabilizer subsystem codes Poulin (2005); Bacon (2006), which leads to a
number of generalizations and further examples. This formalism can be applied
not only to Hamiltonian dynamics, but also to arbitrary unitary, unitary-
projective, and open-system dynamics with the appropriate symmetry. We also
consider conditions on the underlying stabilizer subsystem code that enable
the application of prethermalization arguments Else  _et al._ (2017, 2020)
that ensure a parametrically long-lived quantum memory under dynamics
generated by a symmetric Hamiltonian with symmetry-breaking perturbations that
are small compared to a set of applied fields. In particular, we find that it
is sufficient for the underlying code to be a topological subsystem code
Bombin (2010) that contains a nontrivial global error-detecting subsystem code
with no local dressed logical operators. This condition is also necessary in
the sense that a Hamiltonian suitable for the application of prethermalization
arguments along the lines of Refs. Else  _et al._ (2017, 2020) defines a
topological stabilizer group.

The paper is laid out as follows. In Section II we show that the above-
mentioned family of Hamiltonians based on a $\mathbb{Z}_{2}$ gauge theory with
global symmetries supports an infinite temperature quantum memory. We also
provide explicit numerical examples showing that this family of Hamiltonians
is quantum-chaotic. We then define a closely related class of models that we
argue based on previous work Else  _et al._ (2017, 2020) should exhibit an
approximate quantum memory for (stretched) exponentially long times in the
presence of symmetry-breaking perturbations. We also present numerical results
investigating the effects of perturbations in these models. In Section III, we
consider more general criteria under which stabilizer subsystem codes define
dynamical quantum memories similar to the one constructed in our main example.
Based on this discussion, we also consider further examples of such models,
including one to which the prethermalization arguments of Refs. Abanin  _et
al._ (2017); Else  _et al._ (2017, 2020) cannot be applied. In Section IV we
draw our conclusions and discuss potential future directions.

##  II Main example: 2D $\mathbb{Z}_{2}$ gauge theory with global symmetries

###  II.1 Setup and global symmetry

We consider a square lattice with “gauge” qubits on its links $\ell$ and
“matter” qubits on its vertices $v$ and on its plaquettes $p$ (see Fig. 1).
The total Hilbert space is therefore $\bigotimes_{\ell,v,p}\mathbb{C}^{2}$. We
refer to the full collection of gauge and matter qubits as “physical qubits,”
to distinguish them from the logical qubit that is introduced later. Pauli
operators for all physical qubits are denoted by $X_{r},Y_{r},Z_{r}$, with
$r=\ell,v,p$ specifying the position of the qubit and

$\displaystyle X_{r}=\begin{bmatrix}0&1\\\
1&0\end{bmatrix}\,\,\,\,\,\,\,Y_{r}=\begin{bmatrix}0&-i\\\
i&\,\,\,\,\,0\end{bmatrix}\,\,\,\,\,\,\,Z_{r}=\begin{bmatrix}1&\,\,\,\,\,0\\\
0&-1\end{bmatrix}.$ (1)

We employ boundary conditions on the square lattice inspired by the surface
code Kitaev (2003); Bravyi and Kitaev (1998); Freedman and Meyer (2001);
Fowler  _et al._ (2012), taking “rough” boundaries with dangling links on the
top and bottom and “smooth” boundaries without dangling links on the left and
right sides.

We define a subspace of the total Hilbert space spanned by gauge-invariant
states, namely

$\displaystyle\mathcal{H}=\\{\ket{\psi}\mid
A_{v}\ket{\psi}=B_{p}\ket{\psi}=\ket{\psi}\\},$ (2)

with the mutually commuting gauge transformation generators $A_{v},B_{p}$
defined as

$\displaystyle A_{v}=X_{v}\prod_{\ell\in\ell_{v}}X_{\ell},\indent
B_{p}=Z_{p}\prod_{\ell\in\ell_{p}}Z_{\ell},$ (3)

with $\ell_{p}$ the set of links touching the plaquette $p$ and $\ell_{v}$ the
set of links touching the vertex $v$. The sets $\ell_{v}$ and $\ell_{p}$ each
contain four links for $v,p$ in the bulk of the lattice and three links for
$v,p$ on the boundary. The gauge invariance condition in Eq. (2) ensures that
electric and magnetic excitations of the gauge spins are paired with vertex
and plaquette excitations of the matter spins, respectively. For example, any
state $\ket{\psi}\in\mathcal{H}$ containing an electric gauge excitation on
vertex $v$—i.e., that satisfies
$\prod_{\ell\in\ell_{v}}X_{\ell}\ket{\psi}=-\ket{\psi}$—must also satisfy
$X_{v}\ket{\psi}=-\ket{\psi}$.

We further decompose the Hilbert space $\mathcal{H}$ into “topological
symmetry” sectors associated with eigenvalues of the global spin and phase
flip operators

$\displaystyle S_{X}=\prod_{v}X_{v},$ $\displaystyle S_{Z}=\prod_{p}Z_{p},$
(4)

which commute with each other and with the gauge generators. These operators
count the parity of the number of vertex and plaquette matter excitations,
respectively, and enforcing these topological symmetries therefore breaks
$\mathcal{H}$ up into four disconnected symmetry sectors that we label with
indices $(\pm,\pm)$. Note that these topological symmetry constraints are only
nontrivial in the case of open boundary conditions; for periodic boundary
conditions, $S_{X}=\prod_{v}A_{v}$ and $S_{Z}=\prod_{p}B_{p}$, so that only
the $(+,+)$ sector is consistent with the gauge constraints.

Figure 1:  Depiction of the main example qubit system. (a) The square lattice
of size $(L_{x},L_{y})=(2,3)$ with “gauge” qubits on its links $\ell$ (magenta
squares), and “matter” qubits on its vertices $v$ (blue circles) and on its
plaquettes $p$ (yellow squares). Also shown are the logical-$X$ operator
$\mathcal{X}=\prod_{\ell\in\mathcal{P}_{x}}X_{\ell}$ and the logical-$Z$
operator $\mathcal{Z}=\prod_{\ell\in\mathcal{P}_{y}}Z_{\ell}$ each acting on a
specific subset of links, i.e. for the logical-$X$ operator $\mathcal{X}$ this
subset consists of all dangling links on one of the rough boundaries as
indicated by the (horizontal) path $\mathcal{P}_{x}$, while for the
logical-$Z$ operator $\mathcal{Z}$ the respective subset is made up of all
links on one of the smooth boundaries, i.e. the vertical path
$\mathcal{P}_{y}$. (b) The operators $A_{v}$ and $B_{p}$ acting on a vertex
and a plaquette in the bulk and at the boundary, respectively.

###  II.2 Symmetric and non-symmetric local operators

We now consider the set of gauge-invariant local operators

Terminology | Sec. II | Sec. III
---|---|---
Gauge group |  $G$ [as defined by Eqs. (5)] | $G$
Stabilizer group | $S=\langle S_{X},S_{Z},\\{A_{v}\\},\\{B_{p}\\}\rangle$ | $S=Z(G)$
Bare logical group | $L=\langle{\mathcal{X},\mathcal{Z},S}\rangle/S$ | $L=C(G)/S$
Dressed logical group | $\braket{\mathcal{X},\mathcal{Z},G}/G$ | $C(S)/G$
Table 1: Summary of stabilizer subsystem code concepts. This table collects
the key concepts of subsystem codes—namely the gauge, stabilizer, and logical
groups—and points out how these concepts are instantiated in Secs. II and III.
Here, $\left\langle{\cdot}\right\rangle$ denotes the multiplicative group
generated by the enclosed set of operators, and $Z(G)$ and $C(G)$ denote the
center and centralizer, respectively, of the group $G$.

acting on the Hilbert space $\mathcal{H}$. In the bulk, such operators either
reside on a vertex $v$ or plaquette $p$, or include a link $\ell(v_{1},v_{2})$
connecting nearest-neighbor vertices $v_{1,2}$ or a link $\ell(p_{1},p_{2})$
separating nearest-neighbor plaquettes $p_{1,2}$. These operators are
generated by

$X_{v},\ Z_{p},\ Z_{v_{1}}Z_{\ell(v_{1},v_{2})}Z_{v_{2}},\
X_{p_{1}}X_{\ell(p_{1},p_{2})}X_{p_{2}},$ (5a) products of which include,
e.g., $Y_{v_{1}}Z_{\ell(v_{1},v_{2})}Y_{v_{2}},\
Y_{p_{1}}X_{\ell(p_{1},p_{2})}Y_{p_{2}}\,\dots.$ (5b)

We remark that all of these bulk operators commute with the topological
symmetry generators (4). Gauge invariant operators that fail to commute with
the topological symmetries are generated by terms along the boundary of the
system, up to multiplication by the symmetric bulk terms described above. For
each dangling link $\ell$ along a rough boundary, with adjacent vertex
$v(\ell)$, the operator

$\displaystyle Z_{\ell}Z_{v(\ell)},$ (6a) commutes with $A_{v}$ but
anticommutes with $S_{X}$, and generates further terms such as
$Z_{\ell}Y_{v(\ell)}$, etc. For each link $\ell$ along a smooth boundary, with
adjacent plaquette $p$, the operator $\displaystyle X_{\ell}X_{p(\ell)},$ (6b)

commutes with $B_{p}$ but anticommutes with $S_{Z}$, and generates additional
terms such as $X_{\ell}Y_{p(\ell)}$, etc. We denote by $G$ the set of gauge-
invariant local operators, including (5), that commute with the topological
symmetry generators $S_{X},S_{Z}$, and by $\overline{G}$ the set of gauge-
invariant local operators, including (6), that do not commute with
$S_{X},S_{Z}$.

###  II.3 Encoded logical qubit and connection to subsystem codes

We now show that the Hilbert space $\mathcal{H}$ supports a single encoded
logical qubit when the topological symmetries (4) are enforced. To do this, we
identify a conjugate pair of nonlocal logical operators. The logical-$X$
operator is defined as

$\displaystyle\mathcal{X}=\prod_{\ell\in\mathcal{P}_{x}}X_{\ell},$ (7a) while
the logical-$Z$ operator is defined as
$\displaystyle\mathcal{Z}=\prod_{\ell\in\mathcal{P}_{y}}Z_{\ell}.$ (7b)

Here, $\mathcal{P}_{x}$ is a path including all dangling links on one of the
rough boundaries, while $\mathcal{P}_{y}$ is a path including all links on one
of the smooth boundaries (see Fig. 1). Note that these operators anticommute
with each other and commute with $A_{v},B_{p}$ and $S_{X},S_{Z}$, as expected.
Moreover, from the definitions of the generators (5a) of $G$ and (6) of
$\overline{G}$, we see that $\mathcal{X}$ and $\mathcal{Z}$ commute with all
symmetric gauge-invariant operators in $G$, while at least one of
$\mathcal{X}$ and $\mathcal{Z}$ (up to multiplication by $S_{X}$ and/or
$S_{Z}$) anticommute with any non-symmetric gauge-invariant operator from
$\overline{G}$. Thus, in the presence of the topological symmetries (4), we
can decompose an arbitrary state $\ket{\psi}\in\mathcal{H}$ within a symmetry
sector labeled by eigenvalues of $S_{X}$ and $S_{Z}$ as

$\displaystyle\ket{\psi}=\ket{\psi_{\rm L}}\otimes\ket{\psi^{\prime}},$ (8)

where operators in $G$ act nontrivally only on $\ket{\psi^{\prime}}$ and where
$\ket{\psi_{\rm L}}$ is the state of the logical qubit. $\ket{\psi_{\rm L}}$
is specified by the expectation values
$\braket{\mathcal{X}},\braket{\mathcal{Y}},$ and $\braket{\mathcal{Z}}$, where
$\mathcal{Y}=i\mathcal{X}\mathcal{Z}$. As an aside, we note that there is also
an analog of this setup that encodes more than one logical qubit; we briefly
discuss this setup in Appendix A.

The local generators we have introduced above in fact define a stabilizer
subsystem code Poulin (2005); Bacon (2006), whose stabilizer group is
generated by the gauge transformation generators $A_{v},B_{p}$ and the
topological symmetry operators $S_{X},S_{Z}$, and whose encoded qubit is
$\ket{\psi_{\rm L}}$. (For a precise definition of stabilizer subsystem codes,
we refer the reader to Sec. III; the present discussion is intended to be a
less formal complement to that treatment. The relationships between concepts
discussed in this Section and in Sec. III are summarized in Table 1.) This
subsystem code is not particularly attractive from the point of view of
practical quantum error correction, as the code distance is independent of the
number $n$ of physical qubits. To see this, we note that there exist nonlocal
but few-body operators that commute with all stabilizers but anticommute with
the nonlocal logical operators. An example of such an operator is

$\displaystyle X_{\ell_{\rm TL}}X_{v(\ell_{\rm TL})}X_{v(\ell_{\rm
TR})}X_{\ell_{\rm TR}},$ (9)

where $\ell_{\rm TL}$ and $\ell_{\rm TR}$ are the dangling links at the top-
left and top-right corners of the lattice. This operator commutes with the
gauge generators $A_{v},B_{p}$ and the topological symmetry generators (4) but
anticommutes with $\mathcal{Z}$, and thereby generates an undectected logical
error. In the language of subsystem codes, these are known as “dressed logical
operators”, as they can be realized as products of the “bare logical
operators” $\mathcal{X},\mathcal{Z}$ and operators from the set $G$ of
symmetric gauge-invariant local operators Poulin (2005). The operators from
$G$ are, somewhat confusingly, known as “gauge” operators in the subsystem
code literature. We use quotations to make clear that this is distinct from
the gauge symmetries present in our model, which correspond to stabilizers of
the subsystem code.

Although dressed logical operators like (9) are undesirable from the viewpoint
of quantum error correction, they are less detrimental in the context of
locally generated quantum dynamics. To see this, imagine preparing an initial
state with an arbitrary encoded state $\ket{\psi_{\rm L}}$ and subjecting it
to quantum evolution that respects the structure of the subsystem code; for
example, we can evolve the state using a generic local Hamiltonian $H$
constructed from the generators (5a) of $G$. Because any operator in $G$
commutes with $\mathcal{X}$ and $\mathcal{Z}$, the quantum dynamics of the
initial state preserves the state of the logical qubit. In this way the
encoded qubit is protected by a combination of locality and symmetry; this is
reminiscent of Kitaev’s Majorana wire code Kitaev (2001), where a nonlocal but
few body term that acts simultaneously on both ends of an open wire can enact
a logical operator, but a combination of fermion parity symmetry and locality
protects the encoded information.

This reasoning is in fact quite general, and applies equally well to any
locally-generated quantum dynamics that respects the Hilbert-space
decomposition (8). For example, one could consider dynamics of an initial
state under a random unitary circuit
$\prod_{\mu}e^{-i\theta_{\mu}\mathcal{O}_{\mu}}$, where $\mathcal{O}_{\mu}$ is
any local operator in $G$ (or a linear combination thereof). One can even
consider unitary-projective (or open-system) dynamics, so long as the
projective measurements (or system-bath couplings) are also local operators
drawn from $G$. The only constraints on the dynamics are (i) spatial locality
and (ii) the topological symmetries (4). This is similar to codes defined by
symmetry-protected commuting projector Hamiltonians that protect against
spatially local symmetric errors.

We formulate more precisely and elaborate further on the connections between
subsystem codes and dynamics in Sec. III.

###  II.4 Disentangling unitary

Figure 2:  Disentangling circuit. (a) The action of the disentangling circuit
$U_{v}=\prod_{l\in v}CX_{v\rightarrow\ell}$ on the qubits surrounding a vertex
$v$. (b) The action of the disentangling circuit $U_{p}=\prod_{l\in
p}CX_{v\rightarrow p}$ on the qubits surrounding a plaquette $p$. After
application of the total circuit $U=\prod_{p}U_{p}\prod_{v}U_{v}$ we are left
with matter qubits that are no longer dynamical. Thus, the “gauge” qubits are
the only relevant degrees of freedom in the system.

In the next subsection, we exemplify the above discussion with numerical
results on a concrete model Hamiltonian. To facilitate this study, it is
useful to first apply a unitary disentangling circuit that effectively allows
us to “integrate out” the matter qubits, greatly reducing the number of
degrees of freedom to be simulated. The circuit is a product of CNOT gates
$CX_{i\to j}$, which act on Pauli operators as

$\displaystyle\begin{split}CX_{i\to j}\,X_{i}\,CX_{i\to j}&=X_{i}X_{j}\\\
CX_{i\to j}\,X_{j}\,CX_{i\to j}&=X_{j}\\\ CX_{i\to j}\,Z_{j}\,CX_{i\to
j}&=Z_{i}Z_{j}\\\ CX_{i\to j}\,Z_{i}\,CX_{i\to j}&=Z_{i}.\end{split}$ (10)

For each vertex $v$, we apply the local unitary [see Fig. 2(a)]

$\displaystyle U_{v}=\prod_{\ell\in\ell_{v}}CX_{v\to\ell},$ (11)

which acts on the gauge generators (3) as

$\displaystyle\begin{split}&U_{v}\,A_{v}\,U^{\dagger}_{v}=X_{v}\\\
&U_{v}\,B_{p}\,U^{\dagger}_{v}=B_{p}.\end{split}$ (12)

For each plaquette $p$, we apply the local unitary [see Fig. 2(b)]

$\displaystyle U_{p}=\prod_{\ell\in\ell_{p}}CX_{\ell\to p},$ (13)

which acts on the gauge generators (3) as

$\displaystyle\begin{split}&U_{p}\,A_{v}\,U^{\dagger}_{p}=A_{v}\\\
&U_{p}\,B_{p}\,U^{\dagger}_{p}=Z_{p}.\end{split}$ (14)

The total disentangling circuit,

$\displaystyle U=\prod_{p}U_{p}\,\prod_{v}U_{v},$ (15)

thus maps the gauge-invariant Hilbert space $\mathcal{H}$ of Eq. (2) to

$\displaystyle\tilde{\mathcal{H}}=\\{\ket{\psi}\mid
X_{v}\ket{\psi}=Z_{p}\ket{\psi}=\ket{\psi}\\}.$ (16)

In the transformed Hilbert space, the matter qubits are no longer dynamical
and we can make the replacements $X_{v},Z_{p}\mapsto+1$ $\forall\,v,p$.

We now consider the action of the disentangling circuit on the operators
defining the subsystem code. We first consider the generators (5a) of $G$, the
set of gauge-invariant local operators respecting topological symmetry (4).
The transformed generators $X_{v}$ and $Z_{p}$ can be obtained by inverting
Eqs. (12) and (14):

$\displaystyle\begin{split}U\,X_{v}\,U^{\dagger}&=A_{v}=X_{v}\,\tilde{A}_{v}\equiv\tilde{A}_{v}\\\
U\,Z_{p}\,U^{\dagger}&=B_{p}=Z_{p}\,\tilde{B}_{p}\equiv\tilde{B}_{p},\end{split}$
(17a) where in the final equality on both lines we have replaced the matter-
qubit operators by their eigenvalues in the space $\tilde{\mathcal{H}}$. The
operators $\displaystyle\tilde{A}_{v}=\prod_{\ell\in
v}X_{\ell},\indent\tilde{B}_{p}=\prod_{\ell\in p}Z_{\ell}$ (17b)

are the usual toric-code stabilizers. The remaining generators transform as

$\displaystyle\begin{split}U\,X_{p_{1}}X_{\ell(p_{1},p_{2})}X_{p_{2}}\,U^{\dagger}&=X_{\ell(p_{1},p_{2})}\\\
U\,Z_{v_{1}}Z_{\ell(v_{1},v_{2})}Z_{v_{2}}\,U^{\dagger}&=Z_{\ell(v_{1},v_{2})}.\end{split}$
(18)

Note that in the above expression, the links $\ell(p_{1},p_{2})$ and
$\ell(v_{1},v_{2})$ must lie in the bulk of the lattice, since they are shared
by two plaquettes and vertices, respectively. The generators (6) of
$\overline{G}$, the set of gauge-invariant symmetry-violating operators,
transform as

$\displaystyle\begin{split}U\,X_{\ell}X_{p(\ell)}\,U^{\dagger}&=X_{\ell}\\\
U\,Z_{\ell}Z_{v(\ell)}\,U^{\dagger}&=Z_{\ell},\end{split}$ (19)

where the link $\ell$ lies on a smooth boundary on the top line and a rough
boundary on the bottom line.

Finally, we consider the action of the disentangling circuit on the
topological symmetry generators (4) and on the logical operators (7). From
Eqs. (17), we find that the topological symmetry generators become

$\displaystyle\begin{split}\tilde{S}_{X}&=U\,S_{X}\,U^{\dagger}=\prod_{\ell\in\mathrm{rough}}X_{\ell}\\\
\tilde{S}_{Z}&=U\,S_{Z}\,U^{\dagger}=\prod_{\ell\in\mathrm{smooth}}Z_{\ell},\end{split}$
(20)

where $\ell\in\mathrm{rough(smooth)}$ denotes the set of links belonging to
the rough (smooth) boundaries. This mapping arises because all links belong to
two toric-code stabilizers, except at the boundaries. Meanwhile, inverting
Eqs. (18) and (19) shows that the logical operators (7) are unchanged:

$\displaystyle\begin{split}\tilde{\mathcal{X}}&=U\,\mathcal{X}\,U^{\dagger}=\mathcal{X}\\\
\tilde{\mathcal{Z}}&=U\,\mathcal{Z}\,U^{\dagger}=\mathcal{Z}\,.\end{split}$
(21)

Note that the logical operators are supported on only one rough or smooth
boundary, while the topological symmetry operators are supported on both rough
or smooth boundaries.

Figure 3:  Model with “matter” qubits integrated out. (a) A lattice of size
$(L_{x},L_{y})=(3,3)$ with only the 18 “gauge” qubits is depicted. “Matter”
qubits are suppressed after application of the disentangling circuit of Sec.
II.4. (b) Histogram $P(r)$ of energy spacing ratios $r$ with overlaid curves
showing the analytical forms of the Wigner-Dyson GOE (magenta) and Poisson
distributions (blue). The average value $\bar{r}$ of $P(r)$ is calculated to
be $\approx 0.529$, while the analytical value is $\bar{r}_{\rm GOE}\approx
0.536$. (c) The von-Neumann entropy $S^{vN}$ for all $8,192$ states in the
spectrum for a lattice size of $(L_{x},L_{y})=(2,3)$. The maximum value of
$S^{vN}$ is close to the Page value $S_{\rm Page}\approx 3.85$ (see equation
(25)), indicating that eigenstates near the middle of the spectrum have no
special structure lending evidence to the fact that the Hamiltonian (22) is
generic.

###  II.5 Concrete model Hamiltonian

We now define a concrete model Hamiltonian, constructed using the subsystem-
code generators (5a), that features a single encoded logical qubit as long as
the topological symmetry is enforced. The Hamiltonian, defined in the Hilbert
space $\tilde{\mathcal{H}}$ with matter qubits integrated out (see Sec. II.4
and Fig. 2), is given by

$\displaystyle
H=\sum_{v}h_{v}\,\tilde{A}_{v}+\sum_{p}h_{p}\,\tilde{B}_{p}+\\!\\!\\!\\!\\!\sum_{\ell\notin\mathrm{smooth}}\\!\\!\\!\\!J_{x,\ell}\,X_{\ell}+\\!\\!\\!\\!\sum_{\ell\notin\mathrm{rough}}\\!\\!\\!J_{z,\ell}\,Z_{\ell}.$
(22)

This Hamiltonian is reminiscent of that of the surface code (i.e., the toric
code with boundary conditions as depicted in Fig. 1) in the presence of
magnetic fields in the $x$\- and $z$-directions. However, the model is
constrained such that no $z$-fields appear on the rough boundaries, and no
$x$-fields appear on the smooth boundaries. (We note, however, that one could
include terms involving products of an even number of $X$\- or $Z$-operators
on the smooth or rough boundaries, respectively, without adversely affecting
the encoded qubit.) These constraints ensure that the Hamiltonian (22) obeys
[see Eqs. (20) and (21)]

$\displaystyle[H,\tilde{S}_{X}]=[H,\tilde{S}_{Z}]=[H,\tilde{\mathcal{X}}]=[H,\tilde{\mathcal{Z}}]=0.$
(23)

In other words, irrespective of the values of the parameters
$h_{v},h_{p},J_{x,\ell},J_{z,\ell}$—they can be arbitrarily large or small,
and even vary in space—the model (20) respects the Hilbert-space decomposition
(8) and therefore harbors an encoded logical qubit addressed by the logical
operators $\tilde{\mathcal{X}}$ and $\tilde{\mathcal{Z}}$.

It is important to note that the logical qubit preserved by the Hamiltonian
(22) is not the same as the one that appears in the surface/toric code. In the
latter case, the encoded qubit resides in the ground state manifold and is
only stable to small external fields. Here, instead, the logical operators
that encode the qubit reside on the open boundaries of the lattice, and so are
unaffected by even arbitrarily strong fields in the bulk. This gives rise to a
twofold degeneracy of every energy eigenstate, rather than just the ground
state. Indeed, if the model is defined with periodic rather than open boundary
conditions, the encoded qubit disappears. In this respect, the encoded qubit
in this model more closely resembles the qubits encoded by strong zero modes
in 1D systems Fendley and Schoutens (2005); Fendley (2016); Else  _et al._
(2017); Kemp  _et al._ (2017).

Despite the fact that it respects the Hilbert-space decomposition
$\eqref{eq:psi}$ and therefore encodes a logical qubit, the Hamiltonian (22)
is quite generic by standard measures. To emphasize this, we present numerical
exact diagonalization (ED) results for the model (22) defined on the lattice
depicted in Fig. 1(a). After integrating out all matter qubits, we are left
with a model of interacting gauge qubits as shown in Fig. 3(a). We label
system sizes by the number of plaquettes in the horizontal and vertical
directions, which we call $L_{x}$ and $L_{y}$, respectively. We use model
parameters $J_{x,\ell}=J_{z,\ell}=1$, and random $h_{v}$ and $h_{p}$ drawn
uniformly from the interval $[1-0.2,1+0.2]$. All data shown are for a single
disorder realization.

In Fig. 3(b), we show that the energy level statistics of the model fits
predictions from random matrix theory, as expected for nonintegrable
Hamiltonians D’Alessio  _et al._ (2016). We plot the probability distribution
$P(r)$ of the level spacing ratio Oganesyan and Huse (2007); Pal and Huse
(2010)

$\displaystyle
r_{i}=\frac{\text{min}(E_{i+1}-E_{i},E_{i+2}-E_{i+1})}{\text{max}(E_{i+1}-E_{i},E_{i+2}-E_{i+1})},$
(24)

where $\\{E_{i}\\}$ is an ordered list of the eigenvalues of $H$. The data
shown are for system size $(L_{x},L_{y})=(3,3)$ for energy levels in the
topological symmetry sector $(\tilde{S}_{X},\tilde{S}_{Z})=(+1,+1)$, upon
restricting to states with $\tilde{\mathcal{Z}}$ eigenvalue $+1$ ($32,768$
states in total). The resulting histogram is in excellent agreement with the
analytical expression for $P(r)$ in the Gaussian Orthogonal Ensemble (GOE) of
random matrices Atas  _et al._ (2013). For comparison, the average value
$\bar{r}$ of $P(r)$ in Fig. 3(b) is $\approx 0.529$, while the analytical
value is $\bar{r}_{\rm GOE}\approx 0.536$. Since integrable systems exhibit
Poisson level statistics with $\bar{r}_{\rm Poisson}\approx 0.386$, this is
strong evidence that the Hamiltonian $\eqref{eq:ModelHZ2}$ is generic once all
commuting conserved quantities are accounted for.

In Fig. 3(c), we plot the von Neumann entanglement entropy
$S^{vN}=-\text{tr}(\rho_{A}\ln\rho_{A})$, where
$\rho_{A}=\text{tr}_{B}(\ket{E}\bra{E})$ is the reduced density matrix of half
of the system, for all eigenstates $\ket{E}$ in the
$(\tilde{S}_{X},\tilde{S}_{Z},\tilde{\mathcal{Z}})=(+1,+1,+1)$ sector at
system size $(L_{x},L_{y})=(2,3)$. The entanglement-vs.-energy curve displays
a “domed” structure characteristic of models obeying the ETH. The maximum
value of the curve is close to the “Page value” Page (1993)

$\displaystyle S_{\rm Page}=N_{A}\ln 2-\frac{2^{N_{A}}}{2^{N_{B}+1}}$ (25)

characteristic of a random state with $N_{A}$ qubits in region $A$ and $N_{B}$
qubits in region $B$. For the bipartition used in Fig. 3(c), $N_{A}=7$ and
$N_{B}=6$, so that $S_{\rm Page}\approx 3.85$. This indicates that eigenstates
near the middle of the many-body spectrum have no special structure apart from
the $O(1)$ number of commuting conserved quantities, further substantiating
the fact that the Hamiltonian (22) is generic.

Taken together, these numerical results substantiate the surprising fact that
a commuting pair of global symmetries, $\tilde{S}_{X}$ and $\tilde{S}_{Z}$, is
sufficient to encode a logical qubit in the Hilbert space of an otherwise
generic model. They also strongly suggest that the system obeys the ETH once
the commuting conserved quantities
$(\tilde{S}_{X},\tilde{S}_{Z},\tilde{\mathcal{Z}})$ [or, equivalently,
$(\tilde{S}_{X},\tilde{S}_{Z},\tilde{\mathcal{X}})$] are specified. It is
therefore expected that a quantum quench in which the system is prepared in an
arbitrary state of the form (8) (e.g., a $Z$-basis product state) will
generically yield relaxation to a state that locally resembles a Gibbs
ensemble with fixed values of these conserved quantities D’Alessio  _et al._
(2016). The same phenomenology applies to initial states where $\ket{\psi_{\rm
L}}$ occupies a generic position on the logical Bloch sphere, i.e., where
$\ket{\psi_{\rm L}}$ is the $+1$ eigenstate of a logical operator of the form
$n_{x}\,\tilde{\mathcal{X}}+n_{y}\,\tilde{\mathcal{Y}}+n_{z}\tilde{\mathcal{Z}}$,
where $\tilde{\mathcal{Y}}=i\tilde{\mathcal{X}}\tilde{\mathcal{Z}}$. Such
initial states can be prepared using techniques developed for surface codes,
see e.g. Refs. Brown  _et al._ (2017); Satzinger  _et al._ (2021).

###  II.6 Stability and Prethermalization

Having uncovered the structure of a symmetry-protected quantum memory within
an otherwise quantum-chaotic many-body system, the question naturally arises
of whether this memory can be stabilized against perturbations. Naively, it
would seem that any perturbation to Eq. (22) with energy scale $g$ that
violates the conservation of $\tilde{S}_{X/Z}$ (or equivalently $S_{X/Z}$ in
the language of Eq. 4) would lead to decoherence of the encoded qubit in a
time $t_{*}\sim 1/g$. However, it turns out that the lifetime of the quantum
memory can be extended far beyond this naive limit by leveraging the concept
of prethermalization Abanin  _et al._ (2017); Else  _et al._ (2017). Invoking
the results of Ref. Else  _et al._ (2020), we argue that the lifetime can be
bounded from below by a stretched exponential,

$\displaystyle t_{*}\gtrsim e^{C\sqrt{\nu/g}},$ (26)

where $C$ is a constant and where $\nu\gg g$ is an energy scale, explained
below, that suppresses the creation of excitations that could cause bit-flip
or phase errors in the encoded qubit. The discussion in this Section follows
Refs. Else  _et al._ (2017, 2020), which consider a weakly perturbed surface
code and find the same long-lived encoded qubit. The primary difference
between our work and Refs. Else  _et al._ (2017, 2020) is that our starting
point is the family of nonintegrable Hamiltonians defined in Sec. II.5, which
follows from the subsystem-code structure elucidated at the beginning of Sec.
II.

In order to leverage prethermalization, we embed the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ topological symmetry group generated by
Eq. 4 into a U(1)$\times$U(1) symmetry group with generators

$\displaystyle N_{X}$ $\displaystyle=\sum_{v}X_{v},\indent
N_{Z}=\sum_{p}Z_{p},$ (27a) or equivalently, in the disentangled picture,
$\displaystyle\tilde{N}_{X}=\sum_{v}\tilde{A}_{v},\indent\tilde{N}_{Z}$
$\displaystyle=\sum_{p}\tilde{B}_{p}.$ (27b)

These operators count the number of vertex and plaquette excitations of the
surface code; the original topological symmetry generators $S_{X,Z}$ measure
the parity of $N_{X,Z}$, respectively. It is possible to write down an analog
of Eq. (22) that respects this enlarged topological symmetry,

$\displaystyle
H_{0}=\sum_{v}h_{v}\,\tilde{A}_{v}+\sum_{p}h_{p}\,\tilde{B}_{p}+\\!\\!\\!\\!\sum_{\ell\notin\mathrm{smooth}}\\!\\!\\!\\!J_{x,\ell}\,\left(\frac{1-\tilde{B}_{p_{1}(\ell)}\tilde{B}_{p_{2}(\ell)}}{2}\right)X_{\ell}+\\!\\!\\!\\!\sum_{\ell\notin\mathrm{rough}}\\!\\!\\!J_{z,\ell}\,\left(\frac{1-\tilde{A}_{v_{1}(\ell)}\tilde{A}_{v_{2}(\ell)}}{2}\right)Z_{\ell},$
(28)

where $p_{1,2}(\ell)$ are the two plaquettes that share link $\ell$ and
$v_{1,2}(\ell)$ are the two vertices that share link $\ell$. This model arises
as a projection of Eq. (22) into a sector with a fixed number of vertex and
plaquette excitations. We have checked numerically using the methodology of
Sec. II.5 that the model (28) remains nonintegrable while retaining the same
encoded logical qubit. For simplicity, we henceforth restrict to the case of
uniform parameters $h_{v}=h_{p}=0$ and $J_{x,\ell}=J_{z,\ell}=J$.

Figure 4:  Dynamics of the expectation values of the vertex and plaquette
excitation operators $\braket{\tilde{N}_{X}}$ [top panel] and
$\braket{\tilde{N}_{Z}}$ [bottom panel]. (a) The dynamics of
$\braket{\tilde{N}_{X}}$ [top panel] and $\braket{\tilde{N}_{Z}}$ [bottom
panel] from initial product states with
$(\braket{\tilde{N}_{X}},\braket{\tilde{N}_{Z}})=(3,0)$ and $(0,3)$,
respectively, are shown from time $t=0$ until $t=300$. Parameters used in the
plots are $(L_{x},L_{y})=(2,3)$, $J=0.1$, and $g=0.15$. Yellow, blue, and
magenta curves use parameters $\nu=(2.0,\,3.0,\,4.0)$, respectively. Dashed
curves use $\xi=1$, while solid curves belong to $\xi=\sqrt{3}$. Solid
(dashed) horizontal lines indicate the diagonal ensemble averages
$\braket{\tilde{N}_{X(Z)}}_{\rm DE}$ for $\xi=\sqrt{3}$ $(\xi=1.0)$. (b) We
plot the data for the case $\xi=\sqrt{3}$ shown in (a) again for a smaller
range of time $t$, i.e. $t=0,\ldots,60$. As easily seen now, the expectation
values $\braket{\tilde{N}_{X}}$ and $\braket{\tilde{N}_{Z}}$ display
oscillating behavior around the diagonal ensemble averages
$\braket{\tilde{N}_{X,Z}}_{\rm DE}$ with $\braket{\tilde{N}_{X}}_{\rm
DE}=(3.00015,\,3.00005,\,3.00002)$ and $\braket{\tilde{N}_{Z}}_{\rm
DE}=(2.99992,\,2.99998,\,2.99999)$ for $\nu=(2.0,\,3.0,\,4.0)$, respectively.
On the qualitative side we note that with increasing factor $\nu$, the
oscillations become stronger. (c) We spotlight the behavior of
$\braket{\tilde{N}_{X}}$ and $\braket{\tilde{N}_{Z}}$ for the case of
$\xi=1.0$. Contrarily to the case $\xi=\sqrt{3}$ detailed in (b), we now
observe distinctively different behavior. The computed expectation values
$\braket{\tilde{N}_{X(Z)}}$ do not show any oscillations around their diagonal
ensemble averages. Instead we see a strong discrepancy between
$\braket{\tilde{N}_{X(Z)}}$ and $\braket{\tilde{N}_{X(Z)}}_{\rm DE}$. To give
the eye a guide line we again include the diagonal ensemble averages
$\braket{\tilde{N}_{X(Z)}}_{\rm DE}$ as dashed lines. Their exact values are
$\braket{\tilde{N}_{X}}_{\rm DE}=(3.07146,\,3.03196,\,3.01963)$ and
$\braket{\tilde{N}_{Z}}_{\rm DE}=(2.96784,\,2.97793,\,2.98982)$ for
$\nu=(2.0,\,3.0,\,4.0)$, respectively. Thus, the presented numerical data
shows that emergent U(1)$\times$U(1) symmetry persists to long times $t$ for
irrational $\xi$, but not for integer $\xi$, consistent with the predictions
of the theory of prethermalization.

Next, we consider the effect of symmetry-violating perturbations on the
symmetric model (28). These perturbations are of the form

$\displaystyle
V=g_{x}\sum_{\ell\in\text{smooth}}X_{\ell}+g_{z}\sum_{\ell\in\text{rough}}Z_{\ell}.$
(29)

Physically, these perturbations create individual vertex and plaquette
excitations at the appropriate boundaries of the system. We henceforth
simplify these perturbations by setting $g_{x}=g_{z}=g$. Note that the
perturbations (29) reside on the boundaries of the lattice and break the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ subgroup of the U(1)$\times$U(1)
symmetry. Because the generators of the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
subgroup reside on the boundaries, any local perturbation added to the bulk
will preserve this subgroup, even if it breaks the U(1)$\times$U(1) symmetry.
Since such local U(1)$\times$U(1)-breaking but
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-preserving bulk terms do not affect the
logical qubit (see also Ref. Else  _et al._ (2017)), we opt to focus instead
on the boundary perturbations (29), which are detrimental to the encoded
qubit.

Prethermalization aims to suppress the effect of symmetry-violating
perturbations [e.g., Eq. (29)] by inducing a separation of energy scales
Abanin  _et al._ (2017). To achieve this, we add two large “external fields”
that couple to the two U(1)$\times$U(1) generators; i.e., we consider a
Hamiltonian of the form

$\displaystyle H=\nu_{x}\tilde{N}_{X}+\nu_{z}\tilde{N}_{Z}+H_{0}+V$ (30)

in the limit $\nu_{x,z}\gg g$. This imposes an energetic penalty on violations
of the U(1)$\times$U(1) symmetry. To simplify the analysis, we will write

$\displaystyle\nu_{z}=\xi\,\nu_{x}\equiv\xi\,\nu,$ (31)

where $\xi$ is an order-one number to be specified later. Deviations from
$\tilde{N}_{X,Z}$ conservation can then be captured perturbatively in $g/\nu$
(in the absence of resonances, discussed below) by an effective Hamiltonian
that is U(1)$\times$U(1)-symmetric at each order. The validity of this type of
perturbation theory has been rigorously addressed for the case of U(1)
symmetry Abanin  _et al._ (2017) and for the more general case of $K$
commuting U(1) symmetries Else  _et al._ (2020). In this case, it can be shown
that the perturbation theory is valid up to an order

$\displaystyle n_{*}\sim\left(\frac{\nu}{g}\right)^{1/K},$ (32a) corresponding
to a timescale $\displaystyle t_{*}\gtrsim e^{C(\nu/g)^{1/K}}$ (32b)

on which the $K$ U(1) charges are approximately conserved. For a single U(1)
symmetry, this corresponds to an exponentially long-lived approximate
conservation law. The Hamiltonian (30) falls into the special case $K=2$,
where the stretched exponential timescale (26) is recovered.

The results of Ref. Else  _et al._ (2020) can only be applied when the
external fields coupling to the U(1) charges are irrational multiples of one
another. Otherwise, the U(1)×K-symmetric perturbation theory leading to Eqs.
(32) is spoiled by resonant processes that change the U(1) charges in such a
way that the net energy change due to the external fields vanishes. In the
model (30), this occurs when $\nu_{x}$ and $\nu_{z}$ are rational multiples of
each other, i.e., when $\xi=p/q$ for some relatively prime $p,q\in\mathbb{Z}$
in Eq. (31). In this case, one can have a resonant process in which, e.g.,
$\tilde{N}_{Z}\to\tilde{N}_{Z}+q$ and $\tilde{N}_{X}\to\tilde{N}_{X}-p$, such
that the external-field term $\nu(\tilde{N}_{X}+\xi\tilde{N}_{Z})$ is
invariant. If allowed, such “resonances” would lead to decay of the encoded
qubit on a timescale independent of $\nu$, since the qubit encoded in $H_{0}$
is protected by the U(1)$\times$U(1) symmetry rather than the U(1) subgroup
generated by $q\tilde{N}_{X}+p\tilde{N}_{Z}$. Setting $\xi$ to an irrational
number essentially suppresses these processes, endowing the dynamics under Eq.
(30) for $g\ll\nu$ with a long-lived approximate U(1)$\times$U(1) symmetry.

Numerical results probing the emergent U(1)$\times$U(1) symmetry and the
effect of resonances are shown in Fig. 4. Using ED, we calculate the dynamics
from representative initial product states of $\braket{\tilde{N}_{X}}$ and
$\braket{\tilde{N}_{Z}}$ for several values of $\nu$ for both $\xi=1$ and
$\xi=\sqrt{3}$. For $\xi=1$, $\braket{\tilde{N}_{X}}$ and
$\braket{\tilde{N}_{Z}}$ immediately begin to exhibit strong discrepancies in
comparison with their initial values, indicative of the resonant processes
discussed above. For $\xi=\sqrt{3}$, however, $\braket{\tilde{N}_{X}}$ and
$\braket{\tilde{N}_{Z}}$ do not strongly diverge indefinitely from their
initial values. Contrarily, they begin to oscillate around saturation values
close to their initial values. These saturation values are well-described by
the “diagonal ensemble” Kollar and Eckstein (2008); Rigol  _et al._ (2008);
D’Alessio  _et al._ (2016), defined for an arbitrary operator $\mathcal{O}$ in
the eigenbasis $\ket{E}$ of $H$ as

$\displaystyle\braket{O}_{\rm
DE}=\sum_{E}|\braket{\psi_{0}}{E}|^{2}\braket{E}{O}{E},$ (33)

where $\ket{\psi_{0}}$ is the initial state. Physically, $\braket{O}_{\rm DE}$
is the infinite-time average of $\braket{O(t)}$; the fact that
$\braket{\tilde{N}_{X,Z}}_{\rm DE}$ are so close to the values of
$\braket{\tilde{N}_{X,Z}}$ at time $t=0$, and in fact become closer as $\nu$
increases, is indicative of the emergent U(1)$\times$U(1) conservation law in
the limit ${\nu/g\to\infty}$. In particular, the fact that
$\braket{\tilde{N}_{X,Z}}$ stabilize around these values after an initial
transient suggests that the dynamics of the system can be well approximated by
one that conserves both $\tilde{N}_{X}$ and $\tilde{N}_{Z}$.

Figure 5:  Scaling behavior of relaxation times $\tau_{Z}$ and $\tau_{X}$.
Scaling of the lifetimes $\tau_{Z}$ (magenta data points) and $\tau_{X}$ (blue
data points) with the coupling paramters $\nu$ [top panel (a)] and $g$ [bottom
panel (b)] extracted from scrutinizing the quench dynamics of
$\braket{\tilde{\mathcal{X}}(t)}$ and $\braket{\tilde{\mathcal{Z}}(t)}$
starting from initial product states, i.e. eigenstates of the respective
logical operator. The lifetime $\tau_{Z(X)}$ is then defined to be the time
where $\braket{\tilde{\mathcal{X}}(t)}=\braket{\tilde{\mathcal{X}}}_{\rm DE}$
[$\braket{\tilde{\mathcal{Z}}(t)}=\braket{\tilde{\mathcal{Z}}}_{\rm DE}$] is
true for the first time. The coupling parameter $\nu$ governs the strength of
the operators $\tilde{N}_{X}$ and $\tilde{N}_{Z}$ that count the number of
vertex and plaquette excitations of the surface code whereas the parameter $g$
is associated with the symmetry-violating perturbation $V$ (see (29)). System
size is again $(L_{x},L_{y})=(2,3)$ and initial states are the same as those
used in Fig. 4, and we use $\xi=\sqrt{3}$ to avoid resonances. Data points are
plotted on a log-log scale and fits to power-law scaling are shown as solid
lines. The powers extracted from the fits are $1.959$ ($1.963$) for
$\braket{\tilde{\mathcal{Z}}(t)}$ [$\braket{\tilde{\mathcal{X}}(t)}$], and
$-1.833$ ($-1.864$) for $\braket{\tilde{\mathcal{Z}}(t)}$
[$\braket{\tilde{\mathcal{X}}(t)}$].

To probe the lifetime of the encoded qubit in the presence of perturbations,
we simulate the quench dynamics of $\braket{\tilde{\mathcal{X}}(t)}$ and
$\braket{\tilde{\mathcal{Z}}(t)}$ starting from initial product states that
are eigenstates of one of the logical operators. We calculate the lifetime
$\tau_{X}$ ($\tau_{Z}$) as the first time at which
$\braket{\tilde{\mathcal{X}}(t)}=\braket{\tilde{\mathcal{X}}}_{\rm DE}$
[$\braket{\tilde{\mathcal{Z}}(t)}=\braket{\tilde{\mathcal{Z}}}_{\rm DE}$]. We
then repeat the simulation for a sequence of $g$ and $\nu$ values to extract
the dependence of $\tau_{X,Z}$ on these parameters. The data, shown in Fig. 5,
show a clear power-law scaling in $\nu/g$, rather than the expected stretched-
exponential scaling of Eq. (26).

This deviation from the expected scaling can be understood straightforwardly
as a finite-size effect. In the presence of (approximate) $\tilde{N}_{X,Z}$
conservation, only one type of process can “flip” the state of the logical
qubit: an excitation is created at one of the boundaries, propagates through
the bulk of the system, and is annihilated at the other boundary of the same
(rough/smooth) type. If the excitation propagates between rough (smooth)
boundaries, $\braket{\tilde{\mathcal{X}}}$ ($\braket{\tilde{\mathcal{Z}}}$)
changes sign. This process is second-order in $g$, the energy scale for
quasiparticle creation at the boundaries, and subextensive in the kinetic
energy scale $J$, which propagates the excitation between boundaries. The
relaxation time associated with this process for a system of size
$(L_{x},L_{y})$ is then

$\displaystyle\tau_{X}\sim\frac{\nu}{g^{2}}\left(\frac{\nu}{J}\right)^{L_{y}-2},\indent\tau_{Z}\sim\frac{\nu}{g^{2}}\left(\frac{\nu}{J}\right)^{L_{x}-1}.$
(34)

In this formula, we omit constant factors and have included appropriate powers
of $J$ for dimensional purposes. For sufficiently large $\nu$, these lifetimes
diverge exponentially in the thermodynamic limit $L_{x},L_{y}\to\infty$. This
recovers the familiar fact Kitaev (2003); Bravyi and Kitaev (1998); Fowler
_et al._ (2012); Fendley (2016); Else  _et al._ (2017); Kemp  _et al._ (2017)
that the encoded qubit is only stable up to exponentially small finite-size
corrections (or, equivalently, exponentially long times at finite size). At
the small system size $(L_{x},L_{y})=(2,3)$ considered here (due to runtime
constraints, as each relaxation-time calculation involves evolving the system
over times of up to $\sim 10^{4}J^{-1}$), these timescales both scale as
$\nu^{2}/g^{2}$, consistent with the timescales observed in our numerics.
Nevertheless, since the results of Fig. 4 suggest that the approximate
U(1)$\times$U(1) symmetry is operative, we expect that the regime in which
$\tau\gtrsim t_{*}$ [Eq. (26)] holds should be accessible at larger system
sizes, even if numerical simulation of the system at such sizes becomes
infeasible.

In summary, while our numerical results are hindered by finite-size effects,
they clearly reveal the presence of a long-lived emergent U(1)$\times$U(1)
symmetry despite the presence of symmetry-violating perturbations. These
results are consistent with the existence of an encoded qubit with a lifetime
parametrically long in the inverse of the perturbation energy scale, but
larger-scale numerics would be required to validate the prediction of a
stretched-exponential lifetime in this model.

##  III Subsystem Codes and Dynamics

In this section we formalize our notion of quantum memory in a dynamical
system by giving a definition in terms of noiseless subsystems. We then
explain how subsystem codes provide plentiful nontrivial examples of such
dynamical systems and uncover the appropriate local error detection structure
required for these codes to give rise to prethermal quantum memories when the
discrete stabilizer symmetry is enlarged to a U(1)×K symmetry Else  _et al._
(2020).

###  III.1 Noiseless subsystems

A general, potentially open-system, quantum dynamical evolution on a Hilbert
space $\mathcal{H}$ is described by a completely positive trace preserving
(CPTP) map Nielsen and Chuang (2010) $\mathcal{E}_{t}$ on the space of
operators $\mathcal{B}(\mathcal{H})$ that is a function of a discrete or
continuous time parameter $t$. Here, we are interested in dynamical evolutions
that retain a quantum memory of initial conditions. A very general formulation
of the quantum memory condition is obtained by requiring the dynamical
evolution to be correctable with respect to a chosen quantum code. More
precisely, quantum memory of a subsystem $\mathcal{H}_{L}$ within the full
Hilbert space
$\mathcal{H}=(\mathcal{H}_{L}\otimes\mathcal{H}_{J})\oplus\mathcal{H}_{\perp}$—where
$L,J,$ stand for logical and junk subsystems, respectively, and where $\perp$
denotes the orthogonal complement subspace—is retained if the dynamics under
$\mathcal{E}_{t}$ is correctable. That is, there exists a recovery map
$\mathcal{R}_{t}$ such that Kribs  _et al._ (2005, 2006)

$\displaystyle\text{Tr}_{J}\circ
P\circ\mathcal{R}_{t}\circ\mathcal{E}_{t}\,(\rho)=\text{Tr}_{J}(\rho)\,,$ (35)

for $\rho$ a density matrix supported on
$\mathcal{H}_{L}\otimes\mathcal{H}_{J}$ and $P$ a projector onto
$\mathcal{H}_{L}\otimes\mathcal{H}_{J}$. This definition says that
$\mathcal{H}_{L}$ forms a noiseless subsystem Kribs  _et al._ (2005, 2006) for
$\mathcal{R}_{t}\circ\mathcal{E}_{t}$. In particular, when
$\mathcal{R}_{t}=\mathbbm{1}$ this reduces to $\mathcal{H}_{L}$ forming a
noiseless subsystem for the dynamics $\mathcal{E}_{t}$. For the interesting
subclass of closed-system unitary dynamics, the former condition is too weak
to be nontrivial, since all unitary evolutions can in principle be undone. In
this setting, we therefore focus on the latter definition, where the recovery
map is taken to be $\mathcal{R}_{t}=\mathbbm{1}$ and $\mathcal{H}_{L}$ is a
noiseless subsystem for $\mathcal{E}_{t}$, i.e. some initial quantum
information remains static under the evolution.

While the above definition applies to perfect quantum memory for arbitrarily
long times, it is interesting to loosen this to an approximate quantum memory
with high probability for a sufficiently long time, as we consider in a
certain setting below. It is also easy to come up with trivial examples of
dynamics that retain a quantum memory, such as a single qubit with trivial
evolution. We are interested in nontrivial examples where every qubit
participates in the dynamics to form a quantum memory via a collective quantum
many-body effect. We further want this memory to be stabilizable against
generic uniform local perturbations that remain sufficiently weak.

###  III.2 Stabilizer subsystem codes

To construct specific examples of nontrivial noiseless subsystems, we focus
our attention on stabilizer subsystem codes Poulin (2005); Bacon (2006) (see
also Ref. Knill  _et al._ (2000)). For the remainder of this section we
restrict our attention to qubits and $\mathbb{Z}_{2}$ local degrees of freedom
(i.e., qubits) for simplicity of presentation; the extension to
$\mathbb{Z}_{p}$ for $p$ prime (i.e., qudits with prime dimension) is
straightforward. Further extension to general $\mathbb{Z}_{N}$ is possible,
however the theory of stabilizer subsystem codes for composite dimensions
becomes more complicated.

stabilizer subsystem codes are defined by a group of “gauge” operators $G\leq
P$ that form a subgroup of the Pauli group $P$. We remark that these “gauge”
operators are not related to the gauge transformations of lattice gauge
theory, but rather refer to operators acting on redundant “gauge” qubit
degrees of freedom associated to $\mathcal{H}_{J}$. An auxiliary group of
stabilizer operators $S$ is given by the center of the gauge group,
$S=Z(G)<P$. This induces a decomposition of the Hilbert space

$\displaystyle\mathcal{H}=\bigoplus_{\\{\lambda_{i}\\}}(\mathcal{H}_{L}\otimes\mathcal{H}_{J})_{\\{\lambda_{i}\\}}$
(36)

into isomorphic sectors labelled by a complete set of stabilizer eigenvalues,
specified here by the eigenvalues $(-1)^{\lambda_{i}}$ of a generating set of
stabilizers $s_{i}$. Here $L$ denotes the “logical subsystem” and $J$ denotes
the “junk subsystem” for reasons we now explain. The “gauge” operators act
within each sector as $\mathbbm{1}_{L}\otimes g_{J}$. Operators in the
commutant (i.e., centralizer) of the gauge subgroup $C(G)$ are called bare
logical operators; their actions are only defined up to multiplication with
elements of the stabilizer group. Therefore $L=C(G)/S$ describes the group of
bare logical operators up to equivalence. These bare logical operators act
within each sector as $\ell_{L}\otimes\mathbbm{1}_{J}$. As logical quantum
information is only stored in the logical subsystem $\mathcal{H}_{L}$, a
larger class of “dressed logical” operators are relevant. These dressed
logical operators implement the same set of transformations on
$\mathcal{H}_{L}$ while also potentially disturbing $\mathcal{H}_{J}$ by an
unimportant operator. The algebra of dressed logical operators is given by the
commutant of the stabilizer subgroup, $C(S)$. Operators in $C(S)$ have a well
defined logical action up to multiplication with “gauge” operators. Hence
$C(S)/G$ is the group of dressed logical operators up to equivalence.
Representative dressed logical operators act within each sector as
$\ell_{l}\otimes g_{J}$. The code distance $d$ is given by the minimum Pauli
weight over all representative dressed logical operators. The group $L=C(G)/S$
of bare logical operators is a projective representation of the group
$\mathbb{Z}_{2}^{\times|L|}$, where $|L|$ is twice the number of encoded
qubits. Similarly the group $C(S)/G$ of dressed logical operators, modulo
complex phases, forms a representation of the same group
$\mathbb{Z}_{2}^{\times|L|}$.

The essential features of a stabilizer subsystem code are often packaged into
the notation $[[n,k,r,d]]$ where $n$ denotes the number of physical qubits,
$k$ the number of encoded logical qubits, $r$ the number of gauge qubits, and
$d$ the code distance. In this notation, $[[n,k,0,d]]$ codes are conventional
stabilizer codes. Quantum error correction for stabilizer subsystem codes is
implemented by measuring the gauge generators (in a sequence that accounts for
the fact they may not commute) and using this information to reconstruct the
stabilizer eigenvalues. The measured stabilizer eigenvalues are then used to
compute a correction operator that is applied to return the system to the code
space such that the inferred error operator composed with the correction
operator results in a trivial logical operation.

For concreteness we always fix a choice of generators for the gauge group,
i.e., we write $G=\langle\\{g_{i}\\}\rangle$, where the list of generators
$g_{i}$ may be overcomplete. If the generators can be picked to consist of
purely $X$-type and $Z$-type Pauli operators, the code is known as a CSS
subsystem code Calderbank and Shor (1996); Steane (1996). Our focus is on
systems with generators $g_{i}$ that respect the microscopic locality of an
underlying lattice. Furthermore, we fix a choice of generators for the
stabilizer group, $S=\langle\\{s_{i}\\}\rangle$, which can involve a
combination of local and nonlocal operators. A topological subsystem code
Bombin  _et al._ (2009); Kargarian  _et al._ (2010); Bombin (2010); Suchara
_et al._ (2011); Bravyi  _et al._ (2013) is one for which the stabilizer
generators can all be picked to be local with respect to an underlying lattice
while the code distance $d$ is macroscopic (i.e., $d$ grows as the number of
physical qubits increases).

###  III.3 Stabilizer subsystem code symmetry and dynamics

In this subsection we discuss an interpretation of the stabilizer group of a
subsystem code as an anomalous symmetry. We go on to describe the consequences
that such a symmetry has on closed an open system dynamics whose generators
are symmetric, of which the example in Sec. II is a special case.

The stabilizer group of a nontrivial subsystem code forms a type of anomalous
symmetry. For our purposes here, an anomalous symmetry is defined by the
property that no local symmetric gapped Hamiltonian can have a unique ground
state. More precisely, for nontrivial $\mathcal{H}_{L}$ of dimension
$2^{|L|/2}$, any Hamiltonian that is given by a sum of gauge operators, i.e.,

$\displaystyle H=\sum_{v}h_{v}\,,$ (37)

where the interactions $h_{v}$ are (linear combinations of) gauge operators,
has at least a $2^{|L|/2}$-fold degeneracy of its energy levels as it commutes
with the stabilizer and bare logical operators of the code. For the class of
subsystem codes we consider, namely those with local generators that protect
against local errors, any local Hamiltonian that commutes with the stabilizer
symmetry must also commute with the bare logical operator algebra and hence
must have degenerate energy levels. That is, any local Hamiltonian that
respects the stabilizer symmetry group $S$, which is a linear representation
of $\mathbb{Z}_{2}^{\times|S|}$, automatically respects a larger symmetry
group $C(G)$ that additionally includes the bare logical algebra, which is a
projective representation of $\mathbb{Z}_{2}^{\times|L|}$. Pairs of bare
logical operators that define logical qubits anticommute, as an aside for
readers familiar with the concept the resulting anticommutation relations can
be taken to define a type of mixed anomaly valued in
$H^{2}(\mathbb{Z}_{2}^{\times|L|},U(1))$ associated to the projective
representation of the bare logical algebra Tachikawa (2020).

For topological subsystem codes in 2D Bombin (2010), the stabilizer symmetry
is in fact a 1-form symmetry 111A $k$-form symmetry is generated by operators
acting on submanifolds of codimension $k$. associated to string operators of
nontrivial pointlike superselection sectors Gaiotto  _et al._ (2015). In this
case the anomaly corresponds to the topological braiding phases of these
pointlike sectors. In dimensions greater than two, the stabilizer symmetry of
a topological subsystem code may be a higher $k$-form symmetry Bombín (2015);
Kubica and Beverland (2015); Kubica and Vasmer (2021), or a more
unconventional symmetry related to fracton topological order Devakul and
Williamson (2021).

The time evolution $e^{-iHt}$ generated by a local Hamiltonian that commutes
with a nontrivial stabilizer subsystem code symmetry, and hence takes the form
introduced in Eq. (37), necessarily preserves a quantum memory as it commutes
with the bare logical operators and therefore acts trivially within the
$\mathcal{H}_{L}$ subsystems. More generally, any dynamical evolution
generated by local terms that commute with the type of stabilizer subsystem
symmetry we consider, and are hence linear combinations of gauge operators,
preserve a quantum memory on $\mathcal{H}_{L}$. This includes open-system
dynamics, which can be induced by unitary dynamics on an extended Hilbert
space that commutes with the stabilizer symmetry on the original degrees of
freedom.

For a more precise formulation of this point, consider a family of quantum
channels describing time evolution parameterized by $t$,

$\displaystyle\mathcal{E}_{t}(\rho)=\sum_{i}K_{i}(t)\,\rho\,K_{i}^{\dagger}(t)\,,$
(38)

where the Kraus operators $K_{i}(t)$ are (linear combinations of) gauge
operators that satisfy the trace preserving condition
$\sum_{i}K_{i}^{\dagger}(t)K_{i}(t)=\mathbbm{1}$. For the simple case of
closed-system unitary evolution discussed above we have only a single Kraus
operator $K(t)=e^{-iHt}$ for $H$ a local Hamiltonian of the form described in
Eq. (37). This description also applies more generally, including in the case
of monitored dynamics involving quantum measurements, in which case some Kraus
operators may take the form of projections onto the eigenvalues of gauge
operators, e.g., $K_{i}(t)=\frac{1}{2}(\mathbbm{1}-g_{i})$ for a gauge
operator $g_{i}$. In fact, the quantum memory we are discussing exists even at
the level of individual quantum trajectories, where projective measurements
are made without averaging over the measurement outcomes.

Any dynamics generated by local terms that respect a nontrivial stabilizer
subsystem code symmetry (that corrects local errors) must take the form
introduced in Eq. (38). Hence the symmetry condition implies that the dynamics
preserves a quantum memory, i.e., commutes with the bare logical algebra. The
preservation of the quantum memory can be understood as a consequence of the
mixed anomaly associated to the stabilizer subsystem code symmetry, see above.

We emphasize that, in fixed stabilizer eigenspaces, the dynamics within the
junk subsystem $\mathcal{H}_{J}$ can be totally generic or chaotic, or
alternatively may be integrable such as when only a commuting set of gauge
operators enters the dynamics, or in the simple case of conventional
stabilizer codes where $\mathcal{H}_{J}$ is trivial.

###  III.4 Adding perturbations

In the presence of Hamiltonian perturbations and couplings to the environment
that do not respect the stabilizer symmetry, the dynamics generated by a
stabilizer subsystem code may no longer protect $\mathcal{H}_{\rm L}$ in Eq.
(36) as a noiseless subsystem. To maintain protection of the quantum
information in these subsystems, the most systematic approach is the
implementation of quantum error correction, either through active measures or
by passively relying on a sufficiently low temperature environment Dennis  _et
al._ (2001); Bacon (2006). This is an interesting direction with many facets
that we leave to future work. Here we restrict our focus to passive extension
of quantum memory lifetimes under closed-system unitary dynamics in the
presence of nonsymmetric Hamiltonian perturbations, which may correspond to
control errors. We rely on recently established results on prethermal quantum
memories proved in Ref. Else  _et al._ (2017, 2020).

The results of Ref. Else  _et al._ (2017, 2020) apply to dynamics generated by
Hamiltonians of the form

$\displaystyle H=H_{0}+gV+\nu_{1}\Sigma_{1}+\dots+\nu_{K}\Sigma_{K}\,,$ (39)

where $H_{0}$ is a local Hamiltonian that obeys a U(1)×K symmetry generated by
the operators $\Sigma_{k}$ ($k=1,\dots,K$), and $V$ is a sum of symmetry-
breaking local perturbations [with O(1) operator norm] with $g$ sufficiently
small compared to all the $\nu_{i}$. Note that both $H_{0}$ and $V$
generically contain non-commuting local terms, and that $H_{0}$ can be
nonintegrable. In order for the prethermalization arguments to hold, the U(1)
generators $\Sigma_{k}$ must be sums of commuting local terms—in Sec. III.5.1
we consider an example of a stabilizer subsystem code to which these
prethermalization arguments cannot be applied due to nonlocality of the
$\Sigma_{k}$. We demonstrate below that a sufficient condition guaranteeing
that the $\Sigma_{k}$ are sums of commuting local terms is for the underlying
subsystem code to be a topological subsystem code. In the context of quantum
memory, the crucial U(1)×K symmetry is in fact closely related to a pair of
stabilizer (subsystem) codes that we refer to as the local and global codes,
which we now explain. We refer back to the example in Sec. II throughout the
more general explanation.

First, we specify the local subsystem code via a choice of stabilizer
generators $S=\langle\\{s_{i}\\}\rangle$. As the name suggests, our focus is
on cases where these generators have local support on a spatial lattice for
practical reasons 222However, we consider at least one example where this is
not the case.. These stabilizer generators are given by star and plaquette
terms for the example in Sec. II. There may be relations (i.e., redundancies)
among these generators; we label a generating set of such relations as
$\\{r_{j}\\}$, where the relations $r_{j}$ are defined such that

$\displaystyle\prod_{i\in r_{j}}s_{i}=\mathbbm{1}\,.$ (40)

In the example introduced in Sec. II there are no such relations. Next, we
define the group of gauge operators $G$ to be generated by all local operators
contained in $C(S)$. Finally, the set of bare logical operators for the local
code is given by $L=C(G)/S$. We demand that all elements of $L$ have
macroscopic weight (if any were local those would already be included in $G$).
We assume that the dimension $2^{|L|/2}$ of the logical algebra is at least
two, otherwise there would be no quantum memory even for unperturbed dynamics.
The logical operators are macroscopic string operators in the example covered
in Sec. II. The local stabilizer subsystem code defines a block decomposition
of the Hilbert space according to the eigenvalues of the stabilizer generators
$\frac{1}{2}(\mathbbm{1}-s_{i})={\lambda_{i}}$, see Eq. (36). Here, there may
be nontrivial relations $r_{j}$ due to the the choice of generators being
overcomplete. In this case, the blocks in Eq. (36) labelled by eigenvalues not
satisfying the relation conditions in Eq. (40) are empty. In the example
introduced in Sec. II these sectors correspond to configurations of pinned $e$
and $m$ anyons in the surface code.

We now move on to define the U(1)×K symmetry, and the global stabilizer
subsystem code, in terms of the stabilizers of the local subsystem code. As
above, each stabilizer generator defines an occupation number
$\lambda_{i}\in\\{0,1\\}$ via the eigenvalue of
$\frac{1}{2}(\mathbbm{1}-s_{i})$. Any subset of stabilizer generators
$S_{k}\subseteq\\{s_{i}\\}$ (that involves an increasing number of generators
as the number of physical qubits grows) defines a generator of a U(1) symmetry
via

$\displaystyle\Sigma_{k}:=\sum_{s_{i}\in
S_{k}}\frac{1}{2}(\mathbbm{1}-s_{i})\,.$ (41)

A sufficient condition that guarantees the locality of the $\Sigma_{k}$
defined above is for the underlying local stabilizer subsystem code to be a
topological subsystem code. In such codes, the stabilizers $s_{i}$ are local
and mutually commuting by definition (see Sec. III.2). Conversely, a U(1)
symmetry generated by a sum of mutually commuting local Pauli terms always
defines a topological stabilizer subgroup generated by those terms. The local
code in Sec. II is simply the topological surface code, and the
U(1)$\times$U(1) symmetry corresponds to $e$ and $m$ anyon number
conservation.

The U(1) symmetry generated by $\Sigma_{k}$ enforces conservation of the total
occupation number, $\sum\lambda_{i}^{(k)}$, of the stabilizers included in the
set $S_{k}$. This induces a block decomposition of the Hilbert space,

$\displaystyle\bigoplus_{\\{n_{k}\\}}\Big{(}\mathcal{H}_{L}\otimes\bigoplus_{\\{\lambda_{i}\\}:\sum\lambda_{i}^{(k)}=n_{k}}{\mathcal{H}_{J}}_{\\{\lambda_{i}\\}}\Big{)}_{\\{n_{k}\\}}\,,$
(42)

which is strictly coarser than the decomposition induced by the local
stabilizer code symmetry, introduced in Eq. (36). In particular, all blocks of
the local stabilizer symmetry that satisfy $\sum\lambda_{i}^{(k)}=n_{k}$, for
the stabilizer generators in the set $S_{k}$, are collected into the single
$\\{n_{k}\\}$ block of the U(1)×K symmetry. In the surface code example from
Sec. II the blocks correspond to states with fixed $e$ and $m$ anyon number,
collecting many different configurations with the same numbers of $e$ and $m$
anyons. The gauge operators of the local stabilizer code remain symmetric
under the U(1)×K symmetry. In addition, arbitrary local operators can be
projected onto fixed occupation number eigenspaces of the overlapping local
stabilizers $\frac{1}{2}(\mathbbm{1}-s_{i})$ to produce further symmetric
local operators. For example, the gauge generators of the global stabilizer
code introduced below can be projected in this way to create symmetric local
operators, as was done for the example in section II.6.

Finally, we move on to define the global stabilizer subsystem code. Each set
of stabilizers $S_{k}$ introduced above, that is not a stabilizer relation,
defines a nontrivial $\mathbb{Z}_{2}$ subgroup of both the U(1) symmetry and
the local stabilizer symmetry. This subgroup is generated by a $\pi$ rotation
of the form

$\displaystyle e^{i\pi\Sigma_{k}}=\prod_{i\in S_{k}}s_{i}\,,$ (43)

which enforces conservation mod 2 of the occupation number of stabilizers in
$S_{k}$. If $S_{k}$ is a relation, the above rotation by $\pi$ is trivial i.e.
$e^{i\pi\Sigma_{k}}=\mathbbm{1}$ and the conservation of the occupation of
stabilizers in $S_{k}$ mod 2 is a constraint (also known as a materialized
symmetry Kitaev (2003); Brown and Williamson (2020)). For example, consider
the toric code with periodic boundary conditions Kitaev (2003); in this case,
the global stabilizers $S_{X}$ and $S_{Z}$, consisting of the product of all
star and plaquette stabilizers, respectively, are both relations (i.e.,
$S_{X}=S_{Z}=\mathbbm{1}$). The global stabilizer code is thus trivial (i.e.,
has trivial generators) and hence does not protect a quantum memory, even
under dynamics that respect the U(1)$\times$U(1) symmetry corresponding to
conservation of the total number of “electric” and “magnetic” anyons. To see
this, note that an anyon can move around a noncontractible cycle of the torus,
inducing a nontrivial logical action, via local steps that preserve the
U(1)$\times$U(1) symmetry Else  _et al._ (2017). This is no longer possible
with the surface-code-like open boundary conditions considered in Section II.
There the products over all star and plaquette operators, respectively,
correspond to nontrivial boundary operators.

In the nontrivial case where $S_{k}$ is not a relation, the global stabilizer
group is generated by the operators $\\{e^{i\pi\Sigma_{k}}\\}$ which induce a
block decomposition

$\displaystyle\bigoplus_{\\{\sigma_{j}\\}}\bigg{(}\mathcal{H}_{L}\otimes\bigoplus_{\\{\lambda_{i}\\}:\sum\lambda_{i}^{(j)}=\sigma_{j}}{\mathcal{H}_{J}}_{\\{\lambda_{i}\\}}\bigg{)}_{\\{\sigma_{j}\\}}\,,$
(44)

where $\sum\lambda_{i}^{(j)}=\sigma_{j}$ is only required to hold mod 2, with
the sum taken over the set $S_{j}$ and with
$\sigma_{j}=\frac{1}{2}(\mathbbm{1}-e^{i\pi\Sigma_{j}})$ the eigenvalue of the
global stabilizer. In the example from Sec. II the block decomposition
corresponds to the conservation of both $e$ and $m$ parity. Once again we
define the the gauge group to be generated by all local operators that commute
with all global stabilizers. This group contains all the gauge generators of
the local stabilizer code. The bare logical operators are then given by the
commutant of the gauge group, up to multiplication with stabilizers. Each bare
logical must also be a bare logical of the local stabilizer code. The dressed
logical operators for the global stabilizer code may be significantly lower
weight than those of the local stabilizer code, due to the enlarged gauge
group. In particular, the dressed logical operators may have constant weight,
although by definition they must remain nonlocal otherwise they would have
been included in the gauge group. The global stabilizer code may only be an
error detecting code, even when the local stabilizer code is an error
correcting code. However this is sufficient to preserve a quantum memory in a
noiseless subsystem. For the example in Sec. II the gauge generators include
single site Pauli operators in the bulk, and two body Pauli operators along
the boundary. The dressed logical operators become two body on a pair of well
separated boundaries and the global code is simply an error detecting code for
certain single qubit errors on the boundary. For the examples of subsystem
codes considered in Sec. III.5, the global stabilizers act exclusively on the
boundaries and indeed define single-Pauli error detecting codes there. In
general, the global code should be a nontrivial error detecting code with no
spatially local (dressed) logical operators. This suffices to ensure the
persistence of a prethermal quantum memory under the dynamical evolution with
U(1)×K symmetry breaking perturbations introduced in Eq. (39).

As discussed in section III.3, dynamics generated by a Hamiltonian that is a
sum of gauge generators for the global subsystem code preserves a quantum
memory associated to the noiseless subsystem $\mathcal{H}_{L}$. However, in
the presence of generic local Hamiltonian perturbations we cannot provide a
guarantee that this memory persists. For the stricter setting of dynamics
generated by a local Hamiltonian obeying the full U(1)×K symmetry we can apply
the results of Refs. Else  _et al._ (2017, 2020) to ensure the existence of a
prethermal memory time $t_{*}$ that is a stretched exponential in a parameter
that measures the relative perturbation strength in a Hamiltonian of the form
(39). The results of Refs. Else  _et al._ (2017, 2020) imply that such a
Hamiltonian $H$ can be expanded in a U(1)×K-symmetric perturbation series that
is convergent up to an order proportional to $(\nu/g)^{1/K}$, where
$\nu\equiv\sqrt{\sum^{K}_{i=1}\nu^{2}_{i}}$. This in turn implies that the
evolution $e^{-itH}$ generated by the above Hamiltonian retains the U(1)×K
symmetry, up to small corrections, until a time

$\displaystyle t_{*}\gtrsim e^{C(\nu/g)^{1/K}},$ (45)

where $C>0$ is a constant. In particular this implies that the perturbed
evolution commutes with the global stabilizer symmetry up to small
corrections, which in turn implies the preservation of a quantum memory
associated to a noiseless subsystem. We may also choose to strictly enforce
some subset of stabilizer symmetries, while enforcing others via a
$U(1)^{\times K^{\prime}}$ symmetry that is then allowed to be broken by
sufficiently weak perturbations and similar results follow Else  _et al._
(2017, 2020).

###  III.5 Further examples

The formalism outlined in Secs. III.1–III.4 yields a simple recipe for
defining nontrivial quantum dynamics that preserves quantum information
encoded in the initial state. In the most general case, any local quantum
circuit composed of unitary gates generated by local products of the gauge
generators $g_{i}\in G$ will preserve information encoded in the logical
subsystem $\mathcal{H}_{L}$. Furthermore, one can allow hybrid unitary-
projective circuits in which unitary dynamics is interspersed with projective
local measurements, provided that the measurement operators are also local
products of gauge generators. Alternatively, one can consider dynamics
generated by a local Hamiltonian consisting of a sum over local products of
gauge generators. Such models (and perturbations thereof) are studied in Secs.
II.5 and II.6.

Regardless of their nature (i.e., whether they are Hamiltonian, unitary, or
unitary-projective), the dynamics constructed in this way possess a symmetry
group equal to the stabilizer group $S$. When the generators of the stabilizer
group act on all lattice sites at once, the dynamics preserve a global
symmetry, e.g. the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ global symmetry of
Sec. II, that protects the encoded quantum memory. Depending on the structure
of the stabilizer group, the protecting symmetry can be a global, subsystem,
or higher-form symmetry Gaiotto  _et al._ (2015); Vijay  _et al._ (2016);
Williamson (2016).

In this section, we consider several examples of such families of quantum
dynamics and elucidate their essential features. We present in Appendix B a
high-level discussion of other examples based on a variety of stabilizer codes
in three or fewer spatial dimensions.

####  III.5.1 Bacon-Shor Code

Figure 6:  The Bacon-Shor model. (a) Qubits (magenta squares) living on a
square lattice with gauge generators $X_{i,j}X_{i,j+1}$ and $Z_{i,j}Z_{i+1,j}$
highlighted by dashed rectangles. Examples of the stabilizer generators
$S^{X}_{j}$ and $S^{Z}_{i}$, i.e. $S^{X}_{6}$ and $S^{Z}_{2}$, are highlighted
in yellow, and the logical operators $\mathcal{Z}_{1}$ and $\mathcal{X}_{1}$
each being a product of $Z(X)$-operators along a vertical (horizontal) path
are highlighted in magenta. (b) The energy spectra for two systems of size
$(L_{x},L_{y})=(4,4)$ [top panel] and $(7,3)$ [bottom panel] containing a
total of $16,384$ and $4,096$ states, respectively, are shown. The spectrum
splits into eight (four) parts for $(L_{x},L_{y})=(4,4),(7,3)$ each separated
by a relatively large energy difference. A single sector of the spectrum
contains 512 states for $(L_{x},L_{y})=(4,4)$ and 4,096 states for a
$(7,3)$-system. (c) The histograms of energy level spacing ratios $r$ are
depicted again for the systems of size $(4,4)$ [top panel] and $(7,3)$ [bottom
panel]. In order to lift any accidental degeneracies, mild disorder in the two
bond couplings $J^{x}_{ij}$ and $J^{z}_{ij}$ is employed by uniformly drawing
$J^{x}_{ij},J^{z}_{ij}\in[0.9,1.1]\,\forall\,i,j$ with $1\leq i<L_{x}-1$ and
$1\leq j<L_{y}-1$. To ensure convergence we averaged over 1,000 independent
realizations of disorder for the system of size $(4,4)$ and over 200 disorder
realizations for the $(7,3)$-system. The resulting histogram data closely
resemble a Gaussian orthogonal ensemble (GOE) distribution as expected. The
averaged r-values are $\bar{r}=0.5265$ and $\bar{r}=0.5289$ for
$(L_{x},L_{y})=(4,4)$ and $(7,3)$, respectively.

Consider a system of qubits on the vertices $\bm{r}=(i,j)\in\mathbb{Z}^{2}$ of
an $L_{x}\times L_{y}$ square lattice as shown in Fig. 6(a). We can define a
Hamiltonian based on the Bacon-Shor code Bacon (2006); Shor (1995) as follows:

$\displaystyle
H_{\text{B-S}}=\sum^{L_{x}}_{i=1}\sum^{L_{y}}_{j=1}\left(J^{x}_{ij}\,X_{i,j}X_{i,j+1}+J^{z}_{ij}\,Z_{i,j}Z_{i+1,j}\right).$
(46)

The operators $X_{i,j}X_{i,j+1}$ and $Z_{i,j}Z_{i+1,j}$ generate the gauge
group $G$ of the Bacon-Shor code. The Hamiltonian (46) has an extensive number
of $\mathbb{Z}_{2}$ symmetries generated by the operators

$\displaystyle S^{X}_{j}=\prod^{L_{x}}_{i=1}X_{i,j}X_{i,j+1},\indent\indent
1\leq j<L_{y}$ (47a) and $\displaystyle
S^{Z}_{i}=\prod^{L_{y}}_{j=1}Z_{i,j}Z_{i+1,j},\indent\indent 1\leq i<L_{x},$
(47b)

for a total of $L_{x}+L_{y}-2$ $\mathbb{Z}_{2}$ symmetries (see Fig. 6(a)).
The operators (47) generate the stabilizer group $S$ of the Bacon-Shor code.
Physically, $S^{X}_{j}$ and $S^{Z}_{i}$ generate a subsystem symmetry, i.e.,
one that acts on rigid one-dimensional submanifolds of the two-dimensional
space. This model has one encoded qubit with bare logical operators

$\displaystyle\mathcal{X}_{j}$
$\displaystyle=\prod^{L_{x}}_{i=1}X_{i,j},\indent\indent\mathcal{Z}_{i}=\prod^{L_{y}}_{j=1}Z_{i,j}.$
(48)

Note that, for any $j,j^{\prime}$, $\mathcal{X}_{j}$ can be transformed into
$\mathcal{X}_{j^{\prime}}$ by multiplication by some number of $S^{X}_{j}$,
and similarly for any two $\mathcal{Z}_{i}$ and $\mathcal{Z}_{i^{\prime}}$.
Thus, there is only one encoded logical qubit in this model.

We now study the Bacon-Shor Hamiltonian (46) to illustrate that, like the
model studied in Sec. II.5, it is quantum chaotic when all symmetries and the
state of the logical qubit are resolved. To do this, we again study the
statistics of energy levels within a symmetry sector. We aim to resolve the
extensive number of $\mathbb{Z}_{2}$ symmetries generated by (47). The
resolution of the $(L_{x}-1)$ operators $S^{Z}_{i}$ for $1\leq i<L_{x}$ is
trivial in the convenient $Z$-basis. Further we circumvent the twofold energy
eigenvalue degeneracy brought on by the single encoded qubit by taking into
account the two possible quantum numbers of a single $\mathcal{Z}_{i}$ for an
arbitrarily chosen $i$, i.e. $\mathcal{Z}_{1}$ as depicted in Fig. 6(a). The
resolution of these symmetries reduces the size of the Hilbert space
significantly. The full Hilbert space of $N=L_{x}L_{y}$ qubits,
dim$(\tilde{H}_{B-S})=2^{L_{x}L_{y}}$, shrinks to
dim$(\tilde{H}_{B-S}|\\{S^{Z}_{i}\\},\mathcal{Z}_{i=1})$
$=2^{L_{x}L_{y}-L_{x}}$.

To resolve the symmetries generated by $S^{X}_{i}$, $1\leq i<L_{y}$, which are
not diagonal in the $Z$-basis, we proceed by modifying the Hamiltonian (46) by
adding these operators with large prefactors $\mu_{i}$. The new Hamiltonian
reads

$\displaystyle H=H_{\text{B-S}}+\sum^{L_{y}-1}_{i=1}\mu_{i}S^{X}_{i}$ (49)

with
$\mu_{i}=10\left(1,\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{6},\sqrt{7},\ldots\right)$
for $i=1,\ldots,L_{y}-1$. The $\mu_{i}$ values are chosen to be irrationally
related to one another in order to guarantee that energy levels from each
sector labeled by eigenvalues of the $S^{X}_{i}$ are separated by sizeable
energy gaps. To avoid accidental degeneracies we introduce a disordered set of
bond couplings $J^{x}_{ij},J^{z}_{ij}\in[0.9,1.1]\,\forall\,i,j$ with $1\leq
i<L_{x}-1$ and $1\leq j<L_{y}-1$. We present results for two different systems
of sizes $(L_{x},L_{y})=(7,3)$ and $(4,4)$, the former resulting in eight
energy-separated sectors with each sector containing $512$ states while the
latter consists of four energy sectors with each sector harboring $4,096$
states [see Fig. 6.(b)].

To emphasize the nonintegrable nature of the model, we analyzed the level
statistics for the two different system sizes mentioned above. To ensure
convergence, the histograms shown in Fig. 6(c) contain data from a large set
of independent numerical simulations with different realizations of the random
couplings defined above. We used $1,000$ independent realizations for the
$(4,4)$-system and $200$ independent realizations for the $(7,3)$-system. We
observe that the histograms are in very good agreement with the Wigner-Dyson
distributions. The average values for the level spacing ratio $r$ are found to
be ${\bar{r}=0.5265}$ and $\bar{r}=0.5289$ for systems sizes
${(L_{x},L_{y})=(4,4)}$ and $(7,3)$, respectively. This is quite close to the
exact value of $\bar{r}_{\rm GOE}\approx 0.536$ belonging to the Gaussian
orthogonal ensemble (GOE), as expected for generic real matrices and markedly
different from the corresponding value $\bar{r}_{\rm Poisson}\approx 0.386$
for the Poisson distribution Atas  _et al._ (2013).

The global error detecting code contained within Bacon-Shor is essentially the
same as that in our main example of the surface code discussed in Sec. II,
which we discuss further in Sec. III.5.2 below. The relevant global subgroup
of the stabilizer group $S$ is generated by the products

$\displaystyle S_{X}=\prod_{j=1}^{L_{y}-1}S_{j}^{X}\,,$ $\displaystyle
S_{Z}=\prod_{i=1}^{L_{x}-1}S_{i}^{Z}\,.$ (50)

As in the surface code example, these operators have support only along the
perimeter of the lattice, and the gauge group of the global code is generated
by all local Pauli operators that commute with $S_{X}$ and $S_{Z}$. However,
in this case the global stabilizer generators cannot be written as products of
local, mutually commuting terms, unlike the case of the surface code discussed
in Sec. II. This can be viewed as a consequence of an anomaly of the symmetry
generated by $S^{X}_{j}$ and $S^{Z}_{i}$, albeit one slightly different from
the anomaly discussed in Sec. III.3 that guarantees the existence of a
nontrivial logical algebra. This anomaly guarantees that one cannot write a
Hamiltonian of the form in Eq. (39) with locally generated $\Sigma_{k}$ unless
additional degrees of freedom are added.

To see how the anomaly precludes the application of the prethermalization
arguments, it is informative to attempt to write down a set of local,
commuting U(1) generators $\Sigma_{k}$ like those appearing in Eq. (39). To do
this, it is natural to rewrite

$\displaystyle\begin{split}S^{X}_{j}&\propto\exp\bigg{(}-i\frac{\pi}{2}\sum^{L_{x}}_{i=1}X_{i,j}X_{i,j+1}\bigg{)}\\\
S^{Z}_{i}&\propto\exp\bigg{(}-i\frac{\pi}{2}\sum^{L_{y}}_{j=1}Z_{i,j}Z_{i+1,j}\bigg{)},\end{split}$
(51)

where the proportionality omits a global phase. It is then natural to identify
local U(1) generators

$\displaystyle\begin{split}\Sigma^{X}_{j}&=\frac{1}{2}\sum^{L_{x}}_{i=1}(\mathbbm{1}-X_{i,j}X_{i,j+1})\\\
\Sigma^{Z}_{i}&=\frac{1}{2}\sum^{L_{y}}_{j=1}(\mathbbm{1}-Z_{i,j}Z_{i+1,j}).\end{split}$
(52)

However, these U(1) generators manifestly fail to commute with one another, as
expected from the above discussion of the anomaly. In fact, they fail to
commute with the Hamiltonian (46), so there is no basis to apply the
prethermalization arguments of Refs. Else  _et al._ (2017, 2020). One can,
however, define a pair of genuine, albeit nonlocal, U(1) symmetry generators,
namely

$\displaystyle\begin{split}\Sigma^{X}&=\frac{1}{2}\sum^{L_{y}-1}_{j=1}(\mathbbm{1}-S^{X}_{j})\\\
\Sigma^{Z}&=\frac{1}{2}\sum^{L_{x}-1}_{i=1}(\mathbbm{1}-S^{Z}_{i}).\end{split}$
(53)

These generators are mutually commuting, but they are nonlocal. This also
precludes the application of prethermalization arguments, since they require
local Hamiltonians as input Abanin  _et al._ (2017); Else  _et al._ (2017,
2020).

####  III.5.2 Surface code and $\mathbb{Z}_{2}$ gauge theory

The surface code is defined on a square lattice with rough top and bottom
boundaries and smooth left and right boundaries, with one qubit placed on each
link $\ell$ [see Fig. 3(a)]. Its stabilizer generators are

$\displaystyle A_{v}=\prod_{\ell\ni v}X_{\ell}\,,$ $\displaystyle
B_{p}=\prod_{\ell\in p}Z_{\ell}\,$ (54)

which correspond to weight-four operators in the bulk and weight-three
operators on the boundary. The product of all vertex and plaquette terms
generate a global $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ stabilizer subgroup
symmetry which is suitable for the U(1)$\times$U(1) construction outlined in
Sec. III.4. It is also possible to consider, e.g., the vertex terms as an
unbroken gauge symmetry and use only the plaquette terms to generate a single
U(1) symmetry.

Figure 7:  “H”-shaped surface code lattice with five qubits. (a) The gauge
generators consist of a single $Z_{\ell}$ ($X_{\ell}$)-operator on the bulk
qubit together with pairs of $Z_{\ell}Z_{\ell^{\prime}}$-operators on
consecutive edges along the top and bottom boundaries (magenta shaded) and
pairs of $X_{\ell}X_{\ell^{\prime}}$-operators on consecutive edges along the
left and right boundaries (orange shaded). (b) The global code with
stabilizers $ZZZZ$ ($XXXX$) (blue shaded).

The global $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ stabilizer symmetry serves as
an error detecting subsystem code for an odd number of $Z$ errors along the
top or bottom boundaries, and similarly for an odd number of $X$ errors along
the left or right boundaries. This code corresponds to the generators
considered in section II.4 in the disentangled picture. This is very similar
to the subsystem boundary error detecting code derived from Bacon-Shor above.
However, it contains extra qubits in the bulk that allow the global symmetries
to be generated by products of local commuting terms, which is essential for
the application of the U(1) construction. The symmetric local Hamiltonian
terms can be viewed as gauge generators of this subsystem error detecting
code. The simplest generating set corresponds to single $X_{\ell},Z_{\ell}$
operators on qubits in the bulk together with pairs
$Z_{\ell}Z_{\ell^{\prime}}$ on consecutive edges along the top or bottom
boundary and similarly $X_{\ell}X_{\ell^{\prime}}$ on consecutive edges along
the left or right boundary. A subset of these terms appears in the model
discussed in Sec. II.5. The bare logical operators of the global code can be
chosen as $\mathcal{X}=\prod X_{\ell}$ for links $\ell$ in the top rough
boundary, and $\mathcal{Z}=\prod Z_{\ell}$ for links $\ell$ in the left smooth
boundary. There are also dressed logical operators of weight two given by a
pair of $X$ ($Z$) operators on the left and right (top and bottom) boundaries.
We remark that these operators are spatially nonlocal, and remain so under
multiplication with gauge generators. This global stabilizer subsystem code
is, in some sense, a generalization of the simplest error detecting code on
four qubits, which has stabilizer generators $XXXX,ZZZZ$. In particular, on a
small surface code lattice with five qubits on edges forming an “H” (see Fig.
7), the global code reduces to the four-qubit error detecting code for the
boundary qubits, together with a single trivial bulk qubit whose $X$ and $Z$
operators are gauge generators.

####  III.5.3 Color code

Figure 8:  The color code on the honeycomb lattice. (a) A triangular patch of
the color code on the honeycomb lattice. Plaquettes appear in three colors,
i.e. blue, green, and red. The logical operators $\mathcal{Z}$ and
$\mathcal{X}$ are given by product of $Z_{\ell}$’s and $X_{\ell}$’s along a
boundary path ${\mathcal{P}_{x}}$ and ${\mathcal{P}_{y}}$, respectively. (b)
The generators of the stabilizer group $S_{p}^{Z}=\prod_{v\in p}Z_{v}$ and
$S_{p}^{X}=\prod_{v\in p}X_{v}$ consisting each of six (four) $Z(X)$-operators
acting on the qubits of each hexagon in the bulk (on the boundary) independent
from its color.

The stabilizer group for the color code is generated by

$\displaystyle S_{p}^{X}=\prod_{v\in p}X_{v}\,,$ $\displaystyle
S_{p}^{Z}=\prod_{v\in p}Z_{v}\,,$ (55)

with qubits on vertices of a trivalent lattice with three-colorable plaquettes
taken to be red, green, and blue. The code has four independent global
relations generated by the product of $S_{p}^{X}$ on all red and green
plaquettes and all green and blue plaquettes, and similarly for $S_{p}^{Z}$.
To introduce a logical qubit, we consider a triangular patch of the honeycomb
lattice with one boundary of each color, see Fig. 8(a). The logical $X$
operator is given by the product of $X$ operators on physical qubits along a
single boundary, i.e. $\mathcal{X}=\prod_{v\in\mathcal{P}_{y}}X_{v}$.
Similarly, for the $Z$ logical operator we have
$\mathcal{Z}=\prod_{v\in\mathcal{P}_{x}}Z_{v}$ [see Fig. 8(a)]. Under these
boundary conditions, the global relations become a $\mathbb{Z}_{2}^{\times 4}$
global symmetry group along the boundary. This code is suitable for
application of the prethermalization construction of Sec. III.4 by utilizing
this global $\mathbb{Z}_{2}^{\times 4}$ symmetry to define a global U(1)×4
symmetry. The relevant error detecting boundary subsystem code has gauge
generators $X_{v},Z_{v},$ on all bulk qubits $v$, two body
$X_{v}X_{v^{\prime}},Z_{v}Z_{v^{\prime}}$ gauge generators along all
boundaries away from the corners, and three body $X$ and $Z$ terms at each
corner, see Fig. 8(a). The stabilizers are given by the global
$\mathbb{Z}_{2}^{\times 4}$ boundary symmetry. There are weight-two dressed
logical operators generated by a pair of Pauli $X$ ($Z$) operators on a corner
and the opposite boundary. These dressed logicals are spatially nonlocal, and
cannot be made local under multiplication with gauge generators. The smallest
instance of the global subsystem error detecting code is defined on seven
qubits, with a single trivial bulk qubit that has its $X$ and $Z$ operators
included as gauge generators, together with six boundary qubits that have
gauge generators

$\displaystyle\begin{split}&XXIIIX,\quad IXXXII,\quad IIIXXX,\\\ &ZZIIIZ,\quad
IZZZII,\quad IIIZZZ\end{split}$ (56)

and stabilizer generators

$\displaystyle XIXXIX,\ XXIXXI,\ ZIZZIZ,\ ZZIZZI.$ (57)

##  IV Conclusion

In this work, we have shown that the formalism of noiseless subsystems and
stabilizer subsystem codes can be used to define interesting classes of
dynamics that preserve quantum information encoded in the initial state of a
quantum many-body system. This formalism is sufficiently general to encompass
a variety of quantum dynamics, including unitary and open-system dynamics. In
the context of Hamiltonian systems, which are the focus of the examples
considered in Secs. II.5 and III.5.1, stabilizer subsystem codes can be used
to define families of strongly nonintegrable Hamiltonians with encoded logical
qubits. The dynamical quantum memory in the systems we consider is symmetry-
protected in the sense that the lifetime of the encoded quantum information is
infinite when the stabilizer symmetry of the subsystem code is strictly
enforced. In the presence of perturbations that break the stabilizer symmetry,
we argued that existing results on prethermalization can be leveraged to
provide a parametrically long lifetime of the encoded quantum state, provided
that an appropriate global error-detecting code within the parent subsystem
code can be identified.

One intriguing direction for future work concerns the applicability of the ETH
for Hamiltonians like the ones considered in Secs. II.5 and III.5.1. The
numerical results we have obtained for these Hamiltonians strongly suggest
that these models obey the ETH once the stabilizer symmetry group and the
state of the logical qubit are resolved. However, the ETH is a statement about
matrix elements of local operators, so a study of the statistics of such
matrix elements in eigenstates of these models must be performed in order to
determine whether it holds in these models. In the surface code example
presented in Sec. II.5, it is reasonable to expect that, e.g., single-site
Pauli operators in the bulk of the system will obey the ETH, as these
operators commute with the stabilizer symmetry generators $\tilde{S}_{X}$ and
$\tilde{S}_{Z}$. However, applying such operators on the boundary of the
system can change the stabilizer symmetry sector and flip one or both of the
logical operators. Since the model in Sec. II.5 has pervasive spectral
degeneracies when the stabilizer symmetry is not resolved, it is possible that
these boundary operators do not obey the ETH. The possibility that ETH may
hold in the bulk but not on the boundaries of some systems is an enticing one
that warrants further study.

Another avenue for further exploration is to consider the stability of other
classes of dynamics based on subsystem codes. For example, one can consider
Floquet or even random unitary circuits generated by gauge operators of a
subsystem code and inquire whether the lifetime of the encoded quantum
information under these circuits can be parametrically enhanced in the
presence of symmetry-violating perturbations. One promising direction in these
cases is to consider dynamical-decoupling-like protocols Viola  _et al._
(1999); Tran  _et al._ (2021) using the stabilizer generators.

It would also be interesting to consider the mechanisms discussed in this work
in the setting of open or monitored quantum systems. For example, it would be
worthwhile to study how perturbations affect the dynamics in the setting of
hybrid unitary-projective circuits—for example, measuring stabilizer
generators at a sufficient rate can suppress the propagation of errors,
similar to what was observed for topological codes in Refs. Sang and Hsieh
(2021); Lavasani  _et al._ (2021, 2020). An additional question worth
investigating is whether prethermalization could be invoked to protect the
quantum memory when a system with a Hamiltonian of the form (39) is coupled to
a bath via a small term that does not respect the U(1)×K symmetry.

Another related direction to explore is subsystem codes with dynamically
generated logical qubits, which were recently defined in the context of
Kitaev’s honeycomb model Hastings and Haah (2021); Gidney  _et al._ (2021). In
particular, one can consider a wider family of topological subsystem codes
that may generate analogous dynamical memories. In all of the dynamical
settings mentioned above, it would be interesting to understand the influence
that different choices of qualitatively distinct subsystem codes have on the
dynamics. This includes comparing the behaviors of topological and
nontopological subsystem codes, or even different topological codes with
qualitatively distinct properties, such as self-correction Dennis  _et al._
(2001); Bacon (2006); Bravyi and Haah (2013), single-shot error-correction
Bombin (2015); Brown  _et al._ (2016b), or fractal symmetries Haah (2011); Kim
and Haah (2016); Vijay  _et al._ (2016); Williamson (2016); Prem  _et al._
(2017); Devakul and Williamson (2021).

Finally, we comment on the experimental realizability on present-day quantum
hardware of the physics studied in this work. Recently, two groups have
reported the implementation of repeated quantum error correction in a
distance-three surface code Krinner  _et al._ (2021); Zhao  _et al._ (2021).
To achieve this, both works perform measurements of the surface code
stabilizers $\tilde{A}_{v}$ and $\tilde{B}_{p}$ using one ancilla qubit per
vertex $v$ and plaquette $p$. This experimental setup is sufficient to
implement quantum simulation of the surface-code-inspired Hamiltonian (22).
For example, to realize the unitary gate $e^{i\theta\tilde{B}_{p}}$, one can
use a sequence of CNOT gates with the ancilla qubit $p$ as the target,
followed by a rotation $e^{i\theta Z_{p}}$ of the ancilla and the inverse of
the preceding CNOT sequence (see, e.g., Sec. 4.7.3 of Ref. Nielsen and Chuang
(2010)). In fact, the necessary sequence of CNOTs is the same one needed to
measure the stabilizer $\tilde{B}_{p}$. A similar circuit (up to single qubit
rotations) can be used to realize the gate $e^{i\theta\tilde{A}_{v}}$.
Combined with readily available single-qubit rotations around the $X$ and $Z$
axes, these ingredients are all that is needed to perform quantum simulation
of Eq. (22). One could then prepare a logical state of the surface code using
either the methods of Refs. Krinner  _et al._ (2021); Zhao  _et al._ (2021) or
those of Ref. Satzinger  _et al._ (2021), and then evolve this state under the
Hamiltonian (22), or an appropriate Floquet version thereof, and measure the
expectation values $\braket{\tilde{\mathcal{X}}}$ and
$\braket{\tilde{\mathcal{Z}}}$ as functions of time. The evolution could be
performed with and without the perturbations (29) to deduce their effect on
the logical qubit lifetime. Such a study would provide a nontrivial
application of this surface code architecture even before the large-scale
implementation of error correction on such devices.

###### Acknowledgements.

T.I. acknowledges the hospitality of the Aspen Center for Physics, which is
supported by National Science Foundation grant PHY-1607611. D.W. acknowledges
support from the Simons Foundation and related discussions with Arpit Dua,
Sarang Gopalakrishnan, Michael Gullans, David Huse, Matteo Ippoliti, and
Vedika Khemani.

##  Appendix A Encoding multiple qubits

There exists an analog of the setup detailed in Secs. II.1–II.3 that encodes
more than one logical qubit. To do this, one can define analogs of Eqs. (2),
(3), and (4) for a square lattice with alternating smooth and rough boundaries
(see Fig. 9). Then, one can pass to the disentangled picture in the manner
described in Sec. II.4. Next, one can define operators analogous to the
logical operators (7). For each smooth boundary, one can define operators
$\tilde{\mathcal{Z}}_{i}$ for $i=1,\dots,N_{\rm smooth}$, where $N_{\rm
smooth}$ is the number of smooth boundaries. Likewise, for each rough
boundary, one can define operators $\tilde{\mathcal{X}}_{j}$ for
$i=1,\dots,N_{\rm rough}$, where $N_{\rm rough}$ is the number of rough
boundaries. Assuming for simplicity that $N_{\rm smooth}=N_{\rm rough}=N$, the
topological symmetry generators (20) can then be written as
$\tilde{S}_{X}=\prod^{N}_{j=1}\tilde{\mathcal{X}}_{j}$ and
$\tilde{S}_{Z}=\prod^{N}_{i=1}\tilde{\mathcal{Z}}_{i}$. Upon restricting to
specific eigenspaces of the conserved quantities $\tilde{S}_{X}$ and
$\tilde{S}_{Z}$, this setup encodes $N-1$ logical qubits. An example of a
commuting basis of conjugate logical operators for these qubits is

$\displaystyle\begin{split}\tilde{\mathcal{Z}}_{1},&\indent\tilde{\mathcal{X}}_{1}\\\
\tilde{\mathcal{Z}}_{1}\tilde{\mathcal{Z}}_{2},&\indent\tilde{\mathcal{X}}_{2}\\\
&\vdots\\\
\prod^{N-1}_{i=1}\tilde{\mathcal{Z}}_{i},&\indent\tilde{\mathcal{X}}_{N-1}.\end{split}$
(58)

One can verify by explicit calculation that the operators on each line above
anticommute with one another, while the operators in different lines commute
with one another. Note that, in order for this modified surface code to have
macroscopic distance, we must fix $N$ to be constant while taking the
thermodynamic limit on the lattice. This is because we need the length of each
boundary segment to be macroscopic in order to avoid finite-size effects like
those discussed around Eq. (34).

It is interesting to note that all logical qubits in this setup are protected
by the same $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry. We therefore expect
that the prethermalization analysis of Refs. Else  _et al._ (2017, 2020) would
still hold upon promoting the $Z_{2}\times\mathbb{Z}_{2}$ symmetry to a
U(1)$\times$U(1) symmetry and adding the appropriate fields.

Figure 9:  Surface-code lattice encoding multiple qubits. An example of a
square lattice with alternating rough and smooth boundaries is shown. As in
the main text, qubits live on the links of the lattice. There are four rough
boundary segments and four smooth boundary segments, corresponding to $N=4$.
Operators $\tilde{\mathcal{Z}}_{i}$ and $\tilde{\mathcal{X}}_{i}$ analogous to
Eqs. (7) are indicated by magenta and blue shaded regions, respectively. Each
$\tilde{\mathcal{Z}}_{i}$ or $\tilde{\mathcal{X}}_{i}$ operator consists of
the product of $Z$ or $X$ operators within the shaded region, respectively.

##  Appendix B Additional examples for dimensions $D\leq 3$

Harnessing the framework of stabilizer subsystem codes with local generators
leads to many further examples, some of which we briefly discuss below,
organized by spatial dimension, i.e. one-dimensional (1D), two-dimensional
(2D), and three-dimensional (3D) subsystem codes.

### 1D

The Kitaev wire Kitaev (2001) is a Majorana fermion subsystem code with a pair
of Majorana modes per site $\gamma^{A}_{j},\gamma^{B}_{j}$. The local
stabilizer generators are $i\gamma_{j}^{B}\gamma_{j+1}^{A}$, which couple
neighboring sites. The logical operators are
$\prod_{j}i\gamma_{j}^{A}\gamma_{j}^{B}$ and $\gamma_{1}^{A}$. This local code
is capable of correcting local errors that obey global fermion parity
symmetry. The global subsystem code with stabilizer
$\prod_{j}i\gamma_{j}^{B}\gamma_{j+1}^{A}$ is capable of detecting a local
error affecting the boundary that obeys global fermion parity symmetry, which
is utilized in the U(1) prethermalization construction of Ref. Else  _et al._
(2017).

The above example is related by a Jordan-Wigner transformation to the 1D
repetition code, or Ising model with stabilizer generators $Z_{j}Z_{j+1}$. The
logical operators are $Z_{1}$ and $\prod X_{j}$ (see Fig.10). This local code
is capable of correcting local bit-flip (i.e., $X$) errors. The global code
with stabilizer $\prod Z_{j}Z_{j+1}=Z_{1}Z_{L}$ is capable of detecting a
local bit flip error affecting the boundary spins. Dynamics respecting the
global code symmetry is generated by gauge operators $Z_{j}Z_{j+1}$ and
$X_{i}$ on all sites excluding the boundary spins. This global code was
similarly harnessed for the U(1) prethermalization arguments of Ref. Else  _et
al._ (2017).

In both of the 1D examples above, one of the logical operators corresponds to
a global symmetry, implying that the quantum nature of the memory can only be
probed by nonsymmetric initial states and operators. This poses a particular
challenge in fermionic systems, where global fermion parity symmetry cannot be
broken. This obstruction could be circumvented by coupling the system to an
auxiliary fermionic system in such a way that fermion-parity-violating
processes within the chain could be implemented while preserving the global
fermion parity.

### 2D

In addition to the topological stabilizer code examples in above subsections,
there are topological stabilizer subsystem codes with nontrivial gauge degrees
of freedom that can evolve under the dynamics generated by local gauge
generators that commute with the topological 1-form stabilizer symmetry.
Examples include the subsystem surface code Bravyi  _et al._ (2013), which
obeys a $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ 1-form stabilizer symmetry with
the braiding of the toric code anyons; the gauge color code Bombin  _et al._
(2009); Kargarian  _et al._ (2010); Bombin (2010), which obeys a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ 1-form stabilizer symmetry with the
braiding of the three-Fermion anyon theory; and further related models Suchara
_et al._ (2011). These examples demonstrate that directly enforcing a 1-form
symmetry with a nondegenerate braiding can protect a quantum memory directly
in 2D, despite hosting nontrivial dynamics.

Figure 10:  The repetition code. The one-dimensional repetition code on a
chain of length $L$. The green oval indicates the stabilizer generator
$Z_{j}Z_{j+1}$ while the logical operators are $Z_{1}$ (orange circle) and
$\mathcal{X}=\prod_{j=1}^{L}X_{j}$ (magenta shaded). The global code has
stabilizer $\prod_{j=1}^{L}Z_{j}Z_{j+1}=Z_{1}Z_{L}$ (blue shaded).

### 3D

There are natural generalizations of the surface code and color code to three
dimensions Dennis  _et al._ (2001); Bombin and Martin-Delgado (2007). The
stabilizer group of the surface code consists of a $\mathbb{Z}_{2}$ 1-form
symmetry and a $\mathbb{Z}_{2}$ 2-form symmetry that mutually capture the
braiding of the topological $\mathbb{Z}_{2}$ loop and pointlike excitations,
respectively. Since the 1-form symmetry persists to finite temperatures
Castelnovo and Chamon (2007, 2008), Ref. Else  _et al._ (2017) noted that only
the global symmetry coming from the product of all star terms (which measures
the global parity of pointlike excitations) need be enforced to preserve a
quantum memory at sufficiently low temperatures. The color code is local
unitary equivalent to copies of surface code Kubica  _et al._ (2015) and so
exhibits similar code properties. In addition to the global symmetry
stabilizer subgroups acting nontrivially on boundaries with topological
boundary conditions, in these 3D codes it is also possible to introduce
linelike topological defects along which the global symmetry may be supported
Else  _et al._ (2017); Else and Nayak (2017).

There are subsystem versions of the surface code Kubica and Vasmer (2021) and
color code Bombín (2015); Kubica and Beverland (2015) in 3D which support
nontrivial gauge degrees of freedom. The stabilizers in these codes are 1-form
symmetries, and give rise to global constraints that lead to boundary
symmetries on open boundary conditions suitable for the application of the
prethermalization construction of Sec. III.4. Interestingly, these codes also
display single-shot error-correction Bombin (2015), meaning that the
reconstruction of stabilizer values from a measurement of the gauge generators
is fault-tolerant to a sufficiently low rate of measurement error, even
without repeated rounds of measurement. This may have practical utility for
implementation of the prethermalization construction, specifically for the
fault tolerant measurement of stabilizers at the end of an evolution to check
that the symmetry has not been broken.

A further possibility are fracton topological codes Haah (2011); Vijay  _et
al._ (2016); Williamson (2016); Dua  _et al._ (2019) which have unconventional
symmetries that lead to a number of global relations that can scale in between
a constant and subextensively with linear system size, depending on boundary
conditions. With open boundary conditions these would-be relations instead
generate global stabilizer symmetries, and associated error detecting codes,
that are suitable for the application of the prethermalization construction.
In addition, there are subsystem fracton codes obtained by fractalizing two
dimensional subsystem codes Devakul and Williamson (2021), which support
nontrivial gauge degrees of freedom that evolve even under perfect symmetry
respecting local dynamics, while maintaining a number of logical qubits that
can grow subextensively.

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