# Classification of measurement-based quantum wire in
stabilizer PEPS

Paul Herringer  Department of Physics and Astronomy, University of British
Columbia, Vancouver, Canada  Stewart Blusson Quantum Matter Institute,
University of British Columbia, Vancouver, Canada  Robert Raussendorf
Department of Physics and Astronomy, University of British Columbia,
Vancouver, Canada  Stewart Blusson Quantum Matter Institute, University of
British Columbia, Vancouver, Canada

###### Abstract

We consider a class of translation-invariant 2D tensor network states with a
stabilizer symmetry, which we call stabilizer PEPS. The cluster state, GHZ
state, and states in the toric code belong to this class. We investigate the
transmission capacity of stabilizer PEPS for measurement-based quantum wire,
and arrive at a complete classification of transmission behaviors. The
transmission behaviors fall into 13 classes, one of which corresponds to
Clifford quantum cellular automata. In addition, we identify 12 other classes.

##  1 Introduction

How to harness multi-particle entanglement is a central question in quantum
information processing. Among the many known protocols are quantum steering
[1, 2], distribution of localizable entanglement [3] in quantum networks,
Bell-state distillation in quantum communication [4, 5], and measurement-based
quantum computation (MBQC) [6].

In some of these protocols, symmetry plays an important role. For example, we
observe this feature in measurement-based quantum information processing,
including measurement-based quantum state transmission (also known as quantum
wire) [7, 8], quantum computation on a resource state [6, 9, 10], and
computational phases of quantum matter [11, 8, 12, 13, 14, 15, 16, 17, 18].

Figure 1: Phenomenology of measurement-induced state transformations—state vs.
phase and transmission vs. computation. Top left: In cluster states,
arbitrarily distant qubits can be projected into Bell pairs by local
measurement on other qubits, $a$=[7]. Bottom left: the same feature is
observed in an entire SPT phase surrounding the 1D cluster state, $b$=[8]. Top
right: In spatial dimension two and higher, cluster states give rise to
universal measurement-based quantum computation, $c$=[6]. Bottom right:
Universal measurement based quantum computation persists in entire SPT phases
of matter, $d$=[14, 15, 17, 18]. Centre: All the phenomena displayed rely on
symmetries of the PEPS tensor representing the resource state. Some, but not
all, of these symmetries give rise to Clifford cellular automata.

The present work addresses the role of symmetry in protocols of measurement-
based state transformation. Specifically, we study a class of highly symmetric
2D tensor network states and classify their transmission capacity for quantum
wire. This is a first step toward a classification of schemes of measurement-
based quantum computation in higher spatial dimension, based on symmetry.

To put our work into perspective, below we summarize several measurement-
based, symmetry-aided protocols and describe our classification result in more
detail. See Fig. 1 for a summary of the phenomenology of interest. The
simplest among the phenomena displayed is the distribution of localizable
entanglement, as can be observed in cluster states, for example. Therein, any
pair of qubits can be brought to a Bell-entangled state by locally measuring
other qubits in the state [7]. It was later found that the creation of Bell-
type entanglement, or equivalently quantum wire, works for an entire symmetry-
protected topological (SPT) phase surrounding the cluster state [8]. The
quantum wire is a direct consequence of the symmetry representation in those
phases.

Ramping up the complexity of the quantum processing task, it is also known
that certain special quantum states, such as cluster states and AKLT states on
various two-dimensional lattices, are universal resources for quantum
computation by local measurement [7], [9, 10, 19].

The most complex notion in this area is that of computational phases of
quantum matter [11, 8, 12, 13, 14, 15, 16, 17, 18], which extend the
computational power of individual resource states to entire SPT phases
surrounding them. For suitable SPT phases in dimensions 1 and 2, the
computational power of MBQC resource states is uniform across the phase, i.e.,
all states in a given SPT phase have the same computational power. Indeed,
computational phases of matter exist that support universal quantum
computation [17, 18, 14, 15].

The current understanding of computational phases relies upon symmetric tensor
networks and Clifford quantum cellular automata (QCA) [17, 18, 20]. In this
paradigm, the symmetries of a tensor network are associated with a Clifford
QCA that persists throughout an entire SPT phase, and it is the presence of
this QCA that enables MBQC with any state in the phase. This is the content of
Theorem 2 in [18].

Our motivation for the present work is the observation that many symmetric
tensor network states relevant to quantum information processing do not admit
a description in terms of QCA. These include the GHZ state and the toric code
[21], as described in Section 2.3. And so, we ask: What else is there?

To get a handle on this question, in this paper we investigate measurement-
based quantum wire implemented on symmetric 2D tensor network states. In this
context, we arrive at a complete classification of transmission behaviors. One
of the classes identified corresponds to Clifford QCA, which have their own
(finer) classification [20, 22]. In addition to Clifford QCAs, we describe 12
new classes.

The paper is organized as follows. In Sections 2.1-2.3, we describe the
setting for our investigation of stabilizer PEPS, detail the applications of
stabilizer PEPS in measurement-based quantum information processing, and
provide some examples. In Section 2.4, we state our main result. Section 3 is
devoted to proving this result, and Section 4 concludes.

##  2 Phenomenology

###  2.1 Setting

Our setting is a 2D projected entangled pair state (PEPS) on a square lattice
with cylindrical geometry (Fig. 2). The circumference and depth of the
cylinder are denoted $n$ and $d$, respectively, and the state is translation
invariant in the sense that the local PEPS tensor is the same for every
lattice site.

Figure 2: Cylindrical PEPS.

We study a class of states characterized by local tensors which obey a set of
symmetry constraints of the form

(1)

where $g$ is an element of the symmetry group of the tensor and $A(g)$ through
$E(g)$ are Pauli operators that form a linear representation of $g$. In other
words, the symmetry group of the local tensor is also a stabilizer group.
Therefore, we call these states stabilizer PEPS, and the local tensors
stabilizer tensors. We emphasize that stabilizer PEPS are not only theoretical
but can be prepared in the lab: a stabilizer tensor is equivalent to a 5-qubit
stabilizer state, and two tensors can be contracted by Bell measurements (Fig.
3).

---
(a)
(b)
Figure 3: Physical realization of stabilizer PEPS. (a) A stabilizer tensor is
a 5-qubit stabilizer state. (b) Contraction of two “tensors” by Bell
measurement.

###  2.2 Applications

There are a variety of quantum information processing tasks that can be
accomplished by invoking stabilizer PEPS and local measurements. In order of
increasing complexity, these are (i) quantum wire or creation of long-range
Bell-type entanglement, (ii) quantum computation on a resource state, and
(iii) quantum computation in a suitable SPT phase.

##### Quantum wire

The quantum wire protocol consists of performing an $X$ measurement at every
physical site in the bulk of an $n\times d$ stabilizer PEPS, which creates
Bell-type entanglement between the left and right edges111The choice of $X$ as
the measurement basis is arbitrary; any measurement basis can be accommodated
by changing the action of the local tensor symmetries on the physical leg..
The number of qubits worth of entanglement created in this way is a non-
negative integer $C(n,d)$ such that $0\leq C(n,d)\leq n$. We call this number
the transmission capacity of the stabilizer PEPS, because the creation of
long-range Bell-type entanglement is equivalent to the transmission of quantum
information: given long-range entanglement, quantum information may be
transmitted by the standard teleportation protocol, and given the ability to
transmit quantum information, a Bell pair may be created.

##### MBQC

If the local measurement basis is allowed to vary from site to site, it
becomes possible to do more than simply transmit quantum information from one
end of the cylinder to the other. Indeed, we can perform unitary operations on
the $n$-qubit Hilbert space spanned by the sites around the circumference of
the cylinder. For some resource states, such as the cluster state, this leads
to universal quantum computation.

##### Computational phases

Certain stabilizer PEPS also possess subsystem symmetries, which act on
subsets of the lattice rather than at every site [18]. The family of states
that can be reached from these stabilizer PEPS by local, finite-depth quantum
circuits that respect the subsystem symmetries forms an SPT phase. Some of
these SPT phases are known to have uniform computational power [17, 18]. In
other words, given an appropriate measurement protocol every state in the
phase is equally useful for MBQC.

As a first step towards understanding the phenomenology of quantum information
processing with stabilizer PEPS, the present paper deals with the simplest
task, quantum wire.

###  2.3 Examples

In this section, we present three examples of well-known states that are also
stabilizer PEPS. We detail their computational capability with respect to the
three tasks described above, and highlight the role of symmetry and QCA.

####  2.3.1 Cluster state

The 2D cluster state is central to the study of MBQC because it is useful for
quantum information processing at all three levels of complexity described in
the previous section. At the simplest level, the cluster state has maximal
transmission capacity for quantum wire, $C(n,d)=n$ for any depth. At the next
level, MBQC, we can perform universal quantum computation on the cluster state
via local measurements. Finally, the cluster state lies in an SPT phase called
the cluster phase, and every state in the cluster phase is a universal
resource for MBQC [17].

The cluster state is also a stabilizer PEPS where the local tensor has the
following symmetries:

(2) (3)

To see this explicitly, we can construct the stabilizer generators of the
cluster state from its tensor symmetries (Fig. 4a).

|
---|---
(a) | (b)
|
(c) | (d)
Figure 4: Stabilizer generators for (a) the cluster state, (b, c) the GHZ
state, and (d) the toric code state, all constructed from the corresponding
local tensor symmetries.

The symmetry representation of the cluster state offers an illuminating
perspective on quantum wire. Let us treat a single ring of tensors as a matrix
product operator (MPO) acting on correlation space. Measurement in the $X$
basis destroys the symmetry (3), because it anticommutes with the measurement
basis. The action of the remaining symmetries (2) on the virtual legs define a
local, unitary update rule, that is, a QCA (Fig. 5a and 5b). Performing an $X$
measurement on every spin in the bulk of an $n\times d$ cluster state is
equivalent to applying this QCA $d$ times, which entangles every qubit on the
left edge with every qubit on the right. In this way, we obtain maximal
transmission capacity.

####  2.3.2 GHZ state

The Greenberger-Horne-Zeilinger (GHZ) state [23] is important in the study of
quantum foundations [24], entanglement [25, 26], and SPT phases [27], however,
it is of limited computational use. With a GHZ state, the quantum wire
protocol creates only one qubit worth of entanglement, $C(n,d)=1$ regardless
of the circumference. Consequently, there is only one logical qubit available
for the more complex task of MBQC. Furthermore, the only operations available
for this logical qubit are $Z$ rotations.

As a stabilizer PEPS, the local tensors of the GHZ state satisfy

(4) (5)

With (4) and (5) we may construct the stabilizers of the GHZ state (Fig. 4b
and 4c), just as the cluster state stabilizers are constructed from (2) and
(3). However, upon forming a ring of GHZ tensors we find that the resulting
MPO is not a QCA because it has a non-trivial kernel, i.e. some operators are
mapped to the identity when reading from left to right (Fig. 5c). The image of
the MPO is spanned by just two operators, which permits transmission of a
single qubit (Fig. 5e and 5f).

|  |
---|---|---
(a) | (b) | (c)
|  |
(d) | (e) | (f)
Figure 5: When measured in the $X$ basis, a single layer of a cylindrical
stabilizer PEPS constitutes an MPO acting on correlation space. The MPO acts
to the left, thus information flows from left to right. (a, b) The symmetry
action on the virtual legs of the cluster state tensors defines a QCA with
update rule $T(X_{i})=X_{i-1}Z_{i}X_{i+1}$ and $T(Z_{i})=X_{i}$. (c-f) The
symmetries of the GHZ state define a non-unitary update rule.

####  2.3.3 Toric code

The toric code [21] is a popular quantum code, suited to realizing fault-
tolerant universal quantum computation in planar architectures [28]. All
states in the toric code space have topological order.

In our quantum wire setting, the toric code state has transmission capacity
$C(n,d)=n-1$. In the context of MBQC we can perform some unitary operations,
however, it has been shown that MBQC on the toric code can be simulated
efficiently on a classical computer [29, 30]. We are not aware of any work on
a computational phase based around the toric code; whether or not such a phase
exists remains an open question.

The toric code appears in the class of stabilizer PEPS. Consider a stabilizer
tensor with the following symmetries:

(6) (7)

From (6) and (7), we construct the 4-local stabilizer generators shown in Fig.
4d. By shearing the lattice, it can be seen that these correspond to a
translation-invariant version of the toric code, also known as the XZZX code
[31, 32], obtained by applying Hadamard operators along every other column of
the original toric code.

A ring of toric code tensors gives an MPO that is non-unitary, similar to the
GHZ tensors. The difference is that the kernel of the toric code MPO is
smaller in dimension, which allows a greater quantity of quantum information
to pass through each layer of the cylinder.

Figure 6: Transmission capacity $C(n,d)$ for a stabilizer PEPS falls into one
of 13 classes. Class 12 is the class of QCA, $C(n,d)=n$. Class 0 (not shown)
is the trivial class, $C(n,d)=0$.

The representability of states in the toric code by PEPS is in no
contradiction to their topological order. When states in the toric code are
created unitarily from a tensor product state, long-range action is required.
However, when projective measurement and classical communication are allowed,
short-range quantum operations suffice. For example, states in the toric code
can be obtained by individually measuring every other qubit in a 2D cluster
state. A systematic classification of quantum states under local quantum
operations and classical communication, albeit only in spatial dimension 1,
has been provided in [33].

###  2.4 Results

As illustrated by the above examples, the class of stabilizer PEPS contains
states with significantly different capabilities for quantum information
processing. Focusing on quantum wire, we immediately discern that the QCA-type
maximally transmitting channel found for the cluster state is only one
possible behavior. In other cases, the transmission capacity may be smaller –
as little as one qubit, independent of the circumference and distance. Our
goal is to classify the transmission behaviors that arise within the class of
stabilizer PEPS. Specifically, we ask: given a stabilizer PEPS, how many
qubits worth of quantum information can be transmitted by local measurement,
as a function of circumference and depth? The answer to this question is given
by the following theorem.

######  Theorem 1.

For the class of cylindrical stabilizer PEPS on a 2D square lattice, there are
13 distinct behaviors of quantum wire transmission, characterized by the
transmission capacity $C(n,d)$ as a function of the circumference $n$ and the
depth $d$. These include the trivial class $C(n,d)=0$ and the class
corresponding to Clifford quantum cellular automata, $C(n,d)=n$. The other
eleven classes are represented in Fig. 6.

There are 2649 distinct stabilizer PEPS up to gauge equivalence, thus Theorem
1 is a significant simplification to the phenomenology of stabilizer PEPS.

##  3 Classification of Quantum Wire

The purpose of this section is to prove Theorem 1. The proof splits into two
parts: one analytical and one numerical. First, we reduce the size of our
problem considerably with the following proposition.

######  Proposition 1.

Let $C(A,n,d)$ be the transmission capacity of a cylindrical stabilizer PEPS
constructed from local tensor $A$ with circumference $n$ and depth $d$. If
$C(A_{1},n,d)=C(A_{2},n,d)$ for $n\leq 6$ and $d\leq 6$, then
$C(A_{1},n,d)=C(A_{2},n,d)$ for all values of $n$ and $d$.

With Proposition 1 in hand, the proof of Theorem 1 is a direct consequence of
the following lemma.

######  Lemma 1.

For $n\leq 6$ and $d\leq 6$, the transmission capacity $C(n,d)$ must take one
of the 12 forms represented in Fig. 6, or the trivial form $C(n,d)=0$.

###### Proof.

The proof is by computer. Code to reproduce our results is available at [34].
∎

Below, we set out to prove Proposition 1.

###  3.1 Background, notation, and definitions

The single-qubit Pauli group $\mathcal{P}_{1}$ is the group of observables
generated by the Pauli $X$ and $Z$ operators:
$\mathcal{P}_{1}=\expectationvalue{X,Z}$. The $n$-qubit Pauli group is the
$n$-fold tensor product of $\mathcal{P}_{1}$. An $n$-qubit stabilizer group is
an abelian subgroup $S\subset\mathcal{P}_{n}$ such that $-I\notin S$, and an
$n$-qubit stabilizer state is an $n$-qubit quantum state uniquely specified by
a stabilizer group. Mappings between stabilizer states correspond to elements
of the Clifford group, which is the normalizer of $\mathcal{P}_{n}$ in
$U(2^{n})$, defined here modulo a complex phase:
$\mathcal{C}_{n}=\\{U:U\mathcal{P}_{n}U^{\dagger}=\mathcal{P}_{n}\\}/U(1)$.

We also make use of the symplectic representation of the Pauli group by
mapping elements of $\mathcal{P}_{n}$ to binary vectors
$\xi=(\xi^{X},\xi^{Z})\in\mathbb{F}_{2}^{2n}$, such that
$g(\xi)=\bigotimes_{i=1}^{n}X^{\xi_{i}^{X}}Z^{\xi_{i}^{Z}}\in\mathcal{P}_{n}$
[18]. Multiplication of Pauli operators corresponds to addition of binary
vectors, and commutation relations are captured by a symplectic form
$\omega(\xi,\eta)=\sum_{i=1}^{n}\xi^{X}_{i}\eta^{Z}_{i}+\xi^{Z}_{i}\eta^{X}_{i}$
modulo 2 such that $g(\xi)g(\eta)=(-1)^{\omega(\xi,\eta)}g(\eta)g(\xi)$. The
vector space $\mathbb{F}_{2}^{2n}$ equipped with the symplectic form $\omega$
is called a symplectic vector space, and it is isomorphic to $\mathcal{P}_{n}$
up to phase, i.e., $\mathbb{F}_{2}^{2n}\cong\mathcal{P}_{n}/U(1)$. A
stabilizer group corresponds to an isotropic subspace of $\mathbb{F}_{2}^{2n}$
with respect to $\omega$, and a stabilizer state to a maximally isotropic
subspace.

Consider the $\ell$-qubit stabilizer states. There is a natural mapping from
these states to PEPS tensors with $k$ physical legs of dimension $D=2$ and
$\ell-k$ virtual legs of dimension $\chi=2$. Namely, we assign each qubit of
the stabilizer state to one leg of the tensor and re-frame the stabilizer
group as the symmetry group of the tensor. This completely specifies the
components of the tensor in the same way that a stabilizer state is completely
specified by its symmetries.

######  Definition 1.

A stabilizer tensor is a PEPS tensor whose components are completely specified
by a set of symmetry constraints, such that the same symmetry constraints also
specify a stabilizer state. An $[\ell,k]$ stabilizer tensor has $\ell$ legs in
total and $k$ physical legs. A stabilizer PEPS is a translation-invariant PEPS
where the local tensor is a stabilizer tensor.

In this paper we work with a square 2D lattice, so we consider the $[5,1]$
stabilizer tensors. Two stabilizer tensors $A_{1}$ and $A_{2}$ generate the
same stabilizer PEPS if they are related by a gauge transformation:

$\includegraphics[valign={c},scale={0.75}]{figs/SLC.png}.$ (8)

where $U,V\in\mathcal{C}_{1}$. Up to gauge equivalence, there are 2649
distinct stabilizer PEPS.

We denote a cylindrical stabilizer PEPS $\mathcal{A}(A,n,d)$ with local
stabilizer tensor $A$, circumference $n$, and depth $d$. Upon measurement of
every physical spin in the $X$ basis, we are left with a stabilizer state on
the $2n$ virtual legs at the edges. The stabilizer group for this state is
$S(A,n,d)$. $S$ has a natural bipartition into the two open edges of the
cylinder; we write $S_{L}$ and $S_{R}$ to denote the restriction of $S$ to the
left and right edges, respectively.

The generators of a bipartite stabilizer group such as $S(A,n,d)$ can always
be brought to the following canonical form, which reveals the entanglement
structure of the corresponding stabilizer state [35]:

$\displaystyle\mathcal{Z}_{L}(A,n,d)$ $\displaystyle=\begin{cases}a_{i}\otimes
I\\\ \end{cases}\qquad 1\leq i\leq n-p$ (9) $\displaystyle P_{LR}(A,n,d)$
$\displaystyle=\begin{cases}g_{k}^{L}\otimes g_{k}^{R}\\\
\bar{g}_{k}^{L}\otimes\bar{g}_{k}^{R}\end{cases}\qquad 1\leq k\leq p$
$\displaystyle\mathcal{Z}_{R}(A,n,d)$ $\displaystyle=\begin{cases}I\otimes
b_{j}\\\ \end{cases}\qquad 1\leq j\leq n-p.$

Here $\mathcal{Z}_{L}$ is the center of $S_{L}$ and acts trivially on
partition $R$, while $\mathcal{Z}_{R}$ is the center of $S_{R}$ and acts
trivially on partition $L$. $P_{LR}$ is generated by $p$ locally anticommuting
pairs, meaning that $(g_{k}^{L},\bar{g}_{k}^{L})$ anticommute with each other
but commute with all other $g_{\ell}^{L}$ and $\bar{g}_{\ell}^{L}$ for
$\ell\neq k$. The same relationship holds for $(g_{k}^{R},\bar{g}_{k}^{R})$,
so that $S_{LR}(n,d)$ is abelian when we look at both partitions
simultaneously. We write $P_{L}$ and $P_{R}$ for the restriction of $P_{LR}$
to the left and right partitions, respectively.

Later, it will be convenient to encapsulate all the information about the
canonical form of $S(A,n,d)$ with a single mathematical object. To this end,
we denote the canonical form of $S(A,n,d)$ by $\Phi_{d}(A,n)$. Every
$\Phi_{d}$ with the same $(n,p)$ is equivalent to $p$ Bell pairs encoded by an
$[n,p]$ stabilizer code. Therefore, the transmission capacity $C(A,n,d)$ is
equal to $p$, and $\Phi_{d}$ naturally separate into equivalence classes
labelled by the tuple $(n,p)$.

######  Definition 2.

The transmission capacity $C(A,n,d)$ for a stabilizer PEPS
$\mathcal{A}(A,n,d)$ with local stabilizer tensor $A$, depth $d$, and
circumference $n$ is the number of anticommuting pairs $p$ in the canonical
form of the bipartite stabilizer group $S(A,n,d)$.

###  3.2 Proof of Proposition 1

In this section, we prove Proposition 1. Here, we provide a brief overview of
the proof.

First, we generalize the update rule corresponding to a single layer of a
stabilizer PEPS so that it can deal with non-unitary updates, like those found
for the GHZ state and toric code. Then, we use our generalized update rule to
analyze the concatenation of many individual layers, and show that $C(n,d)$ is
completely determined by two mathematical objects, both of which may be
derived from a single layer of the tensor network. These objects are the
canonical stabilizer generators for a single layer, and the commutation
relations between canonical stabilizer generators for a single layer. We
denote them $\Phi_{1}$ and $\Omega_{1}$, respectively.

Both $\Phi_{1}$ and $\Omega_{1}$ are independent of $d$ but remain functions
of $n$; we remove this dependence by deriving a local generating set for the
stabilizer group of a single layer. Relating the local stabilizer generators
to the canonical stabilizer generators allows us to show that if there are two
different stabilizer tensors $A_{1}$ and $A_{2}$ with
$\Phi_{1}(A_{1},n)=\Phi_{1}(A_{2},n)$ and
$\Omega_{1}(A_{1},n)=\Omega_{1}(A_{2},n)$ for $n\leq 6$, then
$\Phi_{1}(A_{1})=\Phi_{1}(A_{2})$ and $\Omega_{1}(A_{1})=\Omega_{1}(A_{2})$
for all $n$.

####  3.2.1 Non-unitary update rule

Consider the MPO $\mathcal{A}(A,n,1)$. For compactness of notation, we draw
$\mathcal{A}$ as a single tensor with higher bond dimension:

$\includegraphics[valign={c},scale={0.75}]{figs/quasi_1d_no_leg.png}.$ (10)

Note that we have removed the physical legs of every tensor, because
measurement in the $X$ basis contracts these legs with an $X$ eigenstate.
Strictly speaking, the action of $\mathcal{A}$ depends on the measurement
outcomes at each site. However, the only effect that different measurement
outcomes may have on the stabilizer generators for $\mathcal{A}$ is to change
some of their signs. This change does not affect the number of anticommuting
pairs in the canonical form (9), i.e., it does not affect the transmission
capacity. Therefore, without loss of generality we contract every physical leg
with the $\ket{+}$ state. To be clear, any tensor drawn without a physical leg
is assumed to be contracted with $\ket{+}$.

The stabilizer group of $\mathcal{A}(A,n,1)$ is $S(A,n,1)$. Bringing $S$ to
canonical form gives the following symmetries for $\mathcal{A}$:

$\includegraphics[valign={b},scale={0.75}]{figs/calAsymms2.png}.$ (11)

In the spirit of QCA, we use Eq. 11 to define an update rule $T$ such that

$T(a_{i})=I;\quad T(g_{k}^{L})=g_{k}^{R};\quad
T(\bar{g}_{k}^{L})=\bar{g}_{k}^{R}.$ (12)

$T$ is non-unitary unless every generator of $S$ is part of an anticommuting
pair, in which case we recover a QCA by identifying $g_{k}^{L}$ and
$\bar{g}_{k}^{L}$ with encoded Pauli operators $\tilde{X}_{k}$ and
$\tilde{Z}_{k}$, respectively.

Unitarity for any value of $p$ can be restored by modifying the update rule
such that

$T(a_{i})=b_{i},\quad T(g_{k}^{L})=g_{k}^{R};\quad
T(\bar{g}_{k}^{L})=\bar{g}_{k}^{R}.$ (13)

However, note that $T$ is not defined on the full $n$-qubit Hilbert space.
Instead, it maps the subspace $S_{L}$ to the subspace $S_{R}$. If $S_{L}\neq
S_{R}$, then some operators in $S_{L}$ have an image under $T$ that falls
outside the domain of $T$. A simple test to determine whether an operator is
in the domain of $T$ follows directly from the following lemma.

######  Lemma 2.

$S_{L}(A,n,d)$ is the centralizer of $\mathcal{Z}_{L}(A,n,d)$ in
$\mathcal{P}_{n}$, and $S_{R}(A,n,d)$ is the centralizer of
$\mathcal{Z}_{R}(A,n,d)$ in $\mathcal{P}_{n}$.

###### Proof.

Without loss of generality we prove the statement for $S_{L}$ and
$\mathcal{Z}_{L}$, as the same argument applies to the right partition.
Switching to the symplectic representation of $\mathcal{P}_{n}$, we have
$\mathcal{Z}_{L}$ an isotropic subspace and
$S_{L}\subseteq\mathcal{Z}_{L}^{\perp}$. By Eq. 9, the dimensions of these
subspaces are $\dim(\mathcal{Z}_{L})=n-p$ and $\dim(S_{L})=n+p$.

For any symplectic vector space $V$ and subspace $W\subset V$ it is known that
$\dim(V)=\dim(W)+\dim(W^{\perp})$. Adding dimensions, we have
$\dim(\mathcal{Z}_{L})+\dim(S_{L})=2n=\dim(\mathcal{P}_{n})$. Therefore
$S_{L}=\mathcal{Z}_{L}^{\perp}$. In other words, $S_{L}$ centralizes
$\mathcal{Z}_{L}$ in $\mathcal{P}_{n}$. ∎

######  Corollary 1.

For any $x\in\mathcal{P}_{n}$, $x\in S_{L}(A,n,d)$ if and only if $x$ commutes
with $\mathcal{Z}_{L}(A,n,d)$.

###### Proof.

$\mathcal{Z}_{L}$ is the center of $S_{L}$, therefore $x\in S_{L}$ implies
that $x$ commutes with every $y\in\mathcal{Z}_{L}$. By Lemma 2, $S_{L}$ is the
centralizer of $\mathcal{Z}_{L}$ in $\mathcal{P}_{n}$, therefore the converse
is also true. ∎

####  3.2.2 Chains of symmetry

Any symmetry of $\mathcal{A}(A,n,d)$ can be derived by chaining symmetries,
i.e., concatenating $d$ copies of $\mathcal{A}(A,n,1)$. We use the chaining
process to characterize the group $\mathcal{Z}_{R}(A,n,d)$, because one way to
determine the channel capacity is $C(A,n,d)=n-\rank[\mathcal{Z}_{R}(A,n,d)]$.

Elements of $\mathcal{Z}_{R}(A,n,d)$ must act as the identity on the left edge
of $\mathcal{A}(A,n,d)$ and non-trivially on the right edge. Therefore, they
have the following form:

(14)

where $a_{i}\in\mathcal{Z}_{L}(A,n,1)$, $1\leq i\leq n-p$, and $\gamma_{i}$ is
the critical depth such that $T^{\gamma_{i}}(a_{i})\notin S_{L}(A,n,1)$.
Recall that this implies $T^{\gamma_{i}+1}(a_{i})$ is not a symmetry. Reading
off the rightmost column of (14), it is clear that $\mathcal{Z}_{R}(A,n,d)$ is
generated by the set
$\mathcal{T}_{R}(n,d)=\quantity{T(a_{i}),\>\dots,\>T^{\min(d,\gamma_{i})}(a_{i})\>|\>1\leq
i\leq n-p}.$ Note that $\mathcal{Z}_{R}(A,n,d)$ is invariant for
$d\geq\max_{i}(\gamma_{i})$, because every chain in (14) may be extended to
the left by the trivial symmetry, but no chain may be extended to the right
once $d\geq\max_{i}(\gamma_{i})$.

######  Lemma 3.

For any fixed value of $n$, $C(A,n,d)\geq C(A,n,d+1)$.

###### Proof.

$\mathcal{T}_{R}(A,n,d)$ generates $\mathcal{Z}_{R}(A,n,d)$, therefore

$C(A,n,d)=n-\rank[\mathcal{T}_{R}(A,n,d)].$ (15)

Any of the chains (14) may be extended indefinitely to the left by the trivial
symmetry, therefore $\mathcal{T}_{R}(A,n,d)\subseteq\mathcal{T}_{R}(A,n,d+1)$
and $\rank[\mathcal{T}_{R}(A,n,d)]\leq\rank[\mathcal{T}_{R}(A,n,d+1)]$.
Substituting (15) into this inequality gives $C(A,n,d)\geq C(A,n,d+1)$. ∎

######  Lemma 4.

For any fixed value of $n$, if $C(A,n,d_{c})=C(A,n,d_{c}+1)$ at some critical
depth $d_{c}$, then $C(A,n,d)=C(A,n,d_{c})$ for all $d\geq d_{c}$.

###### Proof.

$C(A,n,d_{c})=C(A,n,d_{c}+1)$ requires that $T$ map
$\mathcal{Z}_{R}(A,n,d_{c})$ to itself, therefore $\mathcal{Z}_{R}(A,n,d_{c})$
is fixed for $d\geq d_{c}$. Recall that
$C(A,n,d)=n-\rank[\mathcal{Z}_{R}(A,n,d)]$. Therefore, $C(A,n,d)$ is also
fixed for all $d\geq d_{c}$. ∎

######  Lemma 5.

For any fixed value of $n$, the maximum value of $d_{c}$ is $C(A,n,1)$.

###### Proof.

By Lemmas 3 and 4, $C(A,n,d)$ must decrease when $d$ is increased, up to the
critical value $d_{c}$. If it ever fails to decrease, then we have reached
$d_{c}$ and the transmission capacity must be constant thereafter. $C(A,n,d)$
is a non-negative quantity; therefore, it may decrease at most $C(A,n,1)$
times before reaching zero and becoming constant. ∎

####  3.2.3 Codifying the update rule $T$

Using the Lemmas 3, 4, and 5, we formulate an algorithm to compute $C(A,n,d)$.

Algorithm 1: Transmission Capacity
---
Data: Stabilizer tensor $A$, circumference $n$, depth $d$
1 |  Compute $\Phi_{1}(A,n)$, which determines $\mathcal{Z}_{L}(A,n,1)$ and the update rule $T$
2 |  Initialize $\mathcal{T}_{R}(A,n,1)=\\{T(x)|x\in G\\}$, where $G$ is any generating set for $\mathcal{Z}_{L}(A,n,1)$
3 |  For every $y\in\mathcal{T}_{R}(A,n,1)$, check if $y$ commutes with $\mathcal{Z}_{L}(A,n,1)$. If so, $T(y)$ exists. Add every valid $T(y)$ to $\mathcal{T}_{R}(A,n,1)$ and obtain $\mathcal{T}_{R}(A,n,2)$
4 |  Repeat Step 3 until the desired depth is reached, or until $\mathcal{T}_{R}(A,n,d)=\mathcal{T}_{R}(A,n,d-1)$
5 |  Return $C(A,n,d)=n-\rank[\mathcal{T}_{R}(A,n,d)]$

Algorithm 1 shows that the following mathematical objects are sufficient to
compute $C(A,n,d)$:

  1. (i)

The canonical generators of $S(A,n,1)$, i.e. $\Phi_{1}(A,n)$.

  2. (ii)

The commutation relations between generators of $\mathcal{Z}_{L}(A,n,1)$ and
$S_{R}(A,n,1)$, which we denote $\Omega_{1}(A,n)$.

Recall that $\Phi_{1}$ is characterized by the tuple $(n,p)$, however, there
is considerable freedom to define $\Omega_{1}$ by choosing different
generators for $\Phi_{1}$. Therefore, it will be useful to have a standard
form for $\Omega_{1}$. To this end, we represent $\Omega_{1}$ by a bipartite
graph $G_{\Omega}$ with node sets $\\{a_{i}\\}$ and
$\\{b_{j},g_{k}^{R},\bar{g}_{k}^{R}\\}$. Let the edges of $G_{\Omega}$
indicate anticommutation between operators in different node sets, and let the
biadjacency matrix be $M$, with the following layout:

$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left(\kern
0.0pt\kern-2.5pt\kern-6.66669pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
b_{1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle b_{n-p}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
g_{1}^{R}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\bar{g}_{1}^{R}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
g_{p}^{R}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\bar{g}_{p}^{R}\\\a_{1}\\\\\vdots\\\a_{n-p}$\hfil\kern
5.0pt\crcr}}}}\right)$ (16)

The operations on $M$ that preserve $\Phi_{1}$ are:

  1. (i)

Swap any two rows, or any of the first $(n-p)$ columns.

  2. (ii)

Add any row to any other row, modulo 2.

  3. (iii)

Add any of the first $(n-p)$ columns to any of the last $p$ columns, modulo 2.

  4. (iv)

Reorder the $2\times(n-p)$ blocks of $M_{\Omega_{1}}$ that represent the
anticommuting pairs.

  5. (v)

Within any of the blocks for each anticommuting pair:

    1. (a)

Swap columns.

    2. (b)

Add one column to the other or vice versa, modulo 2.

Using these operations, $M$ may be brought to the following standard form:

$\begin{array}[]{cccccccccc}\left(\rule{0.0pt}{34.14322pt}\right.\matrixquantity{\lambda_{1}&&\\\
&\ddots&\\\
&&\lambda_{n-p}}&\left|\rule{0.0pt}{34.14322pt}\right.&H_{1}&\left|\rule{0.0pt}{34.14322pt}\right.&\cdots&\left|\rule{0.0pt}{34.14322pt}\right.&H_{p}\left.\rule{0.0pt}{34.14322pt}\right)\end{array}$
(17)

where $\lambda_{i}\in\\{0,1\\}$ and $H_{j}$ are $(n-p)\times 2$ block matrices
in RCEF, ordered by rank. The RREF and RCEF of any matrix (or block matrix) is
unique, therefore this standard form is well-defined and we can use it to
compare two stabilizer PEPS.

####  3.2.4 Local generators

The next step is to eliminate the dependence of $\Phi_{1}$ and $\Omega_{1}$ on
$n$. We do this by exploiting the translation invariance of stabilizer PEPS to
find a local form for the generators of $S(A,n,1)$. To be precise, we show
that the generating set for $S(A,n,1)$ has a form where every generator save
one is $k$-local with $k\leq 3$. Because there is only one nonlocal generator,
we slightly abuse terminology and refer to the local form of $S(A,n,1)$.

To obtain a local form of $S(A,n,1)$, we recursively apply (9) to the symmetry
group of the local tensor $A$, using smaller and smaller partitions each time.
Let the 4 virtual legs of $A$ be partitioned into horizontal and vertical
subsystems $H$ and $V$, which are subdivided into
$\\{u,d,l,r\\}=\\{\text{up},\text{down},\text{left},\text{right}\\}$:

(18)

The first step in the recursive application of (9) is to bring $S(A,1,1)$ to
canonical form with respect to the $\\{H,V\\}$ partition. For convenience, we
denote $S(A,1,1)$ by $S_{A}$. The subsystems $H$ and $V$ have two legs each,
so there may be $p=0$, 1, or 2 anticommuting pairs. The canonical form of
$S_{A}$ for each case is:

$\begin{aligned} p&=0\\\ h_{1}&\otimes I\\\ h_{2}&\otimes I\\\ I&\otimes
v_{1}\\\ I&\otimes v_{2}\end{aligned}\qquad\begin{aligned} p&=1\\\ h&\otimes
I\\\ g^{H}&\otimes g^{V}\\\ \bar{g}^{H}&\otimes\bar{g}^{V}\\\ I&\otimes
v\end{aligned}\qquad\begin{aligned} p&=2\\\ g_{1}^{H}&\otimes g_{1}^{V}\\\
g_{2}^{H}&\otimes g_{2}^{V}\\\ \bar{g_{1}}^{H}&\otimes\bar{g_{1}}^{V}\\\
\bar{g_{2}}^{H}&\otimes\bar{g_{2}}^{V}.\end{aligned}$ (19)

Note that symmetries like $h_{1}\otimes I$ decouple the action of $S_{A}$ on
the horizontal legs from its action on the vertical legs. $S(A,n,1)$
automatically inherits these symmetries for any $n$ because tiling them into a
ring becomes trivial. In contrast, symmetries like $g^{V}\otimes g^{H}$ induce
a coupling between the vertical and horizontal legs, in which case we cannot
take the vertical tiling of $A$ for granted.

######  Lemma 6.

If there are $p=0$ anticommuting pairs in the canonical form of $S_{A}$, then
$S(A,n,1)$ has the following local generating set:

$S(A,n,1)=\expectationvalue{h_{1,i},\>h_{2,i}\>|\>i=1,\dots,n}.$ (20)

###### Proof.

When $p=0$, the action of $S_{A}$ on the horizontal legs is decoupled from its
action on the vertical legs, and $S(A,n,1)$ directly inherits the operators
$h_{1,i}$ and $h_{2,i}$ for every site $i$. The set
$\quantity{h_{1,i},\>h_{2,i}\>|\>i=1,\dots,n}$ is abelian with rank $2n$,
therefore it is a generating set for $S(A,n,1)$. ∎

When $p=1$, the situation is more complicated because the action of $S_{A}$ on
the horizontal legs is not decoupled from its action on the vertical legs.
Consequently, the elements of $S(A,n,1)$ depend on how the elements of $S_{A}$
match up on the vertical legs. To determine these vertical tilings, consider
the restriction of $S_{A}$ to subsystem $V$, denoted
$S_{A}^{(V)}=\expectationvalue{v,g^{V},\bar{g}^{V}}.$

By definition, $v$ and $g^{V}$ commute, thus the subgroup
$R_{A}=\expectationvalue{v,g^{V}}\subset S_{A}^{(V)}$ is a 2-qubit stabilizer
group and it may be brought to canonical form with respect to the $\\{u,d\\}$
partition. Depending on the number of anticommuting pairs $q$ in $R_{A}$, we
have the following generating sets.

$\displaystyle q=0:\quad R_{A}=\expectationvalue{\mu\otimes I,I\otimes\delta}$
(21) $\displaystyle q=1:\quad
R_{A}=\expectationvalue{\gamma^{u}\otimes\gamma^{d},\bar{\gamma}^{u}\otimes\bar{\gamma}^{d}},$

where Greek letters are used to differentiate operators which act on
$\\{u,d,l,r\\}$ from those that act on $\\{H,V\\}$, and operators that differ
only by an overbar must anticommute.

######  Lemma 7.

If there are $p=1$ anticommuting pairs in the canonical form of $S_{A}$ and
$q=0$ anticommuting pairs in the canonical form of $R_{A}$, then $S(A,n,1)$
has the following local generating set:

$S(A,n,1)=\expectationvalue{g^{H}_{i},\>h_{i}\>|\>i=1,\dots,n}.$ (22)

###### Proof.

Without loss of generality let $v=\mu\otimes I$, so that
$S_{A}^{(V)}=\expectationvalue{\mu\otimes
I,I\otimes\delta,I\otimes\bar{\delta}}.$ Because $v$ commutes with $g^{V}$ and
$\bar{g}^{V}$, we are free to multiply either of the latter by $v$ without
altering the commutation relations. To account for this fact, we deal with the
quotient group $S_{A}/\\{e,v\\}$. The cosets of $\\{e,v\\}$ are as follows.

$\displaystyle\quantity[g^{V}]_{v}=\\{I\otimes\delta,\mu\otimes\delta\\}$ (23)
$\displaystyle\quantity[\bar{g}^{V}]_{v}=\\{I\otimes\bar{\delta},\mu\otimes\bar{\delta}\\}$
$\displaystyle\quantity[g^{V}\bar{g}^{V}]_{v}=\\{I\otimes\delta\bar{\delta},\mu\otimes\delta\bar{\delta}\\}.$

Note that $\\{\delta,\bar{\delta},\delta\bar{\delta}\\}\in\\{X,Y,Z\\}$ are
independent, giving one coset for each Pauli operator. Clearly, one of
$\delta$, $\bar{\delta}$, or $\delta\bar{\delta}$ must equal $\mu$. Therefore,
without loss of generality let us choose $g^{V}=I\otimes\mu$, and write every
element of $S_{A}^{(V)}$ in terms of $\mu$ and its anticommuting partner
$\bar{\mu}$:

$\displaystyle\quantity[v]_{v}=\\{\mu\otimes I,I\otimes I\\}$ (24)
$\displaystyle\quantity[g^{V}]_{v}=\\{I\otimes\mu,\mu\otimes\mu\\}$
$\displaystyle\quantity[\bar{g}^{V}]_{v}=\\{I\otimes\bar{\mu},\mu\otimes\bar{\mu}\\}$
$\displaystyle\quantity[g^{V}\bar{g}^{V}]_{v}=\\{I\otimes\mu\bar{\mu},\mu\otimes\mu\bar{\mu}\\}.$

Recall that $S(A,n,1)$ is constructed from $S_{A}$ by tiling elements of
$S_{A}$ such that they match on contracted tensor legs. Using (24) it is clear
that the tiling $g^{V}_{i}v_{i+1}$ is valid, because $\mu$ is matched:

(25)

We have $g^{V}_{i}$ coupled to $g^{H}_{i}$ and $v_{i}$ not coupled to anything
on the horizontal partition, therefore $g^{H}_{i}\in S(A,n,1)$ for all $i$, as
shown in Eq. 25. This gives $n$ local generators of $S(A,n,1)$. The remaining
$n$ local generators are given by $h_{i}$, which are trivially elements of
$S(A,n,1)$ because they are decoupled from $V$. ∎

######  Lemma 8.

If there are $p=1$ anticommuting pairs in the canonical form of $S_{A}$ and
$q=1$ anticommuting pairs in the canonical form of $R_{A}$, then the local
generating set of $S(A,n,1)$ has one of two possible forms, depending on
whether $\gamma^{u}=\gamma^{d}$ or $\gamma^{u}\neq\gamma^{d}$.

  1. (i)

If $\gamma^{u}=\gamma^{d}$, then
$S(A,n,1)=\expectationvalue{h_{i},\>g^{H}_{i}g^{H}_{i+1},\>\prod_{i}\bar{g}^{H}_{i}\>|\>i=1,\dots,n}$

  2. (ii)

If $\gamma^{u}\neq\gamma^{d}$, then
$S(A,n,1)=\expectationvalue{h_{i},\>\bar{g}^{H}_{i-1}g^{H}_{i}\bar{g}^{H}_{i+1}\>|\>i=1,\dots,n}$

###### Proof.

The proof is similar to Lemma 7, see Appendix A. ∎

######  Lemma 9.

If there are $p=2$ anticommuting pairs in the canonical form of $S_{A}$, then
$S(A,n,1)$ has the following local generating set:

$S(A,n,1)=\expectationvalue{g^{H}_{1,i}g^{H}_{2,i+1},\>\bar{g}^{H}_{1,i}\bar{g}^{H}_{2,i+1}\>|\>i=1,\dots,n}.$
(26)

###### Proof.

When $p=2$, the restriction of $S_{A}$ to $V$ is
$S_{A}^{(V)}=\\{g_{1}^{V},g_{2}^{V},\bar{g}_{1}^{V},\bar{g}_{2}^{V}\\}$, which
has rank 4. Therefore, we must have $S_{A}^{(V)}=\mathcal{P}_{2}$. Without
loss of generality, we may pick any generating set for $\mathcal{P}_{2}$ that
matches the commutation structure of $S_{A}^{(V)}$. Therefore, let

$g^{V}_{1}=I\otimes X,\quad g^{V}_{2}=X\otimes I,\quad\bar{g}^{V}_{1}=I\otimes
Z,\quad\bar{g}^{V}_{2}=Z\otimes I.$ (27)

Clearly the vertical tilings are $g^{V}_{1,i}g^{V}_{2,i+1}$ and
$\bar{g}^{V}_{1,i}\bar{g}^{V}_{2,i+1}$. On the horizontal partition, these
correspond to $g^{H}_{1,i}g^{H}_{2,i+1}$ and
$\bar{g}^{H}_{1,i}\bar{g}^{H}_{2,i+1}$, respectively, which form a set of $2n$
independent, commuting elements. Therefore, they form a generating set for
$S(A,n,1)$. ∎

######  Lemma 10.

The generating set for $S(A,n,1)$ may always be written in one of four local
forms:

  1. (i)

$\quantity{h_{1,i},\>h_{2,i}\>|\>i=1,\dots,n}$

  2. (ii)

$\quantity{h_{i},\>g^{H}_{i}g^{H}_{i+1},\>\prod_{i}\bar{g}^{H}_{i}\>|\>i=1,\dots,n}$

  3. (iii)

$\quantity{h_{i},\>\bar{g}^{H}_{i-1}g^{H}_{i}\bar{g}^{H}_{i+1}\>|\>i=1,\dots,n}$

  4. (iv)

$\quantity{g^{H}_{1,i}g^{H}_{2,i+1},\>\bar{g}^{H}_{1,i}\bar{g}^{H}_{2,i+1}\>|\>i=1,\dots,n}$

###### Proof.

See Lemmas 6 through 9. The local generating sets from Lemmas 6 and 7 are
isomorphic, so they both fall under (i) in the above. ∎

|
---|---
(i) | (ii)
|
(iii) | (iv)
Figure 7: Examples of each class of local generators from Lemma 10.

####  3.2.5 Bipartite local generators

So far we have presented two forms of $S(A,n,1)$, each with advantages and
disadvantages. The canonical bipartite form directly reveals the channel
capacity of $\mathcal{A}(A,n,1)$ and determines the channel capacity for any
depth via $\Phi_{1}$ and $\Omega_{1}$. However, in general, it may depend on
$n$. The local form removes this dependence on $n$ but does not tell us
anything about channel capacity. In this section, we connect the local form
and the canonical form.

The first step is to partition the local generators into the same left and
right subsets used for the canonical form. When viewed on one partition at a
time, the local generators are no longer guaranteed to be abelian; the degree
to which they fail to be abelian dictates $\Phi_{1}$. Without loss of
generality, consider the left partition. A computer search reveals that up to
gauge equivalence there are nine local generating sets of $S_{L}$ (Table 1),
which we refer to as the standard local generators.

Recall that equivalence classes of $\Phi_{1}$ are labeled by the tuple
$(n,p)$, where $p$ is the number of anticommuting pairs. It follows from (9)
that $p=\rank(S_{L})-n.$ The rank of $S_{L}$ is independent of any particular
generating set, so we may use the local generators to determine $p$ as a
function of $n$. When this is done, we find a one-to-one correspondence
between the non-trivial equivalence classes of $\Phi_{1}$ – that is, classes
where $p\neq 0$ – and the standard local generating sets (Table 1).
Furthermore, by comparing $p(n)$ with $C(A,n,1)$ in Fig. 6 we can see that
each transmission capacity class corresponds to exactly one equivalence class
of $\Phi_{1}$ (Table 2).

$\Phi_{1}$ Class |  Standard Local Generators |  Rank of $S_{L}$ |  $p$
---|---|---|---
a | $\langle Z_{i}\rangle$ | $n$ | 0
a | $\langle X_{i-1}Z_{i}X_{i+1}\rangle$ | $n$ | 0
a | $\langle Z_{i}Z_{i+1},\prod_{i}X_{i}\rangle$ | $n$ | 0
b | $\langle Z_{i},\prod_{i}X_{i}\rangle$ | $n+1$ | 1
c | $\langle Z_{i},X_{i}X_{i+1}\rangle$ | $2n-1$ | $n-1$
d | $\langle Z_{i},X_{i-1}I_{i}X_{i+1}\rangle$ | $\begin{cases}2n-2,\text{ $n$ even}\\\ 2n-1,\text{ $n$ odd}\end{cases}$ | $\begin{cases}n-2\\\ n-1\end{cases}$
e | $\langle Z_{i},X_{i-1}X_{i}X_{i+1}\rangle$ | $\begin{cases}2n-2,\text{ $n$ div. by 3}\\\ 2n,\text{ otherwise}\end{cases}$ | $\begin{cases}n-2\\\ n\end{cases}$
f | $\langle Z_{i},X_{i}X_{i+1},\prod_{i}X_{i}\rangle$ | $\begin{cases}2n-1\quad n\text{ even}\\\ 2n\quad n\text{ odd}\end{cases}$ | $\begin{cases}n-1\\\ n\end{cases}$
g | $\langle Z_{i},X_{i}\rangle$ | $2n$ | $n$
Table 1: Standard local generating sets of $S_{L}$. $\Phi_{1}$ Class |  Transmission Classes
---|---
a | 0
b | 1, 2
c | 4, 6, 7
d | 3, 5
e | 8, 10
f | 9, 11
g | 12
Table 2: Correspondence between transmission classes and equivalence classes
of $\Phi_{1}$.

######  Definition 3.

Let the vector $\overrightarrow{\mathbf{\Phi}}(A)$ be

$\overrightarrow{\mathbf{\Phi}}(A)=\Phi_{1}(A,n)\quad n=1,2,\dots,\infty$ (28)

and define
$\overrightarrow{\mathbf{\Phi}}(A_{1})\sim\overrightarrow{\mathbf{\Phi}}(A_{2})$
if $\Phi_{1}(A_{1},n)\sim\Phi_{1}(A_{2},n)$ for all $n$.

######  Lemma 11.

$\overrightarrow{\mathbf{\Phi}}(A_{1})\sim\overrightarrow{\mathbf{\Phi}}(A_{2})$
if and only if $p(A_{1},n)=p(A_{2},n)$ for values of $n$ such that at least
one is even, one is odd, one is divisible by 3, and one is not divisible by 3.

###### Proof.

If
$\overrightarrow{\mathbf{\Phi}}(A_{1})\sim\overrightarrow{\mathbf{\Phi}}(A_{2})$
then by definition $p(A_{1},n)=p(A_{2},n)$ for all $n$. To prove the converse,
notice that for every standard local generatoring set in Table 1, the
expression for $p$ as a function of $n$ only changes if $n$ switches from even
to odd, or from being divisible by 3 to not divisible by 3. Furthermore, these
dependencies are mutually exclusive, i.e., if $p$ depends on whether $n$ is
even or odd then it does not depend on the divisibility of $n$ by 3, and vice
versa. Therefore, if $p(A_{1},n)=p(A_{2},n)$ for values of $n$ that satisfy
the conditions of the lemma, then $p(A_{1},n)=p(A_{2},n)$ for all $n$, which
implies
$\overrightarrow{\mathbf{\Phi}}(A_{1})\sim\overrightarrow{\mathbf{\Phi}}(A_{2})$.
∎

####  3.2.6 Periodic boundary conditions

The local generators of $S_{L}$ and $S_{R}$ act on a ring of $n$ sites, thus
they are subject to the effects of periodic boundary conditions. These effects
have already played a role in the characterization of
$\overrightarrow{\mathbf{\Phi}}(A)$, e.g. in Table 1 and Lemma 11. The next
object we wish to characterize is $\Omega_{1}(A,n)$, but to do so we require
some discussion of local commutation relations in the presence of PBC.

Consider two local operators $B$ and $C$ acting on a ring of $n$ sites. The
intersection between $B$ and $C$ is the overlap between the support of $B$ and
the support of $C$ for every translation of $C$. Following [22], we define the
intersection between $B$ and $C$ to be regular for a given value of $n$ if its
geometry does not change as $n\to\infty$.

For example, let $B_{i}=X_{i}X_{i+1}X_{i+2}$ and $C_{i}=Z_{i}Z_{i+1}Z_{i+2}$.
When $n$ is large $B_{i}$ and $C_{j}$ slide past each other with no boundary
effects. However, when $n=4$ both ends of $C_{3}$ simultaneously overlap with
$B_{1}$. This type of intersection is not possible when $n\to\infty$, so the
intersection of $B$ and $C$ is irregular for $N=4$.

Regularity of intersection has a direct effect on commutation relations, as
illustrated by the fact that $B_{1}$ and $C_{3}$ commute for $N=4$ and
anticommute for $n\geq 5$. To formalize this phenomenon, we have the following
lemma.

######  Lemma 12.

Consider a $k$-local operator $B_{i}$ and an $\ell$-local operator $C_{j}$
acting on a ring of $n$ sites. The intersection between $B_{i}$ and $C_{j}$ is
irregular for $n<k+\ell-1$ and regular otherwise.

###### Proof.

Let $B_{i}$ have support on sites $i$ through $i+k-1$ and $C_{j}$ have support
on sites $j$ through $j+\ell-1$. The transition between irregular and regular
intersection is given by the case where $B_{i}$ and $C_{k}$ only intersect at
their endpoints, i.e. at sites $k$ and $i$. By inspection, this requires
$n=k+\ell-2$, therefore the intersection is irregular for $n<k+\ell-1$. ∎

######  Corollary 2.

The commutation relations between $B_{i}$ and $C_{j}$ change if and only if
$n$ transitions from the irregular to the regular regime, or vice versa.

###### Proof.

The commutation relations are determined by the overlap between $B_{i}$ and
$C_{j}$, and this overlap only changes if there is a transition from irregular
to regular intersection, or vice versa. ∎

######  Corollary 3.

The commutation relations between the local generators of $S_{L}(A,n,d)$ and
$S_{R}(A,n,d)$ are regular for $n\geq 5$, and irregular for $n<5$.

###### Proof.

Apart from the string-like operators $\prod_{i}\bar{g}_{i}$, every local
generator has support on at most three adjacent sites. Applying the result of
Lemma 12, we have regular intersection for $n\geq 3+3-1=5$. The string-like
operators have periodicity one, so their intersection with the rest of the
island stabilizers is regular for $n\geq 3+1-1=3$. Therefore, the special case
of string-like operators does not violate the general result. ∎

####  3.2.7 Removing the dependence on $n$

Standard Local Generators |  Generators of $\mathcal{Z}_{L}$
---|---
$\langle Z_{i}\rangle$ | $\langle Z_{i}\rangle$
$\langle X_{i-1}Z_{i}X_{i+1}\rangle$ | $\langle X_{i-1}Z_{i}X_{i+1}\rangle$
$\langle Z_{i}Z_{i+1},\prod_{i}X_{i}\rangle$ | $\langle Z_{i}Z_{i+1},\prod_{i}X_{i}\rangle$
$\langle Z_{i},\prod_{i}X_{i}\rangle$ | $\langle Z_{i}Z_{i+1}\rangle$
$\langle Z_{i},X_{i}X_{i+1}\rangle$ | $\langle\prod_{i}Z_{i}\rangle$
$\langle Z_{i},X_{i-1}I_{i}X_{i+1}\rangle$ | $\begin{cases}\langle\prod_{i}Z_{2i},\prod_{i}Z_{2i+1}\rangle&\text{$n$ even}\\\ \langle\prod_{i}Z_{i}\rangle&\text{$n$ odd}\end{cases}$
$\langle Z_{i},X_{i-1}X_{i}X_{i+1}\rangle$ | $\begin{cases}\langle\prod_{i}Z_{3i}Z_{3i+1},\prod_{i}Z_{3i+1}Z_{3i+2}\rangle,\\\ \qquad\qquad\qquad\qquad\text{$n$ div. by 3}\\\ \langle I\rangle,\text{ otherwise}\end{cases}$
$\langle Z_{i},X_{i}X_{i+1},\prod_{i}X_{i}\rangle$ | $\begin{cases}\langle\prod_{i}Z_{i}\rangle&\text{$n$ even}\\\ \langle I\rangle&\text{$n$ odd}\end{cases}$
$\langle Z_{i},X_{i}\rangle$ | $\langle I\rangle$
Table 3: Generators of $\mathcal{Z}_{L}$ for each standard local generating
set of $S_{L}$.

######  Definition 4.

Let the vector $\overrightarrow{\mathbf{\Omega}}(A)$ be

$\overrightarrow{\mathbf{\Omega}}(A)=\Omega_{1}(A,n)\quad n=1,\dots,\infty,$
(29)

and define
$\overrightarrow{\mathbf{\Omega}}(A_{1})\sim\overrightarrow{\mathbf{\Omega}}(A_{2})$
if $\Omega_{1}(A_{1},n)\sim\Omega_{1}(A_{2},n)$ for all $n$.

######  Lemma 13.

$\overrightarrow{\mathbf{\Omega}}(A_{1})\sim\overrightarrow{\mathbf{\Omega}}(A_{2})$
if and only if $\Omega_{1}(A_{1},n)\sim\Omega_{1}(A_{2},n)$ for $n\leq 6$.

###### Proof.

By definition,
$\overrightarrow{\mathbf{\Omega}}(A_{1})\sim\overrightarrow{\mathbf{\Omega}}(A_{2})$
implies $\Omega_{1}(A_{1},n)\sim\Omega_{1}(A_{2},n)$ for all $n$. To prove the
converse, recall that $\Omega_{1}(A,n)$ is the set of commutators of the
elements of $\mathcal{Z}_{L}(A,n,1)$ with the elements of
$\mathcal{Z}_{R}(A,n,1)$ and $P_{R}(A,n,1)$. Furthermore, $\mathcal{Z}_{L}$
dictates the elements of $\mathcal{Z}_{R}$ and $P_{R}$. To see this, note that
$P_{L}$ completes the centralizer of $\mathcal{Z}_{L}$, thus $\mathcal{Z}_{L}$
uniquely determines $P_{L}$. Then, $P_{L}$ uniquely determines $P_{R}$ by
taking the right-hand part of each generator of $P_{L}$. Finally, $P_{R}$
uniquely determines $\mathcal{Z}_{R}$, because $\mathcal{Z}_{R}$ is the center
of $P_{R}$.

In Table 3, we have an expression for the generators of $\mathcal{Z}_{L}$ for
every equivalence class of local tensors. These generating sets show that the
elements of $\mathcal{Z}_{L}(A,n)$ and $\mathcal{Z}_{L}(A,m)$ are equal up to
translation, as long as $n$ and $m$ share the same divisibility by 2 or 3.
Therefore, the elements of every group relevant to $\Omega_{1}(A,n)$ remain
constant up to translation, as long as $n$ retains the same divisibility by 2
or 3.

Having characterized the elements of $\mathcal{Z}_{L}$, $\mathcal{Z}_{R}$ and
$P_{R}$ up to translation, it remains to deal with their commutation
relations. This is not trivial because periodic boundary conditions may play a
role. However, every element of $\mathcal{Z}_{L}$, $\mathcal{Z}_{R}$, and
$P_{R}$ may be written as a product of local generators. Thus, the commutator
$\commutator{x}{y}$ for $x\in\mathcal{Z}_{L}$ and $y\in\mathcal{Z}_{R}\cup
P_{R}$ has a decomposition into pairwise commutators of local generators. By
Corollaries 2 and 3, these pairwise commutators may vary in the irregular
regime of intersection ($n\leq 4$), but they are invariant in the regular
regime of intersection ($n\geq 5$). In other words, for any $n\geq 5$, every
local generator anticommutes with the same finite subset of its neighbors.

It follows that if $n_{0}\geq 5$ and
$\Omega_{1}(A_{1},n_{0})\sim\Omega_{1}(A_{2},n_{0})$, then
$\Omega_{1}(A_{1},n)\sim\Omega_{1}(A_{2},n)$ for all $n$ which have the same
divisibility by 2 and 3 as $n_{0}$. Therefore, to guarantee
$\Omega_{1}(A_{1},n)\sim\Omega_{1}(A_{2},n)$ for all $n$, it suffices to check
just a few values of $n_{0}$ and infer the equivalence of the $\Omega$s for
all other $n$. Coverage of every $n$ in the regular regime can be obtained by
checking at least one value of $n$ even, one value of $n$ odd, one divisible
by 3, and one not divisible by 3. The values $n=5,6$ satisfy these
requirements.

We have made no statement about the irregular regime ($n\leq 4$), thus we must
also check every $n$ in the irregular regime. Putting everything together, we
have that if $\Omega_{1}(A_{1},n)\sim\Omega_{1}(A_{2},n)$ for $n\leq 6$, then
they are equivalent for all $n$. ∎

The standard form of $\Omega_{1}(A,n)$ was computed for $n\leq 6$ and every
stabilizer tensor, and it was found that there are 19 different cases. By
Lemma 13, this implies that there are 19 equivalence classes of
$\overrightarrow{\mathbf{\Omega}}$. With this result, we have a classification
of both $\overrightarrow{\mathbf{\Phi}}$ and
$\overrightarrow{\mathbf{\Omega}}$. The last piece of the puzzle is to show
that if two stabilizer tensors share the same $\overrightarrow{\mathbf{\Phi}}$
and $\overrightarrow{\mathbf{\Omega}}$, then they are guaranteed to share the
same channel capacity for any values of $n$ and $d$.

Upon comparing the $\overrightarrow{\mathbf{\Omega}}$ class and the
transmission class of each stabilizer tensor, it was found that there are 7
different $\overrightarrow{\mathbf{\Omega}}$’s that correspond to transmission
Class 0 and one unique $\overrightarrow{\mathbf{\Omega}}$ for each of the
other transmission classes. Thus, we can assign a unique tuple
$(\overrightarrow{\mathbf{\Phi}},\overrightarrow{\mathbf{\Omega}})$ to the
stabilizer tensors of each non-trivial transmission class. By Algorithm 1,
$\overrightarrow{\mathbf{\Phi}}$ and $\overrightarrow{\mathbf{\Omega}}$ are
sufficient to compute $C(A,n,d)$ for any $n$ and $d$, therefore Proposition 1
must be true for every non-trivial transmission class.

Finally, we deal with the trivial transmission class, i.e. $C(A,n,d)=0$. By
Lemma 3, $C(A,n,1)$ cannot increase with depth, and Tables 1 and 2 show that
every $\overrightarrow{\mathbf{\Phi}}$ associated with transmission Class 0
gives $C(A,n,1)=0$ for all $n$. Therefore, if $A$ is in transmission Class 0,
then $C(A,n,d)=0$ for all $n$ and $d$. This completes the proof of Proposition
1.

##  4 Conclusion

We have reviewed three interesting computational tasks that may be
accomplished via local measurements on stabilizer PEPS: quantum wire, quantum
computation on a resource state, and quantum computation in an SPT phase.
Classifying the capability of stabilizer PEPS with respect to these tasks
would have two beneficial outcomes. First, it would help to place well-known
states like the cluster state into perspective: are they exceptionally useful
for quantum computation, or are they just one species in a zoo of useful
states? Second, by revealing the differences between useful and not-so-useful
classes of states, it might offer insights into the nature of quantum
computation.

In this paper, we have taken the first step in a three-part classification
program by presenting a complete classification of quantum wire in cylindrical
stabilizer PEPS on a 2D square lattice. In particular, we show that the
transmission capacity fits into 13 distinct classes, including the class of
Clifford QCA where $C(n,d)=n$, the trivial class where $C(n,d)=0$, and 11
intermediate classes.

It would also be interesting to explore the transmission capacity of
stabilizer PEPS with different lattice geometries and boundary conditions, but
we leave this direction for future work. With our original geometry, the next
step is to classify the computational power of stabilizer PEPS as resource
states for measurement based quantum computation. To be more precise: given a
stabilizer PEPS, what is the set of unitary gates that may be effected by
local measurement? Finally, one might ask if it is possible to construct an
SPT phase around every stabilizer PEPS. If so, is the computational power
uniform throughout those phases?

## Acknowledgements

This research was undertaken thanks, in part, to funding from the Canada First
Research Excellence Fund, Quantum Materials and Future Technologies Program.
PH is funded by the National Science and Engineering Research Council of
Canada (NSERC). RR is supported through NSERC and US ARO (W911NF2010013).

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##  Appendix A Proof of Lemma 8

See 8

###### Proof of (i).

Recall from (21) that when $q=1$,

$R_{A}=\expectationvalue{\gamma^{u}\otimes\gamma^{d},\bar{\gamma}^{u}\otimes\bar{\gamma}^{d}}.$

If $\gamma^{u}=\gamma^{d}=\gamma$, we have $v=\gamma\otimes\gamma$ and
$g^{V}=\bar{\gamma}^{u}\otimes\bar{\gamma}^{d}$, where $\gamma$,
$\bar{\gamma}^{u}$, and $\bar{\gamma}^{d}$ are different Pauli operators. Let
$\gamma=X$, $\bar{\gamma}^{u}=Z$, and $\bar{\gamma}^{d}=Y$ without loss of
generality. This gives $v=X\otimes X$ and $g^{V}=Z\otimes Y$. The only 2-qubit
Pauli operators that commute with $X\otimes X$ but anticommute with $Z\otimes
Y$ are $X\otimes I$ and $I\otimes X$, which are in the same coset with respect
to $\\{e,v\\}$. Therefore, $\bar{g}^{V}=X\otimes I$ completes the group
$S_{A}^{(V)}$. The cosets with respect to $\\{e,v\\}$ are:

$\displaystyle\quantity[v]_{v}=\\{X\otimes X,I\otimes I\\}$ (30)
$\displaystyle\quantity[g^{V}]_{v}=\\{Z\otimes Y,Y\otimes Z\\}$
$\displaystyle\quantity[\bar{g}^{V}]_{v}=\\{X\otimes I,I\otimes X\\}$
$\displaystyle\quantity[g^{V}\bar{g}^{V}]_{v}=\\{Y\otimes Y,Z\otimes Z\\}.$

It will be convenient to re-label the cosets cyclicly, i.e.
$g^{V}\to\bar{g}^{V}\to g^{V}\bar{g}^{V}\to g^{V}$, because this gives the
vertical tilings $\prod_{i}\bar{g}^{V}_{i}$ and $g^{V}_{i}g^{V}_{i+1}$.
Reading off the action of these tilings on the horizontal partition from (19),
we have $\prod_{i}\bar{g}^{H}_{i}$ and $g^{H}_{i}g^{H}_{i+1}$ as elements of
$S(A,n,1)$. The former accounts for one generator, and the latter for $n-1$
independent generators. Finally, $h_{i}$ provide the remaining $n$ generators.
∎

###### Proof of (ii).

If $\gamma^{u}\neq\gamma^{d}$, then $v=\gamma^{u}\otimes\gamma^{d}$. Without
loss of generality, let $\gamma^{u}=X$ and $\gamma^{d}=Z$ so that $v=X\otimes
Z$. Once again we must find three independent cosets that commute with $v$ but
anticommute pairwise:

$\displaystyle\quantity[v]_{v}=\\{X\otimes Z,I\otimes I\\}$ (31)
$\displaystyle\quantity[g^{V}]_{v}=\\{Y\otimes Y,Z\otimes X\\}$
$\displaystyle\quantity[\bar{g}^{V}]_{v}=\\{X\otimes I,I\otimes Z\\}$
$\displaystyle\quantity[g^{V}\bar{g}^{V}]_{v}=\\{Z\otimes Y,Y\otimes X\\}.$

Upon inspection of (31), the vertical tiling
$\bar{g}^{V}_{i-1}g^{V}_{i}\bar{g}^{V}_{i+1}$ is a local operator. The action
of this operator on the horizontal partition gives
$\bar{g}^{H}_{i-1}g^{H}_{i}\bar{g}^{H}_{i+1}\in S(A,n,1)$. This is a set of
$n$ independent and commuting generators, and $h_{i}$ provide the remaining
$n$ generators. ∎