# Fault-tolerant structures for measurement-based quantum computation on a
network

Yves van Montfort  QuTech, Delft University of Technology, Lorentzweg 1, 2628
CJ Delft, The Netherlands  Sébastian de Bone  QuTech, Delft University of
Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands  QuSoft, CWI, Science
Park 123, 1098 XG Amsterdam, The Netherlands  David Elkouss
<EMAIL_ADDRESS>QuTech, Delft University of Technology, Lorentzweg 1,
2628 CJ Delft, The Netherlands  Networked Quantum Devices Unit, Okinawa
Institute of Science and Technology Graduate University, Okinawa, Japan

###### Abstract

In this work, we introduce a method to construct fault-tolerant _measurement-
based quantum computation_ (MBQC) architectures and numerically estimate their
performance over various types of networks. A possible application of such a
paradigm is distributed quantum computation, where separate computing nodes
work together on a fault-tolerant computation through entanglement. We gauge
error thresholds of the architectures with an efficient stabilizer simulator
to investigate the resilience against both circuit-level and network noise. We
show that, for both monolithic (i.e., non-distributed) and distributed
implementations, an architecture based on the diamond lattice may outperform
the conventional cubic lattice. Moreover, the high _erasure_ thresholds of
non-cubic lattices may be exploited further in a distributed context, as their
performance may be boosted through _entanglement distillation_ by trading in
entanglement success rates against erasure errors during the error-decoding
process. These results highlight the significance of lattice geometry in the
design of fault-tolerant measurement-based quantum computing on a network,
emphasizing the potential for constructing robust and scalable distributed
quantum computers.

##  I Introduction

Large-scale quantum computation with low error rates requires handling noise
in a correct and efficient manner—one would like to _fault-tolerantly_
transmit, store, and process quantum information with quantum computing
hardware. Quantum error-correction encompasses methods to achieve fault
tolerance from faulty hardware. Topological error-correction codes, including
surface codes, are a promising avenue to achieve this goal, as these codes
have high error thresholds against local errors, and only require nearest-
neighbor interactions between qubits in a two-dimensional layout.

Surface codes achieve fault-tolerance by repeatedly combining measurement
outcomes of consecutive rounds of stabilizer measurements [1]. Geometrically,
the addition of a time dimension to a two-dimensional decoding (i.e.,
_syndrome_) graph creates a three-dimensional structure, that may be
interpreted as a noisy quantum channel where logical information is propagated
in the direction of time. For example, the conventional planar or toric
surface code may be constructed from qubits on a square lattice, but its
corresponding decoding graphs are three-dimensional cubic lattices.

_Foliation_ [2] is a method to transform any surface code to a three-
dimensional _cluster state_ that forms the resource for fault-tolerant
_measurement-based quantum computation_ (MBQC), with the same geometry as the
corresponding logical quantum channel of the surface code [3, 4]. Although
foliated codes can be interpreted as having replaced time with another spatial
dimension, the resulting cluster states can be initialized and consumed in
arbitrary directions over time, lifting the rigid duality of time and space of
surface codes. A concrete example is so-called _interleaving_ in fusion-based
quantum computation [5], which bears many resemblances to the measurement-
based architectures considered here.

However, foliation does not exhaust all possible fault-tolerant cluster
states—i.e., there are _non-foliated_ three-dimensional cluster states that
cannot be constructed from a two-dimensional surface code. We study these
types of lattices because they may produce higher fault-tolerant error
thresholds than conventional surface codes, at least when assuming simple
combinations of independent and identically distributed (i.i.d.) single-qubit
and measurement errors [6]. In practice though, non-foliated lattices will not
necessarily produce higher error thresholds, because faulty operations during
cluster state preparation and measurement typically introduce highly coupled
and non-identically distributed errors. Both the complexity of the error
decoder (through the syndrome graph) and the complexity of the error model
(through the quantum circuit) eventually determine the error threshold.

Because state-of-the-art hardware is currently not capable of realizing large
cluster states that can achieve sufficiently low logical error rates, some
research has focused its attention to modular implementations of fault-
tolerant MBQC states [7, 8, 9]. The entire cluster state is prepared from
resource states that are generated by small, separate devices, and entangling
operations between devices create the entanglement to encode the entire
cluster state. Several physical systems are suitable for modular MBQC
architectures via optical interfaces [10].

In this paper, we explore fault-tolerant cluster states under more realistic
noise models compared to previous investigations [6]. This brings their
realization in MBQC closer to reality. We numerically estimate fault-tolerant
thresholds of previously proposed cluster states [6] for both _monolithic_
(i.e., non-distributed) and distributed implementations. We use an efficient
stabilizer simulator and the Union-Find decoder [11] to evaluate the
performance of our architectures. In our distributed noise models, we consider
circuit-level noise during state preparation, measurement noise of single-
qubit measurements, and _network noise_ between nodes introduced by qubits
prepared in the Greenberger-Horne-Zeilinger (GHZ) basis.

We investigate the first step towards including entanglement distillation for
one of the distributed cluster states considered and find that including
distillation is particularly effective in the context of measurement-based
fault tolerance. This is due to the high _erasure_ -type error thresholds of
non-foliated lattices that may shield against probabilistic entanglement
generation resulting from distillation.

##  II Cluster state construction

In this section, we briefly summarize the mathematical description of cluster
states used throughout this paper along the same lines as Fujii [1]. We assume
that the reader is already familiar with definitions of the Pauli group, the
Clifford group and the stabilizer group, which may be found in detail in
Nielsen and Chuang [12].

In Sec. II.1, we introduce relevant concepts of a $\mathbb{Z}_{2}$ chain
complex. In Sec. II.2, we define a fault-tolerant three-dimensional cluster
state using such a chain complex. In Sec. II.3, we describe the error
correction process for three-dimensional cluster states.

###  II.1 $\mathbb{Z}_{2}$ chain complex

A $\mathbb{Z}_{2}$ chain complex starts with a definition of _vector spaces_
$C_{i}$, each constructed over the field $\mathbb{Z}_{2}$ and indexed with the
_dimension_ $i\in\\{0,1,\dots,D\\}$. These vector spaces form a sequence
$C_{D}\rightarrow C_{D-1}\rightarrow\dots\rightarrow C_{0}$, where subsequent
pairs are connected by homomorphisms $\partial_{i}:C_{i}\rightarrow C_{i-1}$
called _boundary operators_.

In this work, we construct chain complexes over a three-dimensional set
$\mathcal{S}=\left(Q,F,E,V\right)$ of cells (i.e. volumes)
$Q=\\{\mathbf{q}_{k}\\}$, faces $F=\\{\mathbf{f}_{k}\\}$, edges
$E=\\{\mathbf{e}_{k}\\}$ and vertices $V=\\{\mathbf{v}_{k}\\}$. Cells, faces,
edges and vertices form the basis elements of the vector spaces $C_{3}$,
$C_{2}$, $C_{1}$ and $C_{0}$, respectively:

${C_{3}}$${C_{2}}$${C_{1}}$${C_{0}}$${\\{\mathbf{q}_{k}\\}}$${\\{\mathbf{f}_{k}\\}}$${\\{\mathbf{e}_{k}\\}}$${\\{\mathbf{v}_{k}\\}}$$\scriptstyle{\partial_{3}}$$\scriptstyle{\partial_{2}}$$\scriptstyle{\partial_{1}}$

An element of $C_{i}$ is called an $i$-chain and notated as $\mathbf{c}_{i}$.
Such a chain can be considered a linear combination of basis elements of
$C_{i}$ with $\mathbb{Z}_{2}$ coefficients. For example, a 1-chain
$\mathbf{c}_{1}\in C_{1}$ is a combination of edges:

$\mathbf{c}_{1}=\sum_{k}z_{k}\mathbf{e}_{k}\equiv\begin{bmatrix}z_{0}&z_{1}&\cdots\end{bmatrix}^{T}\quad\text{where}\quad
z_{k}\in\mathbb{Z}_{2}.$ (1)

Here, we take the vector representation with respect to the basis
$E=\\{\mathbf{e}_{k}\\}$. Analogous definitions hold for the remaining vector
spaces and their basis elements.

Boundary operators $\partial_{i}:C_{i}\rightarrow C_{i-1}$ are linear
operators:

$\partial_{i}\left(\mathbf{c}_{i}+\mathbf{c}_{i}^{\prime}\right)=\partial_{i}\mathbf{c}_{i}+\partial_{i}\mathbf{c}_{i}^{\prime}.$
(2)

For chain complexes defined on $\mathcal{S}$, boundary operators have an
intuitive geometric interpretation. A cell $\mathbf{q}_{k}$ is mapped to the
faces $\\{\mathbf{f}_{j}\\}$ that enclose it, a face $\mathbf{f}_{k}$ is
mapped to the edges $\\{\mathbf{e}_{j}\\}$ that form its boundary and an edge
$\mathbf{e}_{k}$ is mapped to its endpoints $\\{\mathbf{v}_{j}\\}$, which will
always be a pair of vertices on the graph $\left(E,V\right)$. By definition,
two boundary maps applied in succession on a chain $\mathbf{c}_{i}$ produce
the zero map

$\partial_{i-1}\partial_{i}=0,$ (3)

no matter the choice of $\mathbf{c}_{i}$. We equip vector spaces $C_{i}$ with
the standard inner product

$\mathbf{c}_{i}\cdot\mathbf{c}_{i}^{\prime}\equiv\mathbf{c}_{i}^{T}\mathbf{c}_{i}^{\prime},$
(4)

that is, a dot product with addition modulo 2 over the pairwise multiplied
coefficients. The inner product may be geometrically interpreted as a basis-
independent parity measurement of the “overlap” of two $i$-chains. An
$i$-chain $\mathbf{c}_{i}$ with a zero boundary (i.e.,
$\partial_{i}\mathbf{c}_{i}=0$) is called an $i$-_cycle_ (or simply, _cycle_).
Note that $i$-cycles form a group under element-wise addition. A cycle
$\mathbf{c}_{i}$ is called _trivial_ whenever there exists some chain
$\mathbf{c}_{i+1}\in C_{i+1}$, such that
$\mathbf{c}_{i}=\partial_{i+1}\mathbf{c}_{i+1}$. Cycles that cannot be formed
in this way are called _non-trivial_. Trivial cycles form a normal subgroup of
all cycles, such that the quotient groups

$H_{i}\equiv\ker\partial_{i}/\operatorname{Im}\partial_{i+1}$ (5)

divide cycles into equivalence classes that are trivially related. The groups
$H_{i}$ are called _homology groups_ , where two chains $\mathbf{c}_{i}$ and
$\mathbf{c}_{i}^{\prime}$ belong to the same class whenever
$\mathbf{c}_{i}=\mathbf{c}_{i}^{\prime}+\partial_{i+1}\mathbf{c}_{i+1}$.

The _dual complex_ is another sequence
$\overline{C}_{D}\rightarrow\overline{C}_{D-1}\rightarrow\dots\rightarrow\overline{C}_{0}$,
where each $\overline{C}_{i}$ shares the structure of $C_{D-i}$. Associated
with the dual complex are the dual boundaries
$\overline{\partial}_{i}:\overline{C}_{i}\rightarrow\overline{C}_{i-1}$.
Chains in the dual complex are called _dual chains_ or _co_ chains, whereas
those referring to the original complex are _primal_ chains or, more
succinctly, chains. For the three-dimensional lattice $\mathcal{S}$, vertices
map to dual cells $\left(V\rightarrow\overline{Q}\right)$, edges map to dual
faces $\left(E\rightarrow\overline{F}\right)$, faces map to dual edges
$\left(F\rightarrow\overline{E}\right)$, and cells map to dual vertices
$\left(Q\rightarrow\overline{V}\right)$, such that:

${\overline{C}_{0}}$${\overline{C}_{1}}$${\overline{C}_{2}}$${\overline{C}_{3}}$${\\{\mathbf{\overline{v}}_{k}\\}}$${\\{\mathbf{\overline{e}}_{k}\\}}$${\\{\mathbf{\overline{f}}_{k}\\}}$${\\{\mathbf{\overline{q}}_{k}\\}}$$\scriptstyle{\overline{\partial}_{1}}$$\scriptstyle{\overline{\partial}_{2}}$$\scriptstyle{\overline{\partial}_{3}}$

The dual boundaries on $\overline{\mathcal{S}}=\left(V,E,F,Q\right)$ behave
similarly to their primal counterparts. With a slight abuse of notation, a
dual cell $\mathbf{\overline{q}}_{k}=\mathbf{v}_{k}$ is mapped to the faces
$\\{\mathbf{\overline{f}}_{j}\\}$ that enclose it, which correspond to the
primal edges $\\{\mathbf{e}_{j}\\}$ _incident_ to $\mathbf{v}_{k}$. A dual
edge $\mathbf{\overline{e}}_{k}=\mathbf{f}_{k}$ is mapped to the endpoints
$\\{\mathbf{\overline{v}}_{j}\\}$, which are the cells $\\{\mathbf{q}_{j}\\}$
adjacent to $\mathbf{f}_{k}$. Similar to cycles, $i$-_cocycles_ are dual
chains $\mathbf{\overline{c}}_{i}$ that have no coboundary. Dual boundaries
share a similar concept as homology, called _cohomology_. Cohomology groups
are formed in the same way as homology groups (Eq. 5), by replacing cycles
$\ker\partial_{i}$ and trivial cycles $\operatorname{Im}\partial_{i+1}$ by the
dualized versions $\ker\overline{\partial}_{i}$ and
$\operatorname{Im}\overline{\partial}_{i+1}$, such that

$\overline{H}_{i}\equiv\ker\overline{\partial}_{i}/\operatorname{Im}\overline{\partial}_{i+1}.$
(6)

###  II.2 Three-dimensional cluster states

We now make use of chain complexes $C_{3}\rightarrow C_{2}\rightarrow
C_{1}\rightarrow C_{0}$ to construct a fault-tolerant cluster state on a
three-dimensional lattice $\mathcal{S}$, given the following recipe:

  1. 1.

Place qubits on all basis elements of $C_{2}$ and $\overline{C}_{2}$, i.e., on
all faces $\mathbf{f}_{k}$ and edges $\mathbf{e}_{k}$ of the lattice. Pauli
operators on the qubits of an $i$-chain $\mathbf{c}_{i}$ are denoted as
$\sigma(\mathbf{c}_{i})\equiv\prod_{k}\sigma^{z_{k}}$, where
$\sigma\in\\{X,Y,Z\\}$. Qubits with $z_{k}=0$ carry identity, and $z_{k}=1$
carry $\sigma$.

  2. 2.

For each face $\mathbf{f}_{k}$, define a _primal_ stabilizer generator
$g_{k}=X(\mathbf{f}_{k})Z(\partial_{2}\mathbf{f}_{k})$. These operators
generate stabilizers $X(\mathbf{c}_{2})Z(\partial_{2}\mathbf{c}_{2})$ for all
2-chains $\mathbf{c}_{2}\in C_{2}$.

  3. 3.

For each edge $\mathbf{e}_{k}$, define a _dual_ stabilizer
$\overline{g}_{k}=X(\mathbf{e}_{k})Z(\overline{\partial}_{2}\mathbf{e}_{k})$.
These operators generate stabilizers
$X(\mathbf{\overline{c}}_{2})Z(\overline{\partial}_{2}\mathbf{\overline{c}}_{2})$
for all 2-cochains $\mathbf{\overline{c}}_{2}\in\overline{C}_{2}$.

Figure 1: A primal and dual stabilizer generator for a three-dimensional
cluster state as defined in Sec. II.2. The primal generator
$X(\mathbf{f}_{k})Z(\partial_{2}\mathbf{f}_{k})$ is associated with the
highlighted face on the left of the figure. The dual generator
$X(\mathbf{e}_{k})Z(\overline{\partial}_{2}\mathbf{e}_{k})$ is associated with
the highlighted edge on the right of the figure. Stabilizer generators
associated with the other faces and edges are not shown explicitly.

We depict an example of a primal and dual stabilizer generator associated with
a single face $\mathbf{f}_{k}$ and edge $\mathbf{e}_{k}$ in Fig. 1. Note that
each stabilizer generator carries a Pauli-$X$ operator on some face (edge)
qubit and Pauli-$Z$ operators on its direct neighbors on the (dual) boundary.
This stabilizer composition is commonly associated with _cluster_ or _graph_
states [13]. Logical operators are derived from elements of the (co)homology
groups of the underlying chain complex. For the logical identity channel,
logical $X$-type operators correspond with elements of the homology group
$H_{1}$, and logical $Z$-type operators correspond with elements of the
cohomology group $\overline{H}_{1}$. The construction of a fault-tolerant
channel from a cluster state that carries logical information is subtle: for
details we refer the reader to Ref. [1]. We note that logical qubits are
usually introduced by creating lattice boundaries—such as holes inside the
bulk—by switching off stabilizers on these boundaries.

###  II.3 Error correction

For the cluster states described above, we can construct the primal (dual)
_error syndrome_ by measuring out the qubits on the faces (edges) in the
Pauli-$X$ basis. For each face $\mathbf{f}_{k}$ (edge $\mathbf{e}_{k}$), this
leads to a measurement outcome $\mu_{k}\in\\{+1,-1\\}$
($\overline{\mu}_{k}\in\\{+1,-1\\}$). Error syndromes are constructed from
measurement outcomes in the following way:

  1. 1.

For each cell $\mathbf{q}_{k}$, we produce a _primal_ error syndrome $m_{k}$
as the product of measurement outcomes of the qubits that lie on its boundary:
$m_{k}=\prod_{\mathbf{f}_{j}\in\partial_{3}\mathbf{q}_{k}}\mu_{j}$. Note that
$m_{k}$ represents the measurement outcome of the stabilizer
$s_{k}\equiv\prod_{\mathbf{f}_{j}\in\partial_{3}\mathbf{q}_{k}}g_{j}=X(\partial_{3}\mathbf{q}_{k})Z(\partial_{2}\partial_{3}\mathbf{q}_{k})=X(\partial_{3}\mathbf{q}_{k})$.

  2. 2.

For each dual cell $\mathbf{\overline{q}}_{k}$ (i.e., vertex
$\mathbf{v}_{k}$), we produce a _dual_ error syndrome
$\overline{m}_{k}=\prod_{\mathbf{e}_{j}\in\overline{\partial}_{3}\mathbf{v}_{k}}\overline{\mu}_{j}$.
The syndrome represents the measurement outcome of the stabilizer
$\overline{s}_{k}\equiv\prod_{\mathbf{e}_{j}\in\overline{\partial}_{3}\mathbf{v}_{k}}\overline{g}_{j}=X(\overline{\partial}_{3}\mathbf{v}_{k})$.

Note that in the absence of errors, all error syndromes produce outcomes
$m_{k}=+1$ and $\overline{m}_{k}=+1$. Since all cluster state qubits are
measured out in the Pauli-$X$ basis, we can restrict ourselves to
probabilistic Pauli-$Z$ errors—i.e., phase-flips—on the qubits prior to
syndrome measurement. These errors can appear _phenomenologically_ as a result
of i.i.d. sampling, or for example as a result of depolarizing or dephasing
noise after a circuit or network operation. This model also includes
measurement errors, because such an error is equivalent to a probabilistic $Z$
gate before an $X$ basis measurement. Measurement outcomes are then fully
described by (anti)commutation relations of Pauli operators.

For notational convenience, let a particular set of primal error syndromes be
described by a vector $\mathbf{m}$ over the field $\mathbb{Z}_{2}$, where the
$k$th element $m_{k}$ maps measurement outcomes $\\{+1,-1\\}\mapsto\\{0,1\\}$.
Given a set of Pauli-$Z$ errors on dual edges (primal faces) described by a
dual chain $Z(\mathbf{\overline{c}}_{1})$, the entire error syndrome takes the
convenient form

$\mathbf{m}=\overline{\partial}_{1}\mathbf{\overline{c}}_{1}.$ (7)

That is, the primal error syndrome corresponds to the _boundary_ of all the
$Z$-type errors on face qubits $\mathbf{f}_{k}$. Similarly, the dual syndrome
$\overline{\mathbf{m}}=\partial_{1}\mathbf{c}_{1}$ is the boundary of
Pauli-$Z$ errors described by the chain $Z(\mathbf{c}_{1})$.

The decoding problem may now be stated as follows. Given a pair of primal and
dual syndrome outcomes $\\{\mathbf{m},\overline{\mathbf{m}}\\}$, we identify
recovery chains $\mathbf{\overline{r}}_{1}$ and $\mathbf{r}_{1}$ such that
$\overline{\partial}_{1}(\mathbf{\overline{r}}_{1}+\mathbf{\overline{c}}_{1})=0$
and $\partial_{1}(\mathbf{r}_{1}+\mathbf{c}_{1})=0$—i.e., the sum of recovery
and error chains form _cycles_ in the corresponding chain complex. We identify
a _logical failure_ whenever decoding introduces a logical $X$-type and/or
$Z$-type error across the channel, which occurs whenever the sum of a recovery
and error chain forms a _non-trivial_ cycle.

We emphasize that we only evaluate fault tolerance of the cluster state as a
pure _quantum memory_. That is, the noise thresholds that we determine
exclusively assess the state’s ability to protect logical information and do
not, e.g., include the operations required to encode this logical information
or operate on it.

##  III Methods

In the current work, we are interested in crystalline cluster states that are
built from cellulations of flat three-dimensional space. There are various
methods to find such structures. Two approaches that have previously been used
are the splitting procedures on a known (foliated) structure, such as the
cubic cluster state [6], and an algebraic approach based on combinatorial
tiling theory [14]. We briefly discuss the former method below and show some
of the lattices that were found through this method in Fig. 2. Although we
have not used the latter method to construct new cluster states, we emphasize
that the zoo of fault-tolerant cluster states merits further investigation
under noise models considered here.

Below, we first discuss an extension of the concepts of the chain complex as
discussed in Sec. II.1. By adding indices that describe translational
symmetry, we can use the chain complex to describe unit cells of a lattice
that generate a full lattice. We discuss this method in Sec. III.1. In Sec.
III.2, we describe the _cell-vertex splitting_ operation of Nickerson and
Bombín [6] in the context of this unit cell complex—this allows us to
transform unit cells that describe three-dimensional lattices. In Sec. III.3,
we define a _face-edge splitting_ operation that allows us to replace cluster
state qubits with entangled states or Bell measurements. In Sec. III.4, we
introduce the noise models used for monolithic (i.e., circuit-level) and
network noise, and describe our method for generating entanglement in a
distributed cluster state. In Sec. III.5, we discuss how we use the stabilizer
formalism to model and transfer Pauli errors in the circuits that we use to
construct and measure cluster states, and we elaborate on the numerical
aspects of our model and simulations.

Figure 2: Cluster states obtained through splitting. The cubic cluster state
can be created by foliating the standard toric surface code. The diamond
cluster state is obtained through two splits of the primal and dual vertex in
the cubic unit cell. The complex is regular and self-dual, with each face
connected to six edges. The double-edge cubic cluster state is obtained
through multiple simple splits of primal and dual vertices. Each face is
“double”-sided, supporting two different qubit that are connected to eight
surrounding edges.

###  III.1 Unit cell complex

The _unit cell complex_ is the set of basis elements (atoms) together with
their boundary relations (bonds) in the crystal that forms a block with
translation symmetry along the sides of the unit cell. We use the _Miller
index_ notation with square brackets $\left[abc\right]$ or (slightly
unorthodox yet succinct) $\left[\mathbf{r}\right]$ for a translation
$\mathbf{r}\equiv a\mathbf{x}+b\mathbf{y}+c\mathbf{z}$ along the _lattice
vectors_ $\mathbf{x}$, $\mathbf{y}$ and $\mathbf{z}$ that form the sides of
the unit cell. Negative indices are denoted in the usual way as
$\left[\overline{abc}\right]$ or $\left[\mathbf{\overline{r}}\right]$ for a
translation $\mathbf{\overline{r}}\equiv-a\mathbf{x}-b\mathbf{y}-c\mathbf{z}$.

The construction of the unit cell complex follows from the choice of lattice
vectors. The subset of basis elements $\\{\mathbf{b}_{i}\\}$ in $C_{i}$ that
are equivalent under translations $\mathbf{r}$ is mapped to a single
_quotient_ element $\mathbf{q}_{i}$, which serves as a basis vector for a new
vector space $Q_{i}$ over $\mathbb{Z}_{2}$. Similarly, the boundary relation
between elements $\left(\mathbf{b}_{i}\right)_{n}$ and
$\left(\mathbf{b}_{i-1}\right)_{m}$ is mapped to a relation to their quotient
elements $\left(\mathbf{q}_{i}\right)_{n}$ and
$\left(\mathbf{q}_{i-1}\right)_{m}$ in the form of a _quotient_ boundary map
$\partial_{i}^{\left[\mathbf{r}\right]}:Q_{i}\mapsto Q_{i-1}$. Because some of
the boundary relations are present between elements _across_ two unit cells
(such as a face with boundary edges from adjacent unit cells), quotient
boundaries have a Miller index $\left[\mathbf{r}\right]$ that represents the
translation $\mathbf{r}$ required to jump to its neighboring unit cell.
Intracellular boundaries are encoded by $\partial_{i}^{\left[0\right]}$,
whilst intercellular boundaries are encoded by
$\partial_{i}^{\left[abc\right]}$ for a non-zero translation $\mathbf{r}\equiv
a\mathbf{x}+b\mathbf{y}+c\mathbf{z}$. A more detailed description of the
quotient boundaries in the unit cell complex can be found in App. A.

Given a unit cell complex, an _embedding_ is a map that takes each $Q_{i}$ and
the quotient maps $\partial_{i}^{\left[\mathbf{r}\right]}$ to a crystalline
chain complex $C_{3}\rightarrow C_{2}\rightarrow C_{1}\rightarrow C_{0}$, with
lattice dimensions given by the embedding. For a periodic lattice of $N\equiv
N_{\mathbf{x}}\times N_{\mathbf{y}}\times N_{\mathbf{z}}$ unit cells along the
$\mathbf{x}$, $\mathbf{y}$, and $\mathbf{z}$ lattice directions, respectively,
vector spaces of chains take the form

$C_{i}=Q_{i}^{\oplus N}=Q_{i}\otimes L.$ (8)

Here, $L\equiv\mathbb{Z}_{2}^{\oplus N}$ is an $N$-dimensional vector space
over $\mathbb{Z}_{2}$. The $N$ basis vectors of $L$ represent $N$ lattice
points, such that linear combinations in $L$ may be associated with the subset
of lattice points that have non-zero coefficients. Intuitively, the embedding
is realized by repeating the unit cell elements $Q_{i}$ over each lattice
point given by a displacement vector $\mathbf{r}$. We formalize this process
in App. B.

###  III.2 Cell-vertex splitting

The splitting procedure of Nickerson and Bombín [6] may be phrased in terms of
the unit cell complex introduced in Sec. III.1. A visual example of a split in
two dimensions can be seen in App. B, Fig. 14a and b; splitting each face of a
square lattice diagonally results in the triangular lattice shown adjacently.
Here, we review the general case of an $n$-split in three dimensions, with a
simple split following from the case $n=1$. Importantly, the splitting number
$n$ alone is not sufficient to uniquely characterize a split: one should also
specify the new boundary relations between split vertices and edges. Denote
the split vertex with $\mathbf{v}_{0}$. The recipe for an $n$-split is then as
follows:

  1. 1.

Let $E^{\prime}=\set{(\mathbf{v}_{j},\mathbf{v}_{0})}{\mathbf{v}_{j}\in
N_{0}}$ be the set of incident edges on $\mathbf{v}_{0}$ with neighborhood
$N_{0}$. Choose $n$ disjoint subsets $E_{i}^{\prime}\in E^{\prime}$ ($i=1\dots
n$) that will each connect to a new vertex.

  2. 2.

Create $n$ new vertices $\mathbf{v}_{i}$, and connect each $\mathbf{v}_{i}$ to
the incident edges $E_{i}^{\prime}$. The $\mathbf{v}_{0}$ vertex connects to
the remaining edges
$E_{0}^{\prime}=E^{\prime}\setminus\bigcup_{i}E_{i}^{\prime}$, which might be
the empty set.

  3. 3.

Create $n$ new edges $\mathbf{e}_{i}=(\mathbf{v}_{i},\mathbf{v}_{0})$. The
corresponding boundary relations are encoded in
$\partial_{1}^{\left[0\right]}$ of the unit cell complex with Miller index
$[0]$.

  4. 4.

Fix the remaining boundary maps $\partial_{2}^{\left[\mathbf{r}\right]}$. That
is, for each $\mathbf{v}_{i}$ and $\forall\mathbf{r}$, calculate the dual
boundary
$\mathbf{c}_{2}\equiv\sum_{\mathbf{p}}\overline{\partial}_{2}^{\left[\mathbf{p}\right]}\overline{\partial}_{3}^{\left[\mathbf{r-p}\right]}\;\mathbf{v}_{i}$.
By the zero map conditions (Eq. (12) in App. A), the right-hand side should be
zero. That is, we connect faces $\mathbf{f}_{j}\in\mathbf{c}_{2}$ to the newly
created edge $\mathbf{e}_{i}$ with Miller index $\mathbf{\overline{r}}$.

###  III.3 Face-edge splitting

The cell-vertex splitting procedure described before in Sec. III.2 changes
both the number of syndromes and the connectivity between syndromes in the
syndrome graph. We can define an additional splitting operation on faces (dual
edges) of such a complex. We discuss this operation in this section.

Usually, a cluster state as described in Sec. II.2 is constructed with the aid
of C$Z$ gates on qubits initialized in the $\ket{+}$ state. Alternatively, one
may replace C$Z$ gates with other entangling operations that lead to the same
stabilizer states.

Consider the subgraph of a graph state as in Fig. 3. Qubits at odd positions
are marked with integers $i\in\\{1,2,\dots,n\\}$ and their right neighbors at
even positions with a primed index $i^{\prime}\neq n^{\prime}$. In this graph
state, odd qubits have an arbitrary number of neighbors, whereas even qubits
only neighbor the two odd qubits on either side.

Figure 3: A subgraph of $2n-1$ qubits. Qubits marked primed indices are
measured in the $X$ basis. Unmeasured qubits are connected arbitrarily to the
rest of the graph. If one assumes that every measurement outcome $m=0$, the
unmeasured qubits can be initialized in an $n$-partite Bell/GHZ state.

After measuring the even qubits $i^{\prime}$ in the Pauli-$X$ basis, the post-
measurement $m_{i^{\prime}}=0$ graph state stabilizers are

$\begin{split}S_{\text{post}}=\langle&X_{1^{\prime}},X_{2^{\prime}},\dots,X_{n-1^{\prime}},\\\
&\prod_{i=1}^{n}(X_{i}\prod_{j\in N(i)}Z_{j}),\\\
&Z_{1}Z_{2},Z_{2}Z_{3},\dots,Z_{n-1}Z_{n}\rangle.\end{split}$ (9)

Here, $N(i)=\set{j}{(i,j)\in E}$ are the qubits connected to the odd qubit $i$
_outside_ the subgraph depicted in Fig. 3. The disentangled measured qubits
play no further role in the graph state stabilizers. In practice, one may
replace these “virtual” qubits with measurement outcomes $m=0$ and initialize
the unmeasured qubits in the state stabilized by Eq. (9)—the resulting state
is the same. In constructing graph states, C$Z$ gates transform an $X$-type
stabilizer $\prod_{i=1}^{n}X_{i}\mapsto\prod_{i=1}^{n}(X_{i}\prod_{j\in
N(i)}Z_{j})$, whilst leaving the $Z$-type stabilizers untouched. This means
that we can alternatively initialize the odd qubits $i$ as an $n$-qubit
$\Ket{\text{GHZ}_{n}}$ state stabilized by
$\langle\prod_{i=1}^{n}X_{i},Z_{1}Z_{2},Z_{2}Z_{3},\dots,Z_{n-1}Z_{n}\rangle$,
before applying the C$Z$ gates to qubits outside the subgraph. If $n=2$, one
can initialize with the bipartite Bell state $\Braket{X_{1}X_{2},Z_{1}Z_{2}}$.

The above procedure shows how the subgraph in Fig. 3 need not be initialized
through C$Z$ gates, so long as there is a protocol that can create
$\Ket{\text{GHZ}_{n}}$. Our new splitting procedure on faces produces
subgraphs like Fig. 3 in a systematic way. Importantly, the closed-cell
stabilizers of the cluster state stay intact on the split geometry.

Figure 4: Face splitting on a square. (a) Without a split, the square
represents a monolithic cluster state without distributed entanglement. The
corresponding circuit of $\Ket{+}$ state initialization, C$Z$ gates, and
Pauli-$X$ basis measurement is also drawn. (b) A hybrid approach. The
distributed face qubit is split over two parties represented by two faces,
forming a bipartite GHZ or Bell state. The parity check is performed through
two bilocal C$Z$ gates of each face to its neighboring qubits and two local
measurements. (c) A fully distributed approach. A face is split into four
parties that share a GHZ state. Each party performs a single C$Z$ gate to its
neighbor and performs a measurement.

Like a cell-vertex split, a face-edge split subdivides an existing face into
two or more parts, adding new edges to separate the newly created faces. We
give an example of this procedure on a square in Fig. 4. The newly created
edges each support an additional qubit, always laying adjacent to two faces.
Therefore, the subgraph supported by split edges and faces is a chain in the
form of Fig. 3. We may replace the cluster state supported by the $n$
connected faces with an $n$-partite GHZ state. Because the removed qubits
correspond to “virtual” $m=0$ measurement outcomes, their even parity plays no
further role in the evaluation of error syndromes.

Figure 5: Face-edge splitting on a cubic cell. Starting from the left, a
monolithic cluster state is transformed into a distributed cluster state by a
four-partite split for each of its faces (see also Fig. 4). Performing the
same procedure for dual faces (primal edges) produces a cluster state that is
fully distributed, where each network node contains two cluster state qubits.

We may extend this procedure to both the primal and dual complex, producing
GHZ states on both faces and edges. An example based on the cubic cluster
state is given in Fig. 5. Starting from a monolithic architecture, a full
4-partite split of primal faces produces an architecture with nodes containing
five qubits on a single edge and each of the adjacent split faces. Nodes are
entangled with one another by GHZ states on every face. We may perform the
same procedure for dual faces (primal edges), further reducing the number of
qubits in each node to two.

We do not consider this in the rest of the paper but note that, instead of
initializing the qubits introduced by a face-edge split as an entangled state
and measuring them out individually, one can alternatively initialize these
qubits regularly in $\ket{+}$ and measure them out with a joint _(Type-II)
fusion measurement_ [15, 16, 17]. This provides one with a method to transform
(fault-tolerant) cluster states into so-called _fusion networks_ that form the
basis of _fusion-based quantum computing_ [9].

###  III.4 Circuit-level and network noise

In this section, we describe the noise models used for monolithic and
distributed threshold calculations. For the monolithic simulations, the entire
cluster state is built from $\Ket{+}$ state preparation, followed by C$Z$
gates between every connected face-edge pair in the cluster state, concluded
with a Pauli-$X$ basis measurement of every qubit. In this model, we do not
consider the effects of memory decoherence. For circuit-level noise, the
following noise sources are included:

  1. 1.

Noisy state preparation as a classical mixture
$(1-p_{\textrm{p}})\outerproduct{+}{+}+p_{\textrm{p}}\outerproduct{-}{-}$.
That is, with probability $p_{\textrm{p}}$, apply a phase-flip error on
$\Ket{+}$.

  2. 2.

Two-qubit depolarizing noise following every C$Z$ gate. That is, apply the
channel $D(\rho)=(1-p_{\textrm{g}})\rho+p_{\textrm{g}}/15\sum_{A,B}(A\otimes
B)\rho(A\otimes B)^{\dagger}$, where
$(A,B)\in\\{\mathbb{I},X,Y,Z\\}^{2}\setminus(\mathbb{I},\mathbb{I})$ is a pair
of Pauli matrices excluding identity.

  3. 3.

Classical bit-flips following every Pauli-$X$ basis measurement. That is, a
measurement outcome $m$ is flipped to $1-m$ with probability $p_{\textrm{m}}$.
The corresponding noise channel takes on the form
$\widetilde{P}^{\pm}(\rho)=(1-p_{\textrm{m}})P^{\pm}\rho\,P^{\pm}+p_{\textrm{m}}P^{\mp}\rho\,P^{\mp}$,
where $P^{\pm}\equiv\outerproduct{\pm}{\pm}$ are the desired projectors in the
Pauli-$X$ basis.

For the entangled states used in the distributed simulations, we assume that
parties have shared access to Bell states in the _Werner_ or _isotropic_ form

$\begin{split}\rho_{p_{\textrm{n}}}=&\left(1-p_{\textrm{n}}\right)\outerproduct{\Phi^{+}}{\Phi^{+}}+\frac{p_{\textrm{n}}}{3}\outerproduct{\Phi^{-}}{\Phi^{-}}\\\
&+\frac{p_{\textrm{n}}}{3}\outerproduct{\Psi^{+}}{\Psi^{+}}+\frac{p_{\textrm{n}}}{3}\outerproduct{\Psi^{-}}{\Psi^{-}},\end{split}$
(10)

where $\ket{\Phi^{+}}=(\ket{00}+\ket{11})/\sqrt{2}$,
$\ket{\Phi^{-}}=Z\ket{\Phi^{+}}$, $\ket{\Psi^{+}}=X\ket{\Phi^{+}}$, and
$\ket{\Psi^{-}}=X\otimes Z\ket{\Phi^{+}}$. We refer to $p_{\textrm{n}}$ as the
parameter that describes “network noise”. In this paper, we do not take into
account specific physical systems. We justify using Werner states by noting
that a depolarizing channel (i.e., the most general noise channel) converts a
perfect Bell state to a Werner state. Additionally, Werner states can be
considered a general proxy for non-perfect Bell states, because every Bell
state can be _twirled_ [18] into a Werner state using local operations and
classical communication.

To generate GHZ states from these Bell states, we use the very straightforward
method based on local parity measurements [19]. Fundamentally, a GHZ state may
be created from Bell pairs by a local projective measurement of the $ZZ$
parity between two halves of two pairs shared by multiple parties. The
circuits to create a $3$-partite GHZ from 2 Bell states and a $4$-partite GHZ
state from 3 Bell states are drawn schematically in Fig. 6.

Figure 6: Fusion circuits for 3- and 4-partite GHZ states. In each of the
circuits, Bell states are fused through the application of local C$X$ gates
and a subsequent $Z$ basis measurement of the target bits. The conditional
bit-flips ensure that GHZ states have the desired form $\ket{0\dots
0}+\ket{1\dots 1}$.

We note that the circuits in Fig. 6 do not include distillation. Better
quality GHZ states can be generated if the Bell states used to carry out the
projective measurement are pre-distilled, or if GHZ states are distilled after
creation. This, however, introduces a probabilistic factor to the GHZ creation
protocols, which leads to higher numerical complexity in simulating the
circuits.

###  III.5 Numerical tools and considerations

To decode error syndrome graphs, we have implemented a version of the _Union-
Find_ (UF) decoder [11]. This decoder is particularly attractive given its
almost-linear time complexity and ease of implementation for both Pauli and
erasure errors. Despite being a sub-optimal decoder, the phenomenological
threshold for the cubic cluster state ($2.6\%$) is very close to that of the
_Minimum-Weight Perfect-Matching_ (MWPM) decoder ($2.9\%$) [6], which is in
turn not far from the optimal threshold ($3.3\%$) [20]. Furthermore, since the
Union-Find decoder is maximum-likelihood over the erasure channel (due to its
built-in _peeling_ decoder [21]), we expect this performance gap to shrink
further over noisy channels that are a combination of Pauli and erasure
errors. We have specifically implemented the “weighted-growth” version of the
Union-Find decoder, as introduced in the original Union-Find manuscript [11].
Our implementation can be found in the repository of Ref. [22].

Error models per unit cell are constructed with a circuit simulator. Unit
cells are defined according to the description in Sec. III.1, together with
the splitting methods of Secs. III.2 and III.3. The simulator is implemented
as a classical efficient stabilizer simulator. Per operation, the simulator
applies a full error channel as a series of Pauli operators, ending with a
measurement in the $n$-qubit Pauli basis. The choice for Pauli noise is
justified in these simulations because the input states are (convex
combinations of) stabilizer states, the operation noise is typically Pauli
noise and measurement noise is described by classical bit-flips. In App. C, we
describe the details of constructing an error channel. Pauli twirling a state
at the end of a series of non-Clifford operations is equivalent to Pauli
twirling the individual operations before applying them. This allows us to
also use the simulator in situations where one is interested in the Pauli-
twirled version of the full error channel—in that case, it suffices to twirl
the individual channel components before applying them.

The calculated error models are used to sample errors in Monte Carlo style on
qubits of the full crystalline cluster state. These cluster states are
constructed from the associated unit cell complex according to the crystal
embedding procedure described in App. B. Per Monte Carlo sample, we decode the
syndrome graph and assign a logical failure whenever there is a logical error
for a single pair of logical $X$ and $Z$ operators of the channel—see Sec.
II.3 for more details.

##  IV Results

The results are organized into three parts, where the noise models become
increasingly complex. A summary of the most important thresholds found can be
found in Fig. 7.

Secs. IV.1 and IV.2 provide thresholds for various geometries under a
phenomenological noise model. In these sections, we reproduce earlier-known
results and investigate the influence of a lattice boundary. For the
phenomenological noise model, cluster states based on lattices that have a
lower vertex degree are more resilient against errors. However, these cluster
states typically require more two-qubit gates to construct. This aspect is not
regarded by the phenomenological noise model.

To investigate the trade-off between noise resilience and noise introduced by
constructing the cluster state, we investigate scenarios with circuit-level
and network noise in Secs. IV.3 and IV.4. Sec. IV.3 discusses numerical
thresholds for monolithic (i.e., non-distributed) architectures. These results
are compared with the circuit-based error models for cluster states on a
distributed network in Sec. IV.4.

In the last part of this section, in Sec. IV.5, we investigate how the GHZ
success probability of distributed networks can be traded off against a higher
erasure probability to achieve fault-tolerance against larger infidelity of
the entangled states used.

Figure 7: Overview of the threshold values found in our study, for different
lattices and noise models considered. (a) Summary of numerical thresholds for
bit-flip and erasure-type phenomenological noise models of the four lattices
described in this work. (b) Differences of estimated threshold values for the
same lattices and error models as in (a) with boundaries, expressed in
percentage points (p.p.). (c) Monolithic thresholds are gauged against
circuit-level errors with a common error rate as defined in Secs. III.4 and
IV.3, and similarly for distributed thresholds gauged under only circuit-level
noise. Network thresholds are obtained from the model as defined in Sec.
III.4. For the monolithic thresholds, we show only the highest threshold value
found with alternative gate orderings (see Sec. IV.3 in the main text for
details). Numbers above the bars indicate the threshold, with $95\%$
confidence intervals around the value indicated as error bars and as decimals
in parentheses.

###  IV.1 Phenomenological thresholds

Before considering more realistic noise models, we first numerically evaluate
erasure and phenomenological thresholds for several known lattices, which were
constructed using cell-vertex splitting. These are the cubic, diamond,
triamond, and so-called double-edge cubic cluster states. We go beyond the
results in Ref. [6] by determining _fault-tolerant regions_ against erasure
and phenomenological errors. The results are depicted in Fig. 8. We consider a
phenomenological error model. It corresponds to perfect state preparation of
the cluster state followed by measurements that fail to report (i.e., erase)
an outcome with probability $p_{\textrm{e}}$ or flip the outcome with
probability $p_{\textrm{m}}$, both of which are i.i.d. In Fig. 8, we estimate
a fault-tolerant region for the erasure probability and bit-flip probability.
Data points of the fault-tolerant regions are calculated by sweeping over both
error probabilities while keeping their ratio fixed. We use a different
constant of proportionality at each data point on the fault-tolerant boundary.

Figure 8: Fault-tolerant regions of the cubic, diamond, double-edge (d.e.)
cubic, and triamond lattices under phenomenological bit-flip and erasure
noise. (a) Regions are estimated based on individual threshold values. (b)
Thresholds $p_{\textrm{m,th}}$ under pure bit-flip noise as a function of
cluster state valency. From left to right is the cubic, diamond, double-edge
cubic, and triamond lattice. We provide thresholds with and without lattice
boundaries—see Sec. IV.2 for more details. (c) Thresholds
$p_{\textrm{m,th}}/z$ under pure weighted bit-flip noise, where $z$ is the
cluster state valency. (d) Thresholds $p_{\textrm{e,th}}$ under pure erasure
noise. Results agree with known percolation thresholds for the given lattices.
(e) Thresholds $p_{\textrm{e,th}}/z$ under pure weighted erasure noise.

The isolated thresholds found for both types of noise can be found in Fig. 7a.
The thresholds reported are slightly higher than those reported in Ref. [6].
This can be attributed to a slightly different implementation of the Union-
Find decoder—see Sec. III.5 for more details. Because phenomenological noise
does not take into account higher error rates that arise with increasingly
complex preparations of a cluster state, a fairer comparison is made by
weighing each qubit error probability with the cluster state valency $z$,
producing i.i.d. noise with probabilities $zp_{\textrm{e}}$ and
$zp_{\textrm{g}}$ instead. Such a model is called _weighted phenomenological_
error model in Ref. [6]. Because the cubic, diamond, triamond, and double-edge
cubic lattices are regular with valency $z\in\\{4,6,8,10\\}$ on all qubits
respectively, weighted thresholds may be calculated from unweighted thresholds
by simply dividing by $z$.

###  IV.2 Phenomenological thresholds with boundaries

The lattices considered above repeat periodically in all three spatial
directions. It may be difficult to prepare such a cluster state if we take
into account the connectivity of the qubits. We gauged how the performance of
a cluster state is affected by the introduction of boundaries, under the same
phenomenological noise model of both bit-flips with probability
$p_{\textrm{g}}$ and erasure errors with probability $p_{\textrm{e}}$. Each
lattice is introduced to a smooth boundary along the $x=0$ plane, and a rough
boundary along the $y=0$ plane. For the smooth boundary, we remove elements of
the correlation surface defined by a _dual_ logical membrane and its closure
and introduce a boundary on the remaining dangling edges in the dual complex.
The rough boundary is introduced in the same way, by swapping primal and dual
notions.

Figure 9: Sub-threshold logical error rate scaling of cubic, diamond, double-
edge (d.e.) cubic, and triamond lattices with and without boundaries for
phenomenological bit-flip (top row) and erasure (bottom row) noise models.
Data for periodic conditions has round markers with a dashed line. Although
the thresholds are minimally affected, sub-threshold error probabilities of
any lattice with a rough and smooth boundary are about $1.5-2$ times the rate
for the same lattice under periodic boundary conditions.

In Figs. 7b and 8b-e, we show how the threshold values for the
phenomenological and erasure noise models change with the introduction of the
boundaries. The results show that differences in the phenomenological bit-flip
threshold values are insignificant. For the erasure thresholds, only the cubic
and double-edge cubic lattices are slightly affected. The results indicate
that the introduction of boundaries affects the thresholds only minimally, at
least for the lattices and noise models considered here. However, we found
that boundaries _do_ impose significantly weaker _sub-threshold scaling_. The
sub-threshold scaling is the rate at which the logical error rate is
suppressed below the threshold. Fig. 9 shows that boundaryless architectures
have a favorable error probability suppression that is about $50-100\%$ as
effective as the same architecture with boundaries. These results are
consistent with recent work, where it was also shown that the introduction of
boundaries leaves the threshold nearly invariant, but negatively impacts error
rate scaling by roughly the same factor [23].

###  IV.3 Monolithic thresholds

To compare distributed thresholds against monolithic implementations of the
same geometry, we first benchmark monolithic architectures with the circuit-
level noise models from Sec. III.4. In the results that follow, we set all
probabilities $p_{\textrm{p}}=p_{\textrm{g}}=p_{\textrm{m}}\equiv
p_{\textrm{o}}$ equal (see Sec. III.4 for the definition of the parameters),
and sweep a threshold over the value of $p_{\textrm{o}}$. What remains is a
specification of the ordering of C$Z$ gates in the circuits, since pure C$Z$
gates all commute with one another, whereas their noisy versions do not.
Because no qubit can interact with two C$Z$ gates simultaneously, a valid
ordering may be extracted from an edge coloring of the corresponding
$\partial_{2}$ boundary map of the cluster state. This graph is bipartite by
definition. Therefore, the chromatic index (i.e., the minimum number of colors
needed for an edge-coloring) equals the maximum degree of any vertex in the
graph—i.e., the maximum valency of the cluster state [24]. The cubic, diamond,
double-edge cubic, and triamond lattices are all regular, and so their
chromatic indices are 4, 6, 8, and 10, respectively.

For a chromatic index $i$ and a given coloring, there are $i!$ different ways
to order the edges and thus the C$Z$ gates. Rotational and reflection
symmetries of the cubic cluster state imply that there are only two colorings
that correspond to a unique sequence of gates: a “(counter)clockwise” sequence
going around a face, and a “zigzag” sequence jumping to opposite sides first.
For diamond, double-edge-cubic and triamond lattices, the number of orderings
quickly explode as $6!=720$, $8!\approx 4\times 10^{4}$ and $10!\approx
3.6\times 10^{6}$ (not taking into account symmetries). We have not included
an exhaustive search of all orderings and their corresponding thresholds up to
symmetries, but only investigate a subset of orderings.

Figure 10: Monolithic thresholds $p_{\textrm{o,th}}$ for specific gate
orderings. The thresholds are calculated with, for the cubic and diamond unit
cell, a straightforward order of C$Z$ gates according to a
“(counter)clockwise” coloring of the diagram on the right (see arrows in the
diagram for the cubic lattice). We omit details on (counter)clockwise
orderings for the double-edge cubic and triamond lattice. Gates of a single
color are performed simultaneously for all unit cells in the lattice. Each
data point constitutes $50000$ samples. Error bars of the logical error rate
are given as $95\%$ confidence intervals but are too small to be discernible
in most cases. The threshold value is highlighted with a $95\%$ confidence
interval based on its least-squares estimate of a second-order polynomial of
the logical error rate around the threshold value—see App. D for details.

Monolithic thresholds with the (counter)clockwise orderings are shown in Fig.
10. An overview of all monolithic thresholds is included in Fig. 7c. For the
cubic lattice, a zigzag ordering of C$Z$ gates slightly outperforms the
(counter)clockwise ordering. The diamond lattice outperforms a cubic cluster
state, whereas the double-edge cubic lattice drops in performance for the
orderings we considered. In all cases, (counter)clockwise orderings of the
gates (at least in the primal complex) tend to produce lower thresholds than
orderings that skip multiple edges at a time. We can intuitively understand
this result by considering that a single Pauli-$X$ error on a face qubit
spreads through all subsequent C$Z$ gates as correlated Pauli-$Z$ errors to
neighboring edge qubits. In a (counter)clockwise ordering, the Pauli-$Z$
errors produce a single strand that wraps around the face. In a zigzag
orientation, $Z$ error strings can form disconnected chains, such that a
single Pauli-$X$ error produces multiple pairs of syndromes on different sides
of the face. Initial numerical analysis shows that (counter)clockwise
orderings indeed seem to result in, on average, longer error strings. The
triamond lattice was gauged for a single ordering due to its complexity and
performed poorly.

In the limiting case that all gate errors are Pauli-$Z$ errors, the relative
impact of gate order disappears, and thresholds coincide with the weighted
phenomenological thresholds considered above. In Ref. [14], Newman et al.
consider an error model that, after each C$Z$ gate, applies an $X$-type error
on a (primal) face qubit with probability $p_{X}$ and a $Z$-type error on a
(primal) edge qubit with probability $p_{Z}$. Their results show that the
best-performing lattice in terms of threshold moves from higher to lower
valency as Pauli-$X$ errors start to dominate. Our results show a similar
tendency under depolarizing gate noise for the higher valent double-edge cubic
lattice and triamond lattices when compared to the lower valent cubic and
diamond lattices. We can summarize this finding by stating that as the cluster
state valency increases, depolarizing noise incurs a larger cost on the
threshold value. Under these noise models, there exists an optimal threshold
resulting from a trade-off between the complexity of the geometry of the
cluster state, and the structure of the noise created by the circuit. Noise
bias is one example where this trade-off may be abused, by taking advantage of
the architecture of the cluster state in either primal or dual lattice
structures.

The monolithic cluster state thresholds do not compete with surface code
monolithic thresholds, which are estimated at $0.90\%$ and $0.95\%$ under the
same noise model [25]. It should be noted that the numbers for all non-cubic
lattices provided here are likely sub-optimal, as we have only gauged the
performance for a subset of C$Z$ gate orderings. Nevertheless, estimates show
that cluster states defined on the diamond lattice can outperform the cubic
cluster state in the presence of circuit-level noise. For cluster states with
even higher valencies, the cost of initialization negatively impacts the value
of the threshold, consistent with the results under weighted phenomenological
noise models. These results warrant further optimizations of the gate
orderings and comparisons with other lattices.

###  IV.4 Distributed thresholds

In this section, we investigate thresholds for distributed implementations of
the cluster states. We use the face-edge splitting operation of Sec. III.3 to
split faces of the cubic, diamond, and double-edge cubic lattices. The cubic
lattice was gauged for two different splits: one along the diagonals,
producing network nodes with six cluster qubits in a ring and entangled
through Bell states, and one on the entire face, producing 2-qubit nodes that
are entangled through 4-partite GHZ states. Both these architectures also
appear in the context of fusion-based quantum computation in Bartolucci et al.
[9]. The structure of the diamond lattice is more intricate, and we produce
two different architectures that contain only 3-partite GHZ states (the
4-ring) and an architecture with a mixture of Bell states and 3-partite GHZ
states. Architectures for the double-edge cubic lattice resemble the cubic
lattice: the first contains only Bell states, whilst the other shares
4-partite GHZ states. All architectures are drawn schematically in Fig. 11. We
present an investigation on the stabilizer fidelity of the distributed unit
cell implementations of Fig. 11 in App. E.

Figure 11: Distributed architectures for various lattices, obtained through
splitting the faces of the monolithic cluster state. A single-face split leads
to a Bell state, whereas an $n$-split produces a GHZ state. They form
connected components that are distinct nodes in a distributed network. We
identify six different architectures: two each for the cubic, diamond, and
double-edge cubic lattices. The left column shows purely monolithic cluster
states. In the middle column, cluster states are initialized with Bell states
on every face and edge, except for the diamond lattice that also contains
weight-3 GHZ states. Architectures in the right column are initialized with
4-partite GHZ states for the cubic and double-edge cubic lattices, and
3-partite states for the diamond lattice. Architectures in the middle column
tend to have larger nodes than those in the right column.

In the distributed model, we set $p_{\textrm{p}}=p_{\textrm{g}}\equiv
p_{\textrm{o}}$. On top of that, we use entangled states to connect the
cluster states that are separated by the splits. For these states, we assume
that nodes can prepare Werner states with fidelity $1-p_{\textrm{n}}$ and that
GHZ states are generated with the protocols of Fig. 6. In the absence of
network links, this circuit-level model reduces to the model discussed in Sec.
IV.3.

Figure 12: Fusion-based thresholds for the six distributed architectures of
the cubic, diamond, and double-edge cubic lattices. The circuit-level noise
parameter $p_{\mathrm{o}}$ and network error rate $p_{\textrm{n}}$ are both
swept, and a fault-tolerant region is estimated over thresholds of both
parameters. See main text for simulation details.

Using this physical model, we numerically establish thresholds for the
distributed lattices introduced above. In the same way as phenomenological
bit-flip and erasure thresholds, we estimate fault-tolerant regions for error
probabilities $p_{\textrm{o}}$ and $p_{\textrm{n}}$. The results are shown in
Fig. 12 and in the overview of Fig. 7c.

These results show that network error rate thresholds for both the 2-node and
6-ring cubic designs are similar. Even though higher-valent GHZ states tend to
produce lower-quality stabilizers, we suspect that the superior performance of
the 2-node design may be due to the size of its node. Because the 2-node
architecture has only a single C$Z$ gate per node, errors spread to only one
adjacent qubit, as opposed to the 6-ring architecture that moves such an error
to two adjacent qubits. Despite the more significant propagation of errors,
the 6-ring architecture has a higher threshold against gate and measurement
errors (the probability $p_{\mathrm{o}}$) than the 2-node design. Most likely,
this is because measurement errors have a higher influence in the 2-node
architecture, which has more qubits. Both diamond architectures outperform the
cubic lattice by a factor of roughly two. This result is promising, especially
considering that the depolarizing thresholds also outperform cubic
architectures, as was already the case for the monolithic thresholds discussed
above.

For the double-edge cubic architectures, network error rate thresholds drop
again. Nevertheless, the 4-ring design with GHZ states outperforms the bigger
12-node design, which we may explain in the same way as in the cubic case: in
the 4-ring lattice, errors propagate to fewer neighboring qubits compared to
the 12-node architecture. An important difference with the cubic lattices is
that the 4-ring design benefits enough from the lower error spreading to keep
outperforming the 12-node design under pure gate and measurement noise,
despite the additional qubit measurements in the 4-ring implementation.

Thresholds for the circuit-level noise rate $p_{\textrm{o,th}}$ of these
architectures are not optimal, due to the way that the ordering of C$Z$ gates
affects the threshold. We find that, for all architectures considered, the
values found for $p_{\mathrm{o,th}}$ without network noise are similar to the
monolithic thresholds. Similar to the earlier analysis of distributed designs,
comparing distributed to monolithic architectures involves trading off less
error propagation in the distributed architectures versus fewer measurement
errors in the monolithic architecture.

Figure 13: (a) Trade-off between erasure and network error rate. The fault-
tolerant region of the six-ring architecture is drawn (gray), where entangling
links have a network error rate $p_{\textrm{n}}$ and an independent
probability $p_{\textrm{e}}$ to fail. Gates and measurements are assumed to be
noiseless. A failed link causes the edge to be erased—in the absence of
network error, we obtain the phenomenological erasure threshold of the cubic
lattice. The colored lines reflect infidelities and failure rates of
entanglement distillation protocols that use identical copies of a Werner
state with initial fidelity $F_{\textrm{n}}$ to distill a single Bell state
with higher fidelity—i.e., with lower network error rate. Distillation may
bring a non-fault-tolerant architecture into the fault-tolerant regime. Data
marked with triangles correspond to concatenated DEJMPS (“cDEJMPS”)
distillation protocols [26], where the numbers indicate how many Bell states
are used to distill the final state. Data marked with colored circles is based
on a distillation protocol (printed in panel b) that consumes five Bell
states. For this protocol, the data point on the bottom right of each curve
corresponds to a variant where, per Bell state, only coinciding measurement
outcomes are accepted. The data point on the top left of each curve
corresponds to a variant where one accepts all measurement outcomes.
Intermediate protocols accept a subset of non-coinciding measurement results.
(b) Possible circuit for the 5-to-1 distillation protocol. The circuit is
based on the decoding circuit of the 5-qubit error-correction code—see Ref.
[27] for more details. Each line corresponds to half of a Bell state—i.e., the
same circuit has to be applied to the qubits that form the other half of these
Bell states.

###  IV.5 Trade-off GHZ success rate and erasure probability

The results in the previous sections indicate that fault tolerance can be
achieved with network error probabilities below $\sim 2$%—i.e., with Bell
state fidelities above $\sim 98$%. This condition can be challenging from an
experimental perspective. Fortunately, it is possible to boost the fidelity of
entanglement before cluster state preparation through entanglement
distillation. In this process, we can model distillation failures as erasures
on the corresponding qubit(s), and use a suitable decoder, such as the Union-
Find decoder, to deal with these qubit erasures. A suitable decoder can
jointly correct errors and erasures. Phenomenological erasure thresholds can
be roughly one order of magnitude higher than bit-flip error thresholds, so
one may reasonably expect that trading in failed distillation attempts for
higher-quality links will be worth the cost.

We make this argument more quantitative in Fig. 13, using the cubic 6-ring
architecture as an example. In this architecture, every Bell pair is
successfully heralded with probability $1-p_{\textrm{e}}$ and network error
rate $p_{\textrm{n}}$, and discarded with probability $p_{\textrm{e}}$. The
fault-tolerant region is simulated in the same way as before, this time
sweeping over the network and erasure error rates, where circuit-level noise
is not taken into account—i.e., $p_{\textrm{o}}$ is set to $p_{\textrm{o}}=0$.
We can apply an entanglement distillation protocol to distill each Bell state
used in the cluster state by consuming multiple Bell states with initial
fidelity $F_{\textrm{n}}=1-p_{\textrm{n}}$. Distillation protocols that we
consider are (concatenated versions of) the DEJMPS protocol [26] and the
5-to-1 protocol [27] of Fig. 13b. The output fidelities and failure rates of
these distillation protocols are shown in the graph for two different values
of the initial fidelity $F_{\textrm{n}}$. For these protocols, there is a
trade-off between fidelity and failure rate. Importantly, we can directly link
this trade-off to the trade-off between erasure and network error rate—this
makes it possible to move the state inside the fault-tolerant region. For the
distillation protocols considered, the first crossing with this region happens
for an initial network error probability of $p_{\textrm{n}}\approx 5\%$, which
is a five-fold increase of the approximate 6-ring threshold value of $1\%$
without distillation. These results are particularly promising for the diamond
architecture, where both the network error threshold of roughly $2\%$ and the
erasure threshold of $40\%$ are higher than the cubic lattice.

##  V Conclusion

In this paper, we have provided several tools and numerical analyses to
explore fault-tolerant measurement-based quantum computing architectures built
from smaller units, in the context of modular, distributed, or networked
computing. This was achieved with a method that allows us to distribute a
fault-tolerant cluster state over multiple parties. This method augments the
splitting procedure of previous work by Nickerson and Bombín [6]. In the
distributed context, the resulting states will lead to entanglement in the
form of a Bell or GHZ state, such that existing methods for state generation
and distillation may be used to design such a fault-tolerant architecture.

The performance of various three-dimensional cluster state architectures was
studied through numerical evaluation of their fault-tolerant thresholds, which
quantify the rate of specific sources of error below which fault-tolerant
computation is possible. We find that the diamond lattice outperforms the
traditional cubic cluster state for monolithic architectures suffering
standard noise models—i.e., the noise models described in Sec. III.4. It
should be mentioned that even better results may be achieved with different
permutations of the entangling C$Z$ gates during cluster state preparation. In
the cases we have considered, (counter)clockwise orderings of C$Z$ gates tend
to produce lower thresholds.

Furthermore, we have gauged the performance of the same lattices in a
distributed setting. For designs based on the cubic lattice, error thresholds
of the network noise are around $1\%$. We consider two different designs of
distributed cluster states defined on top of a diamond lattice, which
outperform cubic thresholds roughly by a factor of two. Using a cubic
architecture, we show how entanglement distillation may bring a non-fault-
tolerant design into the fault-tolerant regime by trading in network noise for
erasure errors in the cluster state. Combined with favorable erasure
thresholds of the diamond lattice, these results indicate that distributed
fault-tolerant cluster states may outperform topological error-correction
codes, and warrant additional numerical simulations of these distributed
networks in the presence of entanglement distillation.

There are several potential avenues for further investigation. On the one
hand, our circuit-based qubit error models are limited to depolarizing and
erasure-type noise, but one may estimate fault-tolerant thresholds for models
that more accurately represent errors in present-day quantum hardware.
Furthermore, it would be interesting to consider protocols that cannot be
described as a simple sequence of instructions but contain dependencies and
branches that split based on intermediate decisions—as, e.g., protocols that
contain entanglement distillation. Previous results in the field apply pre-
calculated error models (i.e., quantum channels) on a unit cell level, and
randomly sample from the error model during Monte Carlo threshold calculations
[25, 28]. This is possible because the fault-tolerant protocol consists of
identical rounds that are applied over time, where each round is split up and
grouped into multiple sub-rounds that act on disjoint subsets of qubits, so
that the entire protocol may be simulated as distinct sections that are
separated in both space and time. This is not necessarily the case for the
three-dimensional cluster states that we consider here, in which case the
entire cluster state should be simulated at a global level, i.e. without pre-
calculating error models of individual sections [29].

As an alternative to measurement-based quantum computation, one may consider
so-called fusion-based quantum computation [9]. It combines a low circuit
depth with the topological features of cluster states. The fundamental
operations in fusion-based quantum computation are resource-state generation
and fusions. Fault-tolerant fusion-based architectures rely heavily on
resilience against erasure, and the structures considered in this work are a
natural candidate to consider in this alternative framework of computation.

## Acknowledgments

The authors would like to thank Michael Newman, Naomi Nickerson, Tim Taminiau,
and Barbara Terhal for helpful feedback and/or discussions. We gratefully
acknowledge support from the joint research program “Modular quantum
computers” by Fujitsu Limited and Delft University of Technology, co-funded by
the Netherlands Enterprise Agency under project number PPS2007. This work was
supported by the Netherlands Organization for Scientific Research (NWO/OCW),
as part of the Quantum Software Consortium Program under Project
024.003.037/3368. We thank SURF (www.surf.nl) for the support in using the
National Supercomputer Snellius.

##  Appendix A Quotient boundaries in unit cell complex

By the prescription of Sec. III.1, the unit cell complex is described as a
sequence of vector spaces

${Q_{3}}$${Q_{2}}$${Q_{1}}$${Q_{0},}$$\scriptstyle{\partial_{3}^{\left[\mathbf{r}\right]}}$$\scriptstyle{\partial_{2}^{\left[\mathbf{r}\right]}}$$\scriptstyle{\partial_{1}^{\left[\mathbf{r}\right]}}$

with each $Q_{i}$ over the field $\mathbb{Z}_{2}$ and with quotient boundaries
$\partial_{i}^{\left[\mathbf{r}\right]}:Q_{i}\mapsto Q_{i-1}$. Similar to the
equivalence between $\overline{C}_{i}$ and $C_{D-i}$, we define
$\overline{Q}_{i}\cong Q_{D-i}$. One may verify that the dual quotient
boundaries
$\overline{\partial}_{i}^{\left[\mathbf{r}\right]}:\overline{Q}_{i}\mapsto\overline{Q}_{i-1}$
are related to primal boundaries as

$\overline{\partial}_{i}^{\left[\mathbf{r}\right]}=\left(\partial_{D+1-i}^{\left[\mathbf{\overline{r}}\right]}\right)^{T}.$
(11)

Importantly, the translation vector $\mathbf{r}$ is also reversed to
$\mathbf{\overline{r}}$. By dualizing boundary maps of unit cell complex
directly, one may construct a representation of the dual unit cell without
redefining it from the dual crystal. The zero map conditions
$\partial_{i-1}\partial_{i}=0$ take the form of

$\sum_{p}\partial_{i-1}^{\left[\mathbf{p}\right]}\partial_{i}^{\left[\mathbf{r-p}\right]}=0\quad\forall\mathbf{r}.$
(12)

The proof is given in App. B.

It is convenient to represent the underlying unit cell complex as a _labeled_
graph, which is essentially a sparse representation of its boundaries as
_arcs_ and basis elements as _nodes_. (We use nomenclature _nodes_ and _arcs_
for such a graph, to make the distinction between vertices and edges of the
chain complex.) Every basis element $\left(\mathbf{q}_{i}\right)_{n}\in Q_{i}$
is mapped to a node $q_{i,n}$, with two nodes
$q_{i,n}\rightarrow_{\left[\mathbf{r}\right]}q_{i-1,m}$ connected by an
$\left[\mathbf{r}\right]$-labelled arc if the $mn$th matrix element of the
quotient boundary $\partial_{i}^{\left[\mathbf{r}\right]}$ equals one. The
maps $\partial_{i}^{\left[\mathbf{r}\right]}$ thus form the _biadjacency_
matrices between the nodes of $Q_{i}$ and $Q_{i-1}$. We note that this
description, including its labeling, resembles the _vector method_ for
describing three-periodic networks as a quotient graph [30], except that the
nodes of our quotient graph also represent higher-dimensional elements in a
chain complex, such as edges, faces, and cells for a three-dimensional
complex. Examples of the square, triangular, and cubic lattice are given in
Fig. 14.

Figure 14: Unit cell complexes as a labeled graph. Unlabelled edges correspond
to a Miller index containing only zeros. (a) A square lattice. There is one
face, two edges, and one vertex. The face $\mathbf{f}$ is connected twice to
both $\mathbf{e}_{x}$ and $\mathbf{e}_{y}$ within the unit cell ($[00]$) and
outside of it ($[01]$ and $[10]$, respectively). Because the complex is self-
dual, similar relationships hold for its vertex $v$. (b) A triangular lattice.
This lattice can be created by splitting the faces of the square lattice: per
unit cell, the triangular lattice has one extra face and one extra edge
compared to the square lattice. From the asymmetry in the quotient boundary
maps it is clear that this lattice is not self-dual. (c) A cubic lattice. Per
unit cell, there is one cell $\mathbf{q}$; three faces $\mathbf{f}_{x}$,
$\mathbf{f}_{y}$, and $\mathbf{f}_{z}$; three edges $\mathbf{e}_{x}$,
$\mathbf{e}_{y}$, and $\mathbf{e}_{z}$; and one vertex $\mathbf{v}$.

##  Appendix B Crystal embeddings

In Fig. 15, we show an intuitive interpretation of how the vector spaces
$C_{i}$ of the full crystalline chain complex are constructed with the vector
spaces $Q_{i}$ of the unit cell complex of App. A. To construct the boundary
maps $\partial_{i}$ of the full crystal from the quotient boundary maps
$\partial_{i}^{[\mathbf{r}]}$, we let
$\partial_{i}^{\left(\mathbf{n},\mathbf{m}\right)}:C_{i}\mapsto C_{i-1}$ be
the boundary map that applies $\partial_{i}^{\left[\mathbf{n-m}\right]}$ from
cell $Q_{i}^{\left(m\right)}$ to cell $Q_{i-1}^{\left(n\right)}$ and is zero
everywhere else:

$\partial_{i}^{\left(\mathbf{n},\mathbf{m}\right)}=\partial_{i}^{\left[\mathbf{n-m}\right]}\otimes
e_{nm}.$ (13)

Here, $e_{nm}:L\mapsto L$ is a _matrix unit_ , i.e., an $N\times N$ matrix
with a one at indices $n,m$ and zero elsewhere. Then the boundary maps of the
embedding are given as a sum

$\partial_{i}=\sum_{m,n}\partial_{i}^{\left(\mathbf{n},\mathbf{m}\right)}=\sum_{m,n}\left(\partial_{i}^{\left[\mathbf{n-m}\right]}\otimes
e_{nm}\right).$ (14)

Because most $\partial_{i}^{\left[\mathbf{n-m}\right]}$ are zero, we can
substitute $\mathbf{r=n-m}$ and sum $\mathbf{r}$ only over the non-zero maps
$\partial_{i}^{\left[\mathbf{r}\right]}$, leading to

$\partial_{i}=\sum_{m,r}\left(\partial_{i}^{\left[\mathbf{r}\right]}\otimes
e_{m+r,m}\right)=\sum_{r}\partial_{i}^{\left[\mathbf{r}\right]}\otimes\left(\sum_{m}e_{m+r,m}\right).$
(15)

In the second equality, the sum over $\mathbf{m}$ may be carried over to the
right by distributivity of the tensor product over addition. This last term is
a permutation matrix with a single one in each row and column; it represents a
_translation_ of a lattice point $\mathbf{m}$ to $\mathbf{m+r}$. Denote this
term as $T_{\mathbf{r}}\equiv\sum_{m}e_{m+r,m}$, such that the embedding is
given as

$\partial_{i}=\sum_{\mathbf{r}}\partial_{i}^{\left[\mathbf{r}\right]}\otimes
T_{\mathbf{r}}.$ (16)

Intuitively, the crystal boundary is formed by “gluing” the boundaries between
the unit cells $Q_{i}^{\left(\mathbf{m}\right)}$ and
$Q_{i-1}^{\left(\mathbf{m+r}\right)}$ at lattice points $\mathbf{m}$ and
$\mathbf{m+r}$ according to the map $\partial_{i}^{\left[\mathbf{r}\right]}$,
and repeating this process for every non-trivial quotient boundary map. The
zero map conditions for quotient boundaries (Eq. (12)) follow trivially.
Because matrix units multiply as $e_{ij}e_{kl}=\delta_{jk}e_{il}$, the
multiplication of two permutation matrices

$\begin{split}T_{\mathbf{p}}T_{\mathbf{q}}&=\sum_{m,n}e_{m+p,m}e_{n+q,n}=\sum_{m,n}\delta_{m,n+q}e_{m+p,n}\\\
&=\sum_{n}e_{n+p+q,n}=T_{\mathbf{p+q}}\end{split}$ (17)

represents the sum of their translations. Combining this result with the
embedded boundaries (Eq. (16)) directly, the composition of two maps equals

$\partial_{i-1}\partial_{i}=\sum_{p,q}\partial_{i-1}^{\left[\mathbf{p}\right]}\partial_{i}^{\left[\mathbf{q}\right]}\otimes
T_{\mathbf{p+q}}=\sum_{r}\left(\sum_{p}\partial_{i-1}^{\left[\mathbf{p}\right]}\partial_{i}^{\left[\mathbf{r-p}\right]}\right)\otimes
T_{\mathbf{r}},$ (18)

where we have substituted $\mathbf{r}=\mathbf{p}+\mathbf{q}$ in the last
equality. This map is the zero map if and only if the term in brackets is zero
for all $\mathbf{r}$, which is exactly the result stated in Eq. (12).

Figure 15: Abstract interpretation of how the full crystalline chain complex
is created out of the unit cell complex introduced in App. A. (a) We use the
example of the unit cell of the square lattice—see Fig. 14 for more details.
(b) As described in Sec. III.1 of the main text, we use an $N$-dimensional
vector space $L\equiv\mathbb{Z}_{2}^{\oplus N}$ with as basis vectors the $N$
lattice positions of the lattice. (c) The vector spaces $C_{i}$ of the full
crystalline chain complex are realized with the _graph product_ between $L$
and the vector spaces $Q_{i}$ of the unit cell complex. The full boundary maps
$\partial_{i}$ are constructed from the quotient boundary maps
$\partial_{i}^{[\mathbf{r}]}$ according to Eq. (15).

The above recipe for a crystal embedding may be expressed as a composition of
_direct products_ between two graphs, given the following correspondences:

  1. 1.

The matrix $\partial_{i}^{\left[\mathbf{r}\right]}$ is the biadjacency matrix
of the subgraph $G[\partial_{i}^{\left[\mathbf{r}\right]}]$ of the unit cell
complex induced by edges with label $[\mathbf{r}]$. This definition is
consistent with the graph description given in Sec. A.

  2. 2.

The matrix $T_{\mathbf{r}}$ is the biadjacency matrix of the subgraph of the
lattice $H[T_{\mathbf{r}}]$ induced by edges that translate each lattice point
by $\mathbf{r}$. That is, each lattice point is represented by a node
$\mathbf{m}$ and connected by an arc to the translated node
$\mathbf{m}+\mathbf{r}$.

The tensor products $\partial_{i}^{\left[\mathbf{r}\right]}\otimes
T_{\mathbf{r}}$ inside the embedding (Eq. (16)) are direct products of the
corresponding edge-induced subgraphs
$G[\partial_{i}^{\left[\mathbf{r}\right]}]\times H[T_{\mathbf{r}}]$. The sum
over labels $\mathbf{r}$, which adds together adjacency matrices of the
products modulo $2$, composes the edge sets of the corresponding graphs as a
disjunctive union. In this way, the entire crystal complex may be constructed
directly as a graph from a given unit cell and a lattice of arbitrary size.

##  Appendix C Characterization of noisy channels

The general process in our simulations can be described as an $n$-qubit
quantum circuit $C$ composed of Clifford operations, ending in a projective
Pauli basis measurement $P^{\mathbf{m}}$ with outcomes
$\mathbf{m}=\\{m_{1},m_{2},\dots,m_{l}\\}$ of (part) of the evolved state. For
the sake of completeness, we also assume that the circuit operates on an
ancillary input system $A$ with a stabilizer state $\ket{\psi_{0}}$. Such a
circuit might represent an entanglement distillation circuit, operating on a
mixed Bell pair $\rho$ and an ancillary Bell pair to distill it with.
Alternatively, it may represent the action of a measurement-based fault-
tolerant channel, where $\rho$ is the input code space, $\ket{\psi_{0}}$ is
the state of all ancillary qubits in the channel, $C$ represents the C$Z$
gates of the cluster state and $P^{\mathbf{m}}$ is the final $X$-basis
measurement of every ancillary qubit. The output code space is then (up to
normalization) given by $\mathcal{E}_{\mathbf{m}}(\rho)$.

We consider three sources of noise, depicted schematically in Fig. 16. First
of all, noisy ancillary input may not be a pure stabilizer state
$\ket{\psi_{0}}$, but a mixture $\rho_{A}$ of possibly non-stabilizer states.
Secondly, the circuit $C$ consists of imperfect operations, which we assume as
ideal operations followed by a mixture of Pauli gates. Lastly, the projectors
$P^{\mathbf{m}}$ may produce a “wrong” outcome $\widetilde{\mathbf{m}}$, which
we model as a perfect operation $P^{\mathbf{m}}$ followed by classical bit-
flips on $\mathbf{m}$.

Figure 16: A noisy channel that can be characterized efficiently, given that
$\rho_{A}$ is a convex combination of stabilizer states, $\widetilde{C}$ is
the ideal circuit $C$ with Pauli noise, and $\widetilde{P}^{\mathbf{m}}$ are
ideal projectors followed by classical bit-flips of the outcome $\mathbf{m}$.
Under suitable assumptions of the noise models, the
$\widetilde{\mathcal{E}}_{\mathbf{m}}$ operation may be expressed as a mixture
of ideal $\mathcal{E}_{\mathbf{m}}$ and Pauli operations.

Arbitrary noisy ancillary input states $\rho_{A}$ that differ from the
noiseless input $\ket{\psi_{0}}$ cannot be simulated efficiently. If, on the
other hand, $\rho_{A}$ may be approximated as a Pauli channel $\mathcal{P}$
acting on $\ket{\psi_{0}}$, the Pauli operators may be pushed through the
circuit in the same way as Pauli noise coming from imperfect gates. The key
idea is to pre-process $\rho_{A}$ by _twirling_ [18] with the stabilizers
$s_{k}\in S_{0}$, where $S_{0}$ is the stabilizer group describing the state
$\ket{\psi_{0}}$, i.e., to apply a trace-preserving channel $\mathcal{T}$ as

$\mathcal{T}\left(\rho_{A}\right)=\frac{1}{\absolutevalue{S_{0}}}\sum_{s_{k}\in
S_{0}}s_{k}\rho_{A}s_{k}.$ (19)

In the context of Pauli twirling the input state, $\mathcal{T}$ corresponds to
a Pauli channel $\mathcal{P}$ acting on the noiseless input state
$\ket{\psi_{0}}$, with elements of a _destabilizer_ group $D_{0}$ of $S_{0}$
acting as the Pauli operators of the channel. A destabilizer group $D_{0}$
associated with a stabilizer group $S_{0}$ is a subgroup of the full Pauli
group. It has the same size as $S_{0}$ and is generated with a set of
operators that each anti-commute with a different generator of $S_{0}$ and
commute with all other generators of $S_{0}$ [31]. A destabilizer group
$D_{0}$ of $S_{0}$ can be used to decompose $\rho_{A}$ in a basis of states
$\set{d_{k}\ket{\psi_{0}}}_{d_{k}\in D_{0}}$ and write

$\rho_{A}=\sum_{d_{m},d_{p}\in
D_{0}}\lambda_{mp}d_{m}\outerproduct{\psi_{0}}{\psi_{0}}d_{p}.$ (20)

We can now use this to write $\mathcal{T}(\rho_{A})$ as

$\mathcal{T}\left(\rho_{A}\right)=\sum_{d_{k}\in
D_{0}}p_{k}d_{k}\outerproduct{\psi_{0}}{\psi_{0}}d_{k}=\mathcal{P}\left(\outerproduct{\psi_{0}}{\psi_{0}}\right).$
(21)

Here, the prefactors $p_{k}$ are given by
$p_{k}=\lambda_{kk}=\Braket{\psi_{0}}{d_{k}\rho_{A}d_{k}}{\psi_{0}}$. The
trace $\Tr\left[\mathcal{T}\left(\rho_{A}\right)\right]=\sum_{d_{k}\in
D_{0}}p_{k}=1$ is preserved, such that prefactors $p_{k}$ that sum to unity
may be interpreted as the probability of applying some $p_{k}$ in the Pauli
channel $\mathcal{P}$. We see that twirling $\rho_{A}$ over the group $S_{0}$
removes all off-diagonal elements of the state in the
$\set{d_{k}\ket{\psi_{0}}}_{d_{k}\in D_{0}}$ basis.

This shows how we may approximate noisy ancillary input state $\rho_{A}$ as a
mixture of the ideal state $\ket{\psi_{0}}$ that is depolarized by a Pauli
channel $\mathcal{P}$. In the same way, we may twirl non-Pauli noisy processes
occurring during the application of the Clifford circuit $C$ to a Pauli form.
Operators $P_{i}$ can now be propagated through the circuit $C$, forming
another set of Pauli strings $P_{j}=CP_{i}C^{\dagger}$. Pauli operators
$P_{j}$ now appearing after $C$ may each be split up into a string
$P_{j}^{(B)}\equiv P^{\prime}_{j}$ appearing on the ancillary $B$ system and a
string $P_{j}^{\text{(out)}}\equiv P^{\prime\prime}_{j}$ appearing on the
output system as $P_{j}\equiv P^{\prime}_{j}\otimes P^{\prime\prime}_{j}$. The
string $P^{\prime}_{j}$ will commute with some projectors
$P^{m_{k}}P^{\prime}_{j}=P^{\prime}_{j}P^{m_{k}}$ (where
$m_{k}\in\mathbf{m}$), but anticommute with others as
$P^{m_{k^{\prime}}}P^{\prime}_{j}=P^{\prime}_{j}P^{\left(m_{k^{\prime}}+1\right)}$.
The addition of two classical bits $a+b$ here is understood modulo 2, i.e.,
$a+1$ represents a bit-flip of $a$. We summarize both cases as a commutation
relation
$P^{\mathbf{m}}P^{\prime}_{j}=P^{\prime}_{j}P^{\mathbf{m}+\mathbf{m}_{j}}$,
where $\mathbf{m}_{j}$ is a string of errors that represents the bit-flips due
to the individual commutation relations above.

Figure 17: Stabilizer fidelities of monolithic and distributed protocols for
various architectures. The three protocols per architecture correspond to the
three columns of Fig. 11, where the GHZ state is defined as weight-4 for cubic
and double-edge cubic architectures and weight-3 for the diamond lattice, and
a face with a Bell pair split is chosen for the diamond 7-node lattice. The
noise probability $p_{\textrm{o}}$ describes both depolarizing gate noise with
probability $p_{\textrm{g}}$ and faulty measurements with probability
$p_{\textrm{m}}$. The quality of the network link is parameterized by
$p^{\prime}_{\textrm{n}}$ (see main text for details). Since we are only
looking at the errors arising on a single face, the ordering of C$Z$ gates is
not relevant for the stabilizer fidelity of that same face.

One can now derive an expression for the noisy operation
$\widetilde{\mathcal{E}}_{\mathbf{m}}$ as a mixture of ideal operations with
the addition of probabilistic Pauli strings:

$\widetilde{\mathcal{E}}_{\mathbf{m}}(\rho)=\sum_{j}p_{j}P^{\prime\prime}_{j}\mathcal{E}_{\mathbf{m}+\mathbf{m}_{j}}(\rho)P^{\prime\prime}_{j}.$
(22)

The noisy channel $\widetilde{\mathcal{E}}_{\mathbf{m}}$ is a mixture of ideal
operations $\mathcal{E}_{\mathbf{m}+\mathbf{m}_{j}}$ and Pauli noise, with the
mixture arising due to classical bit-flips $\mathbf{m}_{j}$ that act on
$\mathbf{m}$. Because we assumed each operation $\mathcal{E}_{\mathbf{m}}$ to
be efficiently simulatable, the mixture may also be simulated efficiently.

On top of this, to account for faulty measurements, we assume that each
projector $P^{m_{k}}$ has a fixed probability $p_{\mathrm{m}}$ of reporting
the wrong outcome $\widetilde{m}_{k}\equiv m_{k}+1$ (modulo 2), such that the
noisy measurement channel $\widetilde{P}^{m_{k}}$ is given by a mixture

$\widetilde{P}^{m_{k}}(\rho)=\left(1-p_{\mathrm{m}}\right)P^{m_{k}}\rho\,P^{m_{k}}+p_{\mathrm{m}}P^{m_{k}+1}\rho\,P^{m_{k}+1}.$
(23)

For all measurement outcomes $\mathbf{m}=\\{m_{1},m_{2},\dots,m_{l}\\}$ this
corresponds to the channel

$\begin{split}\widetilde{P}^{\mathbf{m}}(\rho)&=\sum_{f}p_{f}P^{\mathbf{m}+\mathbf{m}_{f}}\rho\,P^{\mathbf{m}+\mathbf{m}_{f}},\\\
p_{f}&\equiv(p_{\mathrm{m}})^{h(\mathbf{m}_{f})}(1-p_{\mathrm{m}})^{l-h(\mathbf{m}_{f})},\end{split}$
(24)

where $h(\mathbf{m}_{f})$ is the _Hamming weight_ of the binary string
$\mathbf{m}_{f}$ with measurement errors. The additional mixing of measurement
outcomes does not change the form of the noisy channel as in Eq. (22) but
introduces additional terms
$\mathcal{E}_{\mathbf{m}+\mathbf{m}_{j}+\mathbf{m}_{f}}$ with prefactors that
reflect the probability of applying a particular configuration of measurement
errors:

$\widetilde{\mathcal{E}}_{\mathbf{m}}(\rho)=\sum_{j,f}p_{j}p_{f}P^{\prime\prime}_{j}\mathcal{E}_{\mathbf{m}+\mathbf{m}_{j}+\mathbf{m}_{f}}(\rho)P^{\prime\prime}_{j}.$
(25)

##  Appendix D Statistical error in threshold values

To fit the thresholds, we assume a second-order polynomial model of the
logical error probability $p_{\mathrm{L}}$ around the threshold crossing
$p_{\textrm{th}}$ of the form [32]

$p_{\mathrm{L}}(p,L)=p_{\mathrm{L},\textrm{th}}+c_{1}\left(p-p_{\textrm{th}}\right)L^{1/\nu}+c_{2}\left(p-p_{\textrm{th}}\right)^{2}L^{2/\nu},$
(26)

where $L$ corresponds to the lattice size used with the error probability $p$,
and $p_{\mathrm{L},\textrm{th}}$, $p_{\mathrm{th}}$, $c_{1}$, $c_{2}$, and
$\nu$ are fitting parameters. The fit is obtained through non-linear least-
squares minimization of the residuals
$R_{i}=\left(p_{\mathrm{L}}(p,L)-\hat{p}_{i}\right)/\hat{\sigma}_{i}$ for
every data point $i$ with the observed logical error rate $\hat{p}_{i}$ and
the associated standard deviation $\hat{\sigma}_{i}$. Confidence intervals for
threshold crossings $p_{\textrm{th}}$ are taken from the standard errors of
the least-squares approximation.

In the results of Sec. IV, error bars of the logical error rate are given as
$95\%$ confidence intervals. Sometimes, they are too small to be discernible.
The threshold value is highlighted with a $95\%$ confidence interval based on
its least-squares estimate of a second-order polynomial of the logical error
rate around the threshold value.

##  Appendix E Stabilizer fidelities for distributed setting

In this appendix, we gauge the fidelity of the stabilizer operator supported
by each face, for several face splittings in the distributed setting. For
these results, we model the entangled states as Bell and GHZ states that are
depolarized to a diagonal form. For these states, the primary component
$S_{0}\equiv\Braket{X_{0}\dots X_{n-1},Z_{0}Z_{1},\dots,Z_{0}Z_{n-1}}$ has
probability $1-p^{\prime}_{\textrm{n}}$ and all off-diagonal terms
$\Braket{\pm X_{0}\dots X_{n-1},\pm Z_{0}Z_{1},\dots,\pm Z_{0}Z_{n-1}}$ with
at least one negative sign are uniformly distributed with probabilities
$p^{\prime}_{\textrm{n}}/(2^{n}-1)$. With $n=2$, this state corresponds to the
Werner state of Eq. (10). Furthermore, just as in Sec. IV.4, we assume that
state preparation and measurements invert the state with probability
$p_{\textrm{m}}\equiv p_{\textrm{o}}$, and that C$Z$ gates are followed by a
depolarizing channel with probability $p_{\textrm{g}}\equiv p_{\textrm{o}}$.

For each of the geometries, the stabilizer fidelity of one of the faces is
calculated under this circuit-based noise model. Fidelities are calculated as
the overlap of the resulting mixed state with the ideal cluster state
stabilizer. A strong simulation for each of the circuits provides an exact
closed-form expression of the fidelity. We show these results for each of the
lattices and the three different protocols (monolithic, Bell, and GHZ)
corresponding to the three columns of Fig. 17. Based on these results, we see
that an increase in the number of splittings produces worse stabilizer
fidelities. These differences are more pronounced for the 4-partite GHZ states
when compared to the 3-partite GHZ state in diamond. Distributed architectures
are likely unable to compete with monolithic protocols, unless we combine
entanglement generation protocols with better-quality GHZ states through
entanglement distillation.

## References

  * Fujii [2015] K. Fujii,  _Quantum Computation with Topological Codes: From Qubit to Topological Fault-Tolerance_, SpringerBriefs in Mathematical Physics, Vol. 8 (Springer, Singapore, 2015).
  * Bolt  _et al._ [2016] A. Bolt, G. Duclos-Cianci, D. Poulin, and T. M. Stace, Foliated quantum error-correcting codes, Physical Review Letters 117, 070501 (2016).
  * Raussendorf  _et al._ [2006] R. Raussendorf, J. Harrington, and K. Goyal, A fault-tolerant one-way quantum computer, Annals of Physics 321, 2242 (2006).
  * Barrett and Stace [2010] S. D. Barrett and T. M. Stace, Fault tolerant quantum computation with very high threshold for loss errors, Physical Review Letters 105, 200502 (2010).
  * Bombin  _et al._ [2021] H. Bombin, I. H. Kim, D. Litinski, N. Nickerson, M. Pant, F. Pastawski, S. Roberts, and T. Rudolph,  _Interleaving: Modular Architectures for Fault-Tolerant Photonic Quantum Computing,_ arXiv:2103.08612 [quant-ph] .
  * Nickerson and Bombín [2018] N. Nickerson and H. Bombín,  _Measurement Based Fault Tolerance beyond Foliation,_ arXiv:1810.09621 [quant-ph] .
  * Auger  _et al._ [2018] J. M. Auger, H. Anwar, M. Gimeno-Segovia, T. M. Stace, and D. E. Browne, Fault-tolerant quantum computation with nondeterministic entangling gates, Physical Review A 97, 030301 (2018).
  * Herr  _et al._ [2018] D. Herr, A. Paler, S. J. Devitt, and F. Nori, A local and scalable lattice renormalization method for ballistic quantum computation, npj Quantum Information 4, 1 (2018).
  * Bartolucci  _et al._ [2023] S. Bartolucci, P. Birchall, H. Bombín, H. Cable, C. Dawson, M. Gimeno-Segovia, E. Johnston, K. Kieling, N. Nickerson, M. Pant, F. Pastawski, T. Rudolph, and C. Sparrow, Fusion-based quantum computation, Nature Communications 14, 912 (2023).
  * Awschalom  _et al._ [2018] D. D. Awschalom, R. Hanson, J. Wrachtrup, and B. B. Zhou, Quantum technologies with optically interfaced solid-state spins, Nature Photonics 12, 516 (2018).
  * Delfosse and Nickerson [2021] N. Delfosse and N. H. Nickerson, Almost-linear time decoding algorithm for topological codes, Quantum 5, 595 (2021).
  * Nielsen and Chuang [2000] M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information_ (Cambridge University Press, 2000).
  * Briegel and Raussendorf [2001] H. J. Briegel and R. Raussendorf, Persistent entanglement in arrays of interacting particles, Physical Review Letters 86, 910 (2001).
  * Newman  _et al._ [2020] M. Newman, L. A. de Castro, and K. R. Brown, Generating fault-tolerant cluster states from crystal structures, Quantum 4, 295 (2020).
  * Browne and Rudolph [2005] D. E. Browne and T. Rudolph, Resource-efficient linear optical quantum computation, Physical Review Letters 95, 010501 (2005).
  * Grice [2011] W. P. Grice, Arbitrarily complete Bell-state measurement using only linear optical elements, Physical Review A 84, 042331 (2011).
  * Ewert and van Loock [2014] F. Ewert and P. van Loock, 3/4-efficient Bell measurement with passive linear optics and unentangled ancillae, Physical Review Letters 113, 140403 (2014).
  * Bennett  _et al._ [1996] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Mixed-state entanglement and quantum error correction, Physical Review A - Atomic, Molecular, and Optical Physics 54, 3824 (1996).
  * Nickerson [2015] N. Nickerson,  _Practical Fault-Tolerant Quantum Computing_ , Doctoral thesis, Imperial College London, London, United Kingdom (2015).
  * Ohno  _et al._ [2004] T. Ohno, G. Arakawa, I. Ichinose, and T. Matsui, Phase structure of the random-plaquette Z2 gauge model: Accuracy threshold for a toric quantum memory, Nuclear Physics B 697, 462 (2004).
  * Delfosse and Zémor [2020] N. Delfosse and G. Zémor, Linear-time maximum likelihood decoding of surface codes over the quantum erasure channel, Physical Review Research 2, 033042 (2020).
  * van Montfort  _et al._ [2022] Y. van Montfort, S. de Bone, and D. Elkouss, Data/software underlying the publication: Fault-tolerant structures for measurement-based quantum computation on a network, 4TU.ResearchData, https://doi.org/10.4121/929e24f9-31fa-4816-99fa-3356e272df43 (2024).
  * Bombín  _et al._ [2023] H. Bombín, C. Dawson, R. V. Mishmash, N. Nickerson, F. Pastawski, and S. Roberts, Logical blocks for fault-tolerant topological quantum computation, PRX Quantum 4, 020303 (2023).
  * König [1931] D. König, Graphok es matrixok, Matematikai és Fizikai Lapok 38, 116 (1931).
  * Nickerson  _et al._ [2013] N. H. Nickerson, Y. Li, and S. C. Benjamin, Topological quantum computing with a very noisy network and local error rates approaching one percent, Nature Communications 4, 1756 (2013).
  * Deutsch  _et al._ [1996] D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, Quantum privacy amplification and the security of quantum cryptography over noisy channels, Physical Review Letters 77, 2818 (1996).
  * Goodenough  _et al._ [2023] K. Goodenough, S. de Bone, V. L. Addala, S. Krastanov, S. Jansen, D. Gijswijt, and D. Elkouss,  _Near-Term $n$ to $k$ Distillation Protocols Using Graph Codes,_ arXiv:2303.11465 [quant-ph] .
  * Nickerson  _et al._ [2014] N. H. Nickerson, J. F. Fitzsimons, and S. C. Benjamin, Freely scalable quantum technologies using cells of 5-to-50 qubits with very lossy and noisy photonic links, Physical Review X 4, 041041 (2014).
  * Gidney  _et al._ [2022] C. Gidney, M. Newman, and M. McEwen, Benchmarking the planar honeycomb code, Quantum 6, 813 (2022).
  * Chung  _et al._ [1984] S. J. Chung, T. Hahn, and W. E. Klee, Nomenclature and generation of three-periodic nets: The vector method, Acta Crystallographica Section A: Foundations of Crystallography 40, 42 (1984).
  * Aaronson and Gottesman [2004] S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Physical Review A 70, 052328 (2004).
  * Wang  _et al._ [2003] C. Wang, J. Harrington, and J. Preskill, Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory, Annals of Physics 303, 31 (2003).