††thanks: A professor emeritus

# Interplay between lattice gauge theory and subsystem codes

Yoshihito Kuno  Graduate School of Engineering Science, Akita University,
Akita 010-8502, Japan  Ikuo Ichinose  Department of Applied Physics, Nagoya
Institute of Technology, Nagoya, 466-8555, Japan

###### Abstract

It is now widely recognized that the toric code is a pure gauge-theory model
governed by a projective Hamiltonian with topological orders. In this work, we
extend the interplay between quantum information system and gauge-theory model
from the viewpoint of subsystem code, which is suitable for gauge systems
including matter fields. As an example, we show that $Z_{2}$ lattice gauge-
Higgs model in (2+1)-dimensions with specific open boundary conditions is
nothing but a kind of subsystem code. In the system, Gauss-law constraints are
stabilizers, and order parameters identifying Higgs and confinement phases
exist and they are nothing but logical operators in subsystem codes residing
on the boundaries. Mixed anomaly of them dictates the existence of boundary
zero modes, which is a direct consequence of symmetry-protected topological
order in Higgs and confinement phases. After identifying phase diagram,
subsystem codes are embedded in the Higgs and confinement phases. As our main
findings, we give an explicit description of the code (encoded qubit) in the
Higgs and confinement phases, which clarifies duality between Higgs and
confinement phases. The degenerate structure of subsystem code in the Higgs
and confinement phases remains even in very high-energy levels, which is
analogous to notion of strong-zero modes observed in some interesting
condensed-matter systems. Numerical methods are used to corroborate
analytically-obtained results and the obtained spectrum structure supports the
analytical description of various subsystem codes in the gauge theory phases.

##  I Introduction

Lattice gauge theory (LGT) [1] was invented to describes physical phenomena of
elementary particle physics and succeeded in explaining quark-confinement
phenomenon in strong interactions [2]. The application of the LGT is rich from
condensed matter [3, 4] to quantum information [5, 6, 7]. In particular, some
of quantum phases emergent in LGT are closely related with quantum memory [8,
5, 9, 10]. Their fault tolerance is explained by the notion of topological
order [11, 12] in condensed matter physics. Description of local-gauge
symmetry in the LGT is also related to the notion of stabilizer in quantum
information [13]. Gauss law of the LGT giving a strong constraint on the
Hilbert space can be regarded as stabilizer condition in quantum information
theory [7]. It is also known that some states constrained by Gauss law in LGT
models acquire ability of quantum-error correction [8, 5]. This is done by
restoring a distorted state back to the original one by using the Gauss-law
constraint. (In this sense, deep understanding of LGT has the potential to
lead us into some interdisciplinary discoveries.)

Recently, the lattice gauge-Higgs model [14] has been revisited in some
studies [16, 15] from a viewpoint of condensed matter physics. It was
suggested that Higgs phase can be regarded as a symmetry protected topological
phase (SPT phase) [17, 18], and in (2+1) dimensions ((2+1)-D) with cylinder
boundary conditions, the Higgs and confinement regimes are distinguishable by
observing operators on boundaries [15]. This observation is connected to
concept of one-form symmetry inherent in gauge theory [20], which clarifies
the origin of ’t Hooft loop as well as Wilson loop. In the aspect of quantum
information science, Ref. [21] gave us a hint to notice that lattice gauge-
Higgs models with open boundaries can be a subsystem code [22], that is,
degenerate eigenstates of the lattice gauge-Higgs model behave as encoded
qubits. This suggested an important fact that not only in the ground state but
also almost all higher-energy states in the model correspond to qubits in the
subsystem code, and they can be stable in time evolution with long periods.

It is now widely recognized that the toric code is a pure gauge-theory model
governed by a projective Hamiltonian with topological orders. In this paper,
we shall extend the interplay between quantum information system and gauge-
theory model, in which the gauge degrees of freedom couple with matter fields,
and therefore, Hamiltonian is not projective, i.e., all terms in the
Hamiltonian do not commute with each other. There, the notion of subsystem
code plays an important role. (See Fig. 1.) As a specific example of the above
proposal, we take a further step towards detailed understanding of (2+1)-D
lattice $Z_{2}$ gauge-Higgs models by following the previous works [15, 21].
To this end, we propose an extended version of the model with additional
degrees of freedom of magnetic charge (flux), and also employ certain specific
boundary conditions. Although by a gauge fixing, the model reduces to the
ordinary one, we can introduce suitable order parameters to clarify physical
properties of the gauge-Higgs model.

In the previous study [21], a novel subsystem code was constructed. This work
raises interesting unresolved questions: (I) How the degenerate states of the
subsystem code are understood in gauge-theory and SPT phase point of view?
(II) How ‘wavefunction’ of the states for the subsystem code in each gauge-
theory phase looks like? It is expected that the ‘wavefunction’ clarifies
physical meaning of SPT phase in gauge theory. (III) It is desired to obtain
‘wavefunctions’ corresponding to multiple qubits, and high-energy states in
the subsystem code, as they are expected to be closely related to strong zero
modes discovered in Refs. [23, 24].

In the following, we shall answer the above questions. To this end, we first
clarify the relationship between the subsystem codes and the gauge-Higgs
model. In particular, we explain that the logical operators of the subsystem
codes are nothing but order parameters of the gauge-Higgs model, which are
supplied by the employed open boundary conditions and play a central role in
the present study. The spontaneous symmetry breaking (SSB) observed by the
order parameters [SSB of charge and magnetic flux conservation symmetries]
clarifies phase diagram of the gauge-Higgs model. In this work, the explicit
analytical descriptions of the degenerate encoded qubits in Higgs and
confinement regimes are given. These descriptions elucidate an exact duality
between Higgs and confinement phases, and clarify physical properties of Higgs
and confinement phases from the viewpoint of SSB. Duality in the present
formalism reveals that the confinement phase is an SPT phase as the Higgs
phase. Beyond single-encoded qubit, we shall give an explicit analytical
description of general multiply-encoded qubits in Higgs and confinement
regimes to show the utility of the correspondence between order parameters of
the gauge theory and logical operator in quantum code. This is one of main
findings in this work, corroborated by numerical studies.

We further focus on not only the ground-state multiplet but also on excited
states in the system. As predicted by the recent work [21], the degenerate
spectrum structure of encoded qubit of subsystem code would be maintained in
excited states. We shall verify this prediction exploiting the knowledge of
the corresponding gauge theory. In particular, we find that the degenerate
encoded qubits in Higgs and confinement regimes tend to survive in the entire
spectrum, which is reminiscent of the strong zero mode discussed recently [23,
24]. To corroborate the analytical description for single and doubly-encoded
qubits, we numerically investigate the degeneracy of encoded qubits as
subsystem code in the entire energy spectrum of Higgs and confinement regimes.
These results indicate that the subsystem codes with gauge-theory structure
are generally robust up to high-energy regimes.

Before going into details of the model study, in order to capture the entire
picture of the present proposal, we show schematically the relationship
between the gauge-Higgs model and subsystem codes in Fig. 1. We expect that
this scheme is applicable for wide-range of lattice gauge models including
matter fields, and is an extension of (toric code)-(gauge model)
correspondence. The toric code is a pure $Z_{2}$ lattice-gauge model in
$(2+1)$ D, and its Hamiltonian is composed of stabilizers, all of which are
commutative with each other. This kind of system is sometimes called
projective Hamiltonian system. Figure 1(a) displays this fact. Contrary to the
toric code, the gauge-Higgs model is not a projective system, i.e., all terms
in the Hamiltonian do not commute with each other. [See $H_{\rm GHM}$ in Eq.
(1).] For this case, a suitable notion in quantum information science is
subsystem codes, which are composed of gauge operators, logical operators, and
stabilizers [22]. Please see Fig. 1(b). Hamiltonian of the gauge-Higgs model
consists of the gauge operators instead of stabilizers, whereas Gauss-law
constraints play the role of stabilizers and the logical operators are order
parameters. Details of them will be explained in the main text.

Figure 1: Schematic of (gauge model)-(information code) relationship proposed
in the present work: (a) Toric code and (b) Subsystem code. The toric code is
a projective pure gauge system, Hamiltonian of which is composed of
stabilizers commutative with each other. On the contrary, for ordinary gauge
systems including matter degrees of freedom, all terms of Hamiltonian do not
commute with each other. For such a system, suitable quantum information
notion is supplied by subsystem codes with gauge operators, logical operators,
and stabilizers.

The rest of this paper is organized as follows. In Sec. II, we explain the
target LGT model. We introduce Hamiltonian of (2+1)-D lattice $Z_{2}$ extended
gauge-Higgs model and explain the specific boundary conditions employed in
this work. We discuss symmetries and properties of the model, and then, derive
an effective Hamiltonian by disentangling local-gauge degrees of freedom,
which is an extension of the toric code with perturbations. Section III
exhibits dramatic findings in this work. Firstly, we explain notion of
subsystem code (if the reader wants to know the essence of subsystem code in
detail, see Refs. [22, 21]), and then give detailed discussion on various
properties concerning to gauge-theory ground state of the model. In
particular, physical motivation for introducing the open boundary and
resultant SSB of the global charge symmetry are discussed. Secondly, we give
the analytical description of the encoded qubit of subsystem code in Higgs and
confinement regimes, and further discuss its extension to multiply-encoded
qubit. Thirdly, in addition to the ground-state properties, we discuss
degeneracy structure of excited states. In Sec. IV, we show the results of the
numerical calculations, which corroborate analytical arguments given in Sec.
III. Detailed discussions on the numerical results and phase transition
criticality are given. Section IV is devoted to discussion and conclusion.

Even though we use terminologies of quantum information, we hope that this
article is readable for condensed matter, quantum information, and high-energy
particle physicists.

##  II Model

In this work, we shall study one of the LGT models, the gauge-Higgs model in
two spatial dimensions. Accessible review of the LGT is Ref. [1], and in
particular for the gauge-Higgs mode, review is available in Ref. [4]. As we
explained in introduction, this gauge model is a good example for exemplifying
the subsystem-code formalism of the LGT.

We shall study a (2+1)-D $Z_{2}$ extended gauge-Higgs model, Hamiltonian of
which is given by

$\displaystyle H_{\rm GHM}$ $\displaystyle=$
$\displaystyle\sum_{v}h_{v}X_{v}+\sum_{(p,p^{\prime})}J^{x}_{p,p^{\prime}}X_{p}\sigma^{x}_{p,p^{\prime}}X_{p^{\prime}}$
(1) $\displaystyle+$
$\displaystyle\sum_{p}h_{p}Z_{p}+\sum_{(v,v^{\prime})}J^{z}_{v,v^{\prime}}Z_{v}\sigma^{z}_{v,v^{\prime}}Z_{v^{\prime}}.$

Here, we impose the following double gauge-invariant conditions, Gauss laws,
for the physical subspace,

$\displaystyle
G_{v}|\psi\rangle=|\psi\rangle,\>\>B_{p}|\psi\rangle=|\psi\rangle.$ (2)

where

$\displaystyle G_{v}=X_{v}\prod_{\ell_{v}\in v}\sigma^{x}_{\ell_{v}}\equiv
X_{v}\tilde{G}_{v},$ $\displaystyle B_{p}=Z_{p}\prod_{\ell_{p}\in
p}\sigma^{z}_{\ell_{p}}\equiv Z_{p}\tilde{B}_{p},$ (3)

and $\ell_{v}\in v$ stands for links emanating from vertex (site) $v$, and
$\ell_{p}\in p$ for links composing plaquette (box) $p$. The $Z_{2}$-electric
matter is defined on each vertex $v$, $(X_{v},Z_{v})$, and its magnetic dual,
$(X_{p},Z_{p})$, on each dual vertex $p$ (i.e., plaquette of the original
lattice), where $X_{v}(Z_{v})$ stands for the Pauli matrix
$\sigma^{x}_{v}(\sigma^{z}_{v})$, and similarly for $X_{p}(Z_{p})$ (See Fig.
2). On the other hand, the $Z_{2}$ gauge field is defined on links and denoted
by $(\sigma^{x}_{\ell},\sigma^{z}_{\ell})$, $\sigma^{z}_{v,v^{\prime}}$ denote
a gauge variable on link connecting neighboring vertices $v$ and $v^{\prime}$,
and $\sigma^{x}_{p,p^{\prime}}$ denote a gauge variable on link connecting
neighboring dual vertices $p$ and $p^{\prime}$. The gauge field
$\sigma^{x}_{\ell}$ is related to the electric field $\hat{E}_{\ell}$ as
$\sigma^{x}_{\ell}=e^{i\pi\hat{E}_{\ell}}$, and $\sigma^{z}_{\ell}$ to the
vector potential $\hat{A}_{\ell}$ as
$\sigma^{z}_{\ell}=e^{i\pi\hat{A}_{\ell}}$, and eigenvalues are $\\{0,1\\}$
for both the operators. The electric field and vector potential are conjugate
with each other, and operation of $\sigma^{z}_{\ell}$ produces electric flux
on link $\ell$.

There are two differences between the system given by $H_{\rm GHM}$ [Eq. (1)]
and the ordinary $Z_{2}$ gauge-Higgs LGT:

  1. (I)

Dual matter field couples with the gauge field and the coupling term,
$X_{p}\sigma^{x}_{p,p^{\prime}}X_{p^{\prime}}$ is added to the Hamiltonian
besides the ordinary $Z_{2}$ electric matter-gauge coupling. This $Z_{2}$
degrees of freedom residing on each plaquette $p$, $(X_{p},Z_{p})$,
corresponds to ‘particle’ carrying magnetic flux (magnetic charge), and its
hopping induces fluctuation of the gauge field and, therefore, confinement of
electric charges [9]. The ordinary electric term as well as the magnetic-
plaquette term in the Hamiltonian are replaced with dynamical variables
$(X_{p},Z_{p})$, even though similar dynamics to the ordinary one emerges from
$(X_{p},Z_{p})$.

  2. (II)

By the presence of the magnetic-charge degrees of freedom, an additional local
gauge symmetry emerges such as,

$X_{p}\rightarrow X_{p}V_{p},\;\;\sigma^{x}_{p,p^{\prime}}\rightarrow
V_{p}\sigma^{x}_{p,p^{\prime}}V_{p^{\prime}},\;\;V_{p},V_{p^{\prime}}\in
Z_{2},$ (4)

and we impose additional Gauss-law constraint on the physical state, Eqs. (2)
and (3). One may wonder that the system (1) reduces to the ordinary gauge-
Higgs model by ‘integrating out’ the magnetic charge degrees of freedom via,
e.g., employing unitary gauge of the second local gauge symmetry in Eq. (4).
In fact as we show shortly, disentangling of electric and magnetic particles
generates the ordinary Hamiltonian of the gauge-Higgs model in unitary gauge.
However with specific open boundary conditions, which we shall employ, there
emerges a small but important difference between the gauge-Higgs model (1) and
the ordinary one.

The reason why we employ the above Hamiltonian (1), which exhibits the
electric-magnetic duality manifestly, as a starting model will become clear
later on. The model of Eq. (1) is different from the conventional toric code
[5], in such a way that, the model of Eq. (1) is not composed of stabilizers,
and therefore, is not solvable, while the toric code is projective and
solvable since all terms in the Hamiltonian are commutative with each other,
i.e., stabilizers (See Fig. 1).

Here, we also note that the above Hamiltonian was recently proposed for
describing subsystem quantum code producing fault-tolerant qubit [21],
although the explicit form of $H_{\rm GHM}$ in Eq. (1) was not shown. There,
each term of the Hamiltonian (1) is categorized as “gauge” operator in the
subsystem-code literature [22]. [It is often remarked that the terminology
“gauge” is rather confusing as it has nothing to do with gauge symmetry in
LGT. See later discussion.]

Figure 2: Schematic figure of lattice with rough and smooth boundaries. The
green dashed box represents the top rough boundary and the orange dashed box
represents the left smooth boundary, on which logical operators are defined.

We introduce a square lattice shown in Fig. 2, with specific boundaries named
rough and smooth boundaries. For both boundaries, the form of $G_{v}$ and
$B_{p}$ are explicitly shown, both of which are composed of three links with a
vertex and a plaquette, respectively. Some of the electric-matter-gauge and
magnetic-matter-gauge terms in the Hamiltonian, $H_{\rm GHM}$ in Eq. (1), are
missing on the boundaries. That is, the electric (magnetic) hopping on the
rough (smooth) boundary does not exist. We note that the exact electric-
magnetic duality holds in the system.

The model with the above open boundary conditions has four important
symmetries, generators of which are given by,

$\displaystyle P$ $\displaystyle=$
$\displaystyle\prod_{v}X_{v},\>\>S_{Z}=\prod_{p}Z_{p},$ (5) $\displaystyle
W_{\gamma}$ $\displaystyle=$
$\displaystyle\prod_{\ell\in\gamma}\sigma^{z}_{\ell},\>\>H_{\gamma}=\prod_{\ell\in\gamma}\sigma^{x}_{\ell}.$
(6)

$P$ is the parity of the total electric charge, corresponding to the global
spin flip on each vertex, and similarly, $S_{Z}$ is the parity of the total
magnetic flux per plaquette. The boundary hopping terms are forbidden by the
$P$ and $S_{Z}$ symmetries. $P$ and $S_{Z}$ are global topological symmetries,
whereas $W_{\gamma}$ and $H_{\gamma}$ are one-form symmetry, which has been
extensively studied recently [19, 20]. Although the $J^{x}_{p,p^{\prime}}$ and
$J^{z}_{v,v^{\prime}}$ terms in the Hamiltonian $H_{\rm GHM}$ [Eq. (1)]
explicitly break the one-form symmetries, it was shown that the higher-form
symmetry is generally robust and give non-trivial effect on dynamics of the
system. In the $(2+1)$-D system, $H_{\gamma}$ can be regarded as ’t Hooft loop
(string) dual to Wilson loop (string) $W_{\gamma}$. In particular, the path
$\gamma$ in the one-form symmetry $W_{\gamma}$ ($H_{\gamma}$) is arbitrary. It
is easily verified that by using the Gauss laws in Eq. (3), $P$ ($S_{Z}$) is
expressed as a ’t Hooft (Wilson) ‘loop’ residing on the top and bottom rough
(left and right smooth) boundaries (see Fig. 2), which we call boundary-one-
form operators hereafter. This fact plays an important role in later
discussion on phase diagram of the gauge-Higgs model.

In the context of the present model $H_{\rm GHM}$, the decomposed Hilbert
space of $H_{\rm GHM}$ can be regarded as subsystem code [22], and its basic
discussion is reviewed in [21]. The Hilbert space of the system $H_{\rm GHM}$
are characterized by gauge, stabilizer, and bare logical operators. The
Hilbert space is operated by logical encoded qubit and gauge qubit [21]. The
structure of subsystem code may give an efficient error correction route as
discussed by proposing simple examples [25]. The stabilizer of the subsystem
code is given by the projectors $\\{G_{v},B_{p},P,S_{z}\\}$, “gauge” operators
are those commute with the projectors, corresponding to each terms of $H_{\rm
GHM}$, and logical operators $\\{\tilde{X},\tilde{Z}\\}$ are shown later in
Sec.III. It should be noted that all eigenstates of $H_{\rm GHM}$ is labeled
by $G_{v}=B_{p}=+1$ due to the condition of Eq. (2) while $P,S_{z}=\pm 1$ for
eigenstates. This situation is different from some conventional stabilizer
model such as toric code [5, 7] and cluster model [26, 27, 28], etc., where
eigenvalue of every stabilizers takes $\pm 1$ depending on eigenstates.

Here, to capture the physical properties of $H_{\rm GHM}$ more clearly, we
consider the following unitary transformations (sometimes called circuit
unitary transformation):

$\displaystyle U_{v}$ $\displaystyle=$ $\displaystyle
H\biggl{(}\prod_{v}\prod_{\ell\in v}(CZ)_{v,\ell}\biggr{)}H,$ (7)
$\displaystyle U_{p}$ $\displaystyle=$ $\displaystyle
H\biggl{(}\prod_{p}\prod_{\ell\in p}(CZ)_{p,\ell}\biggr{)}H,$ (8)

where $H$ is the Hadamard transformation on each link and $(CZ)_{i,j}$ is a
controlled $Z$-gate for the site $i$ and link $j$. Applying the above
transformation to $H_{\rm GHM}$, we obtain the following effective
disentangled model

$\displaystyle U_{v}U_{p}H_{\rm GHM}(U_{v}U_{p})^{\dagger}\equiv H_{\rm TC},$
$\displaystyle H_{\rm TC}$ $\displaystyle=$
$\displaystyle\sum_{v}h_{v}\tilde{G}_{v}+\sum_{\ell\notin\mbox{rough}}J^{z}_{\ell}\sigma^{z}_{\ell}$
(9) $\displaystyle+$
$\displaystyle\sum_{p}h_{p}\tilde{B}_{p}+\sum_{\ell\notin\mbox{smooth}}J^{x}_{\ell}\sigma^{x}_{\ell}.$

The above circuit disentanglement corresponds to gauge fixing with unitary
gauge, in which degrees of freedom corresponding to local gauge transformation
are eliminated [29]. We remark that the specific form of the $J^{z}$ and
$J^{x}$-terms originates from the starting Hamiltonian $H_{\rm GHM}$ in Eq.
(1).

The model $H_{\rm TC}$ is nothing but an extended system of toric code [5]
including local perturbations ($J^{z}$ and $J^{x}$-terms) and with rough and
smooth boundaries. $H_{\rm TC}$ with periodic boundary conditions and also
$H_{\rm TC}$ having the full $J^{z}$ and $J^{x}$-terms under the standard open
boundary conditions have been studied in previous works from the viewpoint of
quantum information.

##  III Gauge-theory phase diagram and subsystem quantum code

The model of $H_{\rm GHM}$ and $H_{\rm TC}$ has three distinguishable gauge-
theory phases, i.e., Higgs, confinement and deconfinement (toric
code/topological) phases under the specific type of present open boundary
conditions. In particular, recent work exhibited that the Higgs phase is an
SPT phase protected by the symmetries $P$ and $W_{\gamma}$, and can be
distinguished from the confinement phase by employing open boundaries [15],
contrary to the previous common belief [14]. Furthermore, even in the presence
of explicit breaking of the one-form symmetry $W_{\gamma}$, the SPT phase
survives as long as the gap is open. This is an important contribution for
understanding the gauge model complementing the seminar and influential study
of gauge-Higgs model [14].

In this work, we go a step further and obtain a concrete form of boundary
states in Higgs and confinement phases, which play an essential role in
understanding the gauge-theory phase diagram and properties of the subsystem
code proposed in [21].

Degeneracy of quantum states in the system with suitable open boundary
conditions for subsystem code is governed by the encoded qubit [22]. Logical
operators for the code embedded in the Hamiltonian $H_{\rm TC}$ are given by
(see Fig. 2),

$\displaystyle\tilde{X}=\prod_{\ell\in\mbox{top
rough}}\sigma^{x}_{\ell},\>\>\>\tilde{Z}=\prod_{\ell\in\mbox{left
smooth}}\sigma^{z}_{\ell}.$ (10)

It is easily verified that they anticommute with each other
$\\{\tilde{X},\tilde{Z}\\}=0$ since single link is shared by each rough and
smooth boundaries, and also both of them commute with the Hamiltonians,
$[H_{\rm TC},\tilde{X}]=[H_{\rm TC},\tilde{Z}]=[H_{\rm GHM},\tilde{X}]=[H_{\rm
GHM},\tilde{Z}]=0$. Note that the operators $\tilde{Z}$ and $\tilde{X}$ are
invariant under the disentangling transformation $U_{v}U_{p}$. Thus, these
operators act as logical operators in the both models $H_{\rm GHM}$ and
$H_{\rm TC}$, as pointed out in [21].

We can introduce counterpart of $\tilde{X}$ and $\tilde{Z}$ residing on the
bottom-rough and right-smooth boundaries, respectively, defined such as,

$\displaystyle\tilde{X}^{\ast}=\prod_{\ell\in\mbox{bottom
rough}}\sigma^{x}_{\ell},\>\>\>\tilde{Z}^{\ast}=\prod_{\ell\in\mbox{right
smooth}}\sigma^{z}_{\ell}.$ (11)

As we explained in the above, Gauss laws in Eq. (3) relate the global symmetry
operators $P$ and $S_{Z}$ with the above boundary operators as follows,

$\displaystyle S_{X}\equiv
P=\tilde{X}\tilde{X}^{\ast},\;\;S_{Z}=\tilde{Z}\tilde{Z}^{\ast},$ (12)

where we have introduced the notation $S_{X}$, and the above relation
eloquently tells us emergence of the symmetry fractionalization [17].

As $S_{X}$ and $S_{Z}$ commute with each other and also with the Hamiltonian,
the full space of state can be divided into subspaces with eigenvalue of
$(S_{X},S_{Z})=(\pm 1,\pm 1)$. However, $S_{X}$ and/or $S_{Z}$ can be SSB, and
in that case, $\tilde{X}(\tilde{Z})$ and $\tilde{X}^{\ast}(\tilde{Z}^{\ast})$
are independent operators. On the other hand as $\tilde{X}(\tilde{X}^{\ast})$
and $\tilde{Z}(\tilde{Z}^{\ast})$ anticommute with each other, gapless modes
emerge in the vicinity of boundary. It is also possible that one of the
symmetries generated by $\tilde{X}$ or $\tilde{Z}$ is spontaneously broken by
the condensation of the other as they also play a role of order parameter.
Pattern of the SSB clarifies physical picture of gauge-theory phase, which has
been masked by the local-gauge symmetry so far.

In the following, we discuss the gauge-theory phases by showing explicit form
of boundary states for each phase of the gauge-Higgs model, which has not been
given so far, and then we shall verify the analytical observation by using
numerical methods, exact diagonalization (ED), where we employ useful
numerical package [30, 31].

###  III.1 Gauge-theory phases: Spontaneous symmetry breaking and confinement

In this subsection, we shall take a look at phase diagram of the present gauge
model, $H_{\rm TC}$ and $H_{\rm GHM}$. As we mentioned in the above, the phase
diagram of $H_{\rm TC}$ with periodic boundary conditions was investigated and
it was clarified that there are three ‘phases’, deconfined-topological, Higgs,
and confinement phases. The seminal work in Ref. [14] showed that the Higgs
and confinement phases are connected without thermodynamic singularities, that
is, they are adiabatically connected. However, in the model in a semi-infinite
cylinder geometry, the Higgs and confinement phases are distinct in the
behavior of the boundary excitation [15]. In what follows, we mainly focus on
physical properties in the vicinity of the boundaries, as they are essentially
related to the SPT and also subsystem code. In addition, we shall show that
the investigation of states near boundaries clarifies the SSB of the charge
symmetry, mechanism of quark confinement, and duality between them. This is a
reason why we employ specific boundary conditions.

In what follows, we mostly focus on the model $H_{\rm TC}$, and carry out
numerical calculation for $H_{\rm TC}$ in Sec. IV since the model is simple
and, nevertheless, keeps physical essence of $H_{\rm GHM}$.

We first consider the Higgs regime for large negative $J^{z}_{\ell}$ such as
$-J^{z}_{\ell}\gg|J^{x}_{\ell}|,|h_{v}|,|h_{p}|$. In this regime, link
variables in the bulk are ordered as $\langle\sigma^{z}_{\ell}\rangle\sim 1$,
and this phase is regarded sometimes trivial. However, this long-range order
(LRO) in the bulk indicates $\langle\tilde{Z}\rangle\neq 0$, as the three-link
plaquette terms on the rough boundary shown in Fig. 2 generate ferromagnetic
interactions between $\sigma^{z}$’s on dangling links. The fractionalized-
symmetry operator $\tilde{X}$ operates on edge states nontrivially in that
state. In fact, the existence of the LRO in the 1D rough boundary is
numerically verified as we show later on (See Fig. 7). Therefore, states on
the rough boundaries are approximately given as
$|\uparrow\uparrow\cdots\uparrow\rangle$ or
$|\downarrow\downarrow\cdots\downarrow\rangle$. The operator
$\tilde{X}(\tilde{X}^{\ast})$ interchanges these states,
$|\uparrow\uparrow\cdots\uparrow\rangle\leftrightarrow|\downarrow\downarrow\cdots\downarrow\rangle$.
On the top and bottom boundaries, the above two states can be taken
independently, and therefore, the ground state is expected to be four-fold
degenerate. Here, we should remark that the gauge operators
$\sigma^{z}_{\ell}$ creates (flips in $Z_{2}$ case) electric flux on link
$\ell$, and therefore, its non-vanishing expectation value on the boundary
means strong fluctuations of electric field and the SSB of the charge
symmetry.

Figure 3: (a) Single-encoded qubit system. The system has two rough and smooth
boundaries. (b) Doubly-encoded qubits system. The system has three rough and
smooth boundaries. (c) $N$-encoded qubits system. The system has $N+1$ rough
and smooth boundaries.

Before going to show detailed numerics, we briefly verify this expectation for
$J^{z}_{\ell}=-4$, where the system for numerical calculation is shown in Fig.
3(a). The results are shown in Figs. 4(a) and (b), and we observe four-fold
degeneracy in ground states. Also even for a small but finite value of
$J^{x}_{\ell}=-0.5$, the four-fold degeneracy maintains. As effects of the
one-form symmetry $W_{\gamma}$ is robust, it protects the bulk LRO as long as
the state is in the Higgs phase.

The above result gives very important observation, i.e., the charge symmetry
$P=S_{X}$ is expected to exhibit SSB in the thermodynamic limit. In ordinary
systems of gauge theory with periodic boundary conditions, condensation of
charged operators is masked by the requirement of the local-gauge invariance.
Order parameters, which are expected to signal the SSB of the charge symmetry,
are non-local and charge-neutral objects such as
$\langle\phi_{x}\exp(i\int_{\Gamma}A_{\mu}dx_{\mu})\phi^{\dagger}_{y}\rangle$
with a charged matter field $\phi$, vector potential $A_{\mu}$ and line
$\Gamma$ connecting $\phi_{x}$ and $\phi^{\dagger}_{y}$ [32]. In the present
case, however, the symmetry operator $S_{X}$ is nothing but the charge
operator $P$ by Gauss law, and the SSB of $S_{X}$, $\langle
S_{X}\rangle=\langle P\rangle=0$, strictly indicates the SSB of the global
charge.

In order to avoid confusion and misunderstanding, here we emphasize that the
specific boundary conditions do not induce the SSB but triggers the boundary
SSB, which distinguishes Higgs and confinement phases. A suitable order
parameter and its conjugate, $\tilde{Z}$ and $\tilde{X}$, are supplied by the
boundary conditions, and they clearly describe the boundary SSB [33]. Monte-
Carlo simulation of the system $H_{\rm GHM}$ with the rough-smooth boundary
conditions is interesting and desired to see if bulk or 1D critical behavior
emerges in the specific heat, etc. This is a future problem to be worked
without difficulties.

Similar argument can be applied to the confinement of charged particles
because of the electric-magnetic duality. For
$-J^{x}_{\ell}\gg|J^{z}_{\ell}|,|h_{v}|,|h_{p}|$, the LRO of
$\sigma^{x}_{\ell}$ emerges with finite expectation value of order parameter
$\langle\tilde{X}\rangle\neq 0$ on the smooth boundaries indicating the SSB of
the $\tilde{Z}$-symmetry, $\langle Z\rangle=0$, in the thermodynamic limit.
This means the emergence of condensation of magnetic charge (magnetic flux),
which causes strong fluctuations of the gauge field $\sigma^{z}_{\ell}$, and
charge confinement. Numerical calculations in Figs. 4(c) and 4(d) clearly show
the four-fold degeneracy of the ground state in confinement regime, hence
supporting the above consideration for both Higgs and confinement phases.
Existence of the one-form symmetry $W_{\gamma}(H_{\gamma})$ makes this state
robust against the $J^{x}_{\ell}(J^{z}_{\ell})$-terms in the Hamiltonian. The
above qualitative picture of the Higgs mechanism and quark confinement is not
new, but the present model explicitly shows these mechanisms without any
obscurity.

In the following subsections, we shall consider the model in a finite system
from the viewpoint of subsystem quantum code and logical qubit. We shall
derive the analytical description of the encoded qubit states, and introduce
notion of strong zero modes, which are generated by $(\tilde{X},\tilde{Z})$
and $(\tilde{X}^{\ast},\tilde{Z}^{\ast})$, and play an important role in the
present work.

Figure 4: (a) and (b): Low-lying energy spectra for the deep Higgs phase. (a)
$J^{x}_{\ell}=0$ ($W_{\gamma}$-symmetry exact), (b) $J^{x}_{\ell}=-0.5$
(explicit breaking of $W_{\gamma}$-symmetry). For both cases, $h_{v}=-1$,
$h_{p}=-1$, $J^{z}_{\ell}=-4$. (c) and (d): Low-lying energy spectra for the
deep confinement phase. (c) $J^{z}_{\ell}=0$ ($H_{\gamma}$-symmetry exact),
(d) $J^{z}_{\ell}=-0.5$ (explicit breaking of $H_{\gamma}$-symmetry). For both
cases, $h_{v}=-1$, $h_{p}=-1$, $J^{x}_{\ell}=-4$. The setup of the system for
calculation is shown in Fig. 3(a), in which the total number of link is 18,
and $(L_{x},L_{y})=(2,4)$.

###  III.2 Higgs phase

In this subsection, we consider the deep Higgs regime such as
$J^{z}_{\ell}\to-\mbox{large}$ , $h_{v}=-1$, $h_{p}=-1$, and
$|J^{x}_{\ell}|\ll|J^{z}_{\ell}|$. The bulk LRO $\sigma^{z}_{\ell}\sim 1$
induces an effective Hamiltonian of the boundary spins similar to the
transverse field Ising model (TFIM) as mentioned in [15].

For both top and bottom rough boundaries, the $P$ symmetry is spontaneously
broken in the Higgs phase. Then on the edges, cat states emerge in a finite
system, and therefore, the boundary states are given by

$\displaystyle|\pm\rangle_{\rm
T(B)}\equiv\frac{1}{\sqrt{2}}[|\Uparrow\rangle_{\rm
T(B)}\pm|\Downarrow\rangle_{\rm T(B)}],$ (13)

where ${\rm T(B)}$ stands for the top (bottom) rough boundary,
$|\Uparrow\rangle_{\rm T(B)}$ and $|\Downarrow\rangle_{\rm T(B)}$ are all spin
up and down states on dangling links on each rough boundary,
$|\Uparrow\rangle=|\uparrow\uparrow\cdots\uparrow\rangle$ and
$|\Downarrow\rangle=|\downarrow\downarrow\cdots\downarrow\rangle$. The
boundary states $|\pm\rangle_{\rm T(B)}$ is $Z_{2}$ cat states, the presence
of which is justified by SSB of $P$ symmetry on the top and bottom boundaries

In the Higgs phase, each edge of the system induces two-fold degeneracy, thus,
we expect the ground state of the Higgs phase is four-fold degenerate. The
four degenerate states for the whole system are given as follows in the deep
Higgs limit,

$\displaystyle|G1\rangle\equiv|+\rangle_{\rm T}|-\rangle_{\rm
B}\otimes|\mbox{bulk}\rangle,$ $\displaystyle|G2\rangle\equiv|-\rangle_{\rm
T}|+\rangle_{\rm B}\otimes|\mbox{ bulk}\rangle,$
$\displaystyle|G3\rangle\equiv|+\rangle_{\rm T}|+\rangle_{\rm B}\otimes|\mbox{
bulk}\rangle,$ $\displaystyle|G4\rangle\equiv|-\rangle_{\rm T}|-\rangle_{\rm
B}\otimes|\mbox{ bulk}\rangle,$ (14)

where $|{\rm bulk}\rangle$ represents the bulk states not including links on
rough boundaries. The above explicit form of the four-fold degenerate ground
state for $H_{{\rm TC}}$ is useful for investigation by numerical methods
given later on. Actually, we identify two pairs in a topologically-symmetric
sector. That is, for the sector $(P,S_{z})=(-1,+1)$, the pair of the ground
state is $\\{|G1\rangle,|G2\rangle\\}$ since
$P|G1(2)\rangle=(-1)|G1(2)\rangle$ and $S_{Z}|G1(2)\rangle=(+1)|G1(2)\rangle$.
This pair is an encoded qubit satisfying
$\tilde{Z}|G1(2)\rangle=|G2(1)\rangle$, $\tilde{X}|G1\rangle=(+1)|G1\rangle$
and $\tilde{X}|G2\rangle=(-1)|G2\rangle$. Note that making the boundary cat
state is an essential ingredient to get a qubit controlled by the logical
operators $(\tilde{X},\tilde{Z})$.

The other pair $\\{|G3\rangle,|G4\rangle\\}$ in the four-fold degenerate
ground state is another encoded qubit. This pair belongs to the sector
$(P,S_{z})=(+1,+1)$ since $P|G3(4)\rangle=(+1)|G3(4)\rangle$ and
$S_{Z}|G3(4)\rangle=(+1)|G3(4)\rangle$. They also satisfy
$\tilde{Z}|G3(4)\rangle=|G4(3)\rangle$ and
$\tilde{X}|G3\rangle=(+1)|G3\rangle$ and $\tilde{X}|G4\rangle=(-1)|G4\rangle$.

Due to the robustness of the one-form symmetry, the bulk LRO survives even in
the presence of its explicit breaking terms in the Hamiltonian. As a result,
the above-encoded qubit with cat-state properties is preserved intact. We
shall demonstrate this observation by numerical methods later on.

###  III.3 Confinement phase

We next consider the deep confinement limit, $J^{x}_{\ell}\to-\mbox{large}$,
$h_{v}=-1$, $h_{p}=-1$, and $|J^{z}_{\ell}|\ll|J^{x}_{\ell}|$. The bulk LRO
$\sigma^{x}_{\ell}\sim 1$ induces an effective Hamiltonian for the smooth-
boundary spins similar to TFIM.

For both left and right smooth boundaries, the $S_{Z}$ symmetry is
spontaneously broken in the confinement phase. Then on the smooth boundaries,
the boundary states are given by,

$\displaystyle|\pm_{x}\rangle_{\rm
L(R)}\equiv\frac{1}{\sqrt{2}}[|\Rightarrow\rangle_{\rm
L(R)}\pm|\Leftarrow\rangle_{\rm L(R)}],$ (15)

where the subscript ${\rm L(R)}$ stands for the left (right) smooth boundary,
$|\Rightarrow\rangle_{\rm L(R)}$ and $|\Leftarrow\rangle_{\rm L(R)}$ are all
positive $x$ and negative $x$ spin states on each smooth boundary,
respectively, i.e.,
$|\Rightarrow\rangle=|\rightarrow\rightarrow\cdots\rightarrow\rangle$ and
$|\Leftarrow\rangle=|\leftarrow\leftarrow\cdots\leftarrow\rangle$. The
boundary states $|\pm_{x}\rangle_{\rm L(R)}$ are $Z_{2}$-cat states composed
of $|\Rightarrow\rangle$ and $|\Leftarrow\rangle$, the emergence of which is
induced by SSB of $S_{Z}$ symmetry at the left and right boundaries.

We expect that the ground-state structure is analogue to that of the Higgs
phase. That is, each edge of the system induces two-fold degeneracy and
therefore, the ground state is four-fold degenerate. Furthermore, the four-
fold degenerate states for the whole system are given by

$\displaystyle|C1\rangle\equiv|+_{x}\rangle_{\rm L}|-_{x}\rangle_{\rm
R}\otimes|\mbox{bulk}\rangle,$
$\displaystyle|C2\rangle\equiv|-_{x}\rangle_{\rm L}|+_{x}\rangle_{\rm
R}\otimes|\mbox{ bulk}\rangle,$
$\displaystyle|C3\rangle\equiv|+_{x}\rangle_{\rm L}|+_{x}\rangle_{\rm
R}\otimes|\mbox{ bulk}\rangle,$
$\displaystyle|C4\rangle\equiv|-_{x}\rangle_{\rm L}|-_{x}\rangle_{\rm
R}\otimes|\mbox{ bulk}\rangle.$ (16)

The above explicit form of the four-fold degenerate ground state for $H_{{\rm
TC}}$ in the confinement limit is useful for later investigation by the
numerical methods. Here, we should mention that the ground states in Eqs. (14)
and (16) exhibit a typical form of the subsystem code considered in Ref. [21].

The four states in Eq. (16) again behave as two pairs of qubit. For the sector
$(P,S_{z})=(+1,-1)$, the relevant pair of the ground state is
$\\{|C1\rangle,|C2\rangle\\}$ since $P|C1(2)\rangle=(+1)|C1(2)\rangle$ and
$S_{Z}|C1(2)\rangle=(+1)|C1(2)\rangle$. These paired states form an encoded
qubit, that is, $\tilde{X}|C1(2)\rangle=|C2(1)\rangle$ and
$\tilde{Z}|C1\rangle=+1|C1\rangle$ and $\tilde{Z}|C2\rangle=-1|C2\rangle$.
Similarly, the other paired states $\\{|C3\rangle,|C4\rangle\\}$ also form an
encoded qubit in the sector $(P,S_{z})=(+1,+1)$. The logical operators are
similarly given by $\tilde{Z}$ and $\tilde{X}$.

Here, we should note that the states in Eq. (16) are exactly dual to those of
the Higgs phase in Eq. (14) under the interchange
$\sigma^{x}_{\ell}\Longleftrightarrow\sigma^{z}_{\ell}$ for all links. However
sometimes, this duality is broken explicitly by the geometry of the lattice if
$L_{x}\neq L_{y}$, where $L_{x(y)}$ is the number of links in the horizontal
(vertical) direction, as the rectangular lattice used in our numerical
studies. (See Fig. 3.)

We remark that the wavefunctions in Eqs. (14) and (16) have a standard form of
the subsystem code such as $|\psi_{L}\rangle|\psi^{\prime}\rangle$ [21], where
$|\psi_{L}\rangle$ is the state of the logical qubit and
$|\psi^{\prime}\rangle$ is that of the gauge qubit. In the Higgs and
confinement regimes, the states $|\psi_{L}\rangle$ can be explicitly obtained
in a compact form, whereas it cannot in the deconfinement regime. This is also
the case for more general cases of multiply-encoded qubits and higher-energy
qubits, which we discuss in the subsequent subsections. Therefore, on
constructing subsystem codes in the present model, the Higgs and confinement
regimes are better than the deconfinement regime. The practical representation
of the wavefunctions obtained in the above is useful for investigating
stability and error corrections of the subsystem codes and also may give an
insight into the design of quantum memories robust to decoherence, which are
future problems, although we will briefly mention the fault tolerance of the
present subsystem codes to thermal noise [34].

###  III.4 Multiply-encoded qubits

By manipulating the shape of the boundary, multiply-encoded qubit can be
constructed [21]. For example, a doubly-encoded qubit subsystem code can be
put on three rough and smooth boundaries as shown in Fig. 3(b). In the Higgs
and confinement phases, we can explicitly describe the state of these qubits
using ideas of the gauge theory. In this subsection, we mostly focus on the
deep Higgs regime $J^{z}_{\ell}\to-$large, since duality between Higgs and
confinement phases exists even in this multiply-encoded qubits system.

Doubly-encoded qubits: As we mentioned in the above, we employ the system
displayed in Fig. 3(b). The ground state of $H_{\rm TC}$ for the Higgs and
confinement phase is eight-fold degenerate. Here, these states are decomposed
into two distinct $(P,S_{Z})$ sectors. Each sector includes four-fold
degenerate eigenstates.

In the deep Higgs phase, we find that one set of four-fold degenerate ground
states are given as

$\displaystyle|Q2_{1}\rangle\equiv|+\rangle_{r1}|+\rangle_{r2}|+\rangle_{r3}\otimes|\mbox{bulk}\rangle,$
$\displaystyle|Q2_{2}\rangle\equiv|+\rangle_{r1}|-\rangle_{r2}|-\rangle_{r3}\otimes|\mbox{
bulk}\rangle,$
$\displaystyle|Q2_{3}\rangle\equiv|-\rangle_{r1}|+\rangle_{r2}|-\rangle_{r3}\otimes|\mbox{
bulk}\rangle,$
$\displaystyle|Q2_{4}\rangle\equiv|-\rangle_{r1}|-\rangle_{r2}|+\rangle_{r3}\otimes|\mbox{
bulk}\rangle,$ (17)

where $r_{\ell}$ ($\ell=1,2,3$) represents three rough boundaries in Fig. 3(b)
and $|{\rm bulk}\rangle$ represents the bulk state without including boundary
links. Then, for these four states, two set of logical operators exist [21],
which are given by

$\displaystyle\tilde{X}_{1}$ $\displaystyle=$ $\displaystyle\prod_{k\in
s1}\sigma^{z}_{k},\>\>\tilde{X}_{2}=\prod_{k\in s2}\sigma^{z}_{k},$
$\displaystyle\tilde{Z}_{1}$ $\displaystyle=$ $\displaystyle\prod_{m\in
r1}\sigma^{x}_{m},\>\>\tilde{Z}_{2}=\tilde{Z}_{1}\prod_{m\in
r2}\sigma^{z}_{m}.$ (18)

These operator act on the above four-fold degenerate ground states suitably,
and the states work as a doubly-encoded qubit.

The other four-fold degenerate ground states also work as a doubly-encoded
qubit, and they are explicitly given as

$\displaystyle|Q2_{5}\rangle\equiv|+\rangle_{r1}|+\rangle_{r2}|-\rangle_{r3}\otimes|\mbox{bulk}\rangle,$
$\displaystyle|Q2_{6}\rangle\equiv|+\rangle_{r1}|-\rangle_{r2}|+\rangle_{r3}\otimes|\mbox{
bulk}\rangle,$
$\displaystyle|Q2_{7}\rangle\equiv|-\rangle_{r1}|+\rangle_{r2}|+\rangle_{r3}\otimes|\mbox{
bulk}\rangle,$
$\displaystyle|Q2_{8}\rangle\equiv|-\rangle_{r1}|-\rangle_{r2}|-\rangle_{r3}\otimes|\mbox{
bulk}\rangle.$ (19)

The operators of Eq. (18) again act on the above states as the logical
operators.

In the deep confinement phase, the states dual for the above degenerate ground
states are also regarded as doubly-encoded qubits, and there the logical
operators are interchanged with each other,
$\tilde{X}_{\ell}\longleftrightarrow\tilde{Z}_{\ell}$ $(\ell=1,2)$.

General $N$-encoded qubits: General $N$-encoded qubits can be designed in the
system with many rough and smooth boundaries [21]. One of lattices for that
system is displayed in Fig. 3(c), where to implement $N$-encoded qubits, $N+1$
rough and smooth boundaries are needed and their locations must be alternate.
As in the previous cases, we can give an explicit analytical description of
the $N$-encoded qubit states with the boundary SSB in the deep Higgs and
confinement phases. For the system $H_{\rm TC}$, we expect that the ground
state is $(2\times 2^{N})$-fold degenerate in the deep Higgs phase. Then, the
first set of $2^{N}$-degenerate ground states of $N$-encoded qubits are given
by

$\displaystyle|z_{1},z_{2},\cdots,z_{N}\rangle\equiv\biggl{[}\bigotimes^{N}_{\ell=1}|{z_{\ell}}\rangle\biggr{]}|z_{N+1}\rangle_{r_{\ell}}\otimes|\mbox{bulk}\rangle,$
(20)

where $z_{\ell}=\pm$, $|z_{\pm}\rangle=|\pm\rangle_{r_{\ell}}$, $r_{\ell}$
relabels the $r_{\ell}$-th rough boundary as shown in Fig. 3(c) and

$\displaystyle z_{N+1}=(-1)^{N+\sum^{N}_{\ell=1}(z_{\ell}+1)/2}.$ (21)

The label $z_{\ell}$ ($\ell=1,2,\cdots,N$) denotes up or down state of the
$\ell$-encoded qubits. The above set of $N$-qubits is embedded in
$(P,S_{Z}=+1)$ sector, where
$P=\biggr{[}\prod^{N}_{1}z_{\ell}\biggr{]}z_{N+1}\equiv P_{N}$.

The second set of $2^{N}$-degenerate ground states of $N$-encoded qubits in
ground-state multiplet are similarly given by Eq. (20) with replacing as

$\displaystyle z_{N+1}\to z_{N+1}=(-1)^{N+1+\sum^{N}_{\ell=1}(z_{\ell}+1)/2}.$
(22)

The $2^{N}$-degenerate ground states are embedded in the sector
$(P,S_{Z})=(-P_{N},+1)$.

For these $N$-encoded qubits, the logical operators are given as [21]

$\displaystyle\tilde{X}_{n}=\prod_{k\in
r_{n}}\sigma^{z}_{k},\>\>\tilde{Z}_{n}=\prod^{n}_{m=1}\tilde{Z}^{s}_{m},$ (23)

where $n$ denotes the number of encoded qubit taking $n=1,2,\cdots,N$, $r_{n}$
is $n$-th rough boundary shown in Fig. 3(c), $n=1,2,\cdots,N$, and

$\displaystyle\tilde{Z}^{s}_{m}=\prod_{q\in s_{m}}\sigma^{z}_{q}.$ (24)

Here $s_{m}$ is $m$-th smooth boundary shown in Fig. 3(c), $m=1,2,\cdots,N$.

As in the previous cases, the dual of the above qubits states are obtained
straightforwardly to get $N$-qubit states in the deep confinement phase. Also,
the dual of the logical operators of Eq. (23) act as logical operators on the
dual states.

###  III.5 Excited states and single-encoded qubits in Higgs and confinement
phases

The previous work [21] suggested that the encoded qubit can be embedded in
arbitrary excited states, as natural properties of the subsystem code [22]. In
this subsection, we study an explicit analytical description of the encoded
qubits in excited states. These properties of excited states are closely
related to the strong zero mode [23, 24], which has been discussed in various
contexts.

As shown in the previous section, the states of the qubit in both deep Higgs
and confinement phase are given via the bulk-boundary factorised form such as

$\displaystyle|Q\rangle=|\mbox{boundary}\rangle\otimes|\mbox{bulk}\rangle,$
(25)

where the bulk is a simple product state and the boundary is a cat state.
Here, we assume that the excitation energy of the bulk $\Delta E_{\rm bulk}$
is much larger than that of the boundary $\Delta E_{\rm b}$, $\Delta E_{\rm
bulk}\gg\Delta E_{\rm b}$, which is satisfied for
$|J^{x(z)}_{\ell}|\gg|h_{v}|,|h_{p}|,|J^{z(x)}_{\ell}|$. Then, low-energy
excited states of the encoded qubit are constructed only by the boundary
excitation in $|\mbox{boundary}\rangle$. For example in the deep Higgs phase,
the boundary states are cat states, and therefore, the excited states are
constructed upon the cat state residing on one of the rough boundaries [36],
which are explicitly given by

$\displaystyle|e^{k}_{\pm}\rangle\equiv\frac{1\pm
P}{\sqrt{2}}|\uparrow\rangle_{j_{0}}\otimes|\\{\uparrow\\}_{k}\\{\downarrow\\}_{\ell_{e}-k-1}\rangle_{\noindent
j_{0}},$ (26)

where $j_{0}$ is a link at one end of a rough boundary, $\ell_{e}$ is the
number of the link of one rough boundary, and $0\leq k\leq\ell_{e}-1$. The
state $|\\{\uparrow\\}_{k}\\{\downarrow\\}_{\ell_{e}-k-1}\rangle_{\noindent
j_{0}}$ is a product state composed of $k$ up-spins and $(\ell_{e}-k-1)$ down-
spins, in which location of the domain wall is arbitrary. The low-energy
excited states without bulk excitation can be obtained by replacing the cat
state $|\pm\rangle$ in states of Eq. (14) with the state $|e^{k}_{\pm}\rangle$
in Eq. (26). As an example, excited encoded qubit states constructed from the
states $|G1\rangle$-$|G4\rangle$ in Eq. (14) are given by

$\displaystyle|G1e^{k}\rangle\equiv|e^{k}_{+}\rangle_{\rm
T}|e^{k}_{-}\rangle_{\rm B}\otimes|\mbox{bulk}\rangle,$
$\displaystyle|G2e^{k}\rangle\equiv|e^{k}_{-}\rangle_{\rm
T}|e^{k}_{+}\rangle_{\rm B}\otimes|\mbox{ bulk}\rangle,$
$\displaystyle|G3e^{k}\rangle\equiv|e^{k}_{+}\rangle_{\rm
T}|e^{k}_{+}\rangle_{\rm B}\otimes|\mbox{ bulk}\rangle,$
$\displaystyle|G4e^{k}\rangle\equiv|e^{k}_{-}\rangle_{\rm
T}|e^{k}_{-}\rangle_{\rm B}\otimes|\mbox{ bulk}\rangle.$ (27)

Here, pairs $(\\{|G1e^{k}\rangle,|G2e^{k}\rangle\\})$ and
$(\\{|G3e^{k}\rangle,|G4e^{k}\rangle\\})$ are single-encoded qubits embedded
into excited states in the deep Higgs phase. By this manipulation, various
types of degenerate eigenstates of encoded qubit can be constructed, and also
by using duality, the low-energy excited states constructing an encoded qubit
are obtained for the deep confinement phase.

Furthermore, general higher-excited states are to be described in the
following way. In the case in which pure bulk excitations emerge in a
plaquette or at a vertex not-touching the boundaries, the excited states can
be written as a product state composed of cat states on the boundaries and
bulk excited states. As a whole, we expect that most of the states having
properties of encoded qubit are to be described in the Higgs and confinement
phases.

###  III.6 Entangled encoded qubit states in $H_{\rm GHM}$

In the previous subsections, we showed the explicit form of the logical
encoded qubit states for the disentangled Hamiltonian, $H_{{\rm TC}}$. In
general, the encoded qubit state denoted by $|Q_{{\rm TC}}\rangle$ can be
transformed into the encoded qubit state in the original (2+1)-D $Z_{2}$
extended gauge-Higgs model, $H_{\rm GHM}$, denoted by $|Q_{\rm GHM}\rangle$ as

$\displaystyle|Q_{\rm GHM}\rangle=(U_{v}U_{p})^{\dagger}|Q_{{\rm
TC}}\rangle=U_{p}U_{v}|Q_{{\rm TC}}\rangle,$ (28)

since undoing the disentangling is manipulated by using Gauss-law constraints,
and states are uniquely determined up to the local-gauge symmetries. Under
this transformation, the boundary state and bulk state in the state $|Q_{{\rm
TC}}\rangle$ become moderately entangled. In particular, we expect that the
Higgs phase regime, the encoded qubit state $|Q_{\rm GHM}\rangle$ in the Higgs
ground state multiplet can be short-range entangled state since the state is
expected to be the SPT (2D cluster state) [16, 15], also due to duality, the
confinement is the same. For example, spin operators in dangling links of the
rough boundaries in $H_{\rm TC}$ correspond to gauge-invariant operators in
$H_{\rm GHM}$ via
$\sigma^{z}_{\ell_{v}}\Longleftrightarrow\sigma^{z}_{\ell_{v}}Z_{v}$. This
means, e.g., $|+\rangle_{\rm
TC}\Longleftrightarrow{1\over\sqrt{2}}(|+,+\rangle_{\rm GHM}+|-,-\rangle_{\rm
GHM})$, where notations are self-evident. In such a way, short-range
entanglement emerges between the gauge field and matter fields.

Figure 5: Signature of doubly-encoded qubit. (a) and (b): Low-lying energy
spectra for the deep Higgs phase. (a) $J^{x}_{\ell}=0$($W_{\gamma}$-symmetry
exact), (b) $J^{x}_{\ell}=-0.5$ (explicit breaking of $W_{\gamma}$-symmetry).
For both cases, $h_{v}=-1$, $h_{p}=-1$, $J^{z}_{\ell}=-4$, $v_{1}=0.5$,
$v_{2}=0$. The first and the second pair energies are splitted $\sim 2v_{1}$.
(c) and (d): Low-lying energy spectra for the deep confinement phase. (c)
$J^{z}_{\ell}=0$ ($H_{\gamma}$-symmetry exact), (d) $J^{z}_{\ell}=-0.5$
(explicit breaking of $H_{\gamma}$-symmetry). For both cases, $h_{v}=-1$,
$h_{p}=-1$, $J^{x}_{\ell}=-4$, $v_{1}=0$, $v_{2}=0.5$. The first pair and the
second pair energies split with a gap $\sim 2v_{2}$.

##  IV Numerical study for small systems

In this section, we investigate the ground state properties of the disentangle
Hamiltonian in detail by numerical methods. In particular, to identify the
degeneracy of the encoded qubits of the subsystem code, we add the potential
$V_{\rm pot}=v_{1}P+v_{2}S_{Z}$ to the Hamiltonian $H_{\rm TC}$, and then, the
system is described by $H_{{\rm TC}}+V_{\rm pot}$.

###  IV.1 Subsystem code of single-encoded qubit in low-energy spectrum

We diagonalize the Hamiltonian $H_{{\rm TC}}+V_{\rm pot}$ by employing Quspin
package [30, 31] and calculate low-energy spectrum up to $20$-$30$ th state
from the ground state. In what follows, we set $h_{v}=h_{p}=-1$ and focus on
both the deep Higgs and confinement phases. The lattice structure is shown in
Fig. 3(a), where each rough and smooth boundaries have three and four vertical
links, respectively, and the total number of links is eighteen, i.e.,
$(L_{x},L_{y})=(2,4)$.

Deep Higgs phase: We first study the deep Higgs regime and set the parameters
as $J^{z}_{\ell}=-4$ and $(v_{1},v_{2})=(0.5,0)$. The obtained spectrum for
the $W_{\gamma}$-symmetric case $J^{x}_{\ell}=0$ is shown in Fig. 5(a). We
observe that the four states in the degenerate ground-state multiplet for
$(v_{1},v_{2})=(0,0)$ split into two pairs, each of which is nothing but
single-encoded qubits in the sector $(P,S_{Z})=(+1,-1)$ and $(+1,+1)$,
respectively. The above result agrees with the analytical study in the
previous section, in particular, the emergent sectors. We observe that the
higher-energy states belong to a degenerate multiplet with more than two even-
number states. This degeneracy of the energy spectrum is stable against the
explicit breaking of the $W$-symmetry by the $J^{x}_{\ell}$-term in the
Hamiltonian, as the results for $J^{x}_{\ell}=-0.5$ in Fig. 5(b) indicate that
the ground state and first excited pairs are single-encoded qubits in the
sector $(P,S_{Z})=(+1,-1)$ and $(+1,+1)$, respectively. Also for finite values
of $J^{x}_{\ell}$, the higher-excited states tend to be two-fold degenerate,
i.e., the splitting into two pairs is enhanced by the $J^{x}_{\ell}$-term.

Figure 6: Signature of doubly-encoded qubits: Low-lying energy spectra for the
deep Higgs phase. (a) $J^{x}_{\ell}=0$ ($W$-symmetry exact), (b)
$J^{x}_{\ell}=-0.5$ (explicit breaking of $W_{\gamma}$-symmetry) For both
cases, $h_{v}=-1$, $h_{p}=-1$, $J^{z}_{\ell}=-4$, $v_{1}=0.5$, $v_{2}=0$. The
first and second four-pair energies are splitted $\sim 2v_{1}$. (c) and (d):
Low-lying energy spectra for the deep confinement phase. (c) $J^{z}_{\ell}=0$
($H_{\gamma}$-symmetry exact), (d) $J^{z}_{\ell}=-0.5$ (explicit breaking of
$H_{\gamma}$-symmetry) For both cases, $h_{v}=-1$, $h_{p}=-1$,
$J^{x}_{\ell}=-4$, $v_{1}=0$, $v_{2}=0.5$. The first and second four-pair
energies split with a gap $\sim 2v_{2}$.

Deep confinement phase: We turn to the deep confinement regime, and put
$J^{x}_{\ell}=-4$ and set $(v_{1},v_{2})=(0,0.5)$ to examine duality between
Higgs and confinement regimes. The spectrum for the $H_{\gamma}$-symmetric
case $J^{x}_{\ell}=0$ is shown in Fig. 5(c). We again observe that the four-
fold degenerate ground-state multiplet for $(v_{1},v_{2})=(0,0)$ splits into
two pairs, each of which is a single-encoded qubit in the sector
$(P,S_{Z})=(-1,+1)$ and $(+1,+1)$, respectively. The above result agrees with
the analytical observation in the previous section. The low-energy degenerate
structure is stable against the explicit breaking of $H_{\gamma}$-symmetry by
finite values of $J^{x}_{\ell}$ as shown in Fig. 5(d). The spectrum is almost
the same as that of the deep Higgs regime connected by duality, with a small
but finite discrepancy coming from $L_{x}\neq L_{y}$.

###  IV.2 Subsystem code of doubly-encoded qubit

We next numerically observe doubly-encoded qubits existing in the system shown
in Fig. 3(b), in which the total number of link is eighteen. We employ a
similar procedure, i.e., diagonalize the Hamiltonian $H_{{\rm TC}}+V_{\rm
pot}$ to calculate the low-energy spectrum.

Deep Higgs phase: We set the parameters in the deep Higgs phase,
$J^{z}_{\ell}=-4$ and also $(v_{1},v_{2})=(0.5,0)$. The spectrum for
$W_{\gamma}$-symmetric case $J^{x}_{\ell}=0$ is shown in Fig. 6(a). We observe
a four-fold degenerate ground state multiplet in the sector
$(P,S_{Z})=(-1,+1)$, which are nothing but two encoded qubits, and the first
excited four-fold degenerate states in the sector $(P,S_{Z})=(+1,+1)$. The two
four-fold degenerate states are predicted by the analytical study in the
previous section.

Similarly to the single-encoded qubit system, this low-energy degenerate
structure is stable against the explicit breakdown of $W_{\gamma}$-symmetry by
finite $J^{x}_{\ell}$ in the Hamiltonian. The result for the case with
$J^{x}_{\ell}=-0.5$ is displayed in Fig. 6(b). We observe that the ground-
state and first-excited four-fold degenerate multiplets are intact for finite
$J^{x}$, and belong to the sector $(P,S_{Z})=(-1,+1)$ and $(+1,+1)$,
respectively. Also, for finite $J^{x}_{\ell}$, the higher-excited states tend
to be four-fold degenerate, i.e., the four-fold degeneracy is enhanced.

Deep confinement phase: We turn to the deep confinement regime, where
$J^{x}_{\ell}=-4$ and $(v_{1},v_{2})=(0,0.5)$. The spectrum for the
$H_{\gamma}$-symmetric case $J^{x}_{\ell}=0$ is shown in Fig. 6(c). We again
observe the four-fold degenerate ground state multiplet in the sector
$(P,S_{Z})=(+1,-1)$ as in the deep Higgs regime. These are two encoded qubit
states. The first-excited four-fold degenerate states belong to the
$(P,S_{Z})=(+1,+1)$ sector. These results are in agreement with the analytical
observation in the previous section. This degeneracy structure at low energies
is intact for finite $J^{x}_{\ell}$ as shown in Fig. 6(d). Again for the
doubly-encoded qubit system, the spectrum is almost the same as that of the
deep Higgs regime by duality.

Figure 7: Correlation functions $C^{x}_{\rm b}$ and $C^{z}_{\rm b}$ on
boundaries for single-encoded qubit system [(a)] and doubly-encoded qubit
system [(b)]. We set $(v_{1},v_{2})=(0.5,0)$ and $(0,0.5)$ for the
calculations, $C^{z}_{\rm b}$ and $C^{x}_{\rm b}$.

###  IV.3 Higgs-confinement phase transition

We numerically observe boundary correlation functions to observe the
$S_{X}$($S_{Z}$)-SSB phase transition. For the single-encoded qubit shown in
Fig. 3(a), the correlation functions are given by $C^{z}_{\rm
b}=\langle\sigma^{z}_{0}\sigma^{z}_{3}\rangle$ and $C^{x}_{\rm
b}=\langle\sigma^{x}_{0}\sigma^{x}_{15}\rangle$. The labels of the link are
shown in Fig. 3(a). We calculate $C^{z}_{\rm b}$ and $C^{x}_{\rm b}$ for the
$\tilde{Z}=+1$ sector of the two-fold degenerate ground state, where
$J^{x}_{\ell}$ and $J^{z}_{\ell}$ are parameterized as $J^{z}_{\ell}=4-\delta
J$ and $J^{x}_{\ell}=\delta J$ with varying $\delta J$.

Figure 7(a) exhibits the behavior of the correlations, where we set
$(v_{1},v_{2})=(0.5,0)$ and $(0,0.5)$. We can observe a phase-transition-like
behavior even in the small system. In the deep Higgs phase (small $\delta J$)
$C^{z}_{\rm b}$ has a large finite value, while in the deep confinement phase
(large $\delta J$) $C^{x}_{\rm b}$ is finite. This numerical result implies
the emergence of the SSB of $P(S_{X})$ or $S_{Z}$ symmetry in the
thermodynamic limit.

Similar behavior is observed in the doubly-encoded qubit system shown in Fig.
3(b). We define the correlation function as $C^{z}_{\rm
b}=\langle\sigma^{z}_{0}\sigma^{z}_{3}\rangle$ and $C^{x}_{\rm
b}=\langle\sigma^{x}_{0}\sigma^{x}_{15}\rangle$, where the labels of the link
are shown in Fig. 3(b), and calculate $C^{z}_{\rm b}$ and $C^{x}_{\rm b}$ for
the $(\tilde{Z_{1}},\tilde{Z_{2}})=(+1,+1)$ sector of the four-fold degenerate
ground state. We plot the result in Fig. 7(b) to confirm the emergence of the
SSB of $P(S_{X})$ or $S_{Z}$ symmetry.

From the all data shown in Fig. 7, we conclude that the SSB phase transition
on the boundary occurs at $J^{z}_{\ell}\sim J^{x}_{x}$. This result indicates
that the criticality of the present system coincides with that of the
effective TFIM emerging on the boundary. However, we must be careful in
concluding the existence of the phase transition because the system size is
very small. More elaborated numerical approaches such as quantum Monte-Carlo
simulation (employed to a similar model [35]) or DMRG are desired as a future
work.

###  IV.4 Autocorrelator

We numerically observed the degenerate energy spectra for both the Higgs and
confinement phases. The degeneracy coming from the nature of encoded qubits
survives to highly-excited eigenstates. This behavior of the system seems to
indicate the emergence of a strong zero mode, which was first proposed in [23,
24].

We shall numerically verify the existence of the strong zero mode by observing
the whole energy spectrum of the present model by following the previous
numerical study [36]. In that work, to examine the strong zero mode,
autocorrelator measuring the coherence of logical operators was investigated
in detail. We employ the same methods and calculate the autocorrelator of the
logical operator in the present model, which is given by [36]

$\displaystyle C^{s}_{\rm au}(t)$ $\displaystyle=$
$\displaystyle\langle\psi_{s}|\tilde{X}(t)\tilde{X}(0)|\psi_{s}\rangle$ (29)
$\displaystyle=$
$\displaystyle\sum_{\ell}|\langle\psi_{s}|\tilde{X}|\psi_{\ell}\rangle|^{2}e^{i(E_{\ell}-E_{s})t},$

where $\tilde{X}(t)=e^{it(H_{{\rm TC}}+V_{\rm pot})}\tilde{X}e^{-it(H_{{\rm
TC}}+V_{\rm pot})}$ (we have set $\hbar=1$), and $|\psi_{s}\rangle$ is $s$-th
eigenstates of $H_{{\rm TC}}+V_{\rm pot}$, and $E_{\ell}$ is $\ell$-th energy
for the ascending order.

We study the single-encoded qubit system as shown in the inset in Fig. 8 and
first focus on the deep Higgs phase, where the logical operator $\tilde{X}$ is
defined by Eq. (10). In order to extract $\tilde{Z}=+1$ encoded state, we add
a very small potential $v_{z}\tilde{Z}$ with $v_{z}=10^{-6}$. The numerically
obtained time evolution is shown in Fig. 8(a), where we pick up three
eigenstates, i.e., the ground state with $(P,S_{Z},\tilde{Z})=(-1,+1,+1)$
labeled as $s=0$, a middle excited state $\tilde{Z}=+1$, $s=4000$ and a
highly-excited state $\tilde{Z}=+1$, $s=8000$.

We observe that $C^{s}_{\rm au}(t)$ keeps perfect coherence for a long period
in all eigenstates. This result indicates that the degeneracy originating from
the nature of encoded qubit is observed in the whole energy spectrum, implying
the existence of the strong zero mode. In addition, in the very late-time
behavior of $C^{s}_{\rm au}(t)$, a tiny decay is observed as shown in the
inset panel in Fig. 8(a). This behavior is similar to the decay of the
autocorrelator in the TFIM investigated in [36]. However, the decay is very
small, thus, we expect that the coherence of the autocorrelater in our model
is almost perfect for very long times. This result strongly supports the
expectation that the degenerate structure of encoded qubit is maintained in
the whole energy spectrum of the model.

We further show the calculations of the autocorrelater in the vicinity of the
confinement-Higgs phase transition in Fig. 8(b), where the parameters are set
as $h_{v}=-1$, $h_{p}=-1$, $J^{z}_{\ell}=-2$, $J^{x}_{\ell}=-1.9$,
$v_{1}=0.5$, $v_{2}=0$. We consider eigenstates with $s=0,4000$ and $8000$ as
in the previous case in Fig 8(a). Even though the system is in the critical
regime, the autocorrelater exhibits the same behavior with that in deep Higgs
regime. Therefore, the numerics indicates that the encoded qubit degeneracy in
the whole energy spectrum persists for the entire gauge-theoretical phase
diagram.

Figure 8: Autocorrelator of the swap logical operator $\tilde{X}$. (a) The
case in the deep Higgs phase without $W_{\gamma}$ symmetry, $h_{v}=-1$,
$h_{p}=-1$, $J^{z}_{\ell}=-4$, $J^{x}_{\ell}=-0.5$, $v_{1}=0.5$, $v_{2}=0$.
The upper figure displays the system of numerical simulation, where the total
number of link is $13$. (b) The case in the vicinity of confinement and Higgs
phase transition, $h_{v}=-1$, $h_{p}=-1$, $J^{z}_{\ell}=-2$,
$J^{x}_{\ell}=-1.9$, $v_{1}=0.5$, $v_{2}=0$. The right inset panels for (a)
and (b) show the detailed behavior for late-time evolution of the
autocorrelator.

##  V DISCUSSION AND CONCLUSIONS

In this work, we propose the subsystem-code formalism of generic LGTs. In
order to exemplify the proposal, we studied the interplay between (2+1)-D
lattice $Z_{2}$ extended gauge-Higgs model and subsystem code beyond the
recent works [15, 21].

We gave the explicit analytical description of the encoded qubit state of this
subsystem code. The state description gives dramatic and useful understanding
of gauge-theory phases, Higgs and confinement phases. The analytically-
obtained states exhibit the boundary SSB, which is robust for the explicit
breaking of the $W_{\gamma}$ or $H_{\gamma}$-symmetry in the Hamiltonian since
these symmetries are one-form symmetry. Our work clearly shows the existence
of genuine order parameters for the SSB of the global charge symmetry as well
as charge confinement.

We remark that the previous study [21] does not give explicit form of the
states of the subsystem code, nor discuss the connection between gauge theory
and SPT state. The accomplishment of this work is to elucidate the strong
relationship between the degeneracy and structure of the ground states in the
gauge-Higgs model with the origin of encoded qubits in subsystem code. We gave
some concrete analytical descriptions for not only single but also multiply-
encoded qubit in both Higgs and confinement regimes. We also constructed
highly-excited eigenstates of the encoded qubits in both Higgs and confinement
regimes. We numerically verified the above observations and investigated the
spectra and state structure of the model rather in detail. Even though system
size is small, we obtained satisfactory results for the degeneracy of encoded
qubits in both Higgs and confinement regimes. All the numerical results are in
good agreement with the analytical observations and also theoretical
consideration on the (2+1)-D lattice $Z_{2}$ extended gauge-Higgs model.

Furthermore, by observing the autocorrelation numerically, we found that it
indicates the existence of the strong zero mode in the Higgs regime as
analytical study predicts. The numerical results strongly support the
existence of the degenerate structure of the subsystem code up to high-energy
regime as predicted by [21], and indicate that the subsystem code is robust
even at finite temperatures.

Finally, the present formalism of lattice gauge-Higgs model clarifies that the
confinement phase is an SPT phase as the Higgs phase. SPT string order
parameter is given by $\langle
X_{p}\sigma^{x}\sigma^{x}\cdots\sigma^{x}X_{p^{\prime}}\rangle\neq 0$.

As future interesting topic, we point out the followings:

  1. 1.

Most of the studies on gauge-theoretical systems for constructing qubits
including the troic code and subsystem stabilizer code focus on the
topological phase, which is nothing but the deconfinement phase of the LGT.
The Higgs-confinement phase in the two-dimensional systems under ordinary
boundary conditions is sometimes regarded as a trivial phase with only gapful
excitations [38, 39, 37]. In the present work, however, we showed that an
explicit description of the boundary states is possible and it plays a very
important role in clarifying structure of the encoded qubits. From this point
of view, the explicit description of the boundary states in deconfinement
phase is desired to study the detailed structure of encoded qubits in the
present model, although it might be rather complicated.

  2. 2.

We are interested in effects of disorder. If a kind of disorders is introduced
on rough or smooth boundary, the SSB in Higgs and confinement regimes might
survive up to highly-excited states even in the gauge-Higgs model. In fact,
this conjecture has been investigated from the viewpoint of many-body
localization [40, 41, 42]. Concrete numerical verification for that in certain
one-dimensional models has been reported in recent works [43, 44, 45, 46]. The
confirmation of the conjecture for the LGT studied here is a future problem.

  3. 3.

We also expect that the encoded qubit state discussed in this work can be
produced by a measurement-only circuit [47, 48, 49, 50], where sequential
projective random measurements of each terms in $H_{\rm GHM}$ are applied
[51]. The numerical verification of it can be an important future direction of
study

  4. 4.

Application of the present formalism to higher-dimensional models and other
various types of stabilizer code Hamiltonians related to quantum memory [52]
is an interesting subject. Our observation seems to indicate that a finite-
temperature phase transition of SSB of charge symmetry as well as confinement
belong to the universality class of the corresponding quantum spin model in
one-lower dimensions. Another interesting issue to be clarified is meaning of
the strong zero modes from the viewpoint of the gauge-Higgs model, which are
expected to exist even at (very) finite temperature. Monte-Carlo simulation is
useful for these studies [53, 54].

  5. 5.

Finally, we comment on the experimental realization and related modification
of the model of $H_{\rm GHM}$ [Eq. (1)] focusing on the gauge-invariant
conditions. We are considering quantum simulation of the present models by
using, e.g., ultra-cold atomic gasses, ion straps, etc. It is a hard task for
real experiment to strictly implement the gauge-invariant condition of Eq.
(2). However, a promising method was proposed recently: it employs concept of
local pseudogenerators (LPGs) and adds an energy penalty term of the LPGs to
the model $H_{\rm GHM}$ [55, 56] instead of directly enforcing the Gauss’s law
[Eq. (2)]. The energy penalty term of the LPGs can be suitable for
experimental realization of the Gauss’ law by its simple form, and it
effectively generates the target sector of the constraint of Eq. (2). We
expect that in such experimentally-suitable Hamiltonian modified from $H_{\rm
GHM}$, it is possible to construct the subsystem codes presented in this work
although the toric code model $H_{\rm TC}$ [Eq. (9)] as a gauge-fixing model
does not appear and the exact solutions of the subsystem code in the modified
system would be more complicated than those derived in this work. However,
physical properties of the resultant states are essentially the same with the
original ones. Detailed investigation of this topic is interesting and a
constructive future direction of study.

## Acknowledgements

This work is supported by JSPS KAKEN-HI Grant Number 23K13026 (Y.K.).

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