††thanks: These authors contributed equally††thanks: These authors contributed
equally

# Unveiling the non-Abelian statistics of $D(S_{3})$ anyons via photonic
simulation

Suraj Goel  Institute of Photonics and Quantum Sciences, Heriot-Watt
University, Edinburgh EH14 4AS, UK  Matthew Reynolds  School of Physics and
Astronomy, University of Leeds, Leeds LS2 9JT, UK  Matthew Girling  School of
Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK  Will McCutcheon
Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh
EH14 4AS, UK  Saroch Leedumrongwatthanakun  Institute of Photonics and Quantum
Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK  Vatshal Srivastav
Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh
EH14 4AS, UK  David Jennings  School of Physics and Astronomy, University of
Leeds, Leeds LS2 9JT, UK  Department of Physics, Imperial College London,
London SW7 2AZ, UK  Mehul Malik<EMAIL_ADDRESS>Institute of Photonics and
Quantum Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK  Jiannis K.
Pachos<EMAIL_ADDRESS>School of Physics and Astronomy, University of
Leeds, Leeds LS2 9JT, UK

###### Abstract

Simulators can realise novel phenomena by separating them from the
complexities of a full physical implementation. Here we put forward a scheme
that can simulate the exotic statistics of $D(S_{3})$ non-Abelian anyons with
minimal resources. The qudit lattice representation of this planar code
supports local encoding of $D(S_{3})$ anyons. As a proof-of-principle
demonstration we employ a photonic simulator to encode a single qutrit and
manipulate it to perform the fusion and braiding properties of non-Abelian
$D(S_{3})$ anyons. The photonic technology allows us to perform the required
non-unitary operations with much higher fidelity than what can be achieved
with current quantum computers. Our approach can be directly generalised to
larger systems or to different anyonic models, thus enabling advances in the
exploration of quantum error correction and fundamental physics alike.

Introduction:– The exotic statistics of non-Abelian anyons make them of
interest in fundamental physics [1, 2, 3, 4, 5]. In addition, their resilience
to local perturbations has given rise to several schemes for topological
quantum computing and other applications [6, 7, 8, 9, 10]. This behavior is
key for fault-tolerant quantum computing, making non-Abelian anyons a
potential solution to error problems that limit the scaling of quantum
computers [11, 12, 10]. In the last decade we witnessed an intense effort to
identify signatures of non-Abelian anyons in various physical platforms, such
as FQH liquids at $\nu=5/2$ [13], $p+ip$ topological superconductors [14, 15]
or quantum wires [8]. Unfortunately, the complexity of these systems allows
for alternative interpretation of the observed signatures [16]. The conclusive
characteristic of non-Abelian anyons is their exchange statistics, which is
currently too complex to realise in the laboratory.

At the same time, several investigations have focused on simulating non-
Abelian anyons [17, 18, 19, 20, 21, 22]. These efforts aim to establish the
necessary conditions for observing non-Abelian statistics and addressing
technical challenges in scaling and accuracy. Often, such simulations suffer
from key loopholes. For example, the simulation of Majorana fermions utilizes
a non-local encoding of fermion-like anyonic states in many qubits, with the
help of the Jordan-Wigner transformation. However, this non-local encoding
lacks the desired topological stability against local errors inherent in
anyonic systems.

Here we propose and implement a photonic simulation that demonstrates the core
features of non-Abelian anyon statistics corresponding to the $D(S_{3})$
planar code [11]. Planar codes are both quantum error-correcting codes and
condensed matter systems that host anyonic excitations. Although they require
many-body interactions, their local encoding on spins makes them attractive
for quantum simulations. The simplest version of the planar code is the toric
code that supports Abelian anyons. The toric code has been already simulated
in the laboratory with Josephson junctions [23] and photonic systems [24, 25].

We show that a single qutrit is sufficient to encode the core manipulations of
$D(S_{3})$ non-Abelian anyons and demonstrate their non-Abelian fusion and
braiding properties. The operations required to generate and manipulate anyons
are in general non-unitary matrices that have a unitary action on the anyonic
Hilbert space [26, 27, 28]. The implementation of non-unitary operations is
typically experimentally challenging with current quantum computing
architecture. To overcome this problem we adopt photonic technologies that can
perform non-unitary operations accurately and with high fidelity. This
simulation can be expanded in two directions with advancements in technology.
First, it can be scaled to larger lattice systems, allowing for a broader
range of anyonic operations. Second, Hamiltonian interactions can be added to
provide active fault-tolerance in the topologically encoded quantum
information.

The non-Abelian $D(S_{3})$ anyonic model:– The $D(S_{3})$ model is based on
the group transformations of a triangle, $S_{3}=\\{e,c,c^{2},t,tc,tc^{2}\\}$,
where $e$ is the identity element, $c$ the generator of $2\pi/3$ rotations and
$t$ the generator of reflections. The $D(S_{3})$ planar code consists of a
square lattice where $d=6$ qudits are positioned at its links parameterised by
the group elements of $S_{3}$, as shown in Fig. 1. The Hamiltonian of the
model has mutually commuting plaquette and vertex operators [11]. The ground
state of the model is identified as the vacuum and its anyons are manifested
as localised excitations at the vertices and/or the plaquettes.

This $D(S_{3})$ planar code supports eight different anyonic excitations
labelled by $\\{A,B,C,D,E,F,G,H\\}$ [29, 30, 31]. Particle $A$ corresponds to
the vacuum that fuses trivially with the rest of the anyons. Here we restrict
ourselves to the $\\{A,B,G\\}$ subgroup that is closed under fusion, $B\times
B=A$, $G\times B=G$ and $G\times G=A+B+G$. Moreover, $G$ has non-trivial
braiding statistics. Its corresponding fusion, $F^{G}_{GGG}$, and braiding,
$R^{GG}$, unitary matrices are given by

$F^{G}_{GGG}=\frac{1}{2}\begin{pmatrix}1&1&\sqrt{2}\\\ 1&1&-\sqrt{2}\\\
\sqrt{2}&-\sqrt{2}&0\end{pmatrix},\,\,R^{GG}=\begin{pmatrix}\bar{\omega}&0&0\\\
0&\bar{\omega}&0\\\ 0&0&\omega\end{pmatrix},$ (1)

where $\omega=e^{2\pi i/3}$, which give a non-trivial braiding matrix
$B^{GG}=FR^{2}F^{\dagger}$. The non-Abelian character of the $G$ anyons is
manifested in the non-trivial commutation relation between $F^{G}_{GGG}$ and
$(R^{GG})^{2}$. The $\\{A,B,G\\}$ subgroup should be contrasted to
$\\{A,B,C\\}$, which has similar fusion rules but trivial braiding statistics,
$B^{CC}=\mathbbm{1}$ [30, 32].

Figure 1: Ribbon operators of $G$ anyons. (a) A dual triangle $\tau$ has
support on link $e_{\tau}$ and creates excitations at $p_{1}$ and $p_{2}$
plaquettes. (b) A direct operator $\tau^{\prime}$ has support on link
$e_{\tau^{\prime}}$ and create excitations at $v_{1}$ and $v_{2}$ vertices.
(c,d) Compositions of dual and direct triangles give rise to ribbon operators
$\rho$. Anyonic excitations of ribbon operators are dyons (dotted ovals) that
are positioned at the endpoint plaquettes and vertices. (e) We define the
elementary ribbon $\rho_{0}$ where both dual and direct triangles have support
on the same link $e_{0}$.

It is possible to verify fusion and braiding properties of the planar code
anyons by generating and manipulating the corresponding anyonic excitations.
Such excitations are created from the vacuum state by applying operations on
the links of the lattice. In general, these rotations are given in terms of
ribbon operators $F_{\rho}^{X}$, where $\rho$ is the path along which the
rotations are applied, giving rise to two $X$ anyons at its endpoints. These
ribbon operators, together with their action on the ground state, encode the
anyonic fusion and braiding properties.

To define the ribbon operators we employ the oriented lattice representation
of $D(S_{3})$ shown in Fig. 1. A dual triangle $\tau$ has support on a link
$e_{\tau}$ and is connecting two plaquettes $p_{1}$ and $p_{2}$ adjacent to
the link $e_{\tau}$, as shown in Fig. 1(a). A direct triangle $\tau^{\prime}$
has support on a link $e_{\tau^{\prime}}$ and is connecting two vertices
$v_{1}$ and $v_{2}$ adjacent to the link $e_{\tau^{\prime}}$, as shown in Fig.
1(b). We now assign a six-dimensional Hilbert space $\\{|h\rangle,h\in
S_{3}\\}$, to each link $e_{\tau}$. To every triangle $\tau$ we define an
operator $L^{h}_{\tau_{\text{dual}}}=\sum_{g\in S_{3}}|hg\rangle\\!\langle
g|$, with $h\in S_{3}$, acting on $e_{\tau}$, if $e_{\tau}$ points towards
$v$. Otherwise $L^{h}_{\tau_{\text{dual}}}=\sum_{g\in
S_{3}}|gh^{-1}\rangle\\!\langle g|$. Similarly, we define
$P^{g}_{\tau_{\text{dir}}}=|g\rangle\\!\langle g|$, with $g\in S_{3}$, if
$e_{\tau}$ is clockwise w.r.t. $p$, otherwise
$P^{g}_{\tau_{\text{dir}}}=|g^{-1}\rangle\\!\langle g^{-1}|$. We next define
the composite operators
$F^{h,g}_{\rho}=L^{h}_{\tau_{\text{dual}}}P^{g}_{\tau_{\text{dir}}}$, where
$\rho=\tau_{\text{dir}}\tau_{\text{dual}}$ with $\tau_{\text{dir}}$ a direct
and $\tau_{\text{dual}}$ a dual triangle. Ribbon operators, $F^{X}_{\rho}$,
that give rise to $X$ anyons are built out of such matrix elements,
$F^{h,g}_{\rho}$, acting on qudits [30].

The $A$ and $B$ anyons are constructed from strings of operators corresponding
to direct $\tau$’s. They give rise to anyons positioned at vertices, i.e. they
have $h=e$ with $L^{e}=\mathbbm{1}$. Similar to the toric code’s $e$ and $m$
anyons the $A$ and $B$ anyons can be created and moved around by applying
single qudit unitary operator, $F^{A}_{\tau}$ and $F^{B}_{\tau}$, to a string
of qudits [11], where

$\displaystyle F^{A}_{\tau}\\!$ $\displaystyle=$
$\displaystyle\\!|e\rangle\\!\langle e|+|c\rangle\\!\langle
c|+|c^{2}\rangle\\!\langle c^{2}|+|t\rangle\\!\langle t|+|tc\rangle\\!\langle
tc|+|tc^{2}\rangle\\!\langle tc^{2}|,\,\,\,\,$ $\displaystyle F^{B}_{\tau}\\!$
$\displaystyle=$ $\displaystyle\\!|e\rangle\\!\langle e|+|c\rangle\\!\langle
c|+|c^{2}\rangle\\!\langle c^{2}|-|t\rangle\\!\langle t|-|tc\rangle\\!\langle
tc|-|tc^{2}\rangle\\!\langle tc^{2}|.$ (2)

Note that $F^{A}$ is the identity operator acting on the qudit and both
$F^{A}$ and $F^{B}$ are unitary matrices whose sum,
$F^{A}_{\tau}+F^{B}_{\tau}=2(|e\rangle\\!\langle e|+|c\rangle\\!\langle
c|+|c^{2}\rangle\\!\langle c^{2}|)$, can be described by a single qutrit
operator.

The $G$ non-Abelian anyons are dyonic, i.e. it includes both direct and dual
triangle operators forming ribbons. Dyons are positioned at composite
plaquette and vertex at each of the endpoints of the ribbon, as shown in Fig.
1(c,d,e). A simple way to create a pair of $G$ anyons on neighbouring vertices
and plaquettes is to use the ribbon $\rho=\tau_{\text{dual}}\tau_{\text{dir}}$
[31], as shown in Fig. 1(c). The corresponding ribbon operator is given by

$F^{G}_{\rho}=F^{c,e}_{\rho}+\omega
F^{c,c}_{\rho}+\bar{\omega}F^{c,c^{2}}_{\rho}+F^{c^{2},e}_{\rho}+\bar{\omega}F^{c^{2},c}_{\rho}+\omega
F^{c^{2},c^{2}}_{\rho},$ (3)

where for simplicity we omitted the overall normalisation. One can show that
$F^{G}_{\rho}$ is hermitian but not unitary. Nevertheless, its restriction to
the eigenstates of the $D(S_{3})$ Hamiltonian are unitary [26]. Finally, note
that larger ribbon operators, such as the ones shown in Fig. 1(d), are created
by acting simultaneously on all the relevant qudits with highly entangling
operations [27, 28].

Minimal encoding of non-Abelian anyons:– To simplify the physical requirements
for the simulation of $G$ anyons we identify the smallest possible ribbon that
can encode the anyonic properties. It is possible to define a ribbon
$\rho_{0}$, where the direct and dual triangles have support on the same
qudit, as shown in Fig. 1(e). We define the corresponding ribbon operator as
$F^{G}_{\rho_{0}}\equiv F^{c,e}_{\rho_{0}}+\omega
F^{c,c}_{\rho_{0}}+\bar{\omega}F^{c,c^{2}}_{\rho_{0}}+\text{h.c.}$, where
hermiticity is explicitly imposed. Explicitly, we have

$\displaystyle F^{G}_{\rho_{0}}=|c\rangle\\!\langle
e|+\omega|c^{2}\rangle\\!\langle c|+\bar{\omega}|e\rangle\\!\langle
c^{2}|+\text{h.c.},$ (4)

which acts only on three states.

With the minimal string and ribbon operators $F^{A}$, $F^{B}$ and $F^{G}$ we
can explicitly verify the fusion properties of the $\\{A,B,G\\}$ subgroup of
$D(S_{3})$ by acting on a single qudit with six levels. By direct
multiplication of the operators given in (2) and (4) we can verify their non-
trivial fusion rules. In particular, when two ribbon operators,
$F^{G}_{\rho_{0}}$, act on top of each other then the $G$ anyons at their
endpoints are fused resulting to the ribbon operator of their fusion outcomes,
i.e.

$F^{G}_{\rho_{0}}F^{G}_{\rho_{0}}=F^{A}_{\rho_{0}}+F^{B}_{\rho_{0}}+F^{G}_{\rho_{0}}.$
(5)

This fusion process can be realised with a three level system as only the
states $|e\rangle$, $|c\rangle$ and $|c^{2}\rangle$ are involved.

We next consider the braiding properties of $G$ anyons. In the case of the
toric code the anyonic statistics of $e$ and $m$ anyons is given in terms of
the commutation relations between their ribbon operators
$F^{e}_{\rho_{1}}F^{m}_{\rho_{2}}=R^{em}F^{m}_{\rho_{2}}F^{e}_{\rho_{1}}$,
where $\rho_{1}$ and $\rho_{2}$ are two crossing paths of $e$ and $m$ anyons,
respectively [11]. Due to topological invariance with respect to the exact
shape of the path, the braiding relation can be realised by isolating the
site, $\rho_{0}$, where paths $\rho_{1}$ and $\rho_{2}$ cross each other. As a
result we can take the full system to be site $\rho_{0}$, with the anyons
positioned outside the system’s boundary. Then we have
$F^{e}_{\rho_{1}\to\rho_{0}}=Z_{\rho_{0}}$ and
$F^{m}_{\rho_{2}\to\rho_{0}}=X_{\rho_{0}}$ acting on the same qubit at
$\rho_{0}$, thus obtaining the exchange statistics $R^{em}=-1$ [24, 25].

For the $D(S_{3})$ model the exchange of two $G$ ribbon operators takes the
form [11]

$F^{G}_{\rho_{1}}F^{G}_{\rho_{2}}=R^{GG}F^{G}_{\rho_{2}}F^{G}_{\rho_{1}},$ (6)

where $R^{GG}$ is given by (1). Hence, to determine $R^{GG}$ we need to
implement $F^{G}_{\rho_{1}}F^{G}_{\rho_{2}}$ and
$F^{G}_{\rho_{2}}F^{G}_{\rho_{1}}$ and compare them. Similarly to the toric
code case we employ a single site, $\rho_{0}$ and the minimal ribbon operator
$F^{G}_{\rho_{0}}$, given in (4), acting one it. As the operators we want to
exchange are identical when acting on the single site system we adopt the
following prescription. We first identify $F^{G}_{\rho_{1}}F^{G}_{\rho_{2}}$
with the product of two $F^{G}_{\rho_{0}}$ operators as given in (5). Next, to
determine $F^{G}_{\rho_{2}}F^{G}_{\rho_{1}}$, we employ the exchange of their
building blocks $F^{h,g}_{\rho}$

$F^{h,g}_{\rho_{2}}F^{k,l}_{\rho_{1}}=F^{k,l\bar{g}\bar{h}g}_{\rho_{1}}F^{h,g}_{\rho_{2}},$
(7)

valid for ribbons $\rho_{1}$ and $\rho_{2}$ with one common end. We employ
these relations to compute $F^{G}_{\rho_{2}}F^{G}_{\rho_{1}}$ and then
identify $\rho_{1}\to\rho_{0}$ and $\rho_{2}\to\rho_{0}$ to obtain (see
Supplementary Material)

$\displaystyle
F^{G}_{\rho_{2}}F^{G}_{\rho_{1}}=\bar{\omega}(F^{A}_{\rho_{0}}+F^{B}_{\rho_{0}})+\omega
F^{G}_{\rho_{0}}.$ (8)

Direct comparison of (5) and (8) deduces the desired braiding matrix
$R^{GG}=\text{diag}(\bar{\omega},\bar{\omega},\omega)$ in the $\\{A,B,G\\}$
basis.

Having a minimal system facilitates the simulation of braiding statistics with
current technology. A natural first candidate is to employ current quantum
computers to implement the ribbon operator, $F_{\rho_{0}}^{G}$. As
$F_{\rho_{0}}^{G}$ is non-unitary it cannot be straightforwardly implemented
with unitary quantum logic gates [33]. However, as for any matrix, a unitary
block encoding [34, 35, 36, 37, 38] can be constructed where
$F_{\rho_{0}}^{G}$ is embedded within a larger unitary, $U_{F}$. Through the
preparation and measurement of a subset of qubits, a quantum circuit
describing $U_{F}$ allows for $F_{\rho_{0}}^{G}$ to be applied to the labelled
qutrit state. Based on the singular value decomposition of $F_{\rho_{0}}^{G}$
[39], $U_{F}$ must act on a minimum of 3 qubits (see Supplementary Material).
Up to a rescaling of $F_{\rho_{0}}^{G}$, the success probability $p$ of
implementing the transformation on pure states is bounded as $1/4\leq p\leq
1$. Compiling an explicit $U_{F}$ using the native qiskit transpiler into the
typical device gateset (e.g. single qubit rotations + CNOT) gives a circuit
depth of 74 operations with 20 CNOTs [40]. Using a simplified model of current
device noise, we simulated the circuits applying this $U_{F}$ to states that
maximise and minimise the success probability of applying $F_{\rho_{0}}^{G}$
to the labeled qutrit state (see Supplementary Material). Unfortunately,
current error rates are too high, and we observe low fidelities between the
idealized circuits and the noisy implementations. As an alternative we resort
to a photonic platform that can encode non-unitary operations in a
straightforward way.

Experimental photonic simulation:– Here we show how a photonic simulator can
accurately perform the non-unitary operations needed to realise the fusion and
braiding properties of the $G$ non-Abelian $D(S_{3})$ anyons with mimimal
errors. We experimentally implement the ribbon operator $F^{G}_{\rho_{0}}$ ,
that encodes two $G$ anyons at its endpoints, and its compositions
$F^{G}_{\rho_{1}}F^{G}_{\rho_{2}}$ and $F_{\rho_{2}}^{G}F_{\rho_{1}}^{G}$
given in (4), (5) and (8), respecteively. The experiment consists of three
distinct parts, as shown in Fig. 2(a). First, we generate a qutrit state
encoded in the transverse-spatial degree of freedom of light. We then evolve
the state through the desired operations programmed on our photonic simulator.
Finally, we characterise the implemented operations via quantum process
tomography.

Figure 2: (a) Experimental Setup. A coherent light source (810 nm) is incident
on a phase-only spatial light modulator ($\text{SLM}_{1}$) and then coupled
into a 2 m-long graded-index (GRIN) multi-mode fiber (MMF) with core diameter
$50\leavevmode\nobreak\ \mu m$. The output of the MMF is incident on
$\text{SLM}_{2}$ followed by a CCD camera. The combination of a high-
dimensional mode mixer (MMF) sandwiched between two phase planes
($\text{SLM}_{1}$ and $\text{SLM}_{2}$) serves as a programmable optical
circuit that can encode any non-unitary operations as shown in [41].
Additionally, $\text{SLM}_{2}$ is used for performing projective measurements
required for quantum process tomography (QPT) to check the fidelity of the
implemented transformations
$\mathbf{T}\in\\{F^{G}_{\rho_{0}},F^{G}_{\rho_{1}}F^{G}_{\rho_{2}},F^{G}_{\rho_{2}}F^{G}_{\rho_{1}}\\}$.
(b) Qutrit Encoding. Images showing three-dimensional photonic transverse-
spatial modes in the macro-pixel basis (M0) generated by $\text{SLM}_{1}$.
Modes from all mutually unbiased bases (M1, M2, M3) of the three-dimensional
macro-pixel basis are also shown, which are used for performing QPT.

The qutrit is encoded in a transverse-spatial modal basis consisting of
discrete macro-pixels, as shown in Fig. 2(b). We choose this particular basis
as it can be tailored to perform high-quality projective measurements [42,
43]. Next, the qutrit state is evolved through the $G$ ribbon operator (4) and
its compositions (5) and (8), following which it is mapped onto spatially
separated outcomes that can be measured on a camera. These operations are by
definition non-unitary. Therefore, the task of performing an operation and
sorting the outcomes spatially can be mapped to the problem of state
discrimination between non-orthogonal states. There exist multiple schemes
that perform this task by compromising either efficiency or accuracy of the
discrimination [44]. As we are interested in simulating the anyonic
properties, we choose to enhance the accuracy of the operations at the expense
of efficiency through optical losses. Herein we use the formalism of
unambiguous state discrimination [45, 46, 47, 48], where one employs auxiliary
modes to embed a low-dimensional non-unitary operation within a higher-
dimensional unitary. The outcomes corresponding to the auxiliary modes can be
ignored since they provide no information about the input state, and thus
correspond to loss.

Interestingly, the three operations we aim to implement,
$\mathbf{T}\in\\{F^{G}_{\rho_{0}},F^{G}_{\rho_{1}}F^{G}_{\rho_{2}},F^{G}_{\rho_{2}}F^{G}_{\rho_{1}}\\}$,
are Hermitian and have symmetric overlaps between different columns, i.e.
$|T_{rj}\cdot{T_{r{i\neq j}}}^{*}|=\alpha,\leavevmode\nobreak\
\forall\leavevmode\nobreak\ r$ where $T_{rc}$ corresponds to the $r^{th}$ row
and $c^{th}$ column of the given $\mathbf{T}$ matrix. This allows us to use a
single auxiliary mode to perform these operations [49] as was recently shown
to be experimentally viable with optical circuits [50]. Due to the non-
unitarity, these operations cannot be performed with unit success probability.
Theoretically, the maximum average success probabilities are $50\%$, $37.5\%$
and $75\%$ for $F^{G}_{\rho_{0}}$ , $F^{G}_{\rho_{1}}F^{G}_{\rho_{2}}$ and
$F_{\rho_{2}}^{G}F_{\rho_{1}}^{G}$ , respectively. Using this approach, we
encode our 3-dimensional Hermitian operators into 4-dimensional unitaries to
proceed with this task.

These unitary operations are implemented using the recently demonstrated “top-
down” approach, where an arbitrary optical circuit is embedded within a
higher-dimensional mode-mixer sandwiched between two programmable phase planes
[41]. Our circuit uses a commercial multi-mode fibre (MMF) as a mode-mixer
that is placed between two programmable spatial light modulators (SLMs) as
shown in Fig. 2(a). We use an inverse design technique known as the wavefront-
matching (WFM) algorithm to program the SLMs. The WFM algorithm calculates the
phase plane solutions by iteratively maximising the overlap between a set of
input fields with the desired output ones. After updating the SLMs with the
phase solutions given by the WFM algorithm, we couple a coherent light source
with a wavelength of $810$ nm to characterize the implemented operation. The
statistics of a single-photon qutrit state propagating through the system are
identical to those obtained for a coherent state, allowing us to simplify the
experiment and use a camera for detection [51].

Operation |  Fidelity |  Purity
---|---|---
($\mathbf{T}$) | ($\mathcal{F}(\rho_{\mathcal{T}},\rho_{\widetilde{\mathcal{T}}})$) | ($\mathcal{P}(\rho_{\widetilde{\mathcal{T}}})$)
$F^{G}_{\rho_{0}}$ |  $95.23\pm 0.93\%$ |  $96.04\pm 0.03\%$
$F^{G}_{\rho_{1}}F^{G}_{\rho_{2}}$ |  $94.44\pm 0.85\%$ |  $97.65\pm 0.05\%$
$F_{\rho_{2}}^{G}F_{\rho_{1}}^{G}$ |  $97.59\pm 0.59\%$ |  $94.43\pm 0.06\%$
Table 1: Experimental results for the best case fidelity and purity of the
processes corresponding to the $F^{G}_{\rho_{0}}$ ,
$F^{G}_{\rho_{1}}F^{G}_{\rho_{2}}$ and $F_{\rho_{2}}^{G}F_{\rho_{1}}^{G}$
operations. The error values are reported up to 3 standard deviations and
correspond to systematic misalignment error (see Supplementary Material).

We perform quantum process tomography to quantify the fidelity of the
implemented operations $\mathbf{\widetilde{T}}$ in relation to the ideal
operations $\mathbf{T}$. Note that in addition to both SLMs being used for
implementing the target operation $\mathbf{T}$, $\text{SLM}_{1}$ is used for
generating the complete set of input modes (macropixel MUBs M0-M3, Fig. 2(b))
and $\text{SLM}_{2}$ is used for performing the projective measurements needed
for quantum process tomography (QPT). For each input mode, we measure the
intensity at each of the three designated output modes at the camera, ignoring
the auxiliary output. Next, SLM2 is used to sequentially project the output
into all mutually unbiased bases (MUBs) of the desired output modes. This is
done in a manner similar to how projective measurements are performed with an
SLM and a single mode fibre (SMF) [52], with the center region of the CCD
camera used in place of the SMF. Using these measurements, we construct a
coupling matrix between the complete set of input and output modes. This
coupling matrix is then used to recover the implemented process via QPT, which
we represent via its Choi state,
$\rho_{\mathcal{\widetilde{T}}}=\mathcal{\widetilde{T}}\otimes\mathbb{1}(\rho^{+})$,
where $\rho^{+}$ is the maximally entangled state. The Choi state captures
complete information about the process and can be used to evaluate the purity
and fidelity to the target operations (See Supplementary Material) [41].

We implement 10 realisations of each operation $F^{G}_{\rho_{0}}$ ,
$F^{G}_{\rho_{1}}F^{G}_{\rho_{2}}$ and $F_{\rho_{2}}^{G}F_{\rho_{1}}^{G}$ ,
and reconstruct their processes using the methods described above. Since
different positions of measurement outcomes result in different performance,
we vary the positions of measurement outcomes on the camera in each
realisation in order to realize the best possible implementation of these
operators. Out of all the implementations, the best-case fidelities,
$\mathcal{F}$, and purities, $\mathcal{P}$, of the process for each operation
$F^{G}_{\rho_{0}}$ , $F^{G}_{\rho_{1}}F^{G}_{\rho_{2}}$ and
$F_{\rho_{2}}^{G}F_{\rho_{1}}^{G}$ are shown in Table 1. To visualise the
quality of these operations, it is convenient to use the Kraus representation,
which is an alternative way to represent these processes (see Supplementary
Material). The ideal target operations have only one non-zero Kraus operator,
which we can compare to the leading Kraus operators of the implemented
processes owing to their high purity. Fig. 3 (top row) depicts the leading
Kraus operator of the implemented processes, showing that it agrees well with
the target operations (insets). The full Choi state representation of these
high purity processes is shown in Fig. 3 (bottom row), with the ideal state
shown for comparison.

Figure 3: Experimentally Measured Operators. We obtain the leading Kraus
operator from the tomographed process (top row) and the Choi state
representation for each operator $F^{G}_{\rho_{0}}$ ,
$F^{G}_{\rho_{1}}F^{G}_{\rho_{2}}$ and $F_{\rho_{2}}^{G}F_{\rho_{1}}^{G}$
(bottom row) (see Supplementary Material). Insets show theoretically expected
results in each plot. $|A|$ corresponds to the maximum amplitude for a given
plot and respective inset.

Conclusions:– Non-Abelian anyons present a fascinating and promising avenue
for fault-tolerant quantum computation. Emulating their complex braiding
statistics has so far evaded experimental realisation. In this letter, we have
demonstrated a photonic simulation of the ribbon operators corresponding to
$D(S_{3})$ non-Abelian anyons with fidelities and purities above $94\%$. Our
simulation has certified the minimal requirements and operations necessary to
identify the statistics of these anyons, which can guide future efforts
towards experimentally realising these exotic systems. Moreover, the extension
of our experimental scheme to multi-qudit scalable quantum systems [53, 54,
55] can potentially unlock the applications of non-Abelian anyons in quantum
information processing. For example, two qutrits encoded in a nine-dimensional
photonic system or in a scalable architecture with Josephson junctions or ion
traps [56] could be employed to encode distinguishable $F^{G}_{\rho}$ ribbon
operators where both the $F^{G}_{GGG}$ and $R^{GG}$ matrices can be realised.
This work represents a significant step forward in the experimental study of
non-Abelian anyons and opens the door to further exploration of their
properties and potential applications.

###### Acknowledgements.

We are grateful to Gavin Brennen, Sofyan Iblisdir and James Wootton for
helpful discussions. This work was in part supported by EPSRC Grant No.
EP/R020612/1.

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## Appendix Appendix

x | A | B | C | D | E | F | G | H
---|---|---|---|---|---|---|---|---
A | A | B | C | D | E | F | G | H
B | B | A | C | E | D | F | G | H
C | C | C | A + B + C | D + E | D + E | G + H | F + H | F + G
D | D | E | D + E | A + C + F + G + H | B + C + F + G + H | D + E | D + E | D + E
E | E | D | D + E | B + C + F + G + H | A + C + F + G + H | D + E | D + E | D + E
F | F | F | G + H | D + E | D + E | A + B + F | H + C | G + C
G | G | G | F + H | D + E | D + E | H + C | A + B + G | F + C
H | H | H | F + G | D + E | D + E | G + C | F + C | A + B + H
Table A.I: The fusion rules for $D(S_{3})$, as calculated from the Verlinde formula. Anyon | Irrep | $e$ | $c$ | $c^{2}$ | $t$ | $tc$ | $tc^{2}$
---|---|---|---|---|---|---|---
1 | $\Gamma^{S_{3}}_{1}$ | 1 | 1 | 1 | 1 | 1 | 1
$\Lambda$ | $\Gamma^{S_{3}}_{-1}$ | 1 | 1 | 1 | -1 | -1 | -1
$\Phi$ | $\Gamma^{S_{3}}_{2}$ | $\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$ | $\begin{pmatrix}\bar{\omega}&0\\\ 0&\omega\end{pmatrix}$ | $\begin{pmatrix}\omega&0\\\ 0&\bar{\omega}\end{pmatrix}$ | $\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$ | $\begin{pmatrix}0&\omega\\\ \bar{\omega}&0\end{pmatrix}$ | $\begin{pmatrix}0&\bar{\omega}\\\ \omega&0\end{pmatrix}$
Table A.II: The irreducible representations of $\mathcal{N}_{e}=S_{3}$ [30].

###  A.1 The $D(S_{3})$ quantum double model

We present here the non-Abelian group $S_{3}$, representing the
transformations of a triangle and the $D(S_{3})$ quantum double model. The
$S_{3}$ group is given by

$S_{3}=\\{e,c,c^{2},t,tc,tc^{2}\\}.$ (A.1)

where $c$ is the generator for rotations following $c^{3}=e$, and $t$ the
generator for reflections following $t^{2}=e$ with $e$ the identity element.
The non-Abelian nature of this group is defined by the relation $ct=tc^{2}$.

The quantum double for this group, $D(S_{3})$, consists of eight anyons
labelled $\\{A,B,C,D,E,F,G,H\\}$ [31], with $A$ corresponding to the vacuum.
From the S matrix and Verlinde formula [57], we can find the fusion rules of
this group, given in Table A.I. These fusion rules can be used to encode
quantum information [58]. This information can be manipulated by anyonic
braiding. Hence, quantum computation can be performed by anyonic fusion and
braiding. Below we will investigate the braiding properties of particular
anyonic subgroups of the $D(S_{3})$ model.

###  A.2 The {1, $\Lambda$, $\Phi$} Submodel

The subgroup $\\{A,B,C\\}$ is also referred to as the $\\{1,\Lambda,\Phi\\}$
submodel [58]. This consists of the vacuum and the two charge anyons of
$D(S_{3})$ corresponding to the irreducible representations of the trivial
normaliser, $\mathcal{N}_{e}=S_{3}$: $\Gamma^{S_{3}}_{1}$,
$\Gamma^{S_{3}}_{-1}$ and $\Gamma^{S_{3}}_{2}$ for $1$, $\Lambda$ and $\Phi$
respectively [30]. Here it is important to note that $\omega=e^{\frac{2\pi
i}{3}}$ and $\bar{\omega}$ denotes its conjugate.

From Table A.II we obtain the fusion rules for this model

$1\times
1=\Lambda\times\Lambda=1,\quad\quad\Phi\times\Lambda=\Phi,\quad\quad\Phi\times\Phi=1+\Lambda+\Phi.$
(A.2)

Anyon $\Phi$ is the only non-Abelian one with three fusion channels. It has
been shown how to encode information in these anyonic fusion channels and how
to manipulate it using non-topological operations.

To realise the fusion and braiding rules of the $D(S_{3})$ quantum double
model we first define ribbon operators, $F^{1}$, $F^{\Lambda}$ and $F^{\Phi}$,
that give rise to the corresponding anyons at their endpoints. We can work out
the form of the ribbon operators $F^{X}_{\rho}$ of abelian anyons $X$ at the
endpoints of a path $\rho$ in the following way [59]

$F^{\chi,c}_{\rho}=\sum_{g\in S_{3}}\bar{\chi}(g)F^{\bar{c},g}_{\rho},$ (A.3)

where $c$ is an element of $S_{3}$ and $\chi$ is a character representation of
$S_{3}$. In this case, the $c$ in equation (A.3) follows $c=e$ as $1$ and
$\Lambda$ are of conjugacy class $C_{e}$. The characters are $1$ and $-1$
respectively. Therefore we need all ribbon operators of the group which have
$h=e$, so all possibilities of $F^{e,g}_{\rho},\ g\in S_{3}$. This gives the
ribbon operators for $1$ and $\Lambda$ as

$F^{1}_{\rho}=F^{e,e}_{\rho}+F^{e,c}_{\rho}+F^{e,c^{2}}_{\rho}+F^{e,t}_{\rho}+F^{e,tc}_{\rho}+F^{e,tc^{2}}_{\rho}$
(A.4)

and

$F^{\Lambda}_{\rho}=F^{e,e}_{\rho}+F^{e,c}_{\rho}+F^{e,c^{2}}_{\rho}-F^{e,t}_{\rho}-F^{e,tc}_{\rho}-F^{e,tc^{2}}_{\rho}.$
(A.5)

These describe the ribbon operators $F^{X}$ creating $X$ anyons in terms of
group ribbon operators $F^{h,g}$, which are defined as [31, 60]

$F^{h,g}_{\rho}=L^{h}_{\tau_{dual}}T^{g}_{\tau_{dir}},$ (A.6)

with

$L^{g}:=\begin{cases}\sum_{h\in S_{3}}\ket{gh}\bra{h}&\mathrm{if\ facing\
along\ edge.}\\\ \sum_{h\in S_{3}}\ket{hg^{-1}}\bra{h}&\mathrm{if\ facing\
away.}\end{cases}$ (A.7) $T^{h}:=\begin{cases}\ket{h}\bra{h}&\mathrm{if\
facing\ along\ edge.}\\\ \ket{h}\bra{h^{-1}}&\mathrm{if\ facing\
away.}\end{cases}$ (A.8)

For our derivations, we can assume the ribbons are all facing along an
oriented edge. Using these definitions, we can rewrite the group ribbon
operators as

$F^{e,e}_{\rho}=\ket{e}\bra{e},\quad\quad
F^{e,c}_{\rho}=\ket{c}\bra{c},\quad\quad
F^{e,c^{2}}_{\rho}=\ket{c^{2}}\bra{c^{2}},$ (A.9)
$F^{e,t}_{\rho}=\ket{t}\bra{t},\quad\quad
F^{e,tc}_{\rho}=\ket{tc}\bra{tc},\quad\quad
F^{e,tc^{2}}_{\rho}=\ket{tc^{2}}\bra{tc^{2}},$ (A.10)

which gives

$F^{1}=\ket{e}\bra{e}+\ket{c}\bra{c}+\ket{c^{2}}\bra{c^{2}}+\ket{t}\bra{t}+\ket{tc}\bra{tc}+\ket{tc^{2}}\bra{tc^{2}}=\mathbb{1}$
(A.11)
$F^{\Lambda}=\ket{e}\bra{e}+\ket{c}\bra{c}+\ket{c^{2}}\bra{c^{2}}-\ket{t}\bra{t}-\ket{tc}\bra{tc}-\ket{tc^{2}}\bra{tc^{2}}.$
(A.12)

To find the group ribbon operators corresponding to $F^{\Phi}_{\rho}$, we must
introduce the non-Abelian basis transformation [60]

$F^{RC;\mathbf{uv}}:=\frac{n_{R}}{|\mathcal{N}_{C}|}\sum_{n\in\mathcal{N}_{C}}\bar{\Gamma}^{j,j^{\prime}}_{R}(n)F^{\bar{c}_{i},q_{i}n\bar{q}_{I^{\prime}}},$
(A.13)

with each anyon having a unique $C$, the conjugacy class and $R$, the irrep of
the normaliser corresponding to $C$, $\mathcal{N}_{C}$. The unitary matrices
of the irrep are given as $\Gamma_{R}(g),g\in S_{3}$ and the dimension of this
is given by $n_{R}$. Each element of the normaliser and conjugacy class are
given by $n_{i}$ and $c_{i}$ respectively. The representation $Q_{C}$ is
defined as by $S_{3}/N_{C}$ and each element is given by $q_{i}$. Then
$\mathbf{u}=(i,j)$ and $\mathbf{v}=(i^{\prime},j^{\prime})$ are extra degrees
of freedom that can be traced out to give the final anyonic ribbon operator
[31]. Using this basis transformation, we find

$F^{\Phi}_{\rho}=2F^{e,e}_{\rho}-F^{e,c}_{\rho}-F^{e,c^{2}}_{\rho}$ (A.14)

hence only three group ribbon operators are needed to define the $\Phi$ anyon.
These were defined in (A.10) and can be substituted in to find

$F^{\Phi}_{\rho}=2\ket{e}\bra{e}-\ket{c}\bra{c}-\ket{c^{2}}\bra{c^{2}}.$
(A.15)

Now that the ribbon operators are defined, the braiding evolution can be
found. For the Abelian anyons $1,\Lambda$ the braiding and fusion operations
follow the familiar toric code steps. For the non-Abelian $\Phi$ anyons, it is
simpler to braid ribbon operators in spacetime. For that we consider two loops
$\rho_{1}$ and $\rho_{2}$ spanned by line paths traversed forwards and
backwards so $\rho_{1}$ and $\rho_{2}$ braid with each other. This has the
form [61]

$F^{x}_{\rho_{1}}F^{x}_{\rho_{2}}(F^{x}_{\rho_{1}})^{-1}(F^{x}_{\rho_{2}})^{-1}\ket{\psi}=U\ket{\psi},$
(A.16)

with $U$ being a unitary. To bring this state back to the vacuum and find $U$,
we must know the commutation relation between $F^{x}_{\rho_{1}}$ and
$F^{x}_{\rho_{2}}$ so we can cancel the ribbon operators applied to the same
ribbon. For our $\Phi$ anyons, this means finding the result of swapping each
group ribbon operator in (A.15) using the exchange relation, describing the
swapping of two anyons as having action [61]

$R:\ket{a,b}\longrightarrow\ket{aba^{-1},a},$ (A.17)

which corresponds to the ribbon operator relation [62]

$F^{h,g}_{\rho_{1}}F^{k,l}_{\rho_{2}}=F^{k,l\bar{g}\bar{h}g}_{\rho_{2}}F^{h,g}_{\rho_{1}}.$
(A.18)

For the group ribbon operators of $\Phi$, this exchange gives

$F^{e,g}_{\rho_{1}}F^{e,l}_{\rho_{2}}=F^{e,l}_{\rho_{2}}F^{e,g}_{\rho_{1}}\quad\forall
g,l\in S_{3},$ (A.19)

so all of these commute. This means no phase is gained upon the double
exchange of these anyons, and hence this subgroup has trivial braiding. This
result could be also derived by the equation for the braiding matrix [31]

$B_{ab}={F^{d}_{acb}}^{-1}R^{2}_{ab}F^{d}_{acb},$ (A.20)

using the $F^{\Phi}_{\Phi\Phi\Phi}$ and $R_{\Phi\Phi}$ matrices given by [63]

$F^{\Phi}_{\Phi\Phi\Phi}=\frac{1}{2}\begin{pmatrix}1&1&-\sqrt{2}\\\
1&1&\sqrt{2}\\\ -\sqrt{2}&\sqrt{2}&0\end{pmatrix},\quad
R_{\Phi\Phi}=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\ 0&0&1\end{pmatrix},$ (A.21)

in the basis $\ket{1}$, $\ket{\Lambda}$, $\ket{\Phi}$, giving

$B_{\Phi\Phi}=FR^{2}F^{-1}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}=\mathbb{1}_{3}.$ (A.22)

meaning there is no non-trivial evolution of the anyonic Hilbert space under
braiding.

###  A.3 The {A, B, G} Submodel

A more promising submodel is given by the subgroup $\\{A,B,G\\}$. In this, $A$
and $B$ are the same $1$ and $\Lambda$ as before, but $G$ is a dyon. Therefore
it is not an irrep of $S_{3}$ itself but of its normaliser
$\mathcal{N}_{c}=\\{e,c,c^{2}\\}\cong\mathbb{Z_{3}}$ instead [31]. The $G$
dyon corresponds to irrep $\Gamma^{\mathbb{Z}_{3}}_{\omega}$ shown in Table
A.III.

Anyon | Irrep | $e$ | $c$ | $c^{2}$ | $t$ | $tc$ | $tc^{2}$
---|---|---|---|---|---|---|---
$A$ | $\Gamma^{S_{3}}_{1}$ | 1 | 1 | 1 | 1 | 1 | 1
$B$ | $\Gamma^{S_{3}}_{-1}$ | 1 | 1 | 1 | -1 | -1 | -1
$G$ | $\Gamma^{\mathbb{Z}_{3}}_{\omega}$ | 1 | $\omega$ | $\bar{\omega}$ | 0 | 0 | 0
Table A.III: The irreducible representations of
$\mathcal{N}_{c}=\mathbb{Z}_{3}$ [30].

Using Table A.III we can obtain the fusion rules for this model

$A\times A=B\cdot B=1,\quad\quad G\times B=G,\quad\quad G\times G=A+B+G,$
(A.23)

with $G$ once again the only non-Abelian anyon. This model has identical
fusion rules to the $\\{1,\Lambda,\Phi\\}$ submodel.

We now determine the braiding evolution of the $G$ anyons. We will first use
equation (A.20), for the $F^{G}_{GGG}$ and $R_{GG}$ matrices in the basis
$\ket{A},$ $\ket{B},$ $\ket{G}$,

$F^{G}_{GGG}=\frac{1}{2}\begin{pmatrix}1&1&\sqrt{2}\\\ 1&1&-\sqrt{2}\\\
\sqrt{2}&-\sqrt{2}&0\end{pmatrix}\quad\quad\quad
R_{GG}=\begin{pmatrix}\bar{\omega}&0&0\\\ 0&\bar{\omega}&0\\\
0&0&\omega\end{pmatrix}.$ (A.24)

Substituting these into the braiding matrix equation (A.20) gives

$B_{GG}=\begin{pmatrix}\cos{\frac{2\pi}{3}}&i\sin{\frac{2\pi}{3}}&0\\\
i\sin{\frac{2\pi}{3}}&\cos{\frac{2\pi}{3}}&0\\\ 0&0&e^{2\pi
i/3}\end{pmatrix},$ (A.25)

demonstrating that these dyons exhibit non-trivial non-Abelian braiding.

The $A$ and $B$ ribbon operators have already been derived in equations (A.11)
and (A.12), with $F^{A}_{\rho}=F^{1}_{\rho}$ and
$F^{B}_{\rho}=F^{\Lambda}_{\rho}$. For the $G$ anyon however, we need to use
(A.13) to know which group ribbon operators to derive. Here we note that the
conjugacy class of this dyon is $C_{c}=\\{c,c^{2}\\}$ [30]. Performing this
calculation gives the ribbon operator of $G$ as

$F^{G}_{\rho}=F^{c,e}_{\rho}+\omega
F^{c,c}_{\rho}+\bar{\omega}F^{c,c^{2}}_{\rho}+F^{c^{2},e}_{\rho}+\bar{\omega}F^{c^{2},c}_{\rho}+\omega
F^{c^{2},c^{2}}_{\rho}.$ (A.26)

For simplicity we omitted in this definition a normalisation factor of $1/3$
present in (A.13). From the general ribbon operator equation
$F^{h,g}_{\rho}=L^{h}_{\tau_{dual}}T^{g}_{\tau_{dir}}$ we see that $F^{G}$
needs a dual and a direct triangle to be defined. In general, these are taken
to be defined on different links of the lattice, such that the commutation
relation

$L^{h}_{\tau_{dual}}T^{g}_{\tau_{dir}}=T^{g}_{\tau_{dir}}L^{h}_{\tau_{dual}}$
(A.27)

trivially holds as the $T$ and $L$ operators act on different Hilbert spaces.
Due to that the Hermitian conjugate of $F^{h,g}_{\rho}$ is given by

$\begin{split}(F^{h,g}_{\rho})^{\dagger}&=(L^{h}_{\tau_{dual}}T^{g}_{\tau_{dir}})^{\dagger}=(T^{g}_{\tau_{dir}})^{\dagger}(L^{h}_{\tau_{dual}})^{\dagger}\\\
&=T^{g}_{\tau_{dir}}L^{h^{-1}}_{\tau_{dual}}=L^{h^{-1}}_{\tau_{dual}}T^{g}_{\tau_{dir}}=F^{h^{-1},g}_{\rho}\end{split}$
(A.28)

Here we want to identify the smallest possible system that can encode the
fusion and braiding properties of the $\\{A,B,G\\}$ anyons. It is possible to
define the dual and direct triangles to act on the same Hilbert space, i.e. we
choose them to overlap (see main text). In this case the commutation property
(A.27) does not hold any more and the conjugation relation given in (A.28) is
not satisfied automatically. To impose hermiticity of $F^{G}_{\rho}$ we take
into account that $F^{c^{2},e}=F^{c^{-1},e}=(F^{c,e})^{\dagger}$,
$F^{c^{2},c}=F^{c^{-1},c}=(F^{c,c})^{\dagger}$,
$F^{c^{2},c^{2}}=F^{c^{-1},c^{2}}=(F^{c,c^{2}})^{\dagger}$, so we can rewrite
(A.29) as

$F^{G}_{\rho}=F^{c,e}+\omega F^{c,c}+\bar{\omega}F^{c,c^{2}}+\text{h.c.}.$
(A.29)

Employing (A.6) we have

$F^{c,e}_{\rho}=\ket{c}\bra{e},\quad\quad
F^{c,c}_{\rho}=\ket{c^{2}}\bra{c},\quad\quad
F^{c,c^{2}}_{\rho}=\ket{e}\bra{c^{2}},$ (A.30)

so we eventually obtain

$F^{G}=\begin{pmatrix}0&1&\bar{\omega}\\\ 1&0&\bar{\omega}\\\
\omega&\omega&0\end{pmatrix}$ (A.31)

in the basis $\\{|e\rangle,|c\rangle,|c^{2}\rangle\\}$. The fusion of two $G$
anyons can be reproduced from (A.31) by multiplying the minimal ribbon
operators $F^{G}$ together, as follows

$F^{G}\times F^{G}=\begin{pmatrix}2&1&\bar{\omega}\\\ 1&2&\bar{\omega}\\\
\omega&\omega&2\end{pmatrix}=F^{A}+F^{B}+F^{G}$ (A.32)

Hence, the minimal ribbon operators reproduce the non-Abelian fusion
properties of $G$ anyons.

We therefore look towards the relation upon exchanging the anyons of this
model, described by their $R$ matrices. With the equation for ribbon operator
exchange (A.18) and the definition of the $G$ dyon ribbon operator (A.29), we
can find the relation

$F^{G}_{\rho_{1}}F^{G}_{\rho_{2}}=R_{GG}F^{G}_{\rho_{2}}F^{G}_{\rho_{1}},$
(A.33)

with $R_{GG}$ being the $R$ matrix for exchanging two $G$ dyons given in (1).

To find $R$ from the ribbon operators, we first need to determine
$F^{G}_{\rho_{1}}F^{G}_{\rho_{2}}$ and $F^{G}_{\rho_{2}}F^{G}_{\rho_{1}}$. We
can multiply two $G$ ribbon operators together

$F^{G}_{\rho}F^{G}_{\rho}=(F^{c,e}_{\rho}+\omega
F^{c,c}_{\rho}+\bar{\omega}F^{c,c^{2}}_{\rho}+F^{c^{2},e}_{\rho}+\bar{\omega}F^{c^{2},c}_{\rho}+\omega
F^{c^{2},c^{2}}_{\rho})^{2},$ (A.34)

which gives the result

$\begin{split}F^{G}_{\rho}F^{G}_{\rho}&=2F^{e,e}_{\rho}+2F^{e,c}_{\rho}+2F^{e,c^{2}}_{\rho}+F^{c,e}_{\rho}+\omega
F^{c,c}_{\rho}\\\ &\hskip
28.45274pt+\bar{\omega}F^{c,c^{2}}_{\rho}+F^{c^{2},e}_{\rho}+\bar{\omega}F^{c^{2},c}_{\rho}+\omega
F^{c^{2},c^{2}}_{\rho},\\\
&=F^{A}_{\rho}+F^{B}_{\rho}+F^{G}_{\rho},\end{split}$ (A.35)

as expected from the fusion rules.

From the definitions of the group ribbon operators given in (A.30), this can
be rewritten as

$F^{G}_{\rho_{1}}F^{G}_{\rho_{2}}=\begin{pmatrix}2&\bar{\omega}&\bar{\omega}\\\
1&2&\omega\\\ 1&\omega&2\end{pmatrix}$ (A.36)

in the basis $\ket{e},\ket{c},\ket{c^{2}}$. To find the form of
$F^{G}_{\rho_{2}}F^{G}_{\rho_{1}}$, we calculate to the full expansion of
(A.34) as

$\begin{split}F^{G}_{\rho_{1}}F^{G}_{\rho_{2}}&=F^{c,e}_{\rho_{1}}F^{c,e}_{\rho_{2}}+\omega
F^{c,e}_{\rho_{1}}F^{c,c}_{\rho_{2}}+\bar{\omega}F^{c,e}_{\rho_{1}}F^{c,c^{2}}_{\rho_{2}}+F^{c,e}_{\rho_{1}}F^{c^{2},e}_{\rho_{2}}+\bar{\omega}F^{c,e}_{\rho_{1}}F^{c^{2},c}_{\rho_{2}}+\omega
F^{c,e}_{\rho_{1}}F^{c^{2},c^{2}}_{\rho_{2}}\\\ &+\omega
F^{c,c}_{\rho_{1}}F^{c,e}_{\rho_{2}}+\bar{\omega}F^{c,c}_{\rho_{1}}F^{c,c}_{\rho_{2}}+F^{c,c}_{\rho_{1}}F^{c,c^{2}}_{\rho_{2}}+\omega
F^{c,c}_{\rho_{1}}F^{c^{2},e}_{\rho_{2}}+F^{c,c}_{\rho_{1}}F^{c^{2},c}_{\rho_{2}}+\bar{\omega}F^{c,c}_{\rho_{1}}F^{c^{2},c^{2}}_{\rho_{2}}\\\
&+\bar{\omega}F^{c,c^{2}}_{\rho_{1}}F^{c,e}_{\rho_{2}}+F^{c,c^{2}}_{\rho_{1}}F^{c,c}_{\rho_{2}}+\omega
F^{c,c^{2}}_{\rho_{1}}F^{c,c^{2}}_{\rho_{2}}+\bar{\omega}F^{c,c^{2}}_{\rho_{1}}F^{c^{2},e}_{\rho_{2}}+\omega
F^{c,c^{2}}_{\rho_{1}}F^{c^{2},c}_{\rho_{2}}+F^{c,c^{2}}_{\rho_{1}}F^{c^{2},c^{2}}_{\rho_{2}}\\\
&+F^{c^{2},e}_{\rho_{1}}F^{c,e}_{\rho_{2}}+\omega
F^{c^{2},e}_{\rho_{1}}F^{c,c}_{\rho_{2}}+\bar{\omega}F^{c^{2},e}_{\rho_{1}}F^{c,c^{2}}_{\rho_{2}}+F^{c^{2},e}_{\rho_{1}}F^{c^{2},e}_{\rho_{2}}+\bar{\omega}F^{c^{2},e}_{\rho_{1}}F^{c^{2},c}_{\rho_{2}}+\omega
F^{c^{2},e}_{\rho_{1}}F^{c^{2},c^{2}}_{\rho_{2}}\\\
&+\bar{\omega}F^{c^{2},c}_{\rho_{1}}F^{c,e}_{\rho_{2}}+F^{c^{2},c}_{\rho_{1}}F^{c,c}_{\rho_{2}}+\omega
F^{c^{2},c}_{\rho_{1}}F^{c,c^{2}}_{\rho_{2}}+\bar{\omega}F^{c^{2},c}_{\rho_{1}}F^{c^{2},e}_{\rho_{2}}+\omega
F^{c^{2},c}_{\rho_{1}}F^{c^{2},c}_{\rho_{2}}+F^{c^{2},c}_{\rho_{1}}F^{c^{2},c^{2}}_{\rho_{2}}\\\
&+\omega
F^{c^{2},c^{2}}_{\rho_{1}}F^{c,e}_{\rho_{2}}+\bar{\omega}F^{c^{2},c^{2}}_{\rho_{1}}F^{c,c}_{\rho_{2}}+F^{c^{2},c^{2}}_{\rho_{1}}F^{c,c^{2}}_{\rho_{2}}+\omega
F^{c^{2},c^{2}}_{\rho_{1}}F^{c^{2},e}_{\rho_{2}}+F^{c^{2},c^{2}}_{\rho_{1}}F^{c^{2},c}_{\rho_{2}}+\bar{\omega}F^{c^{2},c^{2}}_{\rho_{1}}F^{c^{2},c^{2}}_{\rho_{2}}.\end{split}$
(A.37)

Using (A.18) we can exchange these ribbons to get

$\begin{split}F^{G}_{\rho_{2}}F^{G}_{\rho_{1}}&=\omega
F^{c,e}_{\rho_{2}}F^{c,e}_{\rho_{1}}+\bar{\omega}F^{c,e}_{\rho_{2}}F^{c,c}_{\rho_{1}}+F^{c,e}_{\rho_{2}}F^{c,c^{2}}_{\rho_{1}}+\bar{\omega}F^{c,e}_{\rho_{2}}F^{c^{2},e}_{\rho_{1}}+\omega
F^{c,e}_{\rho_{2}}F^{c^{2},c}_{\rho_{1}}+F^{c,e}_{\rho_{2}}F^{c^{2},c^{2}}_{\rho_{1}}\\\
&+\bar{\omega}F^{c,c}_{\rho_{2}}F^{c,e}_{\rho_{1}}+F^{c,c}_{\rho_{2}}F^{c,c}_{\rho_{1}}+\omega
F^{c,c}_{\rho_{2}}F^{c,c^{2}}_{\rho_{1}}+F^{c,c}_{\rho_{2}}F^{c^{2},e}_{\rho_{1}}+\bar{\omega}F^{c,c}_{\rho_{2}}F^{c^{2},c}_{\rho_{1}}+\omega
F^{c,c}_{\rho_{2}}F^{c^{2},c^{2}}_{\rho_{1}}\\\
&+F^{c,c^{2}}_{\rho_{2}}F^{c,e}_{\rho_{1}}+\omega
F^{c,c^{2}}_{\rho_{2}}F^{c,c}_{\rho_{1}}+\bar{\omega}F^{c,c^{2}}_{\rho_{2}}F^{c,c^{2}}_{\rho_{1}}+\omega
F^{c,c^{2}}_{\rho_{2}}F^{c^{2},e}_{\rho_{1}}+F^{c,c^{2}}_{\rho_{2}}F^{c^{2},c}_{\rho_{1}}+\bar{\omega}F^{c,c^{2}}_{\rho_{2}}F^{c^{2},c^{2}}_{\rho_{1}}\\\
&+\bar{\omega}F^{c^{2},e}_{\rho_{2}}F^{c,e}_{\rho_{1}}+F^{c^{2},e}_{\rho_{2}}F^{c,c}_{\rho_{1}}+\omega
F^{c^{2},e}_{\rho_{2}}F^{c,c^{2}}_{\rho_{1}}+\omega
F^{c^{2},e}_{\rho_{2}}F^{c^{2},e}_{\rho_{1}}+F^{c^{2},e}_{\rho_{2}}F^{c^{2},c}_{\rho_{1}}+\bar{\omega}F^{c^{2},e}_{\rho_{2}}F^{c^{2},c^{2}}_{\rho_{1}}\\\
&+\omega
F^{c^{2},c}_{\rho_{2}}F^{c,e}_{\rho_{1}}+\bar{\omega}F^{c^{2},c}_{\rho_{2}}F^{c,c}_{\rho_{1}}+F^{c^{2},c}_{\rho_{2}}F^{c,c^{2}}_{\rho_{1}}+F^{c^{2},c}_{\rho_{2}}F^{c^{2},e}_{\rho_{1}}+\bar{\omega}F^{c^{2},c}_{\rho_{2}}F^{c^{2},c}_{\rho_{1}}+\omega
F^{c^{2},c}_{\rho_{2}}F^{c^{2},c^{2}}_{\rho_{1}}\\\
&+F^{c^{2},c^{2}}_{\rho_{2}}F^{c,e}_{\rho_{1}}+\omega
F^{c^{2},c^{2}}_{\rho_{2}}F^{c,c}_{\rho_{1}}+\bar{\omega}F^{c^{2},c^{2}}_{\rho_{2}}F^{c,c^{2}}_{\rho_{1}}+\bar{\omega}F^{c^{2},c^{2}}_{\rho_{2}}F^{c^{2},e}_{\rho_{1}}+\omega
F^{c^{2},c^{2}}_{\rho_{2}}F^{c^{2},c}_{\rho_{1}}+F^{c^{2},c^{2}}_{\rho_{2}}F^{c^{2},c^{2}}_{\rho_{1}},\end{split}$
(A.38)

with each term gaining a phase of either $\omega$ or $\bar{\omega}$ when
compared to the above. We choose now the two paths to be on the top of each
other, $\rho_{1}=\rho_{2}=\rho$ and employ (A.30) and (A.28) to find

$F^{G}_{\rho_{2}}F^{G}_{\rho_{1}}=\begin{pmatrix}2\bar{\omega}&\omega&1\\\
\omega&2\bar{\omega}&1\\\
\bar{\omega}&\bar{\omega}&2\bar{\omega}\end{pmatrix}=\bar{\omega}(F^{A}+F^{B})+\omega
F^{G}.$ (A.39)

Comparing this to (A.36), it is seen that the diagonal terms have gained a
phase $\bar{\omega}$ and the off-diagonal have gained a phase $\omega$. Noting
that the diagonal terms come from the fusion of $G\times G\to A+B$ and the
off-diagonal are from $G\times G\to G$, it is evident that the $A$ and $B$
fusion channels have gained a phase $\bar{\omega}$ and the $G$ has gained a
phase $\omega$. This is the expected result from the $R$ matrix in the anyonic
basis (1).

## Appendix Appendix

###  B.1 Experimental Apparatus

As shown in Fig. 2, our apparatus consists of a coherent light source at $810$
nm launched using a single mode fiber (Thorlabs-780HP) collimated using a set
of lenses with an effective focal length of $L_{1}=58.67$ mm. The beam is
reflected off a spatial light modulator (Hamamatsu LCOS-X10468,
$\text{SLM}_{1}$) and coupled into a multi-mode GRIN fibre (Thorlabs-M116L02)
with a set of lenses with an effective focal length of $L_{2}=22$ mm. The
light from the multi-mode fibre is then incident on another spatial light
modulator ($\text{SLM}_{2}$) with the help of another set of lenses with an
effective focal length of $L_{3}=22$mm. Finally, the modulated light is
measured on a CMOS Camera (XIMEA-xiC USB3.1) after passing through a set of
lenses with an effective focal length of $L_{4}=33.33$ mm.

###  B.2 Inverse design of operations

We encode the desired operations in an optical circuit that is built with a
cascade of optical mode mixers separated by phase layers as described in [41].
For a given set of mode mixers $U_{j}$ and phase layers $P_{j}$, a given
operation $\mathbf{T}$ is encoded as

$\mathbf{T}=\prod_{j=1}^{2}U_{j}P_{j}.$ (B.1)

Here the mode mixers $U_{1}$ and $U_{2}$ correspond to an MMF and a $2f$
lenses system respectively. The phase layers $P_{j}$ correspond to the SLMs.
In order to encode these operations to the given optical circuit, we perform
an inverse design algorithm called wavefront-matching such that a set of input
optical modes are transformed according to the given transformation to a set
of output modes. The inverse design of operations requires measurement of the
transmission matrix (TM) of the MMF, which is done by displaying a set of
random patterns on both $\text{SLM}_{1}$ and $\text{SLM}_{2}$ and subsequently
measuring the intensity of the output speckle patter on the camera. These
random measurements serve as a data-set to train a multi-plane neural network
(MPNN) which reveals the TM of the MMF [64].

The output modes for the implemented operations are chosen such that the
measurement outcomes are spatially separated and can be measured using a
camera. The position of spatially-separated measurement outcomes on the camera
can be arbitrarily selected, however, might result in different performances.
We thus implement multiple realisations of the same operation with different
positions of measurement outcomes on the camera.

###  B.3 Experimentally implemented processes

Experimentally, when attempting to implement the desired operations, there are
additional incoherent effects which must be characterised by determining the
entire quantum process, which describes the map between density operator on
the input space to those on the output. The implemented processes take the
general form
$\mathcal{\widetilde{T}}(\rho)=\sum_{n}^{R}\mathbf{\widetilde{T}}_{n}\rho\mathbf{\widetilde{T}}_{n}^{\dagger}$.
This is the Kraus representation of the process and it is completely positive
(CP) by construction. We do not require that our operations are trace
preserving (TP) because loss may, and indeed must, occur in the system. In
this case the process must be non-trace-increasing, and the Kraus operators,
$\\{\mathbf{\widetilde{T}}_{n}\\}_{n}$, of the process must obey
$\sum_{n}\mathbf{\widetilde{T}}^{\dagger}_{n}\mathbf{\widetilde{T}}_{n}\leq\mathbb{I}$,
ie. no amplification can occur. Loss could be accounted for by introducing an
auxiliary output mode and an additional Kraus operator, and imposing equality
$\sum_{n}\mathbf{\widetilde{T}}^{\dagger}_{n}\mathbf{\widetilde{T}}_{n}=\mathbb{I}$
to arrive at a TP process. The ideal operations correspond to the processes
$\mathcal{T}(\rho)=\mathbf{T}\rho\mathbf{T}^{\dagger}$, which are, again, not
TP. The non-trace-increasing condition is saturated for operations
$\mathbf{T}/\alpha$ when $\alpha$ is chosen to be the largest singular value
of $\mathbf{T}$. In this case the average success probability of the
operations, defined as
$\mathbb{Tr}[(\mathbf{T}/\alpha)(\mathbb{I}/3)(\mathbf{T}/\alpha)^{\dagger}]$,
is given by $1/2,\,3/8,\,3/4$ for $F^{G}_{\rho_{0}}$ ,
$F^{G}_{\rho_{1}}F^{G}_{\rho_{2}}$ , $F_{\rho_{2}}^{G}F_{\rho_{1}}^{G}$ ,
respectively. We perform quantum process tomography (QPT) to characterise the
implemented processes $\mathcal{\widetilde{T}}$ and quantify their fidelity in
relation to the ideal operations $\mathcal{T}$. As an alternative to the Kraus
representation, it is convenient to represent these processes via their Choi
states,
$\rho_{\mathcal{\widetilde{T}}}=\mathcal{\widetilde{T}}\otimes\mathbb{1}(\rho^{+})$,
with $\rho^{+}$ the maximally entangled state. Rather than work with the non-
TP Choi state, we normalise to unit trace, thus providing the appropriate
object for assessing process fidelities whilst neglecting global losses [65].
The purity of the implemented transformation $\mathcal{\widetilde{T}}$ is
calculated as $\mathcal{P}=Tr(\rho_{\widetilde{\mathcal{T}}}^{2})$ , while its
fidelity with respect to the ideal transformation $\mathbf{T}$ is calculated
as,

$\mathcal{F}(\rho_{\mathcal{T}},\rho_{\widetilde{\mathcal{T}}})=\left[\text{Tr}\left(\sqrt{\sqrt{\rho_{\mathcal{T}}}\rho_{\widetilde{\mathcal{T}}}\sqrt{\rho_{\mathcal{T}}}}\right)\right]^{2},$
(B.2)

###  B.4 Quantum process tomography (QPT) of implemented processes

In order to tomograph the implemented operations, we prepare and measure each
vector from an informationally complete set of mutually unbiased bases(MUB)
[66]. The $j^{\text{th}}$ vector in the $\mu^{\text{th}}$ MUB in three
dimensions is given by [67]

$\ket{M^{\mu}_{j}}=\frac{1}{\sqrt{3}}\sum^{2}_{i=0}\omega^{ji+\mu
i^{2}}\ket{i}$ (B.3)

where $\ket{i}$ corresponds to $i^{\text{th}}$ mode in the computational
basis, and $\omega=e^{i\pi/3}$ is the cube root of unity. Experimentally, we
prepare each of the input states,
$\\{\hat{\tau}^{\mu}_{j}=\ket{M^{\mu}_{j}}\bra{M^{\mu}_{j}}\\}_{\mu j}$, as
superpositions of macro-pixels using $\text{SLM}_{1}$ in all MUBs as shown in
Fig. 2(b). For each of the prepared input states, the output of the optical
circuit is projected on all MUB vectors,
$\\{\hat{\Pi}^{\mu}_{j}=\ket{M^{\mu}_{j}}\bra{M^{\mu}_{j}}\\}_{\mu j}$, of the
modes corresponding to the three selected output modes using $\text{SLM}_{2}$.
These measurements are, up to a constant intensity, $N$, proportional to the
expectation values,

$I^{\mu\nu}_{ij}=N\mathbb{Tr}[\hat{\Pi}^{\mu}_{i}\chi(\hat{\tau}^{\nu}_{j})]\,,$
(B.4)

with $\chi$ the quantum process being implemented by the optical circuit
acting on input state $\hat{\tau}^{\mu}_{j}$ being projected onto
$\hat{\Pi}^{\mu}_{j}$, which are sufficient for QPT. In order to recover the
physical quantum process, $\chi$, from this data, we impose positivity through
the positivity of the Choi state
($\rho_{\chi}:=\chi\otimes\mathbb{1}(\rho^{+})$, with $\rho^{+}$ the maximally
entangled state) representation of the process allowing QPT to be performed
via the semi-definite program (SDP),

$\begin{split}\min_{\rho_{\chi},N}\quad&|I^{\mu\nu}_{ij}-N\mathbb{Tr}[\hat{\Pi}^{\mu}_{i}\otimes\hat{(}\tau^{\nu}_{j})^{T}\rho_{\chi}]|^{2}\\\
\textrm{s.t.}\quad&\rho_{\chi}\geq
0\,,\,\mathbb{Tr}[\rho_{\chi}]=1.\end{split}$ (B.5)

In this manner, the recovered Choi state of the process is not TP, and
normalised to unit trace suitable for assessing process fidelities.

###  B.5 Systematic error analysis due to misalignments

In practice, the states prepared and measurements performed deviate from the
ideal cases, which can lead to erroneous process reconstruction. The extent,
and particularly the distribution, of the infidelity of state preparation and
measurement can by characterised experimentally by sampling from independent
prepare and measure experiments, and fitting these data to a model capturing
sources of systematic error in the experimental devices [41]. This model, and
the distributions of different types of systematic error, allow to simulate
data sampling from these imperfections, perform the QPT procedure to obtain a
Monte-Carlo sample of these effects, and arrive at realistic bounds on the
systematic errors present on our process fidelities. These errors upto three
standard deviations are presented in Table. 1 .

## Appendix Appendix

###  C.1 An explicit qubit block encoding based on the SVD

We wish to construct a unitary block encoding [36, 35, 34, 37, 38] of the
ribbon operator $F_{\rho_{0}}^{G}$ in the top-left block of a unitary matrix:

$U_{F}=\left(\begin{array}[]{cc}F_{\rho_{0}}^{G}/\alpha&\star\cr\star&\star\cr\end{array}\right),$
(C.1)

where $\alpha$ is a scaling factor such that in terms of the operator norm
$||F_{\rho_{0}}^{G}/\alpha||\leq 1$, and $\star$ denotes arbitrary matrix
elements that only have that constraint that that the total matrix $U_{F}$ is
unitary.

We now give an explicit unitary block encoding of $F_{\rho_{0}}^{G}$ based on
the singular-value decomposition (SVD) of $F_{\rho_{0}}^{G}$. The SVD is of
the form $F_{\rho_{0}}^{G}/\alpha=U_{1}DU_{2}$ where $D$ is a diagonal matrix
and where both $U_{1}$ and $U_{2}$ are two unitary matrices of the same
dimension as $F_{\rho_{0}}^{G}$. If we have $||F_{\rho_{0}}^{G}/\alpha||\leq
1$ then the following matrix is guaranteed to be unitary [39, 34]:

$U_{F}=\begin{pmatrix}F_{\rho_{0}}^{G}/\alpha&U_{1}\sqrt{\mathds{1}-D^{2}}U_{2}\\\
U_{1}\sqrt{\mathds{1}-D^{2}}U_{2}&-F_{\rho_{0}}^{G}/\alpha\end{pmatrix}$ (C.2)

As $F_{\rho_{0}}^{G}$ is low dimensional, it is straightforward to calculate
$U_{F}$ numerically. This gives:

$U_{F}=\bordermatrix{\leavevmode\nobreak\
&\bra{000}&\bra{100}&\bra{010}&\bra{110}&\bra{001}&\bra{101}&\bra{011}&\bra{111}\cr\ket{000}&0&\frac{1}{2}&\frac{1}{2}\bar{\omega}&0&-\frac{1}{\sqrt{3}}&\frac{1}{2\sqrt{3}}&\frac{\bar{\omega}}{2\sqrt{3}}&0\cr\ket{100}&\frac{1}{2}&0&\frac{1}{2}\bar{\omega}&0&\frac{1}{2\sqrt{3}}&-\frac{1}{\sqrt{3}}&\frac{\bar{\omega}}{2\sqrt{3}}&0\cr\ket{010}&\frac{1}{2}\omega&\frac{1}{2}\omega&0&0&\frac{\omega}{2\sqrt{3}}&\frac{\omega}{2\sqrt{3}}&-\frac{1}{\sqrt{3}}&0\cr\ket{110}&0&0&0&0&0&0&0&1\cr\ket{001}&-\frac{1}{\sqrt{3}}&\frac{1}{2\sqrt{3}}&\frac{\bar{\omega}}{2\sqrt{3}}&0&0&-\frac{1}{2}&\frac{-\bar{\omega}}{2}&0\cr\ket{101}&\frac{1}{2\sqrt{3}}&-\frac{1}{\sqrt{3}}&\frac{\bar{\omega}}{2\sqrt{3}}&0&-\frac{1}{2}&0&\frac{-\bar{\omega}}{2}&0\cr\ket{011}&\frac{\omega}{2\sqrt{3}}&\frac{\omega}{2\sqrt{3}}&-\frac{1}{\sqrt{3}}&0&-\frac{1}{2}\omega&-\frac{1}{2}\omega&0&0\cr\ket{111}&0&0&0&1&0&0&0&0\cr}$
(C.3)

where $\alpha=2$, and where we pad $F_{\rho_{0}}^{G}$ with zeroes.

###  C.2 Success probability of qubit block encoding

The following circuit then allows us apply $F_{\rho_{0}}^{G}$ to the encoded
qutrit state $\ket{\phi}$, using $U_{F}$ within the 3 qubit system:

$\Qcircuit@C=1em@R=1em{\lstick{\ket{0}}&\multigate{1}{U_{F}}\meter\qw\\\
\lstick{\ket{\phi}}\ghost{U_{F}}\rstick{\propto
F_{\rho_{0}}^{G}\ket{\phi}}\qw}$ (C.4)

if the measurement of the first qubit returns 0. Success probably is therefore
dependent on the input state, as we have
$p(0)=\frac{1}{\alpha^{2}}\mathbb{Tr}[(F_{\rho_{0}}^{G})^{{\dagger}}F_{\rho_{0}}^{G}\rho]$
and therefore we have the following bounds on the success probability of the
block-encoding

$\min_{k}(\sigma_{k}^{2})\leq p(0)\leq\max_{k}(\sigma_{k}^{2}),$ (C.5)

where $\sigma_{k}$ are the singular values of $F_{\rho_{0}}^{G}/\alpha$. Given
a minimal required value of $\alpha=2$ for positivity of
$F_{\rho_{0}}^{G}/\alpha$, we have $\\{\sigma_{k}\\}=\\{1,1/2,1/2\\}$ and
therefore the success probability is bounded tightly as

$\frac{1}{4}\leq p(0)\leq 1.$ (C.6)

We can find the states that saturate these bounds through the eigen-
decomposition of $(F_{\rho_{0}}^{G})^{{\dagger}}F_{\rho_{0}}^{G}$, which is
clearly closely related to the above singular values. We find $e_{1}=1$,
$e_{2}=1/4$, $e_{3}=1/4$, with normalized eigenvectors of

$\ket{e_{1}}=\frac{1}{\sqrt{3}}\begin{pmatrix}\bar{\omega}\\\ \bar{\omega}\\\
1\end{pmatrix},\ \ket{e_{2}}=\frac{1}{\sqrt{2}}\begin{pmatrix}-\bar{\omega}\\\
0\\\ 1\end{pmatrix},\ \ket{e_{3}}=\frac{1}{\sqrt{2}}\begin{pmatrix}-1\\\ 1\\\
0\end{pmatrix},$ (C.7)

where $\ket{e_{1}}$ saturates the upper bound, and $\ket{e_{2}}$ and
$\ket{e_{3}}$ the lower bound. Appending five zeroes to these vectors gives
the required three qubit states, $\ket{\psi_{i}}$, in the same basis as
$U_{F}$ in Equation (C.3).

###  C.3 Fidelity of IBM device implementation

We now consider how well a current IBM device might implement the circuits
applying $U_{F}$ on the three eigenstates. Using the standard native gates of
the IBMQ device, here ibmq_belem, we transpile circuits to prepare the
eigenstates, and apply the unitary $U_{F}$. Using the qiskit complier we find
$U_{F}$ requires a circuit depth of 74 operations with 20 CNOTs.

Through IBMQ we can access a simplified model version of the noise that has
been characterised on a device. For ibmq_belem we download the noise model
associated with its native gates, and simulate our circuits with and without
noise. We can then compare the density matrices of the idealised case with the
noisy versions. We denote the ideal unitary channel
$\mathcal{U}_{F}(\cdot):=U_{F}\cdot U_{F}^{\dagger}$, and denote the noisy
version, $\tilde{\mathcal{U}}_{F}$. For the embedded eigenstates,
$\psi_{i}:=|\psi_{i}\rangle\langle\psi_{i}|$, we denote the noisy versions,
$\rho_{i}$. The results of this are shown in Table C.I. We see that the
fidelities between the idealized implementation and simulated noisy
implementation are strictly below $81\%$, suggesting that current quantum
devices cannot achieve the high levels of fidelity of the photonic simulation.

State |  | Initial
---
Purity
Initial Fidelity |  | Final
---
Purity
Final Fidelity
$\psi_{i}$ | $\mathcal{P}(\rho_{i})$ | $\mathcal{F}(\psi_{i},\rho_{i})$ | $\mathcal{P}(\tilde{\mathcal{U}_{F}}(\rho_{i}))$ | $\mathcal{F}(\mathcal{U}_{F},\tilde{\mathcal{U}}_{F})$
i=1 | $92.9\%$ | $96.4\%$ | $66.0\%$ | $80.9\%$
i=2 | $92.7\%$ | $96.3\%$ | $62.3\%$ | $78.4\%$
i=3 | $91.5\%$ | $95.6\%$ | $61.6\%$ | $77.8\%$
Table C.I: Simulated results comparing noisy quantum circuits of the 3 qubit
unitary block encoding of $F_{\rho_{0}}^{G}$ to idealised noise free case. We
use the noise model of the gateset associated to the device ibmq_belem.