# Errors in heralded circuits for linear optical entanglement generation

Reece D. Shaw<EMAIL_ADDRESS>Quantum Engineering Technology Labs,
H.H Wills Physics Laboratory and Department of Electrical and Electronic
Engineering, University of Bristol, Bristol, United Kingdom  Quantum
Engineering Centre for Doctoral Training, Centre for Nanoscience and Quantum
Information, University of Bristol, Bristol, United Kingdom  Alex E. Jones
Quantum Engineering Technology Labs, H.H Wills Physics Laboratory and
Department of Electrical and Electronic Engineering, University of Bristol,
Bristol, United Kingdom  Patrick Yard  Quantum Engineering Technology Labs,
H.H Wills Physics Laboratory and Department of Electrical and Electronic
Engineering, University of Bristol, Bristol, United Kingdom  Anthony Laing
Quantum Engineering Technology Labs, H.H Wills Physics Laboratory and
Department of Electrical and Electronic Engineering, University of Bristol,
Bristol, United Kingdom

###### Abstract

The heralded generation of entangled states underpins many photonic quantum
technologies. As quantum error correction thresholds are determined by
underlying physical noise mechanisms, a detailed and faithful characterization
of resource states is required. Non-computational leakage, e.g. more than one
photon occupying a dual-rail encoded qubit, is an error not captured by
standard forms of state tomography, which postselect on photons remaining in
the computational subspace. Here we use the continuous-variable (CV) formalism
and first quantized state representation to develop a simulation framework
that reconstructs photonic quantum states in the presence of partial
distinguishability and resulting non-computational leakage errors. Using these
tools, we analyze a variety of Bell state generation circuits and find that
the five photon discrete Fourier transform (DFT) Bell state generation scheme
[Phys Rev. Lett. 126 23054 (2021)] is most robust to such errors for near-
ideal photons. Through characterization of a photonic entangling gate, we
demonstrate how leakage errors prevent a modular characterization of
concatenated gates using current tomographical procedures. Our work is a
necessary step in revealing the true noise models that must be addressed in
fault-tolerant photonic quantum computing architectures.

The workhorse of linear optical quantum computation (LOQC) is Hong-Ou-Mandel
(HOM) interference which enables indirect interactions and subsequent
entanglement between single photons [1]. The first proposal for universal
quantum computation with photons was the Knill, Laflamme and Milburn (KLM)
scheme that requires a large number of optical elements, quantum memory, and
phase stability of the resulting interferometer [2]. More recently, prominent
proposals for the implementation of LOQC, which are built upon the
measurement-based paradigm of quantum computing (MBQC), combine small photonic
entangled states through fusion operations. This provides a universal
computational resource on which algorithms can be executed through single-
qubit measurements [3]. The overheads of KLM are now shifted to the generation
and fusion of small entangled resource states (Bell, Greenberger-Horne-
Zeilinger (GHZ)) [4] composed of indistinguishable photons. The most modern
architecture for universal photonic quantum computing, known as fusion-based
quantum computation (FBQC), also relies on the generation of Bell and GHZ seed
states. Seed states are consumed in resource state generators to provide
states used to construct fault-tolerant fusion networks [5, 6, 7].

In practice, no two photons in a device will be perfectly indistinguishable
and no device will operate without error. Physical errors in photonics may
include photon distinguishability, state impurity, loss, and higher-order
source emissions, all of which will degrade the quality of entangled photonic
states. Partial photon distinguishability can lead to unwanted correlations
between the photonic degree of freedom in which we encode our quantum
information, and other internal degrees of freedom [8]. In heralded circuits,
these correlations can result in non-computational leakage, where more than
one photon occupies the modes defining a dual-rail qubit. Despite the
fragility of quantum states to such noise, if physical error rates are below
some threshold then quantum error correcting codes can be used to suppress
logical error rates arbitrarily, permitting fault-tolerant quantum computation
[9]. Such thresholds bound the desired fidelities of photonic resource states
used as primitives in LOQC architectures. For stringent resource estimates on
implementations of quantum error correcting codes in hardware, we would
ideally like a detailed characterization of the noise model prescribed to our
heralded physical resource states. These considerations open up the question
of how small entangled photonic resources are impacted by errors, and the
implications on universal photonic quantum computing architectures more
broadly.

In this paper we develop a simulation framework to model entangled resource
state fidelities in the presence of errors. This employs a continuous-variable
(CV) representation that naturally captures realistic photonic errors before
projection into the Fock space of discrete-variable (DV) entangled states.
Through first quantization and representation theory, namely Schur-Weyl
duality, the true nature of photonic states considering partial
distinguishability and non-computational leakage errors can be revealed [10,
11, 12, 13, 14]. We benchmark entanglement generation circuits that play a key
role in photonic quantum computing schemes, both in the second and first
quantized state representations. Sec. I details photonic state generation in
the ideal case and then Sec. II discusses general photon imperfections and
postselected state reconstruction. The CV backend we utilise for the
simulation of linear optical circuits is described in Sec. III. The
performance of heralded linear optical entanglement generation circuits with
postselection is detailed in Sec. IV. We then analyze our resources in the
first quantized picture in Sec. V before finally discussing Type-II fusion
gate error characterization in Sec. VI.

##  I Ideal Bell State Generation

A common approach to encoding photonic qubits in linear optical devices is
dual-rail encoding, where two photonic modes and one photon can define a
qubit. In the integrated photonics platform, qubits are often encoded over
spatial modes. This allows high fidelity arbitrary single qubit operations
using programmable two-port Mach-Zehnder interferometers and phase shifters.
We denote the qubit basis states through physical Fock states as
${\ket{0}_{L}}\coloneqq{\ket{1}_{1}}{\ket{0}_{2}}$ and
${\ket{1}_{L}}\coloneqq{\ket{0}_{1}}{\ket{1}_{2}}$, where the occupation
indicates if a photon is present in a given spatial mode (subscript). Such
mode occupations where one photon is present across a dual-rail qubit, i.e.,
$\\{\ket{0}_{L},\ket{1}_{L}\\}$, are states in the computational subspace.
There are four maximally entangled Bell states

$\ket{\Psi^{\pm}}=\frac{1}{\sqrt{2}}\left(\ket{01}_{L}\pm{\ket{10}_{L}}\right),$
(1)
$\ket{\Phi^{\pm}}=\frac{1}{\sqrt{2}}\left(\ket{00}_{L}\pm{\ket{11}_{L}}\right),$
(2)

with the states $\ket{\Psi^{\pm}},\ket{\Phi^{\pm}}$ all connected through
single-qubit operations. We focus on heralded states where we infer the
presence of event-ready entangled states, conditioned on the detection of a
subset of state herald photons. This is in contrast to postselection, where
successful gate operation is inferred by the detection of all photons in the
circuit [15]. There are also hard limits on the accessible entangled photonic
states in the postselected regime [16]. It has been shown that heralded linear
optical Bell state generation (HBSG) requires at least four photons, where two
are measured in auxiliary output modes [17, 18].

Figure 1:  Heralded Bell state generation circuits which produce the
$\ket{\Phi^{+}}$ Bell state when successful heralding has occurred. We denote
input photons as multi-coloured to suggest potential imperfections.

There are four candidate linear optical schemes for HBSG which we model in
this paper with errors. Each of these circuits takes a set of
indistinguishable input photons, $\chi$, and projects a subset of these onto a
Bell state. This is realised with a Positive Operator-Valued Measure (POVM)
through appropriate interferometry and measurement. The first interferometer
requires four single photons and eight modes (4P8M) [19]. Based on the
measurement of two photons in two of six state herald patterns, we obtain
$\ket{\Phi^{+}}$ up to permutations of the output modes (whether a permutation
is required will depend on the state herald pattern). Another scheme proposed
in [20] also requires four input photons, this time interfered across six
spatial modes, again with two photons measured (4P6M). A more recent proposal
in [21] provides a more compact means of Bell state generation for the same
number of input photons. This scheme requires a five-mode interferometer with
two photons detected in a single mode (4P5M), exhibiting four-photon
interference, unlike in the 4P8M and 4P6M schemes which only contain two and
three-photon interference respectively. In [21] it is shown that the 4P6M
scheme can be partitioned into two three-photon interference schemes, whereas
the 4P5M scheme cannot be partitioned further. The final Bell state scheme we
analyze is based on a proposal for arbitrary dimensional heralded GHZ state
generation, exploiting the zero transmission law in discrete Fourier
transforms (DFT) [22]. Five photons interfere in a $5\times 5$ DFT unitary
(5P5M), and the detection of three photons in a single mode leaves the
remaining photons in a Bell state. Each of these schemes is shown in Fig.1.
The success probabilities of these HBSGs (for a fixed heralding pattern) are
approximately $0.031,0.074,0.11$ and $0.096$ respectively [23, 24]. Whilst the
success probabilities of these schemes are not equal and lead to different
resource requirements for near-deterministic generation with multiplexing,
here we assume that we have a set of detection events ideally leading to
heralding of our desired Bell state, $\ket{\Phi^{+}}$. In Appendix A we
discuss 3-(qubit)GHZ state generation (6P12M) which can be seen as a natural
extension of the 4P8M HBSG scheme.

##  II Photons in practice

Indistinguishable single photons are a critical resource for photonic quantum
information processing. If photons are partially distinguishable in internal
degrees of freedom such as spectrum, which-path information will degrade
subsequent quantum interference between these photons. Here we introduce
models for partially distinguishable photons and describe how this is
manifested in realistic photon sources. We then discuss source-agnostic state
reconstruction techniques which are usually applied in LOQC.

###  II.1 Photon distinguishability and sources

Two photonic distinguishability structures that have been proposed are the
orthogonal bad-bits (OBB) and random source (RS) models [25]. In the OBB model
each photon’s state is $a_{\gamma}^{{\dagger}}[\psi_{i}]\ket{0}$, with
$a_{\gamma}^{{\dagger}}[\psi_{i}]$ describing a creation operator for path
$\gamma$ where the $i^{th}$ photon’s pure internal state is

$\ket{\psi_{i}}=\mathcal{V}^{1/4}\ket{\xi_{0}}+(1-\sqrt{\mathcal{V}})^{1/2}\ket{\xi_{i}},$
(3)

consisting of a common state $\ket{\xi_{0}}$ and a contribution from an
orthonormal state
$\ket{\xi_{i}}(\innerproduct{\xi_{i}}{\xi_{j}}=\delta_{ij})$. In this model
the HOM visibility for a photon pair is given as
$\mathcal{V}_{ij}=\lvert{{{\innerproduct{\psi_{i}}{\psi_{j}}}}}\rvert^{2}=\mathcal{V},i\neq
j$. In the RS model, photon sources are assumed to produce independent and
identically distributed (i.i.d) photons in internal states $\nu(\vec{x})$ with
probability $P(\vec{x})$, where $\vec{x}$ are variables that characterize the
photon’s internal state. By assuming independent sources we can define a
common average state for each photon through

$\rho=\sum_{\vec{x}}\text{P}(\vec{x})\nu(\vec{x})=\sum_{i}p_{i}\ket{\xi_{i}}\bra{\xi_{i}}.$
(4)

Here, the average visibility for photons from any pair of sources is given by
$\mathcal{V}=\Tr(\rho^{2})=\sum_{i}{p_{i}^{2}}$. For the OBB model, as the
internal state contribution for each photon comes from an orthonormal basis,
they do not interfere with one another. This leads to the OBB and RS models
being equivalent in the high visibility regime where for the OBB model,
$\mathcal{V}\sim 1$, and for the RS model, $p_{0}\gg p_{i\neq{0}}$, since then
the most likely error is to have a single photon with an internal state that
is distinguishable from the others [25].

These models of imperfect photons can be associated more explicitly to
specific photon sources. Common approaches for photon generation use material
non-linearities, producing photon pairs where a signal photon is heralded to
infer the presence of an event-ready idler photon (source heralding). For HBSG
circuits each photon in $\chi$ will require the detection of a signal photon
to ensure the correct number of photons are injected into the circuit.
Examples of spontaneous photon generation processes are parametric down
conversion (SPDC) and four-wave mixing (SFWM), with interaction Hamiltionians
that represent the generator of multimode two-mode squeezing operators as

$\hat{H}=\iint{d\nu_{1}d\nu_{2}}F(\nu_{1},\nu_{2})a_{1}^{{\dagger}}(\nu_{1})a_{2}^{{\dagger}}(\nu_{2})+h.c,$
(5)

with $F(\nu_{1},\nu_{2})$, the photon-pair correlation function known as the
joint-spectral amplitude (JSA). This function can encapsulate the quality of
photon sources based on two-mode squeezers [26] with the Schmidt decomposition
[27, 28] of the JSA given by

$F(\nu_{1},\nu_{2})=\sum_{k}{s_{k}}{\psi_{k}}(\nu_{1}){\phi_{k}}^{*}(\nu_{2}),$
(6)

where $\\{s_{k}\\}$ are positive Schmidt coefficients and
$\\{{\psi_{k}}\\},\\{{\phi_{k}}\\}$ are sets of orthonormal functions.
Describing the biphoton state $\ket{\zeta}$ (low-squeezing regime) through
evolution of the vacuum (by exponentiation of the interaction Hamiltonian)
with broadband mode operators
$\mathcal{A}_{k}^{{\dagger}}=\int{d\nu_{1}{\psi_{k}}(\nu_{1})a_{1}^{{\dagger}}(\nu_{1})}$
and
$\mathcal{B}_{k}^{{\dagger}}=\int{d\nu_{2}{\phi_{k}^{*}}(\nu_{2})a_{2}^{{\dagger}}(\nu_{2})}$
gives

$\begin{gathered}\ket{\zeta}=e^{-i\hat{H}}\ket{vac}\\\
\approx\ket{vac}-iF(\nu_{1},\nu_{2})\iint{d\nu_{1}}{d\nu_{2}}{a_{1}^{{\dagger}}(\nu_{1})}{a_{2}^{{\dagger}}(\nu_{2})}\ket{vac}.\end{gathered}$
(7)

Multiple Schmidt terms lead to an entangled biphoton state, projecting the
idler into a spectrally mixed state $\rho_{i}$, on detection of the signal
photon [26]. Discretizing Eq. 5 provides a singular value decomposition (SVD)
of Eq. 6 for the multimode two-mode squeezing operators in terms of
independent two-mode squeezers acting on orthogonal Schmidt modes. From the
above discussion the coefficients $\\{p_{i}\\}$ for RS photons are equivalent
to $\\{s_{k}^{2}\\}$. The purity $\mathcal{P}$ of $\rho_{i}$ is given as
$\Tr({\rho_{i}}^{2})=\sum_{k}{{s_{k}}}^{4}$ [29].

The other prominent approach to generating photons is exploiting deterministic
emitters. Solid-state systems such as quantum dots (QDs) have opened the path
towards high quality single photon sources due to their high single photon
generation rates and true single photon emission [30, 31]. Phonon-mediated
dephasing can lead to correlations between the phonon and single emitted
photon energies leading to spectral mixedness, analogous to the state
structure in Eq. 7 for spontaneous sources [32]. For single photons emitted
from separate QDs, imperfections due to differing emission frequencies lead to
coherent distinguishability in their internal states analogous to the OBB
model.

Photon quality from both spontaneous and deterministic sources can be improved
through a variety of techniques. Methods include spectral or temporal
filtering [33, 34], at the cost of reduced count rates, distillation
techniques [25, 35], which require many noisy single photon states, and, for
coherent distinguishability (OBB), high temporal resolution detection [36,
37].

###  II.2 Tomography of photonic qubits

Reconstructing an unknown quantum state $\rho$ is known as quantum state
tomography, allowing for the benchmarking of quantum information experiments
[38]. As we can never perfectly distinguish between non-orthogonal states such
as $\ket{0}$ and $\ket{+}$ with a single measurement, state tomography relies
on preparing multiple copies of $\rho$ and expanding it in a complete
Hermitian operator basis for the state space. In the qubit case, this operator
basis is the Pauli group,
$G=\\{\mathds{1},\sigma_{x},\sigma_{y},\sigma_{z}\\}$ where $\mathds{1}$ is
the $2\times 2$ identity matrix and $\sigma_{x},\sigma_{y},\sigma_{z}$ are the
$2\times 2$ Pauli matrices. The density matrix, $\rho_{n,2}$ of $n$ qubits can
be defined as

$\rho_{n,2}=\frac{1}{2^{n}}\sum_{i_{1},\ldots,i_{n}=0}^{3}r_{i_{1},\ldots,i_{n}}\,\mathcal{G}_{i_{1}}\otimes\ldots\otimes{\mathcal{G}_{i_{n}}}.$
(8)

Full tomographic reconstruction requires determining the expectation values of
the observables

$r_{i_{1},\ldots,i_{n}}=\langle{\mathcal{G}_{i_{1}}}\otimes...\otimes{\mathcal{G}_{i_{n}}}\rangle.$
(9)

This requires an exponential number of measurements and so quickly becomes
experimentally unfeasible [39]. For many states that occur in practical
situations however, only a small number of parameters is required to fully
describe the state. This leads to more efficient approaches for tomographic
reconstruction such as for matrix product states [40]. Low entropy (pure or
nearly pure) quantum states may be reconstructed via tomographic techniques
based on compressed sensing [41]. Pure quantum states may also be described
through the set of operators they are stabilized by (a $+1$ eigenstate of),
with these states forming the basis of many quantum error correcting codes
[42].

##  III The CV Backend

The assessment of imperfect heralded resource states in linear optical
circuits requires a robust simulation framework. Following the approach of
Quesada et al., we conditionally prepare heralded non-Gaussian states with
squeezed Gaussian input states in the CV regime [43]. The non-Gaussianity we
refer to here is that of Fock states which can be projected out from Gaussian
states through the POVM generated by interferometry and heralding. This
framework can accommodate both the CV nature of spontaneous squeezed sources
based on nonlinear frequency conversion and also the DV states generated from
single emitters (by using lossless photon-number-resolving detectors (PNRDs)).
Here we describe the simulation methodology and backend design considerations
required to reconstruct non-Gaussian states.

###  III.1 Non-Gaussian state preparation

Heralded non-Gaussian states can be conditionally prepared from a Gaussian
state defined in terms of its displacement vector and covariance matrix
$({\vec{\beta}},\sigma)$ respectively and PNRDs as described in [44]. This
requires computing an expression for the Fock basis matrix elements
$\matrixelement{\textbf{m}}{\hat{\varrho}}{\textbf{n}}$ with
$\textbf{m},\textbf{n}$ representing photon occupations in each of the $l$
modes comprising the Gaussian state $\hat{\varrho}$. Such elements can be
calculated through

$\matrixelement{\textbf{m}}{\hat{\varrho}}{\textbf{n}}=T\times\text{lhaf}(\tilde{A}).$
(10)

The quantities $T,\tilde{A}$, and the loop hafnian (lhaf) matrix function that
counts perfect matchings in weighted graphs with loops are detailed in
Appendix B. The number of steps to compute Eq. 10 is exponential in $l$.
Resource states that we will be assessing in Sec. IV onwards will be defined
over $k\subset l$ modes, based on the detection and heralding of photons in
modes $\textbf{n}_{h}=(\textbf{n}_{1},\textbf{n}_{2},...,\textbf{n}_{l-k})$.
By the partial measurement of $\hat{\varrho}$ on $\textbf{n}_{h}$, we can
reconstruct Fock basis matrix elements over the heralded non-Gaussian state
$k$-mode subspace.

Figure 2: a-c) Photon sources in the ideal, RS and OBB cases which are used in
the CV backend for state reconstruction. d) The ideal reconstruction requires
$|\chi|$ photon sources and state reconstruction is performed via tomography.
e) Reconstructing the logical state from imperfect sources is performed by
dilating the interferometer to account for the internal photon state
information before incoherently summing logically equivalent contributions.

###  III.2 Modelling imperfect sources

####  III.2.1 Random source

To map the RS photon purity to the CV backend we note that the Schmidt
coefficients $\\{{s_{k}}\\}$ can be represented by the probability of
generating a photon pair in a given Schmidt mode as

$\mathcal{P}=\sum_{i=0}^{d-1}{q_{i}}^{2},$ (11)

with $q_{i}$ defined as the normalised probability that a single photon pair
is generated from a squeezed state, $p_{i}$, in the $i^{th}$ Schmidt mode as

$q_{i}=\frac{p_{i}}{\sum_{i=0}^{d-1}{p_{i}}},$ (12)
$p_{i}={\kappa_{i}}^{2}(1-{\kappa_{i}}^{2}),$ (13)

where $\\{p_{i}\\}=\\{s_{k}^{2}\\}$ and $\kappa=\tanh{r}\in[\,0,1]\,$ is the
parameter quantifying the two-mode squeezing. The set of indistinguishable
photons, $\chi$, required for the $m$-mode entanglement generation circuits in
Sec. I can now be modelled, with arbitrary purity $\mathcal{P}$, through i.i.d
distinguishable photon sources, with non-zero photon generation probabilities
across each Schmidt mode. For each photon source required, we herald on a
signal photon. We then simulate the RS mixed idler photon by running $d$
copies of the ideal circuit in parallel and incoherently summing the results
(see Fig. 2b,e). For indistinguishable photons we have one Schmidt mode (as
seen in Fig. 2a,d) so only require one copy of the circuit.

####  III.2.2 Orthogonal bad-bits

In the OBB model, the photon described through
$a_{\gamma}^{{\dagger}}[\psi_{i}]\ket{0}$ is prepared in the pure state
$\ket{\psi_{i}}$, comprising a component shared with all other photons
$\ket{\xi_{0}}$ and a unique distinguishable state $\ket{\xi_{i}}$ (see Eq.
3). The different distinguishable states can be associated with separate modes
the photons can occupy. The states $\ket{\psi_{i}}$ can be prepared by
injecting an indistinguishable photon into a distinguishability unitary
$\mathcal{D}_{i}$ that spreads the photon across these orthogonal modes
defined as

$\mathcal{D}_{i}=\begin{bmatrix}\alpha&\cdots&\beta&&&\\\
\vdots&\ddots&\vdots&&&\\\ \beta&\cdots&-\alpha&&&\\\ &&&1&&\\\ &&&&\ddots&\\\
&&&&&1\\\ \end{bmatrix},$ (14)

where $\alpha,\beta$ represent beamsplitter matrix elements implemented
between the $0^{th}$ and $i^{th}$ modes (see Fig. 2c). Once the
distinguishability has been set, the interferometer is applied to the spatial
modes for a given distinguishable state. Similar to Schmidt contributions in
the RS model, these evolutions occur independently within an
$m({|\chi|}+1){\times}m({|\chi|}+1)$ interferometer (excluding source
heralds), where $({|\chi|}+1)$ is the dimension required to faithfully model
an OBB photonic internal state structure with a complete orthonormal basis.

Figure 3: Performance benchmarking of the HBSG circuits for the two proposed
distinguishability models – RS and OBB.
a) Logical Bell state fidelities $\mathcal{F}_{\rho}$ as a function of the
pairwise photon visibility, $\mathcal{V}$. b) Entanglement of formation
$E(\rho)$ over the same visibility range.

###  III.3 State reconstruction

Tomography provides a computational state benchmark from which to assess the
impact of distinguishability on logical state fidelities. Details of the CV
tomographic reconstruction are provided in Appendix C, with optical rotation
unitaries appended to modes defining logical qubits (as shown in Fig. 2d,e).
Logically equivalent internal state contributions to the same qubit basis are
incoherently summed to reconstruct the state in the computational subspace in
a postselected manner. When performing tomography we require independent
simulations of the linear optical circuit for the logically equivalent source
herald patterns, known as source firings. Naively we would simulate all state
herald patterns conditioned on the set of source firings but we filter these
as described in Appendix D to provide a reduction in the time overhead for
state reconstruction using Eq. 10.

Figure 4: a) Linear optical BSA for discriminating Bell states which operates
with a success probability of $0.5$. b) Optical implementation of the
teleportation of an arbitrary single photonic qubit. The general HBSG utilized
here can be represented by any of the proposed HBSGs. The output measurements
of the BSA permit teleportation without prior knowledge of the state. c)
Teleported single qubit fidelity $\mathcal{F}_{tel}$ of the $\ket{+}$ state as
a function of $\mathcal{F}_{\rho}$ for both RS (two Schmidt) and OBB photon
sources ($\rho$ used as the entangled resource in the protocol).

##  IV Performance benchmarking of circuits with state tomography

The CV backend provides us with a framework to model the effects of
imperfections in heralded entanglement generation circuits. Here we assess the
performance of proposed HBSGs and optical quantum teleportation in the
presence of partial distinguishability errors. In Appendix E we discuss photon
loss and multi-photon contamination in the presence of threshold detection.

###  IV.1 Resource generation

Simulating each HBSG introduced in Sec. I for the RS (two Schmidt) and OBB
models, in Fig. 3a we report the logical state degradation for decreasing
photon visibility, $\mathcal{V}$ (increasing distinguishability). The
physically accessible visibility range for the RS (two Schmidt) model is
$0.5\leq V\leq 1$ with $\mathcal{V}=0.5$ equivalent to two equal Schmidt terms
(maximal spectral mixedness). The minimum $\mathcal{V}$ for $K$ Schmidt modes
is $1/K$. OBB photons are modelled for $0\leq V\leq 1$ where for
$\mathcal{V}=0$ each photon is defined completely in terms of the
distinguishable orthonormal contribution $\ket{\xi_{i}}$. The choice of $K=2$
for the RS simulation enables tractable simulations on a standard laptop. In
Appendix F we report results for the RS model with a filtered waveguide source
requiring $K>2$ Schmidt modes. Whilst adding additional Schmidt modes slows
down state reconstruction, the CV backend can simply account for additional
Schmidt contributions. Each data point comes from a complete simulation of the
interferometer for a given visibility over all permitted source firings. We
compute the computational fidelity, $\mathcal{F}_{\rho}$, based on the state
overlap with ${\ket{\Phi^{+}}}$ as
$\mathcal{F}_{\rho}=\expectationvalue{\rho}{\Phi^{+}}$ with $\rho$ obtained
via standard tomography as described in Sec. III.3 and depicted in Fig. 2e.
For indistinguishable photons with $\mathcal{V}=1$, as expected we obtain
unity fidelities for both the RS and OBB models. In Appendix G we show that
$\mathcal{F}_{\rho}$ values for RS and OBB sources are approximately equal for
each HBSG for the visibility regime where both source models are physically
accessible.

In the high visibility regime (see insets Fig. 3a), the 5P5M scheme shows the
least logical degradation, whereas the 4P8M scheme is the most susceptible to
source photon imperfections. As we move to the highest photon-number
interference exhibited in any of the HBSG schemes, then we observe increased
robustness to partial distinguishability. At lower visibilities, whilst for
the RS model similar relative performance between circuits is maintained, the
4P8M scheme is degraded the least by OBB partial photon distinguishability.
For completeness, we model over the full visibility regime; however, the main
region of interest is the high visibility regime where candidate sources for
scalable photonic quantum computing should operate. Another metric for
assessing the robustness of the HBSGs to photon imperfections is the
entanglement of formation (EoF), an entanglement monotone for bipartite
quantum states [45]. The EoF asymptotically quantifies the number of Bell
states needed to prepare $\rho$ using local quantum operations and classical
communication [46]. As $\rho$ can be realized through a number of pure state
decompositions, the EoF quantifies the average entanglement of the pure state
decomposition minimized over decompositions of $\rho$ as

$E(\rho)=\min\sum_{i}{p_{i}}E(\psi_{i}).$ (15)

In Fig. 3b, the EoF is computed for each circuit, displaying the same relative
robustness as the logical state fidelities for both RS and OBB photons.
$E(\rho)$ is monotonic in the concurrence entanglement measure [47].

###  IV.2 Optical quantum teleportation

The transfer of a quantum state $\ket{\psi}$ from one location to another
disparate location is known as quantum teleportation. Teleportation can be
performed without the owner of $\ket{\psi}$ having any prior knowledge of the
state. The two parties, Alice and Bob, each hold half of a qubit Bell state
[48]. The entire system is in a pure product state
$\ket{\psi}\otimes\ket{\Phi^{+}}$, with no classical correlations or
entanglement between these two states. Alice then performs a Bell state
measurement (BSM) on her half of the entangled state ${\ket{\Phi^{+}}}$ and
the unknown state $\ket{\psi}$ she wants to teleport. Irrespective of the
measurement outcome, a classical communication channel provides Bob with
information to apply the required rotation for perfect reconstruction of
$\ket{\psi}$ on his qubit. By changing the basis of her BSM, Alice can
implement a unitary rotation on $\ket{\psi}$, with this principle forming the
basis of computation in MBQC. Optically, we can perform teleportation with a
linear optical Bell state analyzer (BSA) representing Alice’s measurement
device and a shared Bell state ${\ket{\Phi^{+}}}$ produced by a HBSG. In
photonics, it is impossible to perfectly discriminate Bell states. The
optimal, ancilla-free, probabilistic BSA is shown in Fig. 4a, and operates
with a success probability of $0.5$ [49, 50, 51].

In Fig. 4b we show the optical circuit for teleportation of a single arbitrary
photonic qubit, utilising a general HBSG as the entangled state resource
generator. If we assume each photon required for teleportation is produced
with an imperfect source, then we can compute the fidelity $\mathcal{F}_{tel}$
of a teleported $\ket{+}$ state for each HBSG circuit. In Fig. 4c we model the
performance of each HBSG for optical quantum teleportation with RS and OBB
photon sources in terms of $\mathcal{F}_{\rho}$, assuming correct heralding on
the output of the BSA. As all HBSGs perform comparably for high fidelity
resource states (see Fig. 4c insets), the choice of which generator to use for
a specific teleportation application requires optimization of other parameters
such as circuit success probability or optical depth, relative to the specific
hardware used. For example, whilst minimising the number of photons and modes
leads to the 4P5M HBSG as the desirable candidate, there are trade-offs such
as in heralding on more than one photon in a single mode that must be
considered. In terms of input photon visibility, relative HBSG robustness
reflects that of the state generation circuits. In Appendix H we model the
robustness of each of the sets of antipodal single qubit eigenstates of the
Pauli matrices utilising the 4P8M HBSG as the entangled resource generator.

##  V Distinguishability in first quantization

The discussion so far has been in the second quantized formalism of photonic
creation and annihilation operators. Now we view bosonic Fock states in first
quantization where the impact of particle distinguishability and symmetries is
more explicit. In this section we motivate the limitations of standard
tomographic reconstruction following the discussions by Adamson et al. [13,
14]. We then formalize the mapping between second and first quantized
partially distinguishable photonic states following [11]. Once these details
are laid out we describe how this is implemented in the CV backend, making use
of efficient Schur-Weyl transforms [52, 53], thus revealing the impact on
resource state fidelities.

###  V.1 Limitations in tomographic reconstruction

Quantum state tomography of photons is typically performed in a postselected
manner with the detection apparatus constraining the reconstructed state to
lie in the multi-qubit computational subspace. However, in the presence of
errors, the carefully engineered interference in state generation circuits
degrades, allowing the photonic state to leak into the full bosonic space that
contains non-computational terms. The characterization of such states was
proposed and carried out (for two-photon polarization-encoded states) in [14,
13]. Here a tomographic method is developed that faithfully accounts for state
degradation from both decoherence and internal state distinguishability,
treating each as distinct phenomena, in the first quantized paradigm. We
leverage this state reconstruction technique and analyze the impact on
entanglement generation circuits. Utilising this photonic state description,
we aim to better understand the impact of errors in heralded circuits
generating resources used in architectures for universal LOQC, than with
standard postselected tomographic approaches to state reconstruction.

Figure 5: Decomposition of the CV backend for simulating heralded linear
optical circuits in the presence of imperfections. The unitary Schur-Weyl
transform, $U_{sp,int}^{\otimes{\absolutevalue{\bar{\chi}}}}$ is performed on
the Fock space representation at the output of the transfer matrix to
benchmark logical state degradation relative to the CV tomography method.
Figure 6: Logical state fidelities $\mathcal{F}_{\tilde{\rho}}$ for HBSG
circuits for the RS and OBB source models. First quantized logical states
$\tilde{\rho}$ are reconstructed using the CV backend and QuTip library,
benchmarked relative to states reconstructed from standard tomography $\rho$
(we also plot $\mathcal{F}_{\rho}$ for comparison).

###  V.2 Photonic states - from second to first quantization

The mapping from second to first quantized photonic states requires taking the
standard mode occupation description of second quantized Fock states and
expressing this explicitly in terms of particle wavefunctions. Let us consider
a distinguishable state of two photons in different spatial (system) modes
($1,2$) with orthogonal internal state (label) modes ($A,B$). Here we assume
that the system is the degree of freedom resolved upon detection, whereas the
label is not. This state can be described by creation operators as
${a_{1,A}^{{\dagger}}}{a_{2,B}^{{\dagger}}}\ket{0}$. Obeying bosonic
symmetries, this state is mapped to a symmetrised two-particle state. We can
explicitly construct the state and separate the system and label degrees of
freedom [11] as

$\begin{gathered}\ket{\psi_{12}^{AB}}=\ket{1,A}\vee\ket{2,B},\\\
=\frac{1}{\sqrt{2}}(\ket{1,A}\otimes\ket{2,B}+\ket{2,B}\otimes\ket{1,A}),\end{gathered}$
(16)

where $\vee$ is the symmetric tensor product and each state ket corresponds to
a single particle with Hilbert space
$\mathcal{H}^{(1)}=\mathcal{H}_{S}\otimes\mathcal{H}_{L}$. This state notation
allows us to see how symmetry is distributed across each degree of freedom.
Separating the resolved and unresolved degrees of freedom for the two-particle
system-label state gives

$\ket{\psi_{12}^{AB}}=\frac{1}{\sqrt{2}}(\ket{\psi_{12}^{+}}\otimes\ket{\psi_{AB}^{+}}+\ket{\psi_{12}^{-}}\otimes\ket{\psi_{AB}^{-}}),$
(17)

where the two-particle state for a single degree of freedom is
$\ket{\psi_{ij}^{\pm}}=(\ket{i,j}\pm\ket{j,i})/\sqrt{2}$ with $(-)+$
corresponding to (anti-)symmetric particle states respectively. We observe
that whilst the state is still symmetric overall, it has components in the
anti-symmetric parts of the two-particle joint Hilbert space. Note, if we were
to apply the same mapping procedure to the indistinguishable
${a_{1,A}^{{\dagger}}}{a_{2,A}^{{\dagger}}}\ket{0}$ state then this only
occupies the symmetric two-particle subspace. The extension of the state
outside of the symmetric Hilbert space is only seen when the labels on each
particle are different – distinguishability has led to system-label space
correlations [11, 54].

Assessing the impact of photon distinguishability will require the subsequent
unitary evolution of symmetric states composing the system and label. Schur-
Weyl duality [55, 53] permits separate decompositions of the system and label
degrees of freedom in the total $N$-particle Hilbert space
$\mathcal{H}_{SL}^{(N)}$ as

$\begin{gathered}\mathcal{H}_{SL}^{(N)}=(\mathcal{H}_{S})^{\otimes{N}}\otimes(\mathcal{H}_{L})^{\otimes{N}}\\\
=(\underset{\lambda}{\oplus}\mathcal{H}_{S}^{\lambda})\otimes(\underset{\lambda^{\prime}}{\oplus}\mathcal{H}_{L}^{\lambda^{\prime}}).\end{gathered}$
(18)

The irreps $\lambda,\lambda^{\prime}$, describing subspaces of
$\mathcal{H}_{SL}^{(N)}$ which transform amongst themselves are reviewed in
Appendix I, along with the construction of symmetrised $N$-particle states and
Schur-Weyl duality more broadly. Evolution of symmetric system-label states
through ${U}_{SL}$ decomposes in the same fashion as the $N$-particle Hilbert
space,

$U_{SL}^{\otimes{N}}\simeq\underset{\lambda}{\oplus}(U_{S}^{\lambda}\otimes{U}_{L}^{\lambda}).$
(19)

Here, $\simeq$ suggests equality up to a basis transformation. Known as
unitary-unitary duality [55], unitary actions in the symmetric multi-particle
space will act on irreps independently. This unitary takes a block-diagonal
form in the Schur-Weyl basis with each block corresponding to a unique irrep
[11]. For $U_{SL}^{\otimes{2}}$, matrix elements in the symmetric and anti-
symmetric subspaces are transformed through permanents and determinants of the
unitary transformation respectively [54]. Obtaining the reduced system state
through tracing out the label can now leave the state with population outside
of the symmetric irrep of the system subspace. Through a simple description of
the photonic input state and interferometer in the presence of physical
errors, we now consider non-zero occupation across all accessible irreps.
Furthermore, in comparison to having to construct the transfer matrix over the
whole Hilbert space, we require the computation of a set of smaller matrices
[56].

###  V.3 Schur state fidelities

Photons defining our heralded resource state can be denoted with
$\bar{\chi}=\chi\setminus{h}$, where $h$ is the set of photons required for
state heralding on the set of input photons to the circuit, $\chi$. We can map
our system of $\absolutevalue{\bar{\chi}}$-photon states with internal
dimension $d$ to symmetric $\absolutevalue{\bar{\chi}}$-particle states,
defined over the $\mathcal{H}_{SL}^{(\absolutevalue{\bar{\chi}})}$ space
through the Schur-Weyl transform. For partially distinguishable photonic
states we define the system as the spatial degree of freedom, $sp$, and label
as the internal degree of freedom, $int$. As the CV backend projects out $h$,
we only require the Schur-Weyl transform on $\bar{\chi}$ before tracing out
the internal degree of freedom. The unitary evolution of the symmetric
spatial-internal photonic output state will be given by

$U_{sp,int}^{\otimes{\absolutevalue{\bar{\chi}}}}\simeq\underset{\lambda}{\oplus}(U_{sp}^{\lambda}\otimes{U}_{int}^{\lambda}),$
(20)

where ${U}_{int}^{\lambda}=\mathds{1}$ if we assume interferometry is
independent of the internal states. The Schur-Weyl transform is shown as part
of the CV backend in Fig. 5. To perform this transformation we require the
full Fock representation at the output, capturing both photon terms in the
computational subspace and non-computational terms (where both photons in
$\bar{\chi}$ occupy a single dual-rail qubit). The implementation of Schur-
Weyl transforms will be given by fixed unitary transformations for a given
$\absolutevalue{\bar{\chi}},d$ which can be stored for later use [52]. We use
the QuTip Python library [57] to perform the mapping and in Appendix J we
provide a pseudocode example of the Schur-Weyl transform to show how we
reconstruct reduced logical Schur states, which we generally refer to
hereafter as $\tilde{\rho}$.

Figure 7:  a) 8P16M circuit consisting of two 4P8M HBSGs and an appended Type-
II fusion operation. Successful heralding with indistinguishable input photons
results in projection onto the $\ket{\Phi^{+}}$ state. b) General
concantenated Type-II fusion circuit with HBSGs acting as entangled state
resource generators for successive (T-II F) gate operations.

Computing the corresponding fidelity $\mathcal{F}_{\tilde{\rho}}$ with
${\ket{\Phi^{+}}}$, we benchmark the logical state degradation for each HBSG
with both the RS and OBB models. These results are shown in Fig. 6 along with
comparative data from Fig. 3a (washed-out). For non-unity visibilities we
observe that
$\mathcal{F}_{\tilde{\rho}}<\mathcal{F}_{\rho}\;\forall\;\mathcal{V}$.
Occupation of the non-computational space in both the symmetric and anti-
symmetric subspaces is accounted for faithfully for $\tilde{\rho}$ states in
the Schur-Weyl basis. For $\rho$ (where postselection has occurred) the
computational anti-symmetric contributions are incorrectly assigned to the
computational symmetric subspace and manifest as decoherence. The probability
of being in the computational subspace for the full Fock reconstruction can be
equated to the equivalent terms in $\tilde{\rho}$, defined across the
symmetric and anti-symmetric two-photon subspaces. The disparity between
$\rho$ and $\tilde{\rho}$ persists even in the high visibility regime (see
insets Fig. 6), suggesting that degradation due to bosonic leakage is likely
to impact realistic LOQC implementations. Whilst the 5P5M HBSG is still the
most robust to imperfections, relative circuit performance for tomographic
reconstruction in this regime is not strictly maintained, with the 4P6M
circuit the most sensitive in the first quantized paradigm. These results
encapsulate the underrepresentation of state degradation due to partial
distinguishability and non-computational leakage when performing postselected
state reconstruction.

##  VI Optical Type-II Fusion

The small entangled states modelled can be used as seed states to produce
larger qubit states which are vital resources in LOQC architectures [5] or
repeater networks [58]. Growing these larger states can be achieved via
projective entangling measurements, known as fusion gates. There exist two
fusion gate variants which differ on the number of photons measured, either a
single photon (Type-I) or both (Type-II). By measuring both input photons, the
Type-II gate has an increased tolerance to loss over Type-I, and is therefore
the focus of the following section. Type-II fusion can be implemented by a
partial BSM (as introduced in Sec. IV.2).

For our purposes of comparison, we perform fusion on two Bell states generated
by HBSGs. On successful state heralding, the remaining qubits are left in a
Bell state. Whilst these initial resource states do not permit growth of
larger cluster states, modelling the gate in this manner allows us to isolate
the gate operation. We combine two 4P8M HBSG circuits in parallel, with the
projective Type-II fusion measurement appended to one output qubit from each,
resulting (in the error-free case) in projection onto $\ket{\Phi^{+}}$ on the
unmeasured qubit modes. Whilst the 5P5M HBSG circuit exhibited the highest
robustness in Sec. V, the 4P8M circuit performed comparably and has a lower
simulation overhead in terms of photon number. The full 8P16M linear optical
circuit is presented in Fig. 7a, with ideal success probability of $1.2\times
10^{-4}$ [23, 24].

Figure 8: a) Relative state degradation for $\rho$ and $\tilde{\rho}$ seed
states and subsequent Type-II gate operation with RS (two Schmidt) photon
sources. Inset focuses on post-fusion states for both the analytic expression
$\rho^{(1)}_{RS}$ and $\rho^{(1)}$ which permits non-computational leakage at
the HBSG step, in the high visibility regime. b) Fidelity of closest state
$\rho^{\prime}$ physically accessible from the 4P8M HBSG (optimized) to a
post-initial Type-II fusion state
$\equiv{\rho^{(1)}_{RS},\rho^{(1)},{\tilde{\rho}}^{(1)}}$ (see Fig. 7a). Inset
again focuses on $\rho^{(1)}_{RS},\rho^{(1)}$ states.

###  VI.1 Postselected gate operation

Describing logical state degradation from concatenated fusion operations (see
Fig. 7b.) would ideally take a modular approach, due to overheads in
simulating larger linear optical circuits with partially distinguishable
photons. For a modular characterization, we attempt to generate a post-HBSG
state with the same density matrix as the post-fusion state, i.e. mapping the
fusion operation onto the HBSG output state. More specifically, if by tuning
the visibility of photons into the HBSG we can recover the post-fusion density
matrix, then we use the single 8P16M circuit characterization to simulate the
effect of subsequent fusion operations.

If we assume that postselected state tomography is performed at the output of
the HBSG then the form of the density matrix (in the computational subspace)
can be derived analytically [25]. For the OBB distinguishability model, we see
that the post-fusion density matrix is equivalent to the post-HBSG density
matrix with a visibility remapping of $\mathcal{V}\mapsto\mathcal{V}^{2}$. For
the specific case of fusing two Bell states, the fusion gate has a simple
operation on the density matrix and concatenated fusion gates with OBB photons
can be modelled through the visibility relabelling procedure. For the RS
distinguishability model, however, we observe that this simple relationship
breaks down and, in fact, it is impossible to find a visibility change that
leaves the density matrices the same [59]. This suggests that there is some
action from the fusion gate that is not fully captured by the HSBG, implying
that we cannot simply model multiple gates in this way. The details of the
analytic state derivations for postselected Type-II fusion with RS and OBB
input photons are provided in Appendix K. An important caveat here is that
these state representations have not considered leakage into the wider non-
computational space, thus irrespective of whether the modular characterization
procedure can be implemented, we need to investigate whether this behaviour
extends to seed states characterized in the Schur-Weyl basis.

###  VI.2 Contamination from non-computational leakage

To consider contamination from non-computational leakage, we simulate the full
8P16M circuit in the presence of RS sources (two Schmidt) within the CV
backend. The circuit characterization quantifying state degradation is
provided in Fig. 8a. Here both the second quantized $\rho$ and first quantized
$\tilde{\rho}$ states are characterized, at the post-HBSG and post-fusion
stages denoted by the superscript labels $(0),(1)$ respectively (see Fig. 7a).
For completion, we also plot the postselected analytic $\rho_{RS}$ state
fidelity. For $\rho,\tilde{\rho}$ state reconstruction we required the use of
the BlueCrystal 4 High Performance Computing cluster, as these simulations are
not tractable on a standard laptop. Relative state degradation between
$\rho^{(0)}$ and $\rho^{(1)}$ is larger than that between the corresponding
$\tilde{\rho}$ states. In the Fig. 8a inset we show both the post-fusion
$\rho^{(1)}$ state fidelity and the analytically calculated $\rho^{(1)}_{RS}$,
in the high visibility regime. The discrepancy here is due to the propagation
of non-computational terms into the fusion gate which contaminate the output
state, but are not accounted for in the derivation of $\rho_{RS}$. This
leakage into the wider bosonic space at the intermediate step is accounted for
in the CV backend when simulating the full 8P16M circuit. We discuss these
error mechanisms for first quantized $\tilde{\rho}$ states, quantifying
leakage contributions from both the symmetric and anti-symmetric two-photon
subspaces in Appendix L. An important consideration is that in this example,
the fusion gate only acts on one qubit from each $\rho^{(0)}$ seed resource.
More general resource generation protocols may require a higher depth of gate
operations, resulting in additional unaccounted logical state contamination
[16]. We also include the first quantized Schur state characterization
(denoted as $\tilde{\rho}^{(1)}$) which provides insight into the non-
computational terms remaining post-fusion.

Previously, we have seen that the analytical form of the post-HBSG state does
not allow a one-to-one visibility remapping between post-HBSG and post-fusion
states in the postselected regime with RS photons. To consolidate this belief,
we attempt to generate the post-fusion Bell states by tuning the HBSG input
photon visibility. To find the closest state $\rho^{\prime}$ to the target
post-fusion Bell state (referred to now as $\rho$), we maximize the fidelity
$\mathcal{F}_{\rho,\rho^{\prime}}$. For the analytic density matrices, we
employ a Nelder-Mead optimization algorithm. For the first and second
quantized density matrices, we leverage the CV backend and perform a brute
force parameter sweep. We plot the maximum fidelity for each of these settings
as a function of the input photon visibility to the 8P16M circuit in Fig. 8b.
As expected, we see for all non-unity visibilities we cannot ever recover the
exact target density matrix. This suggests that part of the photonic state
generated after the fusion operation is not reproducable from HBSG [59].
Postselection results in the loss of critical diagnostic error information for
photonic states in the presence of partially distinguishable photons. A
faithful characterization of the noise process in this setting for photonic
states after successive fusions is not permitted with a modular
characterization approach. It is worth noting here, we do not consider
explicit fusion failures, which result in incorrect heralding patterns. These
are heralded errors and can be handled by multiplexing or subsequent quantum
error correction protocols [5].

## Discussion

In this work we have analyzed heralded linear optical circuits in the presence
of physical errors. Leveraging the flexibility of a CV simulation framework,
we have shown that the 5P5M HBSG scheme has the highest robustness to partial
distinguishability errors for near-ideal photons, when constraining the state
to the computational subspace. Efficient optical quantum teleportation depends
strongly on the interference scheme used to generate the entangled resource
consumed in the protocol, exhibiting similar relative robustness to state
generation schemes. Addressing the limitations of postselected state
reconstruction in capturing the leakage of heralded resources into the wider
non-computational bosonic space, we have utilised the first quantized particle
description of photons to model these effects more faithfully. Again, the 5P5M
scheme exhibits the highest robustness to these errors for near-ideal photons.
The first quantized state representation can be used as a diagnostic tool to
elucidate photonic states in computational primitives. Characterization of the
Type-II fusion gate in the presence of partially distinguishable photons
suggests that true state degradation is underrepresented when we perform
postselection. Here, we take a first step in revealing the true noise models
associated with heralded partially distinguishable resource states in linear
optical circuits.

These insights may lead to modifications in proposed architectures for LOQC to
mitigate for such errors, be that from the mechanisms used for resource state
generation or subsequent gate operations. Moving forward, tools elucidating
the contaminative effects of imperfect photon interference should be developed
further. Due to computational overheads, our Type-II fusion simulations used
two Bell states as resources to probe non-computational leakage. Typical
resource states in FBQC such as the 4-star and 6-ring graph states require
more complex generation circuits with Bell and 3-GHZ seed states as initial
resources. Whilst the noise in these processes will be bounded, and errors are
intrinsically local due to the FBQC construction [5], we would need to
optimize the CV backend to handle increasing photon number, even for modular
characterization. One natural extension is to consider the practical
robustness of photon distillation schemes to such noise models [25, 35]. It
may also be fruitful to explore if unitary averaging protocols which have
theoretically demonstrated improvements in the fidelity of cluster states from
Type-II fusion can provide increased resilience to such errors [60]. As
partial distinguishability and resulting non-computational leakage are
faithfully modelled in the first quantized state representation, the
understanding of photonic entangling gates in this setting should continue to
be developed. This could potentially pave the way to reliable modular resource
state characterization tools. More broadly, a vital next step is in mapping
the discussed leakage mechanisms to code-level errors for explicit relation to
fault-tolerant photonic quantum computing architectures.

Acknowledgements The authors would like to thank T. J. Bell, J. F. F. Bulmer,
A. Maïnos and P. S. Turner for useful discussions.
The authors acknowledge support from the Engineering and Physical Sciences
Research Council (EPSRC) Hub in Quantum Computing and Simulation
(EP/T001062/1). Fellowship support from EPSRC is acknowledged by A.L
(EP/N003470/1). R. D. S is supported by the Quantum Engineering Centre for
Doctoral Training, EPSRC grant EP/LO15730/1. Part of this work was carried out
using the computational facilities of the Advanced Computing Research Centre,
University of Bristol - http://www.bristol.ac.uk/acrc/.

## References

  * Hong  _et al._ [1987] C. K. Hong, Z. Y. Ou, and L. Mandel, Measurement of subpicosecond time intervals between two photons by interference, Physical Review Letters 59, 2044 (1987), publisher: American Physical Society.
  * Knill  _et al._ [2001] E. Knill, R. Laflamme, and G. J. Milburn, A scheme for efficient quantum computation with linear optics, Nature 409, 46 (2001), number: 6816 Publisher: Nature Publishing Group.
  * Raussendorf  _et al._ [2003] R. Raussendorf, D. E. Browne, and H. J. Briegel, Measurement-based quantum computation on cluster states, Physical Review A 68, 022312 (2003), publisher: American Physical Society.
  * Gimeno-Segovia  _et al._ [2015] M. Gimeno-Segovia, P. Shadbolt, D. E. Browne, and T. Rudolph, From Three-Photon Greenberger-Horne-Zeilinger States to Ballistic Universal Quantum Computation, Physical Review Letters 115, 020502 (2015), publisher: American Physical Society.
  * Bartolucci  _et al._ [2023] S. Bartolucci, P. Birchall, H. Bombin, 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 (2023), arXiv: 2101.09310.
  * Bartolucci  _et al._ [2021] S. Bartolucci, P. M. Birchall, M. Gimeno-Segovia, E. Johnston, K. Kieling, M. Pant, T. Rudolph, J. Smith, C. Sparrow, and M. D. Vidrighin, Creation of entangled photonic states using linear optics (2021).
  * 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 (2021), arXiv:2103.08612 [quant-ph].
  * Shchesnovich [2015] V. S. Shchesnovich, Partial indistinguishability theory for multi-photon experiments in multiport devices, Physical Review A 91, 013844 (2015), arXiv:1410.1506 [cond-mat, physics:physics, physics:quant-ph].
  * Nielsen and Dawson [2005] M. A. Nielsen and C. M. Dawson, Fault-tolerant quantum computation with cluster states, Physical Review A 71, 042323 (2005), publisher: American Physical Society.
  * Moylett and Turner [2018] A. E. Moylett and P. S. Turner, Quantum simulation of partially distinguishable boson sampling, Phys. Rev. A 97, 062329 (2018).
  * Stanisic and Turner [2018] S. Stanisic and P. S. Turner, Discriminating distinguishability, Physical Review A 98, 10.1103/PhysRevA.98.043839 (2018), arXiv: 1806.01236 Publisher: American Physical Society.
  * Harrow [2005] A. W. Harrow, Applications of coherent classical communication and the schur transform to quantum information theory (2005), arXiv:quant-ph/0512255 [quant-ph] .
  * Adamson  _et al._ [2008] R. B. A. Adamson, P. S. Turner, M. W. Mitchell, and A. M. Steinberg, Detecting hidden differences via permutation symmetries, Phys. Rev. A 78, 033832 (2008).
  * Adamson  _et al._ [2007] R. B. Adamson, L. K. Shalm, M. W. Mitchell, and A. M. Steinberg, Multiparticle state tomography: Hidden differences, Physical Review Letters 98, 10.1103/PhysRevLett.98.043601 (2007).
  * Gubarev  _et al._ [2020] F. V. Gubarev, I. V. Dyakonov, M. Y. Saygin, G. I. Struchalin, S. S. Straupe, and S. P. Kulik, Improved heralded schemes to generate entangled states from single photons, Physical Review A 102, 012604 (2020), publisher: American Physical Society.
  * Adcock  _et al._ [2018] J. C. Adcock, S. Morley-Short, J. W. Silverstone, and M. G. Thompson, Hard limits on the postselectability of optical graph states, Quantum Science and Technology 4, 015010 (2018), publisher: IOP Publishing.
  * Kok  _et al._ [2007] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Linear optical quantum computing with photonic qubits, Rev. Mod. Phys. 79, 135 (2007).
  * Stanisic  _et al._ [2017] S. Stanisic, N. Linden, A. Montanaro, and P. S. Turner, Generating entanglement with linear optics, Physical Review A 96, 043861 (2017), publisher: American Physical Society.
  * Zhang  _et al._ [2008] Q. Zhang, X.-H. Bao, C.-Y. Lu, X.-Q. Zhou, T. Yang, T. Rudolph, and J.-W. Pan, Demonstration of a scheme for the generation of “event-ready” entangled photon pairs from a single-photon source, Physical Review A 77, 062316 (2008), publisher: American Physical Society.
  * Carolan  _et al._ [2015] J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, Universal linear optics, Science 349, 711 (2015), publisher: American Association for the Advancement of Science.
  * Fldzhyan  _et al._ [2021] S. A. Fldzhyan, M. Y. Saygin, and S. P. Kulik, Compact linear optical scheme for Bell state generation, Physical Review Research 3, 10.1103/PhysRevResearch.3.043031 (2021), arXiv: 2105.06306 Publisher: American Physical Society.
  * Paesani  _et al._ [2021] S. Paesani, J. F. F. Bulmer, A. E. Jones, R. Santagati, and A. Laing, A scheme for universal high-dimensional quantum computation with linear optics 10.1103/PhysRevLett.126.230504 (2021), arXiv: 2105.02748.
  * [23] The ideal circuit success probabilities we quote in the main text are computed by considering a fixed state heralding pattern which generates our target resource state ((up to output mode permutations) see Fig. 1). For the 4P8M HBSG there are multiple patterns which generate maximally entangled Bell states which contribute to the overall success probability of the HBSG circuits. These calculations utilise the LOSim Python module (see Ref. [24]).
  * [24] https://github.com/jakeffbulmer/lo_sim.
  * Sparrow [2017] C. Sparrow, Quantum interference in universal linear optical devices for quantum computation and simulation 10.25560/67638 (2017), accepted: 2019-03-14T13:36:05Z Publisher: Imperial College London.
  * Thomas  _et al._ [2020] O. F. Thomas, W. McCutcheon, and D. P. S. McCutcheon, A general framework for multimode Gaussian quantum optics and photo-detection: application to Hong-Ou-Mandel interference with filtered heralded single photon sources 10.1063/5.0044036 (2020), arXiv: 2012.05991.
  * Grice and Walmsley [1997] W. P. Grice and I. A. Walmsley, Spectral information and distinguishability in type-II down-conversion with a broadband pump, Physical Review A 56, 1627 (1997), publisher: American Physical Society.
  * Law  _et al._ [2000] C. K. Law, I. A. Walmsley, and J. H. Eberly, Continuous Frequency Entanglement: Effective Finite Hilbert Space and Entropy Control, Physical Review Letters 84, 5304 (2000), publisher: American Physical Society.
  * Graffitti  _et al._ [2018] F. Graffitti, J. Kelly-Massicotte, A. Fedrizzi, and A. M. Brańczyk, Design considerations for high-purity heralded single-photon sources, Physical Review A 98, 10.1103/PhysRevA.98.053811 (2018), arXiv: 1807.01015 Publisher: American Physical Society.
  * Hoang  _et al._ [2016] T. B. Hoang, G. M. Akselrod, and M. H. Mikkelsen, Ultrafast Room-Temperature Single Photon Emission from Quantum Dots Coupled to Plasmonic Nanocavities, Nano Letters 16, 270 (2016), publisher: American Chemical Society.
  * Yamamoto  _et al._ [2005] Y. Yamamoto, C. Santori, G. Solomon, J. Vuckovic, D. Fattal, E. Waks, and E. Diamanti, Single photons for quantum information systems, Progress in Informatics , 5 (2005).
  * Tiurev  _et al._ [2021] K. Tiurev, P. L. Mirambell, M. B. Lauritzen, M. H. Appel, A. Tiranov, P. Lodahl, and A. S. Sørensen, Fidelity of time-bin-entangled multiphoton states from a quantum emitter, Phys. Rev. A 104, 052604 (2021).
  * Massaro  _et al._ [2019] M. Massaro, E. Meyer-Scott, N. Montaut, H. Herrmann, and C. Silberhorn, Improving spdc single-photon sources via extended heralding and feed-forward control, New Journal of Physics 21, 053038 (2019).
  * Legero  _et al._ [2003] T. Legero, T. Wilk, A. Kuhn, and G. Rempe, Time-resolved two-photon quantum interference, Applied Physics B 77, 797 (2003).
  * Marshall [2022] J. Marshall, Distillation of Indistinguishable Photons, Physical Review Letters 129, 213601 (2022), publisher: American Physical Society.
  * Yard  _et al._ [2022] P. Yard, A. E. Jones, S. Paesani, A. Maïnos, J. F. F. Bulmer, and A. Laing, On-chip quantum information processing with distinguishable photons (2022), arXiv:2210.08044 [quant-ph] .
  * Zhao  _et al._ [2014] T.-M. Zhao, H. Zhang, J. Yang, Z.-R. Sang, X. Jiang, X.-H. Bao, and J.-W. Pan, Entangling different-color photons via time-resolved measurement and active feed forward, Phys. Rev. Lett. 112, 103602 (2014).
  * D’Ariano  _et al._ [2003] G. M. D’Ariano, M. G. A. Paris, and M. F. Sacchi, Quantum Tomography (2003), arXiv:quant-ph/0302028.
  * Thew  _et al._ [2002] R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, Qudit quantum-state tomography, Physical Review A - Atomic, Molecular, and Optical Physics 66, 6 (2002), arXiv: quant-ph/0201052.
  * Cramer  _et al._ [2010] M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, Efficient quantum state tomography, Nature Communications 1, 149 (2010), number: 1 Publisher: Nature Publishing Group.
  * Gross  _et al._ [2010] D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, Quantum state tomography via compressed sensing, Phys. Rev. Lett. 105, 150401 (2010).
  * Gottesman [1997] D. Gottesman, Stabilizer Codes and Quantum Error Correction (1997), arXiv:quant-ph/9705052.
  * Quesada  _et al._ [2019] N. Quesada, L. G. Helt, J. Izaac, J. M. Arrazola, R. Shahrokhshahi, C. R. Myers, and K. K. Sabapathy, Simulating realistic non-gaussian state preparation, Phys. Rev. A 100, 022341 (2019).
  * Sabapathy  _et al._ [2019] K. K. Sabapathy, H. Qi, J. Izaac, and C. Weedbrook, Production of photonic universal quantum gates enhanced by machine learning, Phys. Rev. A 100, 012326 (2019).
  * Wootters [1998] W. K. Wootters, Entanglement of Formation of an Arbitrary State of Two Qubits, Physical Review Letters 80, 2245 (1998), publisher: American Physical Society.
  * 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 54, 3824 (1996), publisher: American Physical Society.
  * Nielsen and Chuang [2011] M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information: 10th Anniversary Edition_ (Cambridge University Press, 2011).
  * Bennett  _et al._ [1993] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Physical Review Letters 70, 1895 (1993), publisher: American Physical Society.
  * Calsamiglia and Lütkenhaus [2001] J. Calsamiglia and N. Lütkenhaus, Maximum efficiency of a linear-optical Bell-state analyzer, Applied Physics B: Lasers and Optics 72, 67 (2001).
  * Braunstein and Mann [1995] S. L. Braunstein and A. Mann, Measurement of the Bell operator and quantum teleportation, Physical Review A 51, R1727 (1995), publisher: American Physical Society.
  * Weinfurter [1994] H. Weinfurter, Experimental Bell-State Analysis, Europhysics Letters 25, 559 (1994).
  * Bacon  _et al._ [2006] D. Bacon, I. L. Chuang, and A. W. Harrow, Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms, Physical Review Letters 97, 170502 (2006), publisher: American Physical Society.
  * Bacon  _et al._ [2005] D. Bacon, I. L. Chuang, and A. W. Harrow, The Quantum Schur Transform: I. Efficient Qudit Circuits (2005), arXiv:quant-ph/0601001.
  * Turner [2016] P. S. Turner, Postselective quantum interference of distinguishable particles (2016), arXiv:1608.05720 [quant-ph].
  * Rowe  _et al._ [2012] D. J. Rowe, M. J. Carvalho, and J. Repka, Dual pairing of symmetry and dynamical groups in physics, Reviews of Modern Physics 84, 711 (2012), publisher: American Physical Society.
  * Stanisic [2020] S. Stanisic, On the quantum information of photons: Entanglement and distinguishability in linear optics (2020).
  * Johansson  _et al._ [2012] J. Johansson, P. Nation, and F. Nori, QuTiP: An open-source Python framework for the dynamics of open quantum systems, Computer Physics Communications 183, 1760 (2012).
  * Azuma  _et al._ [2013] K. Azuma, K. Tamaki, and H.-K. Lo, All photonic quantum repeaters, Nature communications 6 (2013).
  * [59] Here we have fixed the internal state dimension in attempts to recover the post-fusion state in simulation to $d=2$, i.e. two Schmidt terms. The region of interest for state generation is at high photon visibility $\mathcal{V}\sim{1}$, where we only expect two non-zero Schmidt terms with one dominant.
  * Singh  _et al._ [2022] D. Singh, A. P. Lund, and P. P. Rohde, Optical cluster-state generation with unitary averaging (2022), arXiv:2209.15282 [quant-ph].
  * Varnava  _et al._ [2006] M. Varnava, D. E. Browne, and T. Rudolph, Loss tolerance in one-way quantum computation via counterfactual error correction, Phys. Rev. Lett. 97 (2006).
  * Varnava  _et al._ [2008] M. Varnava, D. E. Browne, and T. Rudolph, How Good Must Single Photon Sources and Detectors Be for Efficient Linear Optical Quantum Computation?, Physical Review Letters 100, 060502 (2008), publisher: American Physical Society.
  * Paesani  _et al._ [2020] S. Paesani, M. Borghi, S. Signorini, A. Maïnos, L. Pavesi, and A. Laing, Near-ideal spontaneous photon sources in silicon quantum photonics, Nature Communications 11, 2505 (2020), number: 1 Publisher: Nature Publishing Group.
  * Raussendorf and Briegel [2001] R. Raussendorf and H. J. Briegel, A one-way quantum computer, Phys. Rev. Lett. 86, 5188 (2001).

##  Appendix A 3-GHZ state generation

$3$-GHZ states can be efficiently generated with six single photons.
Generalising the 4P8M HBSG scheme to six photons across twelve modes (6P12M)
and successfully heralding on three photons (with success probability
$3.9\times 10^{-3}$) [23, 24], we obtain a state locally equivalent to the
$3$-GHZ state (see Fig. 9):

$\ket{3-GHZ}=\frac{1}{\sqrt{2}}(\ket{000}_{L}+\ket{111}_{L}).$ (21)

Unlike with Bell states as fusion resources, subsequent fusions with $3$-GHZ
states permit the creation of 4-GHZ (and larger) states, allowing for the
growth of photonic cluster states. A cluster state structure of trees that can
tolerate up to $50\%$ qubit loss was introduced in [61]. The high threshold is
a result of the error model being restricted to qubit loss as the dominant
mechanism. Further work [62] showed that efficient growth was also possible
for non-unity efficiency photon sources and detectors. More specifically, it
is stated that efficient LOQC is possible when $\eta_{s}\eta_{h}>2/3$, where
$\eta_{s},\eta_{h}$ are the source and detector efficiencies respectively. The
loss tolerant protocol of [61] is robust to uncorrelated loss where the
cluster states generated are of an ID (independently-degraded) form. The ID-
GHZ state structure leads to the $\eta_{s}\eta_{h}>2/3$ result stated above.
Here we simulate the 6P12M circuit for generating $3$-GHZ states,
reconstructing the state with the CV backend to investigate the logical state
degradation for RS and OBB photon sources. We also model this circuit with
indistinguishable photons, under the constraint of photon loss and threshold
detection as discussed for HBSGs in Appendix E. Here we see if the ID-GHZ
state form is retained for more general source model constraints with
indistinguishable photons.

Figure 9: 6P12M heralded scheme for GHZ state generation.

In Fig. 10a we report $\mathcal{F}_{\rho}$ for both RS and OBB sources,
displaying equivalence between RS and OBB in the high visibility regime,
similar to the 4P8M HBSG circuit, of which this circuit is a natural extension
of (in terms of interferometry). In [62], the photonic states injected into
the 6P12M interferometer have the form
$\rho=\eta_{s}\ket{1}\bra{1}+(1-\eta_{s})\ket{0}\bra{0}$, which is an
appropriate assumption for lossy, otherwise ideal deterministic single photon
emitters. Imperfect QD single photon emitters or heralded spontaneous photon
sources will lead to the possibility of multi-photon contamination. Attempting
this state generation protocol, perhaps using multiplexing with threshold
detection, can lead to such terms degrading the quality of the output state.
The general expression for the ID-GHZ state that is generated (conditional on
state heralding of three photons) in the 6P12M circuit is given in the
Supplementary Info. of [62], which contains $0(\ket{vac})-3(\ket{GHZ})$-photon
components. We focus on the amplitude of the three-photon
$\ket{GHZ}\bra{GHZ}_{ID-GHZ}$ component which is given by

$(1-f)^{3}\ket{GHZ}\bra{GHZ}_{ID-GHZ},$ (22)

parameterised in terms of the loss rate, $f$, as

$f=1-{\frac{\eta_{s}}{2-\eta_{s}\eta_{h}}}.$ (23)

We compute the population of the $\ket{GHZ}\bra{GHZ}_{thr}$ component relative
to lower photon-number components using the CV backend as $P_{thr}$, now
permitting contamination from an additional photon as described in Sec. E for
HBSGs. $|P_{thr}-P_{ID-GHZ}|/P_{ID-GHZ}$ defines the difference between this
population and the ideal population of the $\ket{GHZ}\bra{GHZ}_{ID-GHZ}$ term
when injecting $\rho=\eta_{s}\ket{1}\bra{1}+(1-\eta_{s})\ket{0}\bra{0}$
photonic states into the 6P12M interferometer. This data is given in Fig. 10b
for varying $\eta_{s},\eta_{h}$ values, with the largest disparity given for
unity-efficiency state and source herald detectors.

Figure 10: Performance benchmarking of the 6P12M circuit. a) Logical 3-GHZ
state fidelities $\mathcal{F}_{\rho}$ as a function of the pairwise photon
visibility, $\mathcal{V}$ for RS and OBB sources. b) Heatmap of
$|P_{thr}-P_{ID-GHZ}|/P_{ID-GHZ}$ with indistinguishable photons in the
presence of threshold detectors and an extra unwanted photon.

##  Appendix B Continuous-variable (CV) preliminaries

In the CV formalism we can define a quantum state, $\hat{\varrho}$, over the
set of modes, $l$, through the associated Wigner function

$W(\vec{\alpha};\hat{\varrho})=\int\frac{d\vec{\xi}}{\pi^{2l}}\Tr[\,\hat{\varrho}\hat{D}(\vec{\xi})]\,\exp({\vec{\alpha}}^{T}\Omega\vec{\xi}),$
(24)

where $\vec{\alpha},\vec{\xi}$ are complex bivectors,
$\hat{D}(\xi)=\exp(\vec{\xi}^{T}\Omega\hat{\zeta})$ is the displacement
operator ($\Omega$, the symplectic form) and $\hat{\zeta}_{j}$ is the
creation-annihilation bivector for $j=1,\ldots,l$.
$\hat{\zeta}_{j}=\hat{a}_{j}$ and
$\hat{\zeta}_{l+j}={\hat{a}}_{j}^{{\dagger}}$ satisfy the canonical
commutation relations $[\,\hat{a}_{i},\hat{a}_{j}]\,=0$ and
$[\,\hat{a}_{i},\hat{a}_{j}^{\dagger}]\,=\delta_{ij}$.

We can characterize a multimode Gaussian state $\hat{\varrho}$ through its
first and second moments; the displacement vector $\vec{\beta}$ with

${\vec{\beta}}_{j}=\Tr[\,\hat{\varrho}\hat{\zeta}_{j}]\,$ (25)

and covariance matrix, $\sigma$ with elements

$\sigma_{jk}={\Tr[\,\hat{\varrho}\\{\hat{\zeta}_{j},\hat{\zeta}_{k}^{\dagger}\\}]\,}/2-{\vec{\beta}}_{j}{\vec{\beta}}_{k}^{*}.$
(26)

Shown in [43], the following quantities:

$X=\matrixquantity[0&\mathbbm{1}_{l}\\\ \mathbbm{1}_{l}&0],$ (27)
$\sigma_{q}=\sigma+\frac{1}{2}\mathbbm{1}_{2l},$ (28)
$T=\frac{\exp(-\frac{1}{2}{\vec{\beta}}^{\dagger}\sigma_{Q}^{-1}\vec{\beta})}{\sqrt{\det(\sigma_{q})\prod_{s=1}^{l}n_{s}!m_{s}!}},$
(29)

lead to the expression for computing Fock matrix elements of $\hat{\varrho}$
as

$\matrixelement{\textbf{m}}{\hat{\varrho}}{\textbf{n}}=T\times\text{lhaf}(\tilde{A}).$
(30)

Elements of $\tilde{A}$ can be defined as

$\tilde{A}_{i,j}=\begin{cases}\bar{A}_{i,j}&\text{ if }i\neq j\\\
\bar{\gamma}_{i}&\text{ if }i=j\end{cases},$ (31)

with

$A=X(\mathbbm{1}_{2l}-\sigma_{Q}^{-}1),$ (32)
$\gamma^{T}=\beta^{{\dagger}}\sigma_{Q}^{-}1.$ (33)

The number of steps to perform the lhaf calculation is $O(lAG^{2l})$ where $G$
is a geometric mean quantity defined in [43].

##  Appendix C CV tomographic reconstruction

To compute the expectation values of Eq. 9 with the CV backend, we can take
the spectral decomposition of Hermitian operators, $G_{i}$, as

$G_{i}=\sum_{j=0}^{d-1}\lambda_{j}^{(i)}\ket{v_{j}^{(i)}}\bra{v_{j}^{(i)}},$
(34)

where $\left\\{\ket{v_{j}^{(i)}}\right\\}$ are the corresponding eigenvectors
and $\lambda_{j}^{(i)}$ are the associated eigenvalues. Rotating from the
eigenvector basis to a computational qudit basis $\\{\ket{j}\\}$ is permitted
through an optical unitary, $U_{i}$, of the form

$U_{i}=\sum_{j=0}^{d-1}\ket{j}\bra{v_{j}}.$ (35)

This rotation is carried out by appending $U_{i}$ to the modes representing
each qudit of the computational subspace state, post transfer matrix. The
probability, $P(k,l..|i,j..)$, of detecting photons in modes $\\{i,j,..\\}$
can be given by

$\Tr[\,\rho\left(\ket{v_{k}^{(i)}}\otimes\ket{v_{l}^{(j)}}\otimes..\right)\left(\bra{v_{k}^{(i)}}\otimes\bra{v_{l}^{(j)}}\otimes..\right)]\,,$
(36)

where probabilities are computed through expressions of the form of Eq. 10 for
$\textbf{m}=\textbf{n}$. From Eq. 36, we can define expectation values
$\langle{G_{i1}}\otimes...\otimes{G_{in}}\rangle$ as

$\sum_{k,l..}\lambda_{k}^{(i)}\lambda_{l}^{(j)}..\times{P(k,l..|i,j..)}.$ (37)

##  Appendix D Filtering of source firings for efficient state reconstruction

When reconstructing states from linear optical entanglement generation
circuits with RS photon sources, there are equivalent valid source herald
patterns that must be independently simulated, with one simulation prescribed
to a single source firing. In the RS model the number of source firings,
$N_{SF}$, is related to the number of input photons, $|\chi|$, and Schmidt
number $K$ as $N_{SF}=K^{|\chi|}$. A given source firing constrains the
distribution of photons across Schmidt modes, allowing us to filter out a
subset of state herald patterns. By summing over photon occupations
$\textbf{m},\textbf{n}$ with the same number of photons in each Schmidt mode,
we reduce the number of lhaf calculations required for a specific Fock matrix
element of the non-Gaussian output state (see Eq. 10). This allows for more
time-efficient full Fock space state reconstruction when $K\geq 2$ compared to
not filtering out any source firings.

##  Appendix E Threshold detectors and photon loss

For the proposed HBSG circuits, we have so far considered PNRDs that measure
the number of photons in an optical mode. In addition to photon
distinguishability, we can assess the impact of threshold detection and photon
loss on generated Bell state fidelities. Photon loss can occur due to photon
scattering, absorption, or inefficiences in detection. Losses can be modelled
through a beam splitter with transmission coefficient $\eta$, where
transmitted photons are detected with an ideal photon detector and reflected
photons are considered lost. The modes supporting photonic qubits may also
suffer from loss, leading to false positives which are not resolved by
detection apparatus. Indistinguishable photons with source (state) detector
efficiencies $\eta_{s(h)}$, are modelled by applying Gaussian single-mode loss
channels $\hat{a}\mapsto\sqrt{\eta_{s(h)}}\hat{a}+\sqrt{1-\eta_{s(h)}}\hat{b}$
to the source (state) herald modes pre(post)-transfer matrix, with $\hat{b}$
representing an ancillary vacuum mode. When loss is present, multi-photon
contamination from parametric photon sources is important, as higher-order
photon terms are not perfectly heralded as they are in the lossless PNRD case.

Figure 11: Heatmap of $\mathcal{F}_{thr}$ for each HBSG circuit with
indistinguishable photons in the presence of threshold detectors and an extra
unwanted photon.

We model the performance of each HBSG by computing the fidelity,
$\mathcal{F}_{thr}$ for varying $\eta_{s},\eta_{h}$ values in Fig. 11. To
express the ignorance of threshold detection to higher-order contributions, we
adapt the CV backend logic to consider terms with higher photon occupations
and incoherently sum over these and valid logical photon mode occupations. We
truncate by only including cases where one of the sources produces two pairs
of photons (squeezing value of $\kappa=0.2$). This choice is well-motivated as
realistic photon sources will operate in the low-squeezing limit which means
higher-order photon terms are exponentially suppressed. Our simulation always
considers the emission of an additional photon meaning that we never expect to
see unity $\mathcal{F}_{thr}$ values. The maximum $\mathcal{F}_{thr}$ value is
$0.96$ for the 4P8M circuit with $\eta_{s}=\eta_{h}=1$. The 4P6M and 4P8M
circuits are affected the most by reduction in $\eta_{s}$ for a fixed
$\eta_{h}$ value. Logical state fidelities are more sensitive to
inefficiencies in the source herald detectors before the transfer matrix than
on the state herald detectors. Whilst state heralding comprises a subset of
the output circuit modes, each idler photon requires the valid heralding from
a signal photon so loss in these modes will then be compounded. For more
compact HBSG schemes requiring multi-photon state heralding on a single PNRD
in the ideal case (4P5M, 5P5M), the entanglement generation protocol is
minimally impacted by inefficiencies in the state herald detectors for a fixed
$\eta_{s}$ value. In Appendix A we analyze the 6P12M GHZ circuit under the
same constraints, along with both RS and OBB photon sources with PNRDs as
discussed for HBSGs in Sec. IV.1.

##  Appendix F HBSG with RS photons for realistic sources

Here we benchmark the proposed HBSG circuits for a realistic photon source to
demonstrate flexibility in the CV backend. Spontaneous processes for
generating photons in waveguides are energy constrained, for example in SFWM
where the pump bandwidth in the frequency domain, $\omega_{pump}$, satisfies
$2\hbar{\omega_{pump}}=\hbar(\omega_{s}+\omega_{i})$. We can optically filter
signal and idler bandwidths to constrain photon generation correlations,
improving the purity of spontaneous sources. The spectral properties of
generated photon pairs are captured by the JSA which quantifies correlation
between signal and idler photons. High purity heralded single photons are
related to separability in the JSA. As the JSA can be expressed as a sum of
orthogonal modes (see Sec. II.1) then the SVD of the JSA provides us with the
set of Schmidt coefficients $\\{{s_{k}}\\}$ that are mapped to RS photons of
purity $\mathcal{P}$. In general, this decomposition has an infinite number of
terms, so to simulate RS photon sources we truncate the SVD of the JSA to
provide us with a finite number of Schmidt modes. To ensure this truncation is
faithful we define a fidelity
$\mathcal{F}_{\rho,\rho^{\prime}}=\Tr(\sqrt{\rho\sqrt{\rho^{\prime}}\rho})^{2}$
where $\rho\equiv{JSA\cdot{JSA^{{\dagger}}}}$ is the untruncated SVD state and
$\rho^{\prime}$ is truncated for a threshold $\epsilon$. We truncate for
$s_{k}\geq\epsilon$, leaving us with a set of non-zero Schmidt coefficients
from which we reconstruct $\rho^{\prime}$. The waveguide simulation assumes
silicon with a central wavelength of $\simeq 1550$ nm and pump bandwidth of
$400$ GHz. We compute $\mathcal{F}_{\rho,\rho^{\prime}}$ in Fig. 12a for
optical filter widths with purity $\mathcal{P}$, for a set of different
thresholds. For each $\epsilon$ we report the corresponding number of non-zero
Schmidt terms $K_{trunc}$ in Fig. 12b. Based on $\epsilon=0.005$ providing
$\max(1-\mathcal{F}_{\rho,\rho^{\prime}})>$1\text{\times}{10}^{-5}$$ over the
whole filter width range, we map the $\\{\rho^{\prime}(\epsilon=0.005)\\}$
Schmidt coefficients to RS sources in our CV backend (requiring at most four
non-zero Schmidt modes) as described in Sec. III.2.1.

Figure 12: a) Fidelity of the SVD state $\mathcal{F}_{\rho,\rho^{\prime}}$
truncated for different Schmidt coefficient thresholds $\epsilon$ relative to
$\rho$, over the purity range $\mathcal{P}$ defined by the filter bandwidth.
b) Corresponding number of non-zero Schmidt terms for the state
$\rho^{\prime}$.

Simulating each HBSG circuit for RS photonic states in
$\\{\rho^{\prime}(\epsilon=0.005)\\}$ over a filter bandwidth ($\omega_{f}$)
range of $150-400$ GHz, we report $\mathcal{F}_{\rho}$ in Fig. 13. This filter
bandwidth range was chosen to reflect realistic telecommunication wavelength-
division multiplexing (WDM) filters. Relative HBSG robustness for RS sources
(two Schmidt) simulated in Sec. IV.1 is maintained over this filter bandwidth.
The maximum discrepancy of $\mathcal{F}_{\rho}$ increases as the filter
bandwidth is increased, reducing RS photon purity. An important note here is
that for photons with $\mathcal{P}\geq 0.96$, at most only three Schmidt modes
are required in simulation to faithfully model these sources to within
$\epsilon=0.005$, with near-ideal spontaneous sources such as those discussed
by Paesani et al., [63] only requiring two non-zero Schmidt terms.

Figure 13:  Logical heralded Bell state fidelity (with tomographic state
reconstruction) $\mathcal{F}_{\rho}$ for each HBSG circuit, initialized with
RS photonic states in $\\{\rho^{\prime}(\epsilon=0.005)\\}$.

##  Appendix G HBSG state fidelity equivalences (RS and OBB)

Here we display the data from Fig. 3a for each HBSG separately for both
proposed distinguishability models, to show that $\mathcal{F}_{\rho}$ is
highly constrained by the pairwise photon visibilities $\mathcal{V}$. In [25]
this was shown for the 4P8M HBSG, whereas here we show this property extends
to the other HBSG circuits which we analyze.

Figure 14:  Relative RS/OBB $\mathcal{F}_{\rho}$ for each HBSG as a function
of the pairwise photon visibility $\mathcal{V}$.

##  Appendix H Optical quantum teleportation of arbitrary single qubit
photonic states

Teleportation of the $\ket{+}$ state was analyzed in detail for both RS and
OBB sources, with each of the four HBSGs acting as the entangled resource
generator, in Sec. IV.2. Whilst the $\ket{+}$ state comprises the standard
initial resource for cluster state generation (pre-CZ gate operation) enabling
MBQC [64], other single qubit states may be of interest in optical quantum
teleportation protocols. Here we simulate teleportation protocols (see Fig.
4b) for the sets of antipodal single qubit eigenstates of the Pauli matrices,
providing coverage of the Bloch sphere. For these simulations the 4P8M HBSG
represents the entangled resource generator and photons are of the RS (two
Schmidt) form.

Figure 15:  Teleported single qubit fidelity $\mathcal{F}_{tel}$ of each of
the antipodal single qubit states as a function of $\mathcal{F}_{\rho}$ for RS
(two Schmidt) photon sources (4P8M HBSG used as the entangled resource in the
protocol).

From Fig. 15 we observe that the $\ket{+}$ state $\mathcal{F}_{tel}$ value
upper bounds the robustness of the $\sigma_{y}$ eigenstates
$\ket{i},\ket{-i}$. For each antipodal pair of states, performance is
equivalent over the entire state fidelity regime. High $\mathcal{F}_{\rho}$
teleportation equivalence between $\\{\ket{+},\ket{-},\ket{i},\ket{-i}\\}$
suggests that the discussion concerning implementations of optical quantum
teleportation with the $\ket{+}$ state holds more generally, for each of these
states.

##  Appendix I Particle symmetries and Schur-Weyl duality

For a single particle with two degrees of freedom $(S,L)$ the Hilbert space
$\mathcal{H}^{(1)}=\mathcal{H}_{S}\otimes\mathcal{H}_{L}$ can be extended for
$N$ particles to $\mathcal{H}^{(N)}=(\mathcal{H}^{(1)})^{\otimes{N}}$.
Requiring that the wavefunction associated with this $N$-particle space,
$\ket{\Psi_{D}}=\otimes_{j=1}^{N}\ket{\varphi_{j}}$ obeys bosonic exchange
symmetry, we can apply the symmetrisation operator $\hat{\mathcal{S}}$ as

$\begin{gathered}\ket{\Psi_{S}}=\hat{\mathcal{S}}\ket{\Psi_{D}}\equiv{\vee_{j=1}^{N}}\ket{\varphi_{j}}\\\
=\mathcal{N}_{\mathcal{S}}\sum\limits_{\sigma\in{S_{N}}}\ket{\varphi_{\sigma(1)}}\otimes\ket{\varphi_{\sigma(2)}}\otimes...\otimes\ket{\varphi_{\sigma(N)}}.\end{gathered}$
(38)

$\vee$ is the symmetric tensor product, $\mathcal{N}_{S}$ is a normalisation
factor, $S_{N}$ is the qudit permutation group with elements $\sigma$ and
$\ket{\Psi_{S}}$ is in the symmetrised $N$-particle Hilbert space:
$\mathcal{H}_{S}^{(N)}={(\mathcal{H}^{(1)})}^{\vee{N}}$. Permutation
symmetries present in system-label states suggest that we should choose a
representation for the two-particle (focused here on Bell states, but
$N$-particle more generally) Hilbert space that naturally encapsulates these
properties. As we are considering the interference and evolution of photons in
linear optical circuits, we also want this representation to behave well under
unitary group operations ($SU(2)$ in this case). A strong connection between
the permutation group and special unitary group actions is seen through Schur-
Weyl duality [55, 53]: for a system of $N$ $d$-dimensional qudits, the
associated Hilbert space $\mathcal{H}^{(N)}={(\mathbb{C}^{d})}^{\otimes{N}}$
decomposes as

${(\mathbb{C}^{d})}^{\otimes{N}}\simeq\bigoplus_{\lambda\vdash{N}}\mathbb{C}^{\\{\lambda\\}}\otimes\mathbb{C}^{(\lambda)}.$
(39)

Here $\mathbb{C}^{\\{\lambda\\}}$ carries the irrep $\lambda$ of the qudit
unitary transformation group, $SU(d)$, $\mathbb{C}^{(\lambda)}$ carries the
irrep $\lambda$ of the qudit permutation group, $S_{N}$, and $\simeq$
indicates a change of basis through the Schur-Weyl transform.

The irreps which simultaneously diagonalize the unitary and permutation
groups, $\lambda$, can be compactly described through Young diagrams.
Attributing $N$ qudits with dimension $d$ spanning the particles given degree
of freedom, we can construct Young tableux, wavefunctions that form a basis
for the $N$-particle Hilbert space. For $N=2$ there are two Young diagrams:
$\scriptsize\yng(2),\scriptsize\yng(1,1)$. If $d=2$ there are four basis
states: the spin triplet

$\begin{gathered}\ket{\scriptsize\young(11)}=\ket{1,1},\\\
\ket{\scriptsize\young(12)}=\ket{1}\vee\ket{2}=\frac{1}{\sqrt{2}}(\ket{1,2}+\ket{2,1}),\\\
\ket{\scriptsize\young(22)}=\ket{2,2},\end{gathered}$ (40)

and spin singlet

$\ket{\scriptsize\young(1,2)}=\frac{1}{\sqrt{2}}(\ket{1,2}-\ket{2,1}).$ (41)

These wavefunctions form the symmetric and anti-symmetric bases for the two-
particle Hilbert space respectively.

For system-label states, distinguishability in the label allows us to access
anti-symmetric irreps, as can be seen for Eq. 16:

$\begin{gathered}\ket{\psi_{12}^{AB}}=\ket{1,A}\vee\ket{2,B}\\\
\mapsto\frac{1}{\sqrt{2}}(\ket{\scriptsize\young(12)}\otimes\ket{\scriptsize\young(AB)}+\ket{\scriptsize\young(1,2)}\otimes\ket{\scriptsize\young(A,B)}).\end{gathered}$
(42)

This state, opposed to the case with the same labels on each particle,
occupies both $\lambda=(2)$ and $\lambda=(1,1)$.

##  Appendix J Schur-Weyl transform implementation in the CV backend

We use the QuTip Python library to carry out the Schur-Weyl transform on
$\absolutevalue{\bar{\chi}}$-particle states reconstructed at the output of
our interferometer. Here we provide psuedocode showing the main algorithmic
steps for obtaining reduced logical Schur states as part of the CV backend for
HBSG circuits.

Algorithm 1 Schur state reconstruction

Input: Dictionary of full Fock representation $rho-full$ over output state
modes, for the set $\bar{\chi}$.
Output: Reduced logical state $\tilde{\rho}$.

$\tilde{\rho}\leftarrow 0$$\triangleright$ Initialise $\tilde{\rho}$

for $\ket{sp,int}^{\otimes{\absolutevalue{\bar{\chi}}}}$ in $rho-full$

for $\bra{sp,int}^{\otimes{\absolutevalue{\bar{\chi}}}}$ in $rho-full$ do

$\rho_{sym}\leftarrow\hat{S}((\ket{sp,int}\bra{sp,int})^{\otimes{\absolutevalue{\bar{\chi}}}})$
$\triangleright$ Symmetrise $\absolutevalue{\bar{\chi}}$-photon state

$\rho_{SW}\leftarrow
U_{sp,int}^{\otimes{\absolutevalue{\bar{\chi}}}}(\rho_{sym})$ $\triangleright$
Evolve $\rho_{sym}$ through Schur-Weyl transform

$\rho_{sp}\leftarrow\Tr_{int}({\rho_{SW}})$ $\triangleright$ Trace out
internal degree of freedom

$\tilde{\rho}\mathrel{+}=\rho_{sp}$ $\triangleright$ Update $\tilde{\rho}$

end for

end for

##  Appendix K 4P8M OBB vs RS postselected Type-II fusion

In Sec. VI we discuss and analyze the Type-II gate operation for both RS (two
Schmidt) and OBB photons. The resources for the gate operation are ideally two
$\ket{\Phi^{+}}$ Bell states, however in the presence of errors these states
are described more generally by $\rho$. For postselected HBSG states, the
general structure of $\rho^{(0)}$ was derived in [25]. This expression depends
on the pairwise photon visibility $\mathcal{V}$ and fourth-order trace overlap
of the internal photonic state $\varrho$. Here we show the analytic
expressions for the post-fusion states $\rho^{(1)}_{OBB/RS}$ resulting from
these postselected resources. For OBB photons, simplifying the general
$\rho^{(0)}$ expression leads to

$\rho_{OBB}^{(0)}=\frac{1}{4}\begin{bmatrix}1+\mathcal{V}&0&0&2\mathcal{V}^{2}\\\
0&1-\mathcal{V}&0\ &0\\\ 0&0&1-\mathcal{V}&0\\\
2\mathcal{V}^{2}&0&0&1+\mathcal{V}\end{bmatrix}.$ (43)

where the $\ket{01}\bra{10},\ket{10}\bra{01}$ coherences vanish for OBB
photons and $\Tr(\varrho^{4})=\mathcal{V}^{2}$. Performing Type-II fusion on
two copies of $\rho^{(0)}_{OBB}$ which occupy spatial modes $1,2$ and $3,4$
respectively, preserves terms of $\rho^{(0)}_{OBB}\otimes{\rho^{(0)}_{OBB}}$
as

$\begin{gathered}\frac{1}{16}((\alpha\ket{00}\bra{00}+\alpha\ket{11}\bra{11}+\beta\ket{01}\bra{01}+\beta\ket{10}\bra{10}+\gamma\ket{00}\bra{11}+\gamma\ket{11}\bra{00})\\\
\otimes\\\
{(\alpha\ket{00}\bra{00}+\alpha\ket{11}\bra{11}+\beta\ket{01}\bra{01}+\beta\ket{10}\bra{10}+\gamma\ket{00}\bra{11}+\gamma\ket{11}\bra{00}))}.\end{gathered}$
(44)

Here we simplify coefficients as $\alpha=1+\mathcal{V}$, $\beta=1-\mathcal{V}$
and $\gamma=2\mathcal{V}^{2}$. Expanding and grouping terms leads to

$\begin{gathered}\frac{1}{16}((\alpha^{2}+\beta^{2})(\ket{00}\bra{00}+\ket{11}\bra{11})+2\alpha\beta(\ket{01}\bra{01}+\ket{10}\bra{10})+\gamma^{2}(\ket{00}\bra{11}+\ket{11}\bra{00})).\end{gathered}$
(45)

We can then express the output state as

$\rho_{OBB}^{(1)}=\frac{1}{4}\begin{bmatrix}1+\mathcal{V}^{2}&0&0&2\mathcal{V}^{4}\\\
0&1-\mathcal{V}^{2}&0&0\\\ 0&0&1-\mathcal{V}^{2}&0\\\
2\mathcal{V}^{4}&0&0&1+\mathcal{V}^{2}\end{bmatrix}.$ (46)

This state is of the same form as $\rho^{(0)}_{OBB}$, but accompanied by a
prefactor of 1/2 success probability for successful fusion and a relabelling
of $\mathcal{V}\mapsto\mathcal{V}^{2}$. This means we can describe the
degradation of postselected $\rho^{(1)}_{OBB}$ post-fusion states through
using lower visibility photons in the HBSG circuit.

For RS (two Schmidt) photons, the output state from a single HBSG of the type
4P8M is

$\rho^{(0)}_{RS}=\frac{1}{4}\begin{bmatrix}1+\mathcal{V}&0&0&\mathcal{V}^{2}+Y\\\
0&1-\mathcal{V}&\mathcal{V}^{2}-Y\ &0\\\
0&\mathcal{V}^{2}-Y&1-\mathcal{V}&0\\\
\mathcal{V}^{2}+Y&0&0&1+\mathcal{V}\end{bmatrix}.$ (47)

We again assume each of the input photons is identical with pairwise
visibility $\mathcal{V}$. The internal RS photon structure is
$\varrho={p_{0}}\ket{k_{0}}\bra{k_{0}}+(1-{p_{0}})\ket{k_{1}}\bra{k_{1}}$. We
have let $Y=\Tr(\varrho^{4})={p_{0}}^{4}+(1-{p_{0}})^{4}$. Performing Type-II
fusion on two copies of $\rho^{(0)}_{RS}$ preserves terms of
$\rho^{(0)}_{RS}\otimes{\rho^{(0)}_{RS}}$ as

$\begin{gathered}\frac{1}{16}((\alpha\ket{00}\bra{00}+\alpha\ket{11}\bra{11}+\beta\ket{01}\bra{01}+\beta\ket{10}\bra{10}+\gamma\ket{00}\bra{11}+\gamma\ket{11}\bra{00}+\delta\ket{01}\bra{10}+\delta\ket{10}\bra{01})\\\
\otimes\\\
{(\alpha\ket{00}\bra{00}+\alpha\ket{11}\bra{11}+\beta\ket{01}\bra{01}+\beta\ket{10}\bra{10}+\gamma\ket{00}\bra{11}+\gamma\ket{11}\bra{00}+\delta\ket{01}\bra{10}+\delta\ket{10}\bra{01}))}.\end{gathered}$
(48)

Here we simplify coefficients as $\alpha=1+\mathcal{V}$,
$\beta=1-\mathcal{V}$, $\gamma=\mathcal{V}^{2}+Y$ and
$\delta=\mathcal{V}^{2}-Y$. Expanding and grouping terms leads to

$\begin{gathered}\frac{1}{16}((\alpha^{2}+\beta^{2})(\ket{00}\bra{00}+\ket{11}\bra{11})+2\alpha\beta(\ket{01}\bra{01}+\ket{10}\bra{10})\\\
+(\gamma^{2}+\delta^{2})(\ket{00}\bra{11}+\ket{11}\bra{00})+2\delta\gamma(\ket{01}\bra{10}+\ket{10}\bra{01})).\end{gathered}$
(49)

Redefining each coefficient in terms of $\mathcal{V}$ and $Y$ results in

$\rho_{RS}^{(1)}=\frac{1}{4}\begin{bmatrix}1+\mathcal{V}^{2}&0&0&\mathcal{V}^{4}+Y^{2}\\\
0&1-\mathcal{V}^{2}&\mathcal{V}^{4}-Y^{2}&0\\\
0&\mathcal{V}^{4}-Y^{2}&1-\mathcal{V}^{2}&0\\\
\mathcal{V}^{4}+Y^{2}&0&0&1+\mathcal{V}^{2}\end{bmatrix}.$ (50)

This state is of the same form as $\rho^{(0)}_{RS}$ with $1/2$ successful
fusion probability and a relabelling of $\mathcal{V}\mapsto\mathcal{V}^{2}$
and $Y\mapsto{Y^{2}}$. Remapping the initial RS photons used for the 4P8M HBSG
with parameters $\mathcal{V}^{\prime},{Y}^{\prime}$ requires

$\mathcal{V}^{\prime}={q_{0}}^{2}+(1-{q_{0}})^{2}\equiv{\mathcal{V}^{2}}=({p_{0}}^{2}+(1-{p_{0}})^{2})^{2},$
(51)
$Y^{\prime}={q_{0}}^{4}+(1-{q_{0}})^{4}\equiv{Y^{2}}=({p_{0}}^{4}+(1-{p_{0}})^{4})^{2}.$
(52)

The only physical values which satisfy these mappings simultaneously are
$p_{0}=1,q_{0}=1$ and $p_{0}=0,q_{0}=0$ which are the noiseless cases. This
leads to the expectation of not being able to perfectly reconstruct
$\rho^{(1)}_{RS}$ from a HBSG with partially distinguishable RS (two Schmidt)
input photons as discussed in Sec. VI.2 and modelled in Fig. 8b, when
performing postselection at the post-HBSG stage.

##  Appendix L Non-computational leakage in Schur states

To construct states $\tilde{\rho}$ attributed to Bell states, we require the
Schur-Weyl transform on two ququarts (two dual-rail encoded qubits). The first
quantized basis states $\\{\ket{i,j},\ket{\phi^{\pm}_{i,j}}\\}$ form the two-
photon system space with a symmetric $10\times 10$ sector (upper left) and
$6\times 6$ anti-symmetric sector (lower right). Here the ket labels $i,j$
indicate the spatial mode occupied by each photon, with computational and non-
computational terms distributed across both the symmetric and anti-symmetric
subspaces. Logical state degradation when characterizing states in the Schur-
Weyl basis can be attributed to non-zero occupation of Schur-Weyl basis states
which don’t comprise the $\ket{\Phi^{+}}$ state. For ideal HBSG operation
$\ket{\Phi^{+}}=\frac{1}{\sqrt{2}}(\ket{\phi^{+}_{1,3}}+\ket{\phi^{+}_{2,4}})$
(see Fig. 16a). When $\mathcal{V}<1$ (RS two Schmidt) we have non-zero
occupation in the anti-symmetric sector, both in computational states such as
$\\{\ket{\phi^{-}_{1,3}},\ket{\phi^{-}_{2,4}}\\}$ and non-computational states
where two photons occupy the same dual-rail qubit modes,
$\\{\ket{\phi^{-}_{1,2}},\ket{\phi^{-}_{3,4}}\\}$ (see Fig. 16b). The latter
terms contribute to the leakage of $\tilde{\rho}$ out of the computational
subspace and aren’t captured by standard postselection. Whilst we only show
$\tilde{\rho}$ matrices for RS photons here for concision, similar leakage
mechanisms are exhibited for $\mathcal{V}<1$ OBB photons.

Figure 16: Reduced logical $\tilde{\rho}$ Schur states for the 4P8M HBSG (RS
two Schmidt) with $\mathcal{V}=1,0.5$ in a,b) respectively. In the presence of
partial distinguishability, leakage into the anti-symmetric two-photon
subspace (lower right $6\times 6$ sector) is seen. This subspace consists of
both computational and non-computational terms with non-zero occupation for
non-ideal photons.