# Loss-tolerant architecture for quantum computing with quantum emitters

Matthias C. Löbl  Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr
Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen Ø,
Denmark<EMAIL_ADDRESS>Stefano Paesani  Center for Hybrid Quantum
Networks (Hy-Q), The Niels Bohr Institute, University of Copenhagen,
Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark  NNF Quantum Computing
Programme, Niels Bohr Institute, University of Copenhagen, Denmark.
<EMAIL_ADDRESS>Anders S. Sørensen  Center for Hybrid Quantum
Networks (Hy-Q), The Niels Bohr Institute, University of Copenhagen,
Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark<EMAIL_ADDRESS>
###### Abstract

We develop an architecture for measurement-based quantum computing using
photonic quantum emitters. The architecture exploits spin-photon entanglement
as resource states and standard Bell measurements of photons for fusing them
into a large spin-qubit cluster state. The scheme is tailored to emitters with
limited memory capabilities since it only uses an initial non-adaptive
(ballistic) fusion process to construct a fully percolated graph state of
multiple emitters. By exploring various geometrical constructions for fusing
entangled photons from deterministic emitters, we improve the photon loss
tolerance significantly compared to similar all-photonic schemes.

##  1 Introduction

Measurement-based quantum computing requires the generation of large graph
states (cluster states) followed by measurements on them [1, 2, 3]. This
approach is particularly promising for photonic systems since quantum
operations can be implemented with linear optics and photon detectors only.
The required photonic cluster states can be created from small resource states
by connecting them through so-called fusion processes [4, 5, 6]. This
approach, however, has several challenges: first, fusions are probabilistic
and consume photonic qubits [7]. Furthermore, photons travel at immense speed,
which necessitates long delay lines to implement conditional feedback
operations [8]. Most critically, however, photons are easily lost which puts
stringent bounds on the required photon efficiency. Recent schemes require
efficiencies $>97\%$ [5, 9] which is above the typical values of photonic
platforms [10, 11, 12, 13]. Furthermore, schemes using few-photon resource
states require boosting [14, 15] the fusion success probability with ancillary
photons [5, 6, 9], which further complicates the experimental implementation.
Improved architectures have been developed based on larger initial resource
states [9], but even few-photon states can only be made with a low probability
using the typically employed parametric down conversions sources [16, 10, 11].
New approaches are thus required to make the generation of large-scale cluster
states experimentally feasible.

Fortunately, there are promising new methods to generate large resource states
[17, 18]. In particular quantum emitters, such as quantum dots, even enable
generating them in a deterministic and thus scalable way [19, 20, 21, 22, 23].
At the same time, these emitters have high photon efficiencies on-chip [24,
25] and end-to-end [12, 13]. Indeed, the largest photonic resource states that
have ever been generated are GHZ-states made with a quantum emitter [26].
Several resource states that can be generated with a single emitter are shown
in Fig. 1(a). Fig. 1(b) illustrates a lattice on which photonic resource
states are geometrically arranged and fused. Besides purely photonic resource
states, quantum emitters can also generate states where a stationary spin is
entangled with the photons [27, 28, 29]. Here, we exploit these properties to
construct an architecture, which uses star-shaped resource states (locally
equivalent to GHZ states) with a central spin qubit. The spin is entangled
with several photonic leaf qubits where the connections represent the
entanglement properties (see Fig. 1(c)). From these resource states, we
generate a large spin cluster state via rotated type-II fusions (Bell
measurements on the photons) [30]. We propose an implementation with
semiconductor quantum dots [31, 32], but the scheme can also be applied to
atoms [26] or color centers [33]. By using spin qubits as the building blocks
of the cluster state, we remove the need to implement feedback on flying
qubits as well as unheralded loss of the qubits in the final cluster state.
The latter poses a challenge to purely photonic approaches [34]. The proposed
hybrid approach combines the advantages of spin-based and photon-based
platforms: spins in quantum dots are excellent photon emitters with coherence
times [35, 36] much longer than qubit initialization, readout, and
manipulation [37, 38, 39], but so far no clear strategy existed for how to
scale these systems. In our proposal, the photons provide a fast link between
the static spins enabling full-scale quantum computing.

Previous proposals for generating large spin-spin entanglement use a repeat-
until-success strategy [40, 41, 42] that creates an immense overhead in the
number of qubits [40] and requires long coherence times of the qubits. This
overhead can be circumvented to some extent with more than one qubit per
emitter, one for storage and one for generating the entanglement [43, 44, 45],
but this introduces additional experimental complexity. Our cluster-state
generation does not use repeat-until-success and all fusions are performed in
one shot (ballistically), enabling a high overall clock speed. In contrast to
previously proposed ballistic schemes [5, 6], our architecture can operate
loss-tolerantly without boosting [14, 15] the fusion success probability with
ancillary photons. It only requires rotated type-II fusions [30, 46] where
success, failure, and photon loss are heralded on the detection pattern. All
these features keep the experimental overhead low and make our approach
particularly feasible.

To optimize the tolerance to photon loss, we explore several lattices on which
star-shaped resource states are arranged and fused. Since there are no
locality constraints for entanglement generated by Bell measurements on
photons, we consider lattices in several dimensions and search for ideal
lattices with a discrete optimization algorithm.

Figure 1: (a) A quantum emitter in a waveguide can used to generate different
types of resource states [19, 21]. (b) Several star-shaped photonic resource
states are arranged on a simple cubic lattice. Fusions of the resource states
are performed by using the photons on the leaves of the resource states.
Sufficiently many successful fusions generate a large connected cluster state.
(c) Several star-shaped resource states with a quantum emitter spin as the
central qubit can be fused into a distributed spin graph state. (d) As the
fusions succeed with a finite probability, parts of the desired graph state
are missing. The state can be renormalized into a cluster state with a well-
defined lattice (here a square lattice) via Y and Z measurements [47].

##  2 Building lattices by fusing resource states

Our approach starts by arranging resource states on a fusion lattice as
illustrated in Fig. 1(c). Here the resource state is represented as a star-
shaped graph with the central spin in the middle and photon as the leaves of
the star. It is defined by:

$\prod_{j=1}^{N}C_{sj}\left(\ket{+}_{s}\otimes\ket{+}^{\otimes N}\right),$ (1)

where the first qubit is a spin ($s$) and the $N$ other qubits are photons.
The state $\ket{+}=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$ is the $+1$ eigenstate
of the Pauli X operator 111We use the following definition of the Pauli-
matrices: $X=\big{(}\begin{smallmatrix}0&1\\\ 1&0\end{smallmatrix}\big{)}$,
$Y=\big{(}\begin{smallmatrix}0&-i\\\ i&0\end{smallmatrix}\big{)}$,
$Z=\big{(}\begin{smallmatrix}1&0\\\ 0&-1\end{smallmatrix}\big{)}$. $C_{sj}$
represents a controlled-$Z$ gate between the spin and photon number $j$ (see
Appendix 8.1 for details). The central spin qubit of the star-shaped state
will be part of the final cluster state and the photons on the leaves are used
to perform fusions with other resource states. The fusion that we consider
succeeds with a probability of $p_{s}=0.5$ [30, 46] in which case a connection
between two central qubits is established [5] (see Appendix 8.2 for the
required setup, known as rotated type-II fusion). When the fusion fails, there
is no connection and the leaves are erased. When enough fusions succeed, a
large connected entangled state is created, which is a resource for
measurement-based quantum computing [1, 4]. The required success probability
of the fusions can be quantified by a so-called percolation threshold [4, 5,
6]: when the success probability of fusion is below the bond percolation
threshold of the lattice, the generated graph state consists of many small
pieces and is useless for quantum computing (that applies for instance to the
two-dimensional honeycomb lattice [48]). Above the percolation threshold, a
cluster state with a large connected component spanning the entire lattice is
generated. Such a graph state can then be renormalized into a lattice [47] by
local Pauli measurements [49, 50] as illustrated in Fig. 1(d). From here
quantum computation can be performed by measurements on the generated graph
[1].

However, photon losses make it challenging to generate a large-scale
percolated cluster state as they lead to a mixed quantum state [50]. To retain
a pure state, the neighborhood of every lost qubit is removed from the graph
state by measurements in the $Z$-basis [5]. When fusion photons are lost, the
fusion heralds the loss but not which of the two fusion qubits was lost.
Therefore, the neighborhoods of both fusion qubits are removed from the graph
state [5] (see Fig. 2(a)).

In our model, every edge of the fusion lattice has two fusion photons and a
loss thus occurs with probability $1-(1-p_{loss})^{2}$ per edge, where
$p_{loss}$ is the probability that an individual photon is lost ($p_{loss}$ is
assumed to be the same for all photons). In contrast, fusion success occurs
with probability $(1-p_{loss})^{2}\cdot p_{s}$ and fusion failure with
$(1-p_{loss})^{2}\cdot(1-p_{s})$ (both without loss). With this model, we
compute percolation thresholds for photon loss: only when the photon
efficiency $\eta=1-p_{loss}$ is above the percolation threshold
$\lambda_{c}^{\eta}$, a percolated graph state with a large connected
component is created.

Our simulations generally consist of three steps: we first build a lattice
defining the used resource states and the performed fusions, we then apply the
described model for fusion failure and photon loss, and finally we evaluate if
the resulting graph state percolates the lattice (if it is a spanning
cluster).

Figure 2: (a) Loss on a fusion lattice where star-shaped resource states are
geometrically arranged and fused. When a fusion photon is lost, the
neighborhoods of both fusion photons are removed from the graph state by
Z-measurements to retain a pure quantum state. (b) Percolation simulation for
estimating the tolerance to photon loss. The simulated lattice is two-
dimensional and has been optimized for a low percolation threshold. Every
curve is an average of over $10^{3}$ simulations. Above a percolation
threshold $\lambda_{c}^{\eta}$, the probability for a connection between the
edges of a $d$-dimensional lattice of size $L^{d}$ approaches unity as
$L\rightarrow\infty$. (c) Percolation thresholds $\lambda_{c}^{\eta}$ (minimum
required value for $\eta=1-p_{loss}$). The thresholds are obtained by
extrapolating the simulation results of lattices with finite sizes $L$ (see
part (b)) towards infinity using the method from Ref. [51]. The considered
lattices are hypercubic (hc), diamond, body-centered cubic combined with hc
(bcc+hc), face-centered cubic combined with hc (fcc+hc), as well as lattices
obtained by an optimization ($L_{2d}^{a}$, $L_{3d}^{a}$). The simulations are
plotted for all-photonic star-shaped resource states (dashed lines) as well as
resource states with a quantum emitter spin as a static (st) central qubit
(solid lines).

##  3 Lattice construction

The loss tolerance of the constructions described above depends on which
lattice the resource states are geometrically arranged and fused. Therefore,
we consider various fusion lattices and analyze their tolerance to photon
loss. Our lattice construction is based on the $d$-dimensional hypercubic
lattice $\mathbb{Z}^{d}$ [51, 52]. Every point of the lattice represents the
central qubit of a star-shaped resource state and the edges represent which
fusions are performed. Fig. 1(b) corresponds to a simple cubic fusion lattice
for instance. We represent all the connections to the neighbors of a lattice
point (qubit) by integer vectors $\vec{z}\in\mathbb{Z}^{d}$ that illustrate
the corresponding geometric differences. We consider graphs with a local
neighborhood such that for every connection vector $\vec{z}$, its maximum
integer value per dimension is restricted. Initially, the maximum integer
value per dimension is set to $k=1$ ($\vec{z}\in\Set{0,\pm
1}^{d}\backslash\Set{0}$). Lattices of this type are for instance the
hypercubic or the fcc lattice. Further lattices such as the $d$-dimensional
brickwork representation of the diamond lattice [5, 51] can be obtained by
removing particular edges from these lattices. A more detailed description of
all these lattices can be found in Appendix 8.3. To validate our
implementation of these lattices, we compare the corresponding classical site-
and bond-percolation thresholds [51, 52] in Appendix 8.4.

Note that, in practice, the construction of these high-dimensional lattices is
obtained in a standard set-up by collapsing them in the (3+1)-dimensional
space. Such a construction is suitable for the photonic platform as the long-
range links that are generated in such a collapse can be implemented with
minimal loss via optical fibers.

##  4 Simulating photon loss

As a metric for the loss tolerance of a graph state construction, we use the
percolation probability (a cluster spanning from one edge of the simulated
lattice to the other) when fusion failures and photon losses are
probabilistically applied (see Fig. 2(a)). A corresponding percolation
simulation of a two-dimensional lattice is shown in Fig. 2(b). In all
simulations, we consider the emitter to be the central qubit of the star-
shaped resource state (see Fig. 1(c)). This type of qubit cannot be lost, in
contrast to the fused photonic qubits. The percolation thresholds for several
multi-dimensional lattices are shown in Fig. 2(c), and the corresponding
values can be found in Appendix 8.5. For the best lattices, the percolation
thresholds $\lambda_{c}^{\eta}$ are below $0.94$ showing that a photon loss
probability $p_{loss}$ of about $1-\lambda_{c}^{\eta}>6\%$ can be
tolerated.222A detailed description of the algorithm that we use to
efficiently perform the percolation simulations for photon loss is provided in
Ref. [53]. Associated numerical results for the size of the largest connected
cluster component are given in Appendix 8.6.

For completeness, we also simulate the loss tolerance when using purely
photonic resource states where the central qubits can also suffer unheralded
loss [34], reported as dashed lines in Fig. 2(c). Note that, in contrast to
our spin-based approach, the obtained loss thresholds for the all-photonic
cases only provide necessary conditions for creating a useful cluster state
after lattice renormalization [4, 47].

For the considered lattices, Fig. 2(c) shows that the hypercubic lattices
perform best in dimensions 3 and 4 whereas the diamond lattices are ideal for
higher dimensions. A remarkable feature is the dependence of the percolation
thresholds on the dimension. For the hypercubic (hc) lattices and the lattices
fcc+hc, bcc+hc, we observe an optimum dimension where the corresponding fusion
lattice has the lowest percolation threshold. For the diamond lattice, it is
not obvious where the optimum is but we expect such an optimum: with
increasing dimension, the vertex degree of the corresponding lattices
increases. On the one hand, a higher vertex degree is desirable because more
fusion attempts lead to a higher chance of establishing connections in the
cluster state. On the other hand, in the presence of photon loss, a too high
vertex degree (more fusion photons) is problematic since the loss of a fusion
qubit makes the central qubit useless (it must be measured in the Z-basis, see
Fig. 2(a)). We further analyze this point in Appendix 8.6 in the context of
the largest connected cluster state component. A dimension where the
percolation threshold of a certain lattice type reaches a minimum is a feature
of the fragile nature of entanglement. This differs from classical bond- and
site-percolation where the percolation thresholds always decrease when adding
more bonds to the same lattice.

##  5 Lattice optimization

Fig. 2(c) illustrates that different lattices show quite different
performances under photon loss. Therefore, we search for ideal lattices in a
given dimension by a discrete optimization algorithm. As before, we virtually
place vertices on a hypercubic lattice $\mathbb{Z}^{d}$ and represent the
connections by a set $E$ of integer vectors:
$E\subseteq\Set{\vec{z}\in\mathbb{Z}^{d}:\lvert z_{i}\rvert\leq
k}\setminus\Set{0}$, where $k\in\mathbb{N}^{+}$ bounds the maximum number of
steps that a connection vector $\vec{z}$ makes per dimension. The lattice
representation by connection vectors allows constructing lattices with so-
called complex neighborhoods [54, 55] yet it is more general and flexible than
the typical approach of specifying lattices by nearest neighbor connections
[54, 55]. Therefore, the chosen representation is particularly suited for
optimizations within a large space of geometries. We only consider lattices
where all nodes have an identical neighborhood, so the presence of a vector
$\vec{z}\in E$ implies that there is also a connection to a neighbor in the
opposite direction $-\vec{z}\in E$. Therefore, all lattices that we consider
have an even vertex degree.

A flow chart of our algorithm is shown in Fig. 3(a). The algorithm starts from
a lattice where every node only has two connection vectors $\pm\vec{z}_{*}$
($E=\\{\vec{z}_{*},-\vec{z}_{*}\\}$) that are randomly chosen from the set
$\Set{\vec{z}\in\mathbb{Z}^{d}:\lvert z_{i}\rvert\leq k}\setminus\Set{0}$.
Next, two random vectors $\pm\vec{z}$ are taken from a reservoir of vectors
$R\equiv\Set{\vec{z}\in\mathbb{Z}^{d}:\lvert z_{i}\rvert\leq
k}\setminus\Set{0,\pm\vec{z}_{*}}$ and added to the lattice (resp. moved from
$R$ to $E$). If the percolation threshold $\lambda_{c}^{\eta}\left(E\right)$
improves by adding these vectors, the algorithm adds further vectors from $R$.
In contrast, if the percolation threshold gets worse, the algorithm removes
the two most recently added vectors from $E$ and adds another pair of vectors
$\pm\vec{z}\in R$ instead. If adding any vector from $R$ just makes the
percolation threshold worse, the algorithm terminates. In this case, $E$
likely represents a good lattice that cannot be improved by adding any other
vector remaining in $R$ 333Note, that one could instead start from a lattice
with all possible vectors and remove vectors randomly. However, when starting
from a lattice with a very high vertex degree, the algorithm often does not
succeed. The reason is that the percolation threshold can only be determined
with limited accuracy which can cause the algorithm to terminate too early..

Figure 3: (a) Flow chart of the algorithm used to optimize the loss tolerance
of fusion lattices. The lattice is represented by a set of nodes on
$\mathbb{Z}^{d}$ together with connection vectors between the nodes specified
by the set $E$. The algorithm adds pairs of connection vectors $\pm\vec{z}\in
R$ to $E$ trying to improve the percolation threshold
$\lambda_{c}\left(E\right)$. (b) Two-dimensional fusion lattice $L_{2d}^{a}$
with high loss tolerance. In the abstract representation of the fusion lattice
(upper part), every connection is a fusion, and every node is the center of a
resource state. For one exemplary star-shaped resource state, an explicit
representation of the fusions is shown (lower part). The red and blue graph
structures represent the fusions connected to two qubits in the center of the
plot. Every node on the two-dimensional square lattice (gray) has the same
connection pattern. (c) Optimized fusion lattice $L_{3d}^{a}$ in three
dimensions. Every qubit (node) is linked to eight neighbors by fusions
(edges). An exemplary pair of connection vectors $\pm\vec{z}$ is illustrated
in the lower part.

In two and three dimensions, we find lattices that show better performance in
comparison to the simple lattices studied above. These lattices ($L_{2d}^{a}$
and $L_{3d}^{a}$) are shown in Fig. 3(b,c) and the vectors representing the
different lattices are given in Appendix 8.7. They can already tolerate up to
6.5% photon loss when the standard type-II fusions only succeed with
$p_{s}=0.5$. The lattice $L_{2d}^{a}$ has an interesting spider structure with
some local connection plus some long legs making connections to nodes that are
further away. We believe that this lattice has a high loss tolerance because
it imitates properties of higher dimensional lattices. In a high-dimensional
lattice, a loss may break a connection along a certain dimension, but
connections in the other dimensions find a way around it. The long connections
of the spider-lattice may play a similar role by bridging local interruptions
caused by photon loss. In four dimensions and with a photon as the central
qubit, the algorithm finds a lattice with a performance that is almost
identical to the 4d hypercubic lattice. Remarkably, when the central qubit is
a spin, the algorithm finds exactly the 4d hypercubic lattice (resp. an
identical lattice, up to distortions).

##  6 Discussion

We have proposed a scheme for generating large spin cluster states for
measurement-based quantum computing. The basic building blocks are star-shaped
resource states that can be generated on demand by various types of quantum
emitters with a spin [19, 21], a resource that is experimentally in reach [20,
26, 23, 22]. We fuse these resource states performing all fusions
simultaneously (ballistically) allowing a fast clock rate for generating
large-scale entanglement. We only use non-boosted rotated type-II fusion [30],
and find fusion lattices in two and higher dimensions that can tolerate about
$6-8\%$ photon loss. This loss tolerance is an improvement by more than a
factor of two compared to similar ballistic schemes [5] that are all-photonic
and an order of magnitude compared to intrinsically fault-tolerant schemes [9,
56] even though those schemes assume boosted fusion and some even employ
larger resource states. The reason for the improvement is a combination of our
lattice optimization and the fact that some adaptiveness is moved to the
required lattice renormalization step [4, 47]. Alternatively, repeat-until-
success schemes can be used to further improve the loss-tolerance by a divide-
and-conquer strategy [40, 42] which, however, creates a significant overhead
since large parts of the lattice construction may have to be repeated in case
of fusion failure. Furthermore, the need to do operations sequentially puts a
high demand on quantum memories. Another proposal [43] applies a repeat-until-
success scheme to generate entanglement between one type of quantum emitter
spins which is swapped to a second type of spins upon success. Such a highly
adaptive scheme can improve the loss tolerance to arbitrary values, but
increases the requirements on the emitters, since they now need to contain two
spins. These schemes can therefore only be applied to certain emitters, such
as NV-centers [45], but are e.g. not directly applicable to quantum dots. A
suitable compromise might be a minimally adaptive repeat-until-success scheme
where each entanglement attempt can be repeated if it is unsuccessful.
Allowing a fixed number $n_{r}$ of fusion attempts would decrease the overall
rate by the constant factor $n_{r}$, but would increase the fusion success
probability to $1-1/2^{n_{r}}$ in the absence of losses (see Ref. [53]). In
contrast to such approaches, the scheme discussed in this paper optimizes the
loss tolerance while at the same time keeping the requirement on the photon
emitters to a minimum.

Our results can guide experimental work towards scalable quantum computing
with quantum emitters and provide performance thresholds that such emitters
need to meet. To further reduce the experimental requirements we see several
possible extensions of our work: (1) We only consider a certain type of
lattices and more general classes of lattices could be simulated by using
combinatorial tiling theory [57] or quasi-periodic tilings [58]. (2) In our
approach, lattices are built from star-shaped (GHZ) states. Using more complex
resource states might further improve the loss tolerance and potentially also
enable the exploitation of loss and error-tolerant sub-spaces [59]. However,
generating such states is more involved and might require direct spin-spin
gates between emitters [60, 61, 62]. (3) Our scheme uses quantum emitter spins
as the qubits and thus will require a large number of emitters. It would be
interesting to estimate this number and investigate methods to reduce it by
recycling qubits, keeping only part of the cluster state active during
computation [34] (4) Here we focus on loss tolerance. Integrating the approach
with techniques for quantum error correction against logical errors can yield
a fully fault-tolerant architecture. In this regard, it is encouraging that
fault tolerance is related to similar percolation concepts [63, 64, 65]
facilitating its integration with the current approach. Furthermore, error
correction benefits from high-dimensional structures, which also favors the
developed architecture [66, 67, 68].

##  7 Ackowledgements

We are grateful for discussions with Love A. Pettersson, Yu-Xiang Zhang, and
Peter Lodahl. We are grateful for financial support from Danmarks
Grundforskningsfond (DNRF 139, Hy-Q Center for Hybrid Quantum Networks). S.P.
acknowledges funding from the Cisco University Research Program Fund (nr.
2021-234494), from the Marie Skłodowska-Curie Fellowship project QSun (nr.
101063763), from the VILLUM FONDEN research grant VIL50326, and support from
the NNF Quantum Computing Programme.

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##  8 Appendix

###  8.1 Generating the resource states

Figure A4: (a) Pulse sequence to generate a linear cluster state with
polarization encoding [19]. By leaving out the second $\pi/2$-pulse (light
green) one creates a GHZ state. When starting with spin $\ket{\uparrow}$, a
photon with $\sigma_{+}$ polarization is emitted. When starting with spin
$\ket{\downarrow}$, a photon with $\sigma_{-}$ polarization is emitted. (b)
Corresponding pulse sequence for generating a linear cluster state or GHZ
state with time-bin encoding [21]. When starting in the state
$\ket{\uparrow}$, a photon $\ket{e}$ is emitted in the early time-bin. When
starting in the state $\ket{\downarrow}$, no photon is generated in the early
time-bin, yet in the late time-bin $\ket{l}$ after the spin has been flipped.

As we have pointed out in the main text quantum emitters with a spin degree of
freedom can be used to generate GHZ states or linear cluster states. This
concept has been the subject of various theoretical proposals [17, 19, 18, 69,
21] and some experimental realizations have recently been achieved [20, 22,
23, 26]. In this work, star-shaped resource states with a central spin qubit
are used and we therefore explain how such states can be created. The typical
idea is to make use of a level scheme where an optical $\pi$-pulse on the
emitter in the $\ket{\uparrow}$ state leads to a photon in state $\ket{0}$
which is orthogonal to the photon state $\ket{1}$ that is generated when
starting with the spin state $\ket{\downarrow}$. The typical protocol uses
polarization encoding [19, 20, 26, 22, 23] (see Fig. A4(a)), but other degrees
of freedom such as time-bin encoding can be used as well [21, 69] (see Fig.
A4(b)). To generate a resource state, the spin is initialized in the state
$\ket{+}_{s}=1/\sqrt{2}(\ket{\uparrow}+\ket{\downarrow})_{s}$. Applying $N$
$\pi$-pulses therefore leads to the following GHZ state:

$\frac{1}{\sqrt{2}}\left(\ket{\uparrow}_{s}\otimes\ket{0}^{\otimes
N}+\ket{\downarrow}_{s}\otimes\ket{1}^{\otimes N}\right)$ (A2)

Applying a single-qubit Hadamard gate $H$ to all photonic qubits transforms
this state into:

$\displaystyle\frac{1}{\sqrt{2}}\left(\ket{\uparrow}_{s}\otimes\ket{+}^{\otimes
N}+\ket{\downarrow}_{s}\otimes\ket{-}^{\otimes N}\right)$ (A3)
$\displaystyle=\prod_{j=1}^{N}C_{sj}\left(\ket{+}_{s}\otimes\ket{+}^{\otimes
N}\right),$ (A4)

with $\ket{\pm}=1/\sqrt{2}(\ket{0}\pm\ket{1})$ and $C_{sj}$ representing a
controlled-$Z$ gate between the spin and photon number $j$. This state is
exactly a star-shaped graph state with the spin as the central qubit.

Graph states and GHZ states are stabilizer states [70, 49], and creating a
star-shaped resource state from a GHZ state can thus be explained by
transforming stabilizers. The stabilizer generators 444The stabilizers
$\mathcal{S}$ of the stabilizer state $\ket{G}$ are all operators from the
Pauli-group such that $\forall_{S\in\mathcal{S}}S\ket{G}=+1\ket{G}$. The
stabilizer generators $\mathcal{S}_{g}$ are a subset of $N$ (number of qubits)
independent elements of $\mathcal{S}$ that generate $\mathcal{S}$
($\text{rank}(\mathcal{S})=\text{rank}(\mathcal{S}_{g})=N$) [49]. of the GHZ
state (A2) are:

$\braket{X\otimes X^{\otimes N},Z\otimes Z_{j}},$ (A5)

where the first operator acts on the spin qubit and the index $j\in\\{1..N\\}$
labels the corresponding photon. Applying $H^{\otimes N}$ to the GHZ state
transforms its stabilizers in the Heisenberg picture. This leads to new
stabilizer generators of the form

$\braket{X\otimes Z^{\otimes N},Z\otimes X_{j}},$ (A6)

which are the stabilizers of the star-shaped resource state with a central
spin qubit and $N$ photonic leaves 555That these operators are the stabilizers
generators follows from the X-Z-rule [71]: Applying the operator $X$ to one
qubit in a graph state is identical to applying the operator $Z$ to all its
neighbors. Thus, applying both operations simultaneously acts as identity on
the graph state as $X\cdot X=Z\cdot Z=1$.

Note that the state in Eq. (A2) can be transferred to a purely photonic GHZ
state by measuring the spin in the $X$-basis. Measuring $\ket{+}_{s}$ projects
the system into the corresponding $N$-photon GHZ state. When measuring
$\ket{-}_{s}$, an additional $Z$-gate needs to be applied to an arbitrary
photon to retain the same photonic GHZ state. Therefore, photonic star-shaped
resource states can be generated with the above method as well.

###  8.2 Fusion circuits

In this section, we describe the fusion operations that we assume for fusing
star-shaped resource states (such as the two states in Fig. A5(a)). The
process is known as type-II fusion [30] which, upon success, corresponds to
two simultaneous parity measurements involving the detection of two photons.
For pedagogical reasons we first describe two fusion operations that may be
useful for other applications [72], yet do not correspond to the fusion model
in Figs. 1(b,c). We then will give a detailed explanation of a rotated Bell
measurement that has exactly the desired properties [5].

The fusion circuit in Fig. A5(b) is a Bell measurement where the states
$\ket{\Psi^{\pm}}=\frac{1}{\sqrt{2}}(\ket{0_{A}1_{B}}\pm\ket{1_{A}0_{B}})$ are
measured upon a specific detection pattern. Sending, for instance, the state
$\ket{\Psi^{+}}$ to the depicted fusion circuit results in the detection
patterns 13 or 24, which are only measured for this particular Bell basis
state (here e.g. 13 corresponds to clicks in detectors 1 and 3).
$\ket{\Psi^{+}}$ and $\ket{\Psi^{-}}$ are eigenstates of $-Z_{A}Z_{B}$ and
$\pm X_{A}X_{B}$ with eigenvalues $+1$ (and these operators are known as the
stabilizers of the states). A successful fusion operation corresponds to a
projection on $\ket{\Psi^{+}}$ or $\ket{\Psi^{-}}$ and thus is a simultaneous
measurement of the stabilizers $-Z_{A}Z_{B}$ and $\pm X_{A}X_{B}$. The states
$\ket{\Phi^{\pm}}=\frac{1}{\sqrt{2}}(\ket{0_{A}0_{B}}\pm\ket{1_{A}1_{B}})$
cannot be distinguished by the circuit as they both result in the same
detection patterns 11, 22, 33, or 44666Note that we assume photon-number
resolving detectors here such that the detection pattern 11, for instance, can
be distinguished from a single photon in mode 1 and a lost photon elsewhere..
Their detection pattern corresponds to an independent measurement of both
qubits in the Z-basis. Failure of the fusion operation thus corresponds to the
measurement of the operators $Z_{A}\mathds{1}_{B}$ and $\mathds{1}_{A}Z_{B}$.

Simultaneously measuring $Z_{A}Z_{B}$ and $X_{A}X_{B}$ connects graph
components that were detached before. However, it does not connect star-shaped
resource states in the desired way, since it produces the graph state
illustrated in Fig. A5(b), which is not locally equivalent to the desired
graph in Fig. A5(d) [73, 74]. (Note that the derivation for the corresponding
graph transformation is very similar to the derivation of the rotated fusion
that we will discuss in the course of this section.)

Figure A5: (a) The four Bell basis states are stabilized by the operators $\pm
XX$ and $\pm ZZ$. We consider the effect of fusing the qubits A and B in the
graph that is shown on the right. We consider three different circuits in the
following. (b) Upon success (s), the depicted circuit (left half) measures the
Bell states $\ket{\Psi^{\pm}}$ and therefore the corresponding stabilizers.
The Bell states $\ket{\Phi^{\pm}}$ result in the same detection patterns and
therefore cannot be distinguished (fusion failure). In the failure case (f),
the circuit measures both qubits in the $Z$-basis. Fusion success and failure
transform the graph state as illustrated on the right. To obtain the displayed
graph state, the gate $H$ (Hadamard) needs to be applied after the fusion on
the qubit highlighted in red. (c) Rotated fusion with the gate $H$ applied to
the first qubit before the fusion. (d) Modified fusion that can be used to
merge star-shaped resource states into cluster states.
$R=\big{(}\begin{smallmatrix}1&0\\\ 0&i\end{smallmatrix}\big{)}$ is applied
post-fusion to one qubit (red) to obtain the shown graph state.

The effect of the fusion circuit can be modified by adding single qubit gates
before the Bell measurement [72, 5]. When applying such a gate $U$, the
stabilizers $S_{i}$ of the graph state get rotated in the Heisenberg picture:
$S_{i}\rightarrow US_{i}U^{{\dagger}}$. To understand the effect on the graph
state, one can analyze the effect of the Bell measurement on the rotated
stabilizers. Alternatively one can rotate the Bell measurement as well as the
stabilizers by applying $U^{{\dagger}}$ from the left and $U$ from the right.
This operation leaves the stabilizers the same
($U^{{\dagger}}US_{i}U^{{\dagger}}U=S_{i}$) and rotates the Bell measurement:
when $U$ acts on qubit A, the measured operators $X_{A}X_{B}$ and $Z_{A}Z_{B}$
become $U^{{\dagger}}X_{A}U\otimes X_{B}$ and $U^{{\dagger}}Z_{A}U\otimes
X_{B}$. Upon failure, the measured operators become
$U^{{\dagger}}Z_{A}U\otimes\mathds{1}_{B}$ and $\mathds{1}_{A}Z_{B}$

A typical modification is rotating the Bell measurement by adding a single
Hadamard gate before one of the two qubits [75, 72]. Upon fusion success, the
operators $X_{A}Z_{B}$ and $Z_{A}X_{B}$ are then measured which corresponds to
the desired fusion operation on the graph state [75, 72] (see Fig. A5(c)).
Upon failure, however, it corresponds to measuring qubit $A$ in the $X$\- and
qubit $B$ in the $Z$-basis. The $X$-measurement of one fusion qubit also
measures the central qubit of the corresponding star-shaped state in the
$Z$-basis777The reason is that the resource state in Fig. A5(a) before the
fusion has a stabilizer of the form $X_{A}Z_{a_{2}}$. Measuring $X_{A}$
therefore automatically measures $Z_{a_{2}}$ as the product $X_{A}Z_{a_{2}}$
must be equal to $+1$. [50]. This $Z$-measurement destroys one star-shaped
resource state as illustrated in Fig. A5(c). For this reason, the
corresponding fusion circuit is not suited for our purposes.

A suitable fusion circuit has been given in Refs. [5, 46] for polarization-
encoded qubits. We give a detailed explanation of a corresponding circuit for
dual-rail encoding which is based on transforming stabilizers upon the
measurements [76, 77] (fusions). The circuit is shown in Fig. A5(d) where the
gate $R=\sqrt{Z}=\big{(}\begin{smallmatrix}1&0\\\ 0&i\end{smallmatrix}\big{)}$
followed by a Hadamard gate $H=\frac{X+Z}{\sqrt{2}}$ is applied to one of the
fusion qubits 888$R$ interchanges Pauli $X$, $Y$ and leaves $Z$ the same:
$RXR^{{\dagger}}=Y$, $RYR^{{\dagger}}=-X$, $RZR^{{\dagger}}=Z$ [78]. $H$
interchanges Pauli $X$, $Z$ and leaves $Y$ the same: $HXH^{{\dagger}}=Z$,
$HYH^{{\dagger}}=-Y$, $HZH^{{\dagger}}=X$ [78]. The circuit in Fig. A5(d)
applies $K^{{\dagger}}:=H\cdot R$ to fusion qubit $A$. This gate cyclically
interchanges the Pauli matrices: $K^{{\dagger}}XK=-Y$, $K^{{\dagger}}YK=-Z$,
$K^{{\dagger}}ZK=X$ ($KXK^{{\dagger}}=Z$, $KYK^{{\dagger}}=-X$,
$KZK^{{\dagger}}=-Y$). Different gate choices are possible. Ref. [5] uses
$R:=\frac{X-Y}{\sqrt{2}}=i\cdot\exp\left(-i\frac{\pi}{2}\cdot(n_{x}X+n_{y}Y+n_{z}Z)\right)$
with $\vec{n}=\frac{1}{\sqrt{2}}\left(\vec{e}_{x}-\vec{e}_{y}\right)$ which
transforms the Pauli operators in the same way (up to signs):
$RXR^{{\dagger}}=-Y$, $RYR^{{\dagger}}=-X$, $RZR^{{\dagger}}=-Z$. Here, we can
neglect the signs as they do not affect the structure of the stabilizers
(resp. transform the graph state to a graph basis state [71]). The signs can
be compensated post-fusion in measurement-based quantum computing or by
applying a Pauli $Z$-gate to all qubits $i$ for which the stabilizer $S_{i}$
has a minus sign ($ZXZ^{{\dagger}}=-X$).. Applying this gate
$K^{{\dagger}}:=H\cdot R$ to one of the fusion qubits has the effect of
transforming the Bell measurement from $X_{A}X_{B}$, $Z_{A}Z_{B}$ to
$Z_{A}X_{B}$, $Y_{A}Z_{B}$ upon success and from $Z_{A}\mathds{1}_{B}$,
$\mathds{1}_{A}Z_{B}$ to $Y_{A}\mathds{1}_{B}$, $\mathds{1}_{A}Z_{B}$ upon
failure.

Before the fusion, the graph state depicted in Fig. A5(a) has the following
stabilizers:

$\displaystyle S_{a_{1}}=X_{a_{1}}Z_{a_{2}}$ (A7) $\displaystyle
S_{a_{2}}=X_{a_{2}}Z_{a_{1}}Z_{a_{3}}Z_{A}$ (A8) $\displaystyle
S_{a_{3}}=X_{a_{3}}Z_{a_{2}}$ (A9) $\displaystyle S_{A}=X_{A}Z_{a_{2}}$ (A10)
$\displaystyle S_{b_{1}}=X_{b_{1}}Z_{b_{2}}$ (A11) $\displaystyle
S_{b_{2}}=X_{b_{2}}Z_{b_{1}}Z_{b_{3}}Z_{B}$ (A12) $\displaystyle
S_{b_{3}}=X_{b_{3}}Z_{b_{2}}$ (A13) $\displaystyle S_{B}=X_{B}Z_{b_{2}}$ (A14)

Upon fusion success, the circuit in Fig. A5(d) measures $Z_{A}X_{B}$ and
$Y_{A}Z_{B}$ so only stabilizers commuting with these operators remain
invariant. This obviously applies to all stabilizers with no support on qubits
$A$, $B$. From the four remaining stabilizers, two that commute with both
measured operators can be obtained by multiplying $S_{a_{2}}$ with $S_{B}$ as
well as $S_{A}$ with $S_{B}$ and $S_{b_{2}}$. These multiplications yield the
following remaining stabilizers:

$\displaystyle X_{a_{2}}Z_{a_{1}}Z_{a_{3}}Z_{b_{2}}Z_{A}X_{B}$ (A15)
$\displaystyle Z_{a_{2}}Y_{b_{2}}Z_{b_{1}}Z_{b3}X_{A}Y_{B}$ (A16)
$\displaystyle X_{a_{1}}Z_{a_{2}}$ (A17) $\displaystyle X_{a_{3}}Z_{a_{2}}$
(A18) $\displaystyle X_{b_{1}}Z_{b_{2}}$ (A19) $\displaystyle
X_{b_{3}}Z_{b_{2}}$ (A20)

Omitting the Pauli matrices on the destructively measured photonic qubits $A$
and $B$ yields the following stabilizers:

$\displaystyle\tilde{S}_{a_{2}}=X_{a_{2}}Z_{a_{1}}Z_{a_{3}}Z_{b_{2}}$ (A21)
$\displaystyle
R_{b_{2}}\tilde{S}_{b_{2}}R^{{\dagger}}_{b_{2}}=Z_{a_{2}}Y_{b_{2}}Z_{b_{1}}Z_{b3}$
(A22) $\displaystyle\tilde{S}_{a_{1}}=X_{a_{1}}Z_{a_{2}}$ (A23)
$\displaystyle\tilde{S}_{a_{3}}=X_{a_{3}}Z_{a_{2}}$ (A24)
$\displaystyle\tilde{S}_{b_{1}}=X_{b_{1}}Z_{b_{2}}$ (A25)
$\displaystyle\tilde{S}_{b_{3}}=X_{b_{3}}Z_{b_{2}}$ (A26)

Applying the gate $R_{b_{2}}$ after the Bell-measurement changes $Y_{b_{2}}$
into $X_{b_{2}}$ and leaves the rest identical. The resulting operators
${S}_{i}$ are the stabilizer generators of the graph state in Fig. A5(d).

Upon fusion failure, the circuit measures the operators $Y_{A}\mathds{1}_{B}$
and $\mathds{1}_{A}Z_{B}$. Multiplying the stabilizer
$X_{a_{2}}Z_{a_{1}}Z_{a_{3}}Z_{A}$ from Eq. A8 with $X_{A}Z_{a_{2}}$ from Eq.
A10 yields $Y_{a_{2}}Z_{a_{1}}Z_{a_{3}}Y_{A}$ which is commuting with the
measurement pattern of the fusion failure. The stabilizers commuting with the
measurement pattern are (omitting the measured fusion qubits):

$\displaystyle\tilde{S}_{a_{1}}=X_{a_{1}}Z_{a_{2}}$ (A27)
$\displaystyle\tilde{S}_{a_{3}}=X_{a_{3}}Z_{a_{2}}$ (A28)
$\displaystyle\tilde{S}_{b_{1}}=X_{b_{1}}Z_{b_{2}}$ (A29)
$\displaystyle\tilde{S}_{b_{3}}=X_{b_{3}}Z_{b_{2}}$ (A30) $\displaystyle
R_{a_{2}}\tilde{S}_{a_{2}}R^{{\dagger}}_{a_{2}}=Y_{a_{2}}Z_{a_{1}}Z_{a_{3}}$
(A31) $\displaystyle\tilde{S}_{b_{2}}=X_{b_{2}}Z_{b_{1}}Z_{b_{3}}$ (A32)

When applying the gate $R_{a_{2}}$ after the fusion, these stabilizers
correspond to the graph state for the failure mode of Fig. A5(d).

Note that the central qubits $a_{2}$ and $b_{2}$ might be affected by previous
fusions. However, the above calculation remains unaffected when the initial
stabilizers (before the fusion) contain the operators $Y_{a_{2}}$ or
$Y_{b_{2}}$ from previous fusions (instead of $X_{a_{2}}$ or $X_{b_{2}}$).
When another fusion involving the leaf qubits $a_{1}$, $a_{3}$, $b_{1}$, or
$b_{3}$ is performed, then $Y_{a_{2}}$ or $Y_{b_{2}}$ may change back to
$X_{a_{2}}$ resp. $X_{b_{2}}$. To obtain a graph state, only the qubits that
have a stabilizer containing a Pauli $Y$ rather than a Pauli $X$ matrix need a
gate $R$ after the fusions. As all these gates can be performed after all
fusions are done, the need to apply $R$-gates (resp. changing the measurement
basis correspondingly for the subsequent measurement-based quantum computing
[1]) does not compromise the concept of applying all fusion operations at once
(ballistically).

###  8.3 Definition of lattices

Figure A6: Construction of different lattices by adding connection vectors to
the lattice $\mathbb{Z}^{d}$. The illustration shows three-dimensional
lattices, but the construction works in any dimension. In the illustration,
every connection vector $\vec{z}$ fulfills $|z_{i}|\leq k=1$ for all its
elements, resp. $\vec{z}\in\\{0,\pm 1\\}^{d}$. At the same time, the number of
dimensions in which the connection vector is non-zero is fixed to a specific
value $n=\sum_{i=1}^{d}|z_{i}|$. With this definition, the hypercubic lattice
corresponds to $n=1$, the fcc lattice corresponds to $n=2$, and the bcc
lattice corresponds to $n=d$. For all these cases, an exemplary connection
vector $\vec{z}$ is given in red.

In the main text, we have studied the loss tolerance of several lattices on
which star-shaped resource states are geometrically arranged and fused. In
this section, we describe how the corresponding lattices are mathematically
defined. Our source code implementing these lattices is openly available at
Ref. [79].

Graphs with a periodic lattice structure can be created by locally connecting
vertices and there are several techniques to create various classes of
lattices [80, 51, 52]. In the main text, we have explained a lattice
construction that starts with arranging unconnected vertices on the hypercubic
lattice $\mathbb{Z}^{d}$. We then specify a set of integer vectors associated
with the connections between the vertices. For most lattices that we consider,
all nodes/vertices are equivalent (so a connection vector
$\vec{z}\in\mathbb{Z}^{d}$ has a corresponding partner $-\vec{z}$ in the
opposite direction). When the maximum integer value per dimension is
restricted by $|z_{i}|\leq k$, all connection vectors are in the set
$\Set{\vec{z}\in\mathbb{Z}^{d}:\lvert z_{i}\rvert\leq k}\setminus\Set{0}$ and
thus there are in total $(2k+1)^{d}-1$ different connection vectors.

Even when just one step per dimension is allowed ($k=1$), various lattices can
be constructed by combining connection vectors, as illustrated in Fig. A6: the
hypercubic lattice is made by all $2d$ connection vectors with a non-zero
element in only one dimension ($\sum_{i=1}^{d}|z_{i}|=1$); the fcc lattice is
made by all $2d(d-1)$ connection vectors with a non-zero element in exactly
two dimensions ($\sum_{i=1}^{d}|z_{i}|=2$); and the bcc lattice is made by all
$2^{d}$ connection vectors with a non-zero step in every dimension
($\sum_{i=1}^{d}|z_{i}|=d$). Intuitively speaking, the hypercubic lattice
corresponds to every node being connected to its first nearest neighbor, while
the fcc lattice corresponds to every node being connected to its second
nearest neighbor.

With this method, we construct all lattices from Ref. [51] except the Kagome
lattice as well as lattices with complex neighborhoods (see e.g. Refs. [52,
81, 82]). In our simulations, we consider the hypercubic, the bcc, and the fcc
lattice as well as combinations of them 999The $d$-dimensional bcc lattice
alone consists of $2^{d-1}$ mutually unconnected partitions, and the fcc
lattice alone has $d(d-1)$ mutually unconnected components. Combining these
lattices with the hypercubic lattice makes them fully connected.. In three
dimensions, for instance, the combination of hypercubic (simple cubic) and fcc
lattice leads to the lattice NN+2NN [52].

Except for the diamond lattice, all the lattices that we consider can be
constructed with the above method. A graph with the structure of the
generalized diamond lattice can be created by removing edges from the
hypercubic lattice [51]. In this construction, all hypercubic edges pointing
in the first dimension ($|z_{1}|=1$) stay untouched. Every second edge with a
connection vector stepping in any of the other dimensions is removed. In
particular, when $\vec{z}$ connects the vertices $V^{A}$ and $V^{B}$ with
$V^{A}$ having the smaller coordinate in the dimension $l$
($V^{A}_{l}<V^{B}_{l}$, $|z_{l}|=1$, $l>1$), the corresponding edge
$e=(V^{A},V^{B})$ is removed iff $\text{mod}(V^{A}_{l},2)=0$.

###  8.4 Classical percolation simulations

$\lambda_{c}^{site}$ | d=2 | d=3 | d=4 | d=5 | d=6
---|---|---|---|---|---
diamond | 0.6968(7) | 0.4302(19) | 0.2968(27) | 0.2197(39) | 0.1733(14)
|  0.6971(2)[51] |  0.4301(2)[51] |  0.2978(2)[51] |  0.2252(3)[51] |  0.1799(5)[51]
hc | 0.5929(3) | 0.3119(4) | 0.1973(10) | 0.1402(5) | 0.1079(13)
|  0.59274(10)[83] |  0.3116(1)[52] |  0.19688561(3)[84] |  0.14079633(4)[84] |  0.109016661(8)[84]
fcc | 0.5928(8) | 0.1995(3) | 0.0842(2) | 0.0430(2) | 0.0252(6)
|  0.59274(10)[83] |  0.1994(2)[51, 52] |  0.0842(3)[51] |  0.0431(3)[51] |  0.0252(5)[51]
bcc | 0.5928(9) | 0.2460(6) | 0.1028(13) | 0.0426(25) | 0.0184(5)
|  0.59274(10)[83] |  0.2458(2)[51, 52] |  0.1037(3)[51] |  0.0446(4)[51] |  0.0199(5)[51]
fcc+hc | 0.4069(10) | 0.1376(1) | 0.0616(5) | 0.0334(2) | 0.0207(2)
|  0.40725395(3)[85] |  0.1372(1)[52] |  |  |
bcc+hc | 0.4074(9) | 0.1360(5) | 0.0587(6) | 0.0290(1) | 0.0147(4)
|  0.40725395(3)[85] |  $\sim$0.136 101010Reviewed site-percolation threshold for the NN+3NN lattice, private communication, K. Malarz, 2023, 0.1420(1) [52] |  |  |
$\lambda_{c}^{bond}$ | d=2 | d=3 | d=4 | d=5 | d=6
diamond | 0.6529(10) | 0.3893(7) | 0.2709(18) | 0.2072(15) | 0.1646(43)
|  0.652703645…[48] |  0.3893(2)[51, 86] |  0.2715(3)[51] |  0.2084(4)[51] |  0.1677(7)[51]
hc | 0.5000(5) | 0.2491(2) | 0.1602(7) | 0.1185(6) | 0.0940(8)
| 1/2 |  0.2488126(5)[87] |  0.1601312(2)[55] |  0.11817145(3)[84] |  0.09420165(2)[84]
fcc | 0.4996(4) | 0.1202(1) | 0.0495(2) | 0.0271(3) |
| 1/2 |  0.1201635(10)[87] |  0.049517(1)[88] |  0.0271813(2)[89] |
bcc | 0.5003(7) | 0.1801(2) | 0.0741(1) | 0.0320(12) | 0.0142(2)
| 1/2 |  0.1802875(10)[87] |  0.074212(1)[88] |  0.033(1)[51] |
fcc+hc | 0.2505(5) | 0.0753(2) | 0.0359(1) | 0.0213(1) |
|  0.25036834(6)[85] |  0.0752326(6)[55] |  0.035827(1)[88] |  |
bcc+hc | 0.2505(10) | 0.0921(2) | 0.0457(3) | 0.0244(3) | 0.0122(5)
|  0.25036834(6)[85] |  0.0920213(7)[55] |  |  |
Table A1: Results of classical site-percolation ($\lambda_{c}^{site}$, upper
part) and bond-percolation ($\lambda_{c}^{bond}$, lower part) simulations of
various lattices together with corresponding literature values. Note that some
of the lattices are identical, for instance, hc and bcc are the same in two
dimensions.

In our simulation, we first build a certain lattice, second, apply a
particular percolation model, and third analyze the resulting graph structure.
To verify that our implementation of various lattices in the first step is
correct, we perform classical bond- and site-percolation simulations of these
lattices and compare the resulting values to the literature [51, 52]. The
site-percolation thresholds, $\lambda_{c}^{site}$, resulting from these
simulations are shown in the upper part of Table A1. The simulated lattices
are mainly the lattices described in section 8.3. Additionally, we simulate
the lattices 2NN+3NN and NN+2NN+3NN [52] (lattices with complex neighborhoods)
for which we find $\lambda_{c}^{site}=0.1039(6)$ and
$\lambda_{c}^{site}=0.0978(1)$, respectively (in reasonable agreement with the
literature values $0.1036(1)$, $0.0976(1)$ [52]).

To compensate for finite size effects in our simulations, we use the approach
from Ref. [51] but more elaborate methods that do not rely on cluster spanning
could be used instead [81]. Most percolation thresholds agree very well with
the literature values. The only exceptions are the six-dimensional bcc and
diamond lattices. The deviation by a few standard deviations might be caused
by an underestimation of the systematic error (by us and/or the literature
[51]) when using the method from Ref. [51] to compensate for finite size
effects.

The lower part of Tab. A1 shows corresponding bond-percolation simulations. In
the case of no photon loss, the resulting percolation thresholds,
$\lambda_{c}^{bond}$, specify the fusion success probability that is minimally
required for creating a large percolating cluster state [56]. Also for the
bond-percolation thresholds, we find a good agreement with the literature
values.

###  8.5 Percolation thresholds

In Fig. 2 of the main text, we plot percolation thresholds for an architecture
where cluster states are built from star-shaped resource states. In Table A2
we give the corresponding numerical values for the case that the central qubit
of the star-shaped resource states is either a spin or a photon. The
percolation thresholds for the purely photonic architecture are higher
compared to the case of a central spin qubit because of unheralded photon
loss. When unheralded loss is present, the corresponding percolation
thresholds present only a necessary condition for quantum computing after
lattice renormalization, because unheralded loss cannot be foreseen during the
renormalization/path-finding [90].

lattice | d=2 | d=3 | d=4 | d=5 | d=6
---|---|---|---|---|---
diamond | $-$ | 0.9639(1) | 0.9314(4) | 0.9194(3) | 0.9145(11)
hc | 1.0 | 0.9435(1) | 0.9300(1) | 0.9281(3) | 0.9292(8)
fcc+hc | 0.9703(1) | 0.9574(1) | 0.9638(1) | 0.9691(2) |
bcc+hc | 0.9703(1) | 0.9465(1) | 0.9522(1) | 0.9591(1) |
fcc | $-$ | 0.9513(1) | 0.9581(1) | 0.9647(8) |
bcc | $-$ | 0.9422(2) | 0.9454(1) | 0.9573(7) |
lattice | d=2 | d=3 | d=4 | d=5 | d=6
diamond | $-$ | 0.9729(2) | 0.9475(3) | 0.9368(11) | 0.9329(6)
hc | 1.0 | 0.9561(1) | 0.9446(2) | 0.9428(3) | 0.9437(1)
fcc+hc | 0.9768(1) | 0.9653(1) | 0.9703(1) | 0.9746(5) |
bcc+hc | 0.9768(1) | 0.9575(1) | 0.9616(3) | 0.9671(2) |
fcc | $-$ | 0.9605(1) | 0.9657(1) | 0.9710(3) |
bcc | $-$ | 0.9546(1) | 0.9563(2) | 0.9656(4) |
Table A2: Percolation thresholds $\lambda_{c}^{\eta}$ for the loss-tolerance
of fusion lattices in several dimensions (with rotated type-II fusion and
star-shaped resource states). In the upper part, the central qubit of every
star-shaped resource state is a spin that cannot suffer unheralded loss. In
the lower part, the star-shaped resource states are purely photonic and the
central qubits can suffer unheralded loss. For the two-dimensional diamond
brickwork lattice (honeycomb), no percolation is possible since the bond-
percolation threshold of the lattice is above the fusion probability of 0.5
[30]. The two-dimensional simple cubic (square) lattice has a bond percolation
threshold of exactly 0.5 and $\eta=1.0$ is therefore required.

###  8.6 Size of the largest connected component

Figure A7: Size of the largest connected component relative to the initial
number of star-shaped resource states in the fusion lattice. The largest
component size is plotted against the efficiency $\eta=1-p_{loss}$. Smaller
lattice sizes are indicated by a lighter color, but the curves almost overlap
for the chosen lattice sizes. (a) Simulation with a spin qubit as the central
qubit of every star-shaped resource state. The dashed lines illustrate how the
number of active qubits in the six-dimensional hypercubic and diamond lattices
decreases in the regime of low photon loss. (b) The corresponding simulations
for purely photonic resource states.

In the main text, we have presented percolation thresholds that specify the
loss tolerance of different fusion lattices. When the efficiency
$\eta=1-p_{loss}$ is above this threshold, the created graph state has a large
connected component that grows linearly with the system size. From a practical
point of view, it is interesting to know the size of the largest connected
component of the graph. We simulate the largest connected component size as a
function of $\eta$ using periodic boundary conditions (to reduce finite size
effects). The results of these simulations are presented in Fig. A7. When a
precise number for $\eta$ is known in an experiment, such simulations show
with which fusion lattice the largest connected graph state can be obtained. A
smaller percolation threshold does not necessarily mean that the lattice
performs better for all values of $\eta$. An example is the four-dimensional
diamond lattice which has a slightly higher percolation threshold than the
four-dimensional hypercubic lattice, yet performs better in terms of component
size in a certain range of $\eta$.

An interesting point is that the size of the largest connected component
decreases linearly for very low losses. The slope of this decrease is directly
connected to the vertex degree $d_{v}$ of the lattice. Consider a fusion
lattice made of star-shaped resource states with a central spin qubit in a
quantum emitter. If $|V|$ is the number of spins, then the number of fusion
photons is $d_{v}|V|$. On average, the number of lost photons is then
$d_{v}\cdot|V|\cdot p_{loss}$. Every lost photon on a fusion edge $E$ will
make both spins connected to $E$ useless (as they are measured in the
Z-basis). For very low photon losses, there are practically no spin qubits
that are affected by two photon losses simultaneously and the overall number
of active spin qubits in the graph thus becomes $|V|\cdot(1-2\cdot d_{v}\cdot
p_{loss})$. The slope with which the relative size of the largest connected
component decreases at low losses is, therefore, $-2\cdot d_{v}\cdot
p_{loss}$. We have illustrated this slope for the six-dimensional hypercubic
lattice and the six-dimensional diamond lattice in Fig. A7(a). The six-
dimensional hypercubic lattice has a vertex degree of $d_{v}=12$ and the
corresponding slope matches the simulation well. For the diamond lattice, the
match is less good which we attribute to its lower vertex degree of $d_{v}=8$
in combination with the finite fusion success probability. Fusion failure
leads to missing bonds and the size of the largest connected component can
therefore be smaller than the number of active spins (some spins may be
unconnected even for zero loss).

When the photon loss increases, the slope of the largest connected component
as a function of loss can change for two reasons. (1) For more losses, the
chance that the same spin qubit is affected by two or more different photon
losses increases. Consequently, fewer spin qubits are affected and the slope
bends towards less negative values (compared to $-2\cdot d_{v}\cdot p_{loss}$)
lowering the percolation threshold. This effect can be seen for lattices with
high vertex degrees such as the five-dimensional fcc+hc lattice. The high
vertex degree increases the chance that a qubit is affected by two losses. (2)
When a lattice is already heavily affected by photon loss, both loss and the
finite fusion probability may cause a large component to be split into two
pieces. In this case, the size of the largest component rapidly decreases
which causes the slope to become more negative (compared to $-2\cdot
d_{v}\cdot p_{loss}$) making the percolation threshold higher. This effect can
be seen for the three-dimensional lattices in Fig. A7.

###  8.7 Results of lattice search

lattice | vectors $\\{\vec{z}^{(j)}\\}$ representing lattice | $\lambda_{c}^{\eta}$
---|---|---
$L_{2d}^{a}$ | {[5,7], [7,4], [0,3], [7,-6], [7,-5]} | 0.9344(1)
$L_{2d}^{b}$ | {[1,2], [6,-6], [1,4], [4,-5]} | 0.9362(3)
$L_{2d}^{c}$ | {[4,-3], [7,1], [4,4], [7,2]} | 0.9352(1)
$L_{3d}^{a}$ | {[1,0,1], [1,-1,-1], [1,1,0], [1,1,-1]} | 0.9326(2)
$L_{3d}^{b}$ | {[1,2,-2], [2,0,1], [1,-1,-1], [2,-1,0]} | 0.9307(5)
$hc_{4d}$ | {[1,0,1,0], [1,0,0,1], [1,0,0,0], [1,-1,0,-1]} | 0.9295(2)
dimension | vectors $\\{\vec{z}^{(j)}\\}$ representing lattice | $\lambda_{c}^{\eta}$
d=2 | {[1,1], [6,-7], [3,1], [7,-2]} | 0.9493(2)
d=3 | {[1,1,-1], [0,1,1], [1,-1,1], [1,-1,0]} | 0.9468(1)
d=4 | {[1,-1,0,1], [1,-1,-1,1], [1,1,0,-1], [1,-1,1,-1], [1,1,0,0]} | 0.9444(7)
Table A3: Fusion lattices with particularly high tolerance to photon loss and
the corresponding percolation thresholds. For the upper part of the table, the
central qubit of each star-shaped resource state is a spin. For the lower
part, the resource states are purely photonic.

We have performed a search for particularly loss-tolerant fusion lattices with
a greedy optimization described in the main text. Every lattice is represented
by a set of connection vectors $\\{\vec{z}^{(j)}\\}$ that specify connections
between lattice points $\mathbb{Z}^{d}$. Note that the presence of a
connection vector $\vec{z}^{(j)}$ means that there is also a connection along
the vector $-\vec{z}^{(j)}$ since we only consider lattices where all nodes
are equivalent. So if we give say 4 different vectors $\vec{z}^{(j)}$, then
$d_{v}=2\cdot 4$.

The results of this search in dimensions two, three, and four are given in
Table A3 together with the corresponding percolation thresholds. First, we
consider the case that the central qubits of the resource states are spins. In
two dimensions, we allow connection vectors with $|\vec{z}^{(j)}_{i}|\leq 7$
and we find the lattices $L_{2d}^{a}$, $L_{2d}^{b}$, $L_{2d}^{c}$ (with
$L_{2d}^{a}$ being shown in Fig. 3(b) of the main text). In three dimensions,
we allow all connection vectors with $|\vec{z}^{(j)}_{i}|\leq 1$ and we find
the lattice $L_{3d}^{a}$ which is shown in Fig. 3(c) of the main text.
Allowing $|\vec{z}^{(j)}_{i}|\leq 2$ leads to the slightly better lattice
$L_{3d}^{b}$. The optimized four-dimensional lattice (with
$|\vec{z}^{(j)}_{i}|\leq 1$) is equivalent to the hypercubic lattice up to
distortions. Finally, we also consider purely photonic resource states. The
results of these simulations are shown in the lower part of Table A3.