note-name = , use-sort-key = false

# Topological Quantum Computation on a Chiral Kondo Chain

Tianhao Ren<EMAIL_ADDRESS>Condensed Matter Physics and Materials Science
Division, Brookhaven National Laboratory, Upton, New York 11973, USA  Elio J.
König  Max-Planck Institute for Solid State Research, 70569 Stuttgart, Germany
Alexei M. Tsvelik<EMAIL_ADDRESS>Condensed Matter Physics and Materials
Science Division, Brookhaven National Laboratory, Upton, New York 11973, USA

###### Abstract

We describe the chiral Kondo chain model based on the symplectic Kondo effect
and demonstrate that it has a quantum critical ground state populated by non-
Abelian anyons. We show that the fusion channel of two arbitrary anyons can be
detected by locally coupling the two anyons to an extra single channel of
chiral current and measuring the corresponding conductance at finite
frequency. Based on such measurements, we propose that the chiral Kondo chain
model with symplectic symmetry can be used for implementation of measurement-
only topological quantum computations, and it possesses a number of distinct
features favorable for such applications. The sources and effects of errors in
the proposed system are analyzed, and possible material realizations are
discussed.

##  I Introduction

The design of quantum devices and harvesting their properties for quantum
technological utilization is a prime application of present day condensed
matter research. In this context, the topological quantum computation Kitaev
(2003); Nayak et al. (2008); Pachos (2012); Lahtinen and Pachos (2017) is
particularly appealing, because it allows for intrinsically robust storage and
processing of information by braiding of anyons. The latter are low-energy
quasiparticles which display generic exchange statistics, an irrational
Hilbert space dimension (known as “quantum dimension”), and occur in strongly
interacting many-body systems.

Existing proposals for platforms to realize topological quantum computation
include fractional quantum Hall states using non-Abelian anyonic excitations
Nayak et al. (2008) and one-dimensional wire networks using Majorana zero
modes Kitaev (2001); Sarma et al. (2015). Besides, solitary anyons located at
the impurity site are predicted to exist in the ground states of various
quantum critical Kondo models and experimental observations of quantum
criticality Potok et al. (2007); Keller et al. (2015); Karki et al. (2023)
indirectly support this claim. Topological quantum computation, however,
require multiple anyons. It was suggested by Sela et al. Lopes et al. (2020);
Gabay et al. (2022); Lotem et al. (2022) that multiple anyons may exist in a
chiral multichannel Kondo chain where multiple impurities are placed on the
chiral edge of a topologically nontrivial bulk. In the absence of
backscattering, the chiral electrons do not generate the
Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction between the impurities, and
the critical ground states propelled by the Kondo screening develop without
hindrance. The resulting Hilbert space of the ground states is spanned by
fusion of entangled anyons located at the impurity sitesKomijani (2020). The
braiding of anyons can be performed without their physical permutation by
means of measurement-only topological quantum computation Bonderson et al.
(2008a).

However, the existing schemes for the construction of quantum gates, such as
the ones described in Refs. Lopes et al., 2020; Gabay et al., 2022; Lotem et
al., 2022, are based on the multichannel Kondo models where the quantum
critical state is notoriously fragile and hence requires fine tuning of the
energy levels and the exchange interactions of different channels. This is the
main problem we try to address here.

In this paper we describe a chiral Kondo chain model possessing non-Abelian
anyons in its ground state. We also describe a measurement-only protocol for
universal quantum computations. As a resolution of the problem of stability
with respect to perturbations, it was suggested Li et al. (2023); König and
Tsvelik (2023) that instead of the Kondo model with unitary symmetry to use
the one with symplectic symmetry Flint et al. (2008) where the quantum
critical state is stable with respect to the most common perturbations. As for
the implementation, we envisage that the role of quantum impurities will be
played by mesoscopic devices consisting of a mesoscopic superconducting
Coulomb box in proximity to several quantum dots.

The paper is organized as follows. In Sec. II we review the symplectic Kondo
effect, which is central to our construction of the chiral Kondo chain model.
In Sec. III we define the chiral Kondo chain model with symplectic symmetry
and show its integrability. In Sec. IV we describe the low energy properties
of the chiral Kondo chain model and show that it possesses localized non-
Abelian anyons in its ground state. In Sec. V we discuss how the information
encoded by the anyons can be extracted by measuring the correlation functions.
In Sec. VI we describe the implementation of the chiral Kondo chain with
symplectic symmetry as a platform for universal quantum computations. In Sec.
VII we provide a summary of our results. Several basic ingredients of anyonic
quantum computation are reviewed in the appendices to make the paper self-
contained.

##  II Review of the Symplectic Kondo Effect

We repeat some details described in our previous publications Li et al.
(2023); König and Tsvelik (2023) for the symplectic Kondo effect. The
symplectic impurity spin is implemented by means of a superconducting island
coupled to $k$ quantum dots:

$\begin{split}\mathcal{H}_{\mathrm{dot}}&=E_{C}\left(2N_{C}+n_{d}-N_{g}\right)^{2}\\\
&-\frac{\Delta}{2}\sum_{i=1}^{k}\sum_{\sigma\sigma^{\prime}}\left[e^{-\mathrm{i}\phi}d_{i\sigma}^{\dagger}\left[\sigma_{y}\right]_{\sigma\sigma^{\prime}}d_{i\sigma^{\prime}}^{\dagger}+\text{H.c.}\right],\end{split}$
(1)

where $n_{d}=\sum_{i=1}^{k}\sum_{\sigma}d_{i\sigma}^{\dagger}d_{i\sigma}$ is
the total charge in the resonant zero energy states provided by the quantum
dots and $N_{C}=-\mathrm{i}\partial_{\phi}$ counts the number of Cooper pairs
on the island. $E_{C}$ is the charging energy of the island, $N_{g}$ is the
gate charge for the island, and $\Delta$ is the pairing amplitude. The labels
$i=1,\ldots,k$ refer to the resonant zero energy states coupled to the
superconducting island. The configuration of such a system with $k=2$ is shown
in Fig. 1.

Figure 1: A sketch of the proposed Kondo device for $k=2$. The red rectangle
is a superconducting island with charging energy $E_{C}$. The green dots are
quantum dots with discrete single-particle spectrum of energy levels in
proximity to the island. Electrons from the dots can tunnel to the chiral
edges (in yellow) containing chiral electrons, and the chiral edges are
supported by the appropriate topological insulators (in blue).

It is physically realizable when Cooper pairs are converted into two electrons
only locally at the $k$ resonant levels, which is satisfied when the distance
between the quantum dots exceeds the superconducting coherence length. We will
focus on the particle-hole invariant point $N_{g}=1$. In the limit
$0<E_{C}-\Delta\ll E_{C},\Delta$, the ground state of the quantum dot is in
the odd parity sector, $2k$-fold degenerate, and labelled by the channel and
the spin indices of the singly occupied level $\ket{i,\sigma}$. For $t\ll
E_{C}-\Delta$, the coupling to the leads

$\mathcal{H}_{t}=\sum_{i=1}^{k}\sum_{\sigma}\left[tc^{\dagger}_{i\sigma}(x=0)d_{i\sigma}+\text{H.c.}\right]$
(2)

is taken into account perturbatively by means of two virtual processes
corresponding to transitions into one of the two lowest excited states with
one additional and one missing charge. The addition of these two virtual
processes leads to the following symplectic Kondo model:

$\mathcal{H}=\mathrm{i}v_{F}\int\mathrm{d}x\sum_{\alpha}c_{\alpha}^{\dagger}\partial_{x}c_{\alpha}+g\sum_{A\in\mathfrak{g}}J^{A}(0)S^{A},$
(3)

where $\alpha=(i,\sigma)$ is a multi-index including channel and spin indices,
$g=4t^{2}/(E_{C}-\Delta)$ is the coupling strength, $c_{\alpha}(x)$ are
fermionic annihilation operators,
$J^{A}(x)=\sum_{\alpha\alpha^{\prime}}c_{\alpha}^{\dagger}(x)\tau^{A}_{\alpha\alpha^{\prime}}c_{\alpha^{\prime}}(x)$
with $\tau^{A}$ being generators of the Lie algebra
$\mathfrak{g}=\mathfrak{sp}(2k)$ in the fundamental representation, and the
impurity spin $S^{A}$ also transforms in the fundamental representation of
$\mathfrak{g}=\mathfrak{sp}(2k)$.

The fact that the current $J^{A}$ belongs to the $Sp_{1}(2k)$ theory can be
checked by writing down the bulk Hamiltonian in terms of $4k$ species of
Majorana fermions. The latter model is equivalent to the level-1 chiral
$O(4k)$ Wess-Zumino-Novikov-Witten (WZNW) model. Then we use the conformal
embedding Verstegen (1991); Kimura (2021):

$O_{1}(4k)=Sp_{k}(2)\oplus Sp_{1}(2k),$ (4)

to represent the Hamiltonian of free fermions as the sum of two WZNW
Hamiltonians. It follows that Eq. (3) can be written as a sum of two commuting
parts:

$\begin{split}\mathcal{H}&=\Big{\\{}\mathcal{H}_{\text{WZNW}}[Sp_{k}(2)]\Big{\\}}\\\
&+\Big{\\{}\mathcal{H}_{\text{WZNW}}[Sp_{1}(2k)]+g{\textstyle\sum}_{A\in\mathfrak{g}}J^{A}(0)S^{A}\Big{\\}}.\end{split}$
(5)

The central charge of the $Sp_{1}(2k)$ WZNW model is determined as

$c=\frac{k(2k+1)}{k+2}.$ (6)

We can check this by looking at a few simple examples: for $k=1$,
$Sp(2)=SU(2)$ and the central charge is 1, so we have $Sp_{1}(2)=SU_{1}(2)$;
for $k=2$, $Sp(4)=O(5)$ and the central charge is 5/2 (five Majorana
fermions), so we have $Sp_{1}(4)=O_{1}(5)$.

The symplectic multichannel Kondo model defined in Eq. (3) can be exactly
solved König and Tsvelik (2023), and its low energy properties can be derived
accordingly. We will review these results in Sec. III, and here we want to
mention that the most important observation will be that the model possesses
localized non-Abelian anyons in its ground state, which can be utilized to
realize topological quantum computations.

In real situations there are always perturbations which break the degeneracy
of the energy levels provided by the quantum dots, and the most likely one
comes from the impact of a local chemical potential:

$\delta\mathcal{H}=\mu_{i}\sum_{\sigma}d^{+}_{i\sigma}d_{i\sigma},$ (7)

where $i=1,\ldots,k$ label the quantum dots. On the other hand, potentially
important perturbations are those which directly couple to the generators of
the algebra, which satisfy

$(S^{A})^{T}=-\sigma^{y}S^{A}\sigma^{y},$ (8)

where the Pauli matrix $\sigma^{y}$ acts on the spin indices. However, it is
easy to see that the perturbation in Eq. (7) does not satisfy Eq. (8) and
hence does not belong to the potentially important category, as was already
discussed in Ref. König and Tsvelik, 2023. Hence the presence of the
symplectic symmetry will greatly weaken the influence of the level disorder,
the latter being the bane of all anyon scenarios [sela].

[sela]Through private communication with E. Sela we learned about preliminary
numerical renormalization group data for the chiral Kondo chain model with
symplectic symmetry by M. Lotem, E. Sela et al., which suggests that both the
level disorder in Eq. (7) and the Kondo anisotropy in Eq. (34) are relevant
perturbations. Our conclusion, however, is that the level disorder does not
directly couple to the impurity spins, and the Kondo anisotropy is exactly
marginal being equivalent to a coupling of chiral currents. As is stated in
Sec. IV.2, a particularly simple argument favoring our conclusion can be given
for the $k=2$ case.

We may also imagine that in real situations the tunneling in Eq. (2) will be
edge dependent. This will give rise to anisotropy of the Kondo couplings
$g\rightarrow g_{A}$. It is well known, however, that renormalization group
flows, especially for for models with level $k=1$ Kac-Moody currents tend to
converge, restoring the symmetry at low energies Lin et al. (1998); Konik et
al. (2002). In integrable models such as the anisotropic Kondo ones this
effect can be traced beyond the perturbation theory. At the end of Sec. IV we
will provide further arguments in favor of the stability of the critical point
with respect to anisotropic Kondo couplings in the case of symplectic symmetry
[sela].

##  III The Chiral Kondo Chain Model and its Integrability

At the center of our project is the chiral Kondo chain model based on the
symplectic Kondo model of Eq. (3):

$\begin{split}\mathcal{H}=&\mathrm{i}v_{F}\int\mathrm{d}x\sum_{\alpha}c_{\alpha}^{\dagger}\partial_{x}c_{\alpha}+\sum_{j=1}^{N_{s}}g_{j}\sum_{A\in\mathfrak{g}}J^{A}(x_{j})S^{A}(x_{j}),\end{split}$
(9)

where $\mathfrak{g}=\mathfrak{sp}(2k)$ and the single impurity spin is
generalized to multiple impurity spins. For simplicity, we will henceforth set
the Fermi velocity $v_{F}=1$. This type of models can be directly realized in
topological electronic systems with gapped bulk where the boundary electronic
modes are chiral, as discussed in Sec. II. In this section, we show that the
chiral Kondo chain model defined in Eq. (9) is integrable for an arbitrary
number of impurity spins, which provides us a handle to analyze its low energy
properties.

The case of a single impurity spin $(N_{s}=1)$ and $k=1$ was solved previously
Andrei (1980); Vigman (1980); Tsvelick and Wiegmann (1983); Andrei et al.
(1983). For chiral models it is straightforward to generalize the solution to
the case with an arbitrary number of impurity spins and channels due to the
link between the Kondo and chiral Gross-Neveau (GN) models, established in the
pioneering paper by Andrei (1980). In this paper, the model defined by Eq. (9)
is treated as a limit of the chiral GN model, whose solution for an arbitrary
number of right- and left-moving fermions $N_{R},N_{L}$ was obtained in Ref.
Andrei and Lowenstein, 1979. The difference between the Kondo chain model and
the Lorentz invariant GN model is that in the former case the left-moving
particles have zero velocity. Although in Ref. Andrei, 1980 this method was
used for the Kondo problem with $SU(2)$ symmetry, it is completely general and
holds for other Lie group symmetries. In particular, the solution for the
symplectic symmetry can be extracted from the results obtained for the chiral
$Sp(2k)$ GN model Ogievetsky et al. (1987).

Basically, the integrability of the model in Eq. (9) lies in the fact that the
$N$-body scattering problem for an arbitrary number of particles $N$ can be
represented as a succession of two-body scattering problems. This is possible
provided that eigenvalues are invariant under permutations of the momenta of
the particles, which is fulfilled when the two-body $S$-matrices satisfy the
Yang-Baxter equations shown in Fig. 2. For models with linear dispersions, the
particle velocity acquires discrete set of values (in the given case it will
be $v=1$ for the electrons and $v=0$ for the impurity). However, the
integrability requires that the $S$-matrices $S(v,v)$ and $S(v,0)$ must lie on
a continuous curve $S(\lambda)$ where the latter function satisfies the Yang-
Baxter equations.

Figure 2: The Yang-Baxter equations for two-body scatterings, and the
integrability for the chiral Kondo chain model. The line intersections stand
for the scattering matrices. The lower panel illustrates the relation between
the products of the monodromy matrices of the multi-impurity problem. In this
panel, we differentiate the electrons and impurity spins by solid and dashed
lines. The solid lines carry zero rapidity and the dashed ones carry rapidity
$1/g_{j}$, where $g_{j}$ is the coupling for $j$-th impurity.

For particles in the fundamental representation of the $Sp(2k)$ symmetry, the
solution is Berg et al. (1978); Reshetikhin (1985); Jimbo (1986):

$\left[S_{\alpha\beta}^{\bar{\alpha}\bar{\beta}}\right](\lambda)=\frac{\left[\mathrm{i}\lambda(\mathrm{i}\lambda+k+1)\delta_{\alpha\beta}\delta_{\bar{\alpha}\bar{\beta}}+(\mathrm{i}\lambda+k+1)\delta_{\alpha\bar{\beta}}\delta_{\beta\bar{\alpha}}-\mathrm{i}\lambda\epsilon_{\alpha}\epsilon_{\beta}\delta_{\alpha,2k+1-\beta}\delta_{\bar{\alpha},2k+1-\bar{\beta}}\right]}{(\mathrm{i}\lambda+1)(\mathrm{i}\lambda+k+1)},$
(10)

where $\epsilon_{\alpha}=1$ for $\alpha=1,\ldots,k$, and
$\epsilon_{\alpha}=-1$ for $\alpha=k+1,\ldots,2k$, and we have adopted the
notation that $\alpha=1,\ldots,k$ correspond to multi-index $(i,-)$, and
$\alpha=k+1,\ldots,2k$ correspond to multi-index $(i,+)$ to make contact with
the literature on integrable models. It indeed contains the solution of the
Schrödinger equation for the model defined in Eq. (9) as a particular case.
Identifying $S(v,v^{\prime})=S(\lambda=(v-v^{\prime})/g_{j})$ we obtain

$\begin{split}&\left[S_{\alpha\beta}^{\bar{\alpha}\bar{\beta}}\right](1,1)=\left[S_{\alpha\beta}^{\bar{\alpha}\bar{\beta}}\right](\lambda=0)=\delta_{\alpha\bar{\beta}}\delta_{\beta\bar{\alpha}},\\\
&\left[S_{\alpha\beta}^{\bar{\alpha}\bar{\beta}}\right](1,0)=\left[S_{\alpha\beta}^{\bar{\alpha}\bar{\beta}}\right](\lambda=1/g_{j})=\delta_{\alpha\beta}\delta_{\bar{\alpha}\bar{\beta}}\\\
&-\mathrm{i}g_{j}\Big{[}\delta_{\alpha\bar{\beta}}\delta_{\beta\bar{\alpha}}-\epsilon_{\alpha}\epsilon_{\beta}\delta_{\alpha,2k+1-\beta}\delta_{\bar{\alpha},2k+1-\bar{\beta}}\Big{]}+O(g_{j}^{2}).\end{split}$
(11)

Equipped with the solution for the two-body scattering matrix, we can solve
the many-body Schrödinger equation for particles on a ring of length
$L=Na_{0}$ with periodic boundary conditions:

$e^{\mathrm{i}k_{j}L}\Psi=S(v_{j},v_{1})S(v_{j},v_{2})\cdots
S(v_{j},v_{N_{R}+N_{L}})\Psi,$ (12)

where $j=1,\ldots,N_{R}$, $\Psi$ is a suitably chosen component of the
electron wave function, $N_{R}=N_{e}$ is the number of electrons,
$N_{L}=N_{s}$ is the number of impurity spins, and $v=1,0$ for electrons and
impurity spins respectively. The corresponding energy is
$E=\sum_{j=1}^{N_{R}}k_{j}$. Using the general classification of Eq. (12) for
all simple Lie algebras given by Reshetikhin and Wiegmann Ogievetsky et al.
(1987); Reshetikhin and Weigmann (1987); Reshetikhin (1985), we arrive at the
following Bethe ansatz equations for $\mathfrak{g}=\mathfrak{sp}(2k)$ with
$k=2$:

$\begin{split}\left[e_{2}\left(\lambda_{a}^{(1)}\right)\right]^{N_{e}}\prod_{j=1}^{N_{s}}e_{q}\left(\lambda_{a}^{(1)}+1/g_{j}\right)&=\prod_{\begin{subarray}{c}b=1,b\neq
a\end{subarray}}^{M_{1}}e_{2}\left(\lambda_{a}^{(1)}-\lambda_{b}^{(1)}\right)\prod_{c=1}^{M_{2}}e_{-2}\left(\lambda_{a}^{(1)}-\lambda_{c}^{(2)}\right)\\\
1&=\prod_{\begin{subarray}{c}b=1,b\neq
a\end{subarray}}^{M_{2}}e_{4}\left(\lambda_{a}^{(2)}-\lambda_{b}^{(2)}\right)\prod_{c=1}^{M_{1}}e_{-2}\left(\lambda_{a}^{(2)}-\lambda_{c}^{(1)}\right)\end{split}$
(13)

and with $k\geqslant 3$:

$\begin{split}\left[e_{2}\left(\lambda_{a}^{(1)}\right)\right]^{N_{e}}\prod_{j=1}^{N_{s}}e_{q}\left(\lambda_{a}^{(1)}+1/g_{j}\right)&=\prod_{\begin{subarray}{c}b=1,b\neq
a\end{subarray}}^{M_{1}}e_{2}\left(\lambda_{a}^{(1)}-\lambda_{b}^{(1)}\right)\prod_{c=1}^{M_{2}}e_{-1}\left(\lambda_{a}^{(1)}-\lambda_{c}^{(2)}\right),\\\
1&=\prod_{\begin{subarray}{c}b=1,b\neq
a\end{subarray}}^{M_{i}}e_{2}\left(\lambda_{a}^{(i)}-\lambda_{b}^{(i)}\right)\prod_{c=1}^{M_{i-1}}e_{-1}\left(\lambda_{a}^{(i)}-\lambda_{c}^{(i-1)}\right)\prod_{c=1}^{M_{i+1}}e_{-1}\left(\lambda_{a}^{(i)}-\lambda_{c}^{(i+1)}\right),\\\
1&=\prod_{\begin{subarray}{c}b=1,b\neq
a\end{subarray}}^{M_{k-1}}e_{2}\left(\lambda_{a}^{(k-1)}-\lambda_{b}^{(k-1)}\right)\prod_{c=1}^{M_{k-2}}e_{-1}\left(\lambda_{a}^{(k-1)}-\lambda_{c}^{(k-2)}\right)\prod_{c=1}^{M_{k}}e_{-2}\left(\lambda_{a}^{(k-1)}-\lambda_{c}^{(k)}\right),\\\
1&=\prod_{\begin{subarray}{c}b=1,b\neq
a\end{subarray}}^{M_{k}}e_{4}\left(\lambda_{a}^{(k)}-\lambda_{b}^{(k)}\right)\prod_{c=1}^{M_{k-1}}e_{-2}\left(\lambda_{a}^{(k)}-\lambda_{c}^{(k-1)}\right),\end{split}$
(14)

where the involved notations are explained as follows. The index
$i=1,\ldots,k$ enumerates the simple roots of the Lie algebra
$\mathfrak{g}=\mathfrak{sp}(2k)$, including the long roots $i=1,\ldots,k-1$
and the short root $i=k$, as shown on the Dynkin diagram in Fig. 3.

Figure 3: The Dynkin diagram for $\mathfrak{sp}(2k)$. The filled dots
represent the long roots, the open dot represents the short root. For $k=2$,
we only have one filled dot and one open dot connected by a double line.

The parameters $\lambda^{(i)}_{a}$ are called rapidities and $M_{i}$ is the
number of rapidities corresponding to the $i$-th root. Each $\lambda^{(i)}$
interacts with the set of rapidities $\lambda^{(i^{\prime})}$ of the same root
$i^{\prime}=i$ and of adjacent roots $i^{\prime}=i\pm 1$, as shown on the
Dynkin diagram in Fig. 3. The function $e_{n}(x)$ is defined as

$e_{n}(x)=\frac{x+\mathrm{i}n/2}{x-\mathrm{i}n/2}$ (15)

and encodes the information of the scattering matrix. The subscript $n$ for
adjacent roots is determined according to the lines of the Dynkin diagram in
Fig. 3: $n=-1$ if the roots are connected by a single line and $n=-2$ if they
are connected by a double line. The subscript for the same root is positive
and determined by its length: $n=2$ for the long roots and $n=4$ for the short
root. The driving term on the left hand side of the first equation in (13) and
(14) includes two factors, where the first factor has subscript $2$. If the
subscript of the second factor $q=2$, we have the usual Fermi liquid ground
state. Here we have $q=1$ instead, which corresponds to the quantum critical
ground state. The position of the driving term among the Bethe ansatz
equations (labelled by $m$) is determined by the condition $l=2(m-1)+q$, where
$l$ is the occupation number of the orbital degree of freedom. Since we have a
single orbital for each species of fermions, we set $l=1$, leading to $m=1$,
thus the driving term is placed on the left hand side of the first Bethe
ansatz equation.

Notably, the above formulae are written for the general case where impurities
have different couplings. The net result of this anisotropy is that instead of
a single Kondo temperature we will have a certain distribution which will not
affect the low energy behavior Lotem et al. (2023).

##  IV Low Energy Properties of the Chiral Kondo Chain Model

From the viewpoint of thermodynamics, the difference between the case with a
single impurity and the case with a multi-impurity chain is minimal: the
effect of impurities is additive, and the corresponding part of the free
energy for the chain is just multiplied by the factor $N_{L}=N_{s}$. The exact
solution provides us with an additional handle to manipulate the system: we do
not need to restrict ourselves to the quantum critical point, as was done in
literature Lopes et al. (2020); Gabay et al. (2022). However, the main result
is in agreement with these papers. Specifically, the ground state entropy is
$S(T=0)=N_{s}\ln d_{k}$, where

$d_{k}=2\cos\left(\frac{\pi}{k+2}\right),$ (16)

is the anyonic quantum dimension Affleck and Ludwig (1991a). This indicates
that in the low energy limit the symplectic chiral Kondo chain can be
described as a set of anyons located at the impurity sites. Below we will
first summarize the main results for a single impurity obtained from the exact
solution. Then we will discuss the connection to the conformal field theory
(CFT) approach and identify the operator content at the quantum critical point
(QCP) for further developments.

###  IV.1 Thermodynamics from Exact Solution

We summarize the main results for a single impurity obtained from the exact
solution König and Tsvelik (2023). The generalization to the chiral Kondo
chain is simply an additive effect. In the absence of external magnetic
fields, the free energy is

$\begin{split}F\stackrel{{\scriptstyle T\ll T_{K}}}{{\simeq}}&-\frac{\pi
c}{12}\frac{T^{2}}{E_{F}}+\frac{1}{N_{e}}\left[-T\ln\left(d_{k}\right)-\frac{\pi
c_{k}}{12}\frac{T^{2}}{T_{K}}-\cdots\right.\\\ &\left.-\text{const.}\times
T\left(\frac{T}{T_{K}}\right)^{k+1}\ln\left(T_{K}/T\right)\right]\end{split}$
(17)

where $T_{K}\sim E_{F}\exp\left(-\frac{\pi}{(k+1)g}\right)$ is the Kondo
temperature, $c_{k}=c/\left[2\cos\left(\frac{\pi}{2k+2}\right)\right]$ with
$c$ the central charge for $Sp_{1}(2k)$ theory in Eq. (6). The impurity free
energy in the square bracket contains a zero temperature entropy consistent
with the anyonic quantum dimension in Eq. (16). This ground state degeneracy
is the same as that for the $k$-channel $SU(2)$ Kondo effect Lopes et al.
(2020). However, different from the $k$-channel $SU(2)$ Kondo effect, the
leading finite temperature corrections are Fermi liquid like. A similar
behavior is also true for the zero temperature impurity magnetization in the
presence of a small local Zeeman field $H_{i}\ll T_{K}$, which takes the form

$\begin{split}M_{i}=&\alpha_{1}H_{i}/T_{K}+\alpha_{3}\left(H_{i}/T_{K}\right)^{3}+\cdots\\\
&+\beta\left(H_{i}/T_{K}\right)^{k+1}\times\begin{cases}1,&k\text{ odd }\\\
\ln\left(T_{K}/H_{i}\right),&k\text{ even }\end{cases}\end{split}$ (18)

where $\alpha_{1,3,\ldots}$ and $\beta$ are dimensionless constants, and
$i=1,\ldots,k$ label the quantum dots. The fact that the leading order
corrections for low energy thermodynamics are Fermi liquid like has important
implications on the operator content at the QCP, which we will discuss below.

###  IV.2 The CFT Analysis

The CFT analysis of the $Sp_{1}(2k)$ theory for a general $k$ is contained in
Ref. Kimura, 2021, and the special case of $k=2$ ($Sp(4)\simeq SO(5)$) is
discussed in Ref. Liberman et al., 2021; Mitchell et al., 2021 via the Abelian
bosonization technique. Here we will summarize the main points and make
connection to the results obtained from exact solutions.

The $Sp_{1}(2k)$ WZNW theory has $k$ primary fields $\Phi_{a}$ with scaling
dimensions Bouwknegt and Schoutens (1999)

$h_{a}=\frac{a(2k+2-a)}{4(k+2)},\quad a=1,\ldots,k,$ (19)

transforming according to the representations $\Lambda_{a}$ of the Lie algebra
$\mathfrak{sp}(2k)$ with dimensionalities

$\operatorname{dim}\Lambda_{1}=2k,\quad\operatorname{dim}\Lambda_{a}=C_{2k}^{a}-C_{2k}^{a-2},\quad
a\geqslant 2,$ (20)

where $C^{a}_{2k}=(2k)!/[a!(2k-a)!]$ is the binomial coefficient. The fusion
rules for $Sp_{1}(2k)$ primaries can be expressed as Walton (1990); Cummins
(1991)

$\begin{split}a\otimes
a^{\prime}=&|a-a^{\prime}|,~{}|a-a^{\prime}|+2,~{}\ldots,\\\
&\operatorname{min}\\{a+a^{\prime},2k-a-a^{\prime}\\},\end{split}$ (21)

where $a=0,1,\ldots,k$, including the vacuum. For $k=1$, there is only one
primary with $h=1/4$, which is obviously the $SU(2)$ matrix field with entries
$\exp[\mathrm{i}\sqrt{2\pi}\phi]$, $\exp[\mathrm{i}\sqrt{2\pi}\theta]$, where
$(\phi,\theta)$ is a set of conjugate variables. For $k=2$, there are two
primaries, one with $h_{1}=5/16$ being a product of five Ising model conformal
blocks, and the other with $h_{2}=1/2$ being a Majorana fermion.

To figure out the properties of the QCP, we follow the approach of Ref.
Affleck, 1990; Affleck and Ludwig, 1991b, where the vacuum is redefined by
absorbing the impurity such that the description of the system is simplified.
The WZNW Hamiltonian of the $Sp_{1}(2k)$ theory can be expressed in the
Sugawara form Witten (1984); James et al. (2018):

$\begin{split}&\mathcal{H}_{\text{bulk}}=\frac{2\pi}{k+2}\sum_{n\geq
0}\sum_{A\in\mathfrak{g}}J^{A}_{-n}J^{A}_{n},\\\
&J_{n}^{A}=\frac{1}{2\pi}\int_{-L/2}^{L/2}\mathrm{d}x~{}J^{A}(x)e^{2\mathrm{i}\pi
xn/L},\end{split}$ (22)

where the current operator satisfies the following Kac-Moody algebra:

$[J^{A}_{n},J_{m}^{B}]=\mathrm{i}f^{ABC}J^{C}_{n+m}+\frac{1}{2}n\delta_{AB}\delta_{n+m,0}.$
(23)

Then the Kondo term can be absorbed into the Hamiltonian through the
redefinition of the currents Affleck and Ludwig (1991b); Affleck (1990):

${\cal J}_{n}^{A}=J^{A}_{n}+(k/2+1)gS^{A},$ (24)

such that we can “complete the square” in the expression of the Hamiltonian
Affleck (1990). The QCP is obtained by requiring that the Kac-Moody algebra of
Eq. (23) remains unchanged, and this is achieved at $g(k/2+1)=1$. This
absorption leads to the redefinition of the vacuum through the fusion with the
impurity spin, the spin being a $2k\times 2k$ matrix corresponding to the
primary field $\Phi_{1}$. This is known as the “fusion ansatz”, which we
briefly review in Appendix A. As a result of such fusion the impurity spin
operators experience “transmutation” - at the critical point they are
expressed in terms of the bulk fields located at the impurity sites. Our goal
now is to establish this correspondence for the symplectic symmetry.

The distinct feature of the $Sp_{1}(2k)$ theory is the high scaling dimensions
of the irrelevant operators perturbing the QCP Li et al. (2023). This is
consistent with the exact solution results in Eq. (17) and (18), where the
non-analytic terms appear in high orders of the expansion in $T/T_{K}$ and
$H_{i}/T_{K}$. This is unlike the $SU(2)$ Kondo effect and, as we will show
later, constitutes an advantage of the symplectic Kondo effect as a platform
for quantum information applications.

To elaborate on this point, we consider the case $k=2$ where the CFT consists
of five Majorana fermions ($Sp_{1}(4)$ = $O_{1}(5)$) and the two primaries are
a product of five Ising model conformal blocks and a Majorana fermion. The
ground state entropy suggests the existence of a Majorana zero mode. From five
Majorana fermions $\chi_{1},\ldots,\chi_{5}$ and one Majorana zero mode $\xi$,
we can construct the perturbation Liberman et al. (2021)

$\delta\mathcal{H}\sim\xi\chi_{1}(0)\cdots\chi_{5}(0),$ (25)

which generates the non-analytic correction to the specific heat in Eq. (17):

$\delta C_{V}\sim(T^{3}/T_{K}^{2})\ln(T_{K}/T).$ (26)

From Eq. (18) we can see that the magnetic field $H_{i}$ is coupled to the
operator with scaling dimension $\Delta=3/2$ Liberman et al. (2021); Li et al.
(2023), which generates the non-analytic correction to the magnetization

$\delta M_{i}\sim(H_{i}/T_{K})^{3}\ln(T_{K}/H_{i}).$ (27)

The correction in Eq. (27) suggests that for $k=2$ the non-analytic
contribution to the impurity spin comes from the operator containing three
Majorana fermions, that is from a descendant of the primary field $\Phi_{2}$.
The $k=2$ case is particularly illuminating, which provides a clear
demonstration of the stability of the QCP with respect to the anisotropy of
the Kondo exchange interaction (see later in Eq. (34)) . Indeed, as long as
the interaction includes only $Sp_{1}(4)$ Kac-Moody currents, we still have a
theory of odd number of Majoranas which is bound to be a non-Fermi liquid one.

We generalize this conjecture for general $k$ (cf. Lopes et al., 2020 for the
case $SU(2)_{k}$) and suggest that the transmuted impurity spin is given by
the sum of two operators and their descendants - the $Sp_{1}(2k)$ current
which generates the Fermi liquid like corrections and the descendant of the
primary field $\Phi_{2}$ which generates the non-analytic corrections:

$S_{\text{imp},j}=J(x_{j})+\Gamma_{j}J_{-1}\Phi_{2}(x_{j})+\cdots,$ (28)

where the dots stand for the descendants, and we introduce the impurity site
label $j$ for the generalization to the multi-impurity case. The second term
is a product of an operator $\Gamma$ acting on the anyonic Hilbert space and a
descendant of the primary field $\Phi_{2}$ with scaling dimension $h_{2}$. The
primary field $\Phi_{2}$ is chiral and has a noninteger scaling dimension. As
a result, the pair correlation function of $J_{-1}\Phi_{2}$ at the same site
behaves as $\tau^{-2-2h_{2}}$. However, the correlation function of the
physical spin must be even in time, which fixes the time dependence of the
correlation function of $\Gamma$:

$\begin{split}&\left<\Gamma_{j}(\tau)\Gamma_{j}(0)\right>\sim e^{\mathrm{i}\pi
h_{2}\operatorname{sgn}\tau}\\\
\Rightarrow~{}&\left<S^{A}_{\text{imp},j}(\tau)S^{B}_{\text{imp},j}(0)\right>\sim\delta^{AB}\left(\frac{1}{|\tau|^{2}}+\frac{\Xi}{|\tau|^{2+2h_{2}}}\right),\end{split}$
(29)

where $\Xi$ is a constant coefficient. On the other hand, the anyonic property
of the operator $\Gamma$ is determined by its fusion rules, and the latter
ones point to its correspondence to the primary field $\Phi_{1}$ in the fusion
ansatz. To get an insight we consider two relevant cases $k=2$ and $k=3$,
using the $Sp_{1}(2k)$ fusion rules in Eq. (21). For $k=2$, we have $\Phi_{a}$
with $a=0,1,2$ and the fusion rules for them are

$0\times a=a,~{}1\times 1=0+2,~{}1\times 2=1,~{}2\times 2=0.$ (30)

Therefore we have the following identification with the system of Ising anyon
Nayak et al. (2008):

$0\to\mathbf{1},\quad 1\to\sigma,\quad 2\to\psi,$ (31)

and $\Gamma$ plays the role of $\sigma$, the Ising anyon. This is also in
agreement with the anyonic quantum dimension $d_{2}=\sqrt{2}$ from the ground
state entropy. For $k=3$, we have $\Phi_{a}$ with $a=0,1,2,3$ and the fusion
rules for them are

$0\times a=a,~{}3\times a=(3-a),~{}1\times 1=2\times 2=0+2,~{}1\times 2=1+3.$
(32)

Here the set of rules $3\times a=(3-a)$ plays the special role of the
automorphism of the fusion algebra, which maps $3$ to $0$ and $1$ to $2$, and
we have the following identification with the system of Fibonacci anyon Trebst
et al. (2008); Génetay Johansen and Simula (2021):

$0,3\to\bm{1},\quad 1,2\to\tau.$ (33)

Consequently, $\Gamma$ plays the role of $\tau$, the Fibonacci anyon We note
that $\Phi_{1}$ and $\Phi_{2}$ are primary fields, while $\Gamma$ is the
Fibonacci anyon. They share the same fusion rules but they are not identified
with each other., whose properties are briefly reviewed in Appendix B. Again,
this is in agreement with the anyonic quantum dimension $d_{3}=(1+\sqrt{5})/2$
from the ground state entropy.

Equation (28) is also instrumental for the proof of the stability of the QCP
with respect to the anisotropy of the Kondo exchange interaction, which
generalizes the argument given above for the case $k=2$. Indeed, if we add a
perturbation

$\delta\mathcal{H}=\sum_{j}\delta g_{AB}\cdot J^{A}(x_{j})S^{B}(x_{j}),$ (34)

then, after “completing the square” and replacing $S^{A}(x_{j})$ by Eq. (28)
at criticality, we find that the leading term of Eq. (34) is marginal and the
coupling to the anyons is irrelevant. This suggests that the QCP is robust
with respect to the anisotropy of the Kondo exchange interaction[sela].

##  V Extraction of Information from the Correlation Functions

The anyons residing at the impurity sites, which are supported by the chiral
Kondo chain model with symplectic symmetry, can be utilized for measurement-
only quantum computations Bonderson et al. (2008a). The essential ingredients
of the measurement-only quantum computation protocol are briefly reviewed in
Appendix C, where we can see that the building blocks of the protocol are the
projective measurements of the fusion channel of anyons. Such projective
measurements can be carried out by measuring the correlation functions of the
chiral Kondo chain model. The general theory for such correlation functions is
described in Ref. Gabay et al., 2022 and in this section we specialize it to
the symplectic case that we have been considering.

Information of the total fusion channel of all anyons between two spatial
points can be extracted from the correlation function of the $Sp_{1}(2k)$
primary fields, the latter can be obtained under the multi-fusion ansatz Gabay
et al. (2022) and is briefly reviewed in Appendix A. On the other hand, for
the purpose of quantum computations, what is needed is the fusion channel of
two arbitrary anyons, which can be extracted from the pair correlation
function of the impurity spins. We have conjectured that the impurity spin
immersed in the chiral fermions gets decorated by an operator $\Gamma_{j}$
acting on the anyonic Hilbert space, such that their correlation encodes the
information of the fusion channel of the two anyons localized on the two
impurity sites. In case of the $Sp_{1}(2k)$ theory, the conjecture takes the
form in Eq. (28), so for $t_{i}>t_{j}$ we have

$\begin{split}&\left\langle
S^{A}_{\text{imp},i}(t_{i})S_{\text{imp},j}^{B}(t_{j})\right\rangle_{a_{ij}}\sim\\\
&\delta^{AB}\left\\{\frac{1}{(t_{ij}-x_{ij}+\mathrm{i}0)^{2}}+\frac{\mathcal{F}_{k}\left(a_{ij}\right)}{\left(t_{ij}-x_{ij}+\mathrm{i}0\right)^{2+\frac{2k}{k+2}}}\right\\},\end{split}$
(35)

where $t_{ij}=t_{i}-t_{j},x_{ij}=x_{i}-x_{j}$, $a_{ij}$ denotes the fusion
channel of the two anyons, and $\mathcal{F}_{k}(a_{ij})$ encodes the
dependence on $a_{ij}$. Here the subscript $i,j$ label the impurity sites. The
task remains is to find the expression for $\mathcal{F}_{k}(a_{ij})$, or at
least, to show the dependence of $\mathcal{F}_{k}(a_{ij})$ on $a_{ij}$.

We here look at two relevant cases $k=2$ and $k=3$ for illustrations. For
$k=2$, $\Phi_{2}$ has scaling dimension $1/2$ and corresponds to the Majorana
fermion. Consequently, $\Gamma_{i}$ corresponds to the Majorana zero mode
$\xi_{i}$, with the commutation relations of the Clifford algebra:

$\\{\xi_{i},\xi_{j}\\}=\delta_{ij},~{}~{}\xi^{2}_{i}=1/2.$ (36)

On the other hand, we have identified $\Gamma$ with the Ising anyon in Sec.
IV.2, from the fusion rules of $\Phi_{1}$. The two conclusions are in indeed
consistent, where the Majorana zero mode is identified with the Ising anyon,
and the bi-anyonic fusion channels can be constructed as

$\begin{split}&f_{ij}=\frac{\xi_{i}+\mathrm{i}\xi_{j}}{\sqrt{2}},~{}f^{\dagger}_{ij}=\frac{\xi_{i}-\mathrm{i}\xi_{j}}{\sqrt{2}}~{}\Rightarrow~{}\mathrm{i}\xi_{i}\xi_{j}=f^{\dagger}_{ij}f_{ij}-1/2;\\\
&\bm{1}\to\ket{0},\quad\psi\to f^{\dagger}_{ij}\ket{0}.\end{split}$ (37)

We can see that $2\mathrm{i}\xi_{i}\xi_{j}=\mathcal{P}_{ij}$ is the parity
operator, and the fusion channels
$\ket{a_{ij}}=\ket{0},\ket{a_{ij}}=f^{\dagger}_{ij}\ket{0}$ are its
eigenstates with eigenvalues $-1$ and $+1$ respectively. Hence a measurement
of the impurity spin correlation function of Eq. (35) is indeed a projective
measurement of the bi-anyonic fusion channel, with
$\mathcal{F}_{2}(\bm{1})\sim\mathrm{i}$ and
$\mathcal{F}_{2}(\psi)\sim-\mathrm{i}$. The essence is that
$\mathcal{F}_{2}(\bm{1})$ and $\mathcal{F}_{2}(\psi)$ differ by a sign.

Alternatively, we infer the behaviors of $\mathcal{F}_{k}(a_{ij})$ in another
way that can be generalized to $k>2$. Since $\Gamma$ share the same fusion
rule with $\Phi_{1}$, it is natural to assume that their correlation functions
share the same dependence on the fusion channel. The correlation functions of
$\Phi_{1}$ with two impurity spins in between is obtained in Appendix A as

$\left<\Phi_{1}(z_{i})\Phi_{1}(z_{j})\right>\sim\frac{1}{(z_{i}-z_{j})^{2h_{1}}}\frac{S^{1}_{c_{ij}}/S^{1}_{0}}{S^{0}_{c_{ij}}/S^{0}_{0}},$
(38)

where $z\equiv\tau-\mathrm{i}x$ is the complex coordinate, and $c_{ij}$ takes
the value 0 or 2, which comes from the fusion rule $1\times 1=0+2$. The
modular $S$-matrix element $S^{a^{\prime}}_{a}$ for the $Sp_{1}(2k)$ theory is
given by

$S_{a}^{a^{\prime}}=\sqrt{\frac{2}{2+k}}\sin\left[\frac{\pi(a+1)\left(a^{\prime}+1\right)}{2+k}\right].$
(39)

Consequently, we can extract from Eq. (38) the fusion channel-dependent part,
and assign it to $\mathcal{F}_{k}(a_{ij})$:

$\mathcal{F}_{k}(a_{ij})=\frac{S^{1}_{c_{ij}}/S^{1}_{0}}{S^{0}_{c_{ij}}/S^{0}_{0}},$
(40)

where the $\mathcal{F}_{k}(a_{ij})$ is normalized such that
$\mathcal{F}_{k}(\bm{1})=1$ ($c_{ij}=0\to\bm{1}$, the vacuum). For $k=2$, we
have the fusion channels $c_{ij}=0\to a_{ij}=\bm{1}$ and $c_{ij}=2\to
a_{ij}=\psi$, as can be seen from Eq. (31), then Eq. (40) gives us

$\mathcal{F}_{2}(\bm{1})=1,\quad\mathcal{F}_{2}(\psi)=-1.$ (41)

This is essentially the same as that obtained from the Majorana zero mode
algebras, where $\mathcal{F}_{2}(\bm{1})$ and $\mathcal{F}_{2}(\psi)$ differ
by a sign.

For $k=3$, $\Phi_{2}$ has scaling dimension $3/5$ and we do not know a priori
its zero mode $\Gamma$. However, we have identified $\Gamma$ with the
Fibonacci anyon $\tau$ in Sec. IV.2, from the fusion rules of $\Phi_{1}$. We
are not able to calculate the correlation function of the Fibonacci anyons in
case of coincident positions, so we resort to Eq. (40). By identifying
$c_{ij}=0\to a_{ij}=\bm{1}$ and $c_{ij}=2\to a_{ij}=\tau$ from Eq. (33), we
obtain

$\mathcal{F}_{3}(\bm{1})=1,\quad\mathcal{F}_{3}(\tau)=\frac{1}{2}(-3+\sqrt{5}).$
(42)

Different from the case of $k=2$, now $\mathcal{F}_{3}(\bm{1})$ and
$\mathcal{F}_{3}(\tau)$ differ by sign and modulus, which should be easier to
detect experimentally.

A possible measurement of the correlation function of the impurity spins can
be done by locally coupling the two impurity spins to an extra single channel
of chiral current with switches, as shown in Fig. 4, and measuring the
corresponding conductance at finite frequency.

Figure 4: Measurement of the impurity spin (denoted as the star in the figure)
correlation function through an extra single channel of chiral current with
switches.

Different from the channels responsible for creating the degenerate ground
states, this extra single channel is controlled by the switches such that each
time it couples only to those two impurity spins, whose correlation function
is to be measured. Later in Sec. VI.3, we will argue that this extra current
does not effect the original anyonic structure of the system.

To couple to the impurity spins, the extra current must belong to the
$\mathfrak{sp}(2k)$ Kac-Moody algebra. This can be achieved by various means,
for instance, one can use spin polarized current. The following discussion
does not depend on the specific arrangement, we just assume that the extra
current transforms as some generator of the $\mathfrak{sp}(2k)$ algebra and
hence by symmetry can be coupled to at least one projection of the impurity
spins, say $A$. Then the Hamiltonian at criticality for the whole system
including the extra current is

$\begin{split}&\mathcal{H}_{\text{tot}}=\mathcal{H}+\pi\int\mathrm{d}x~{}J^{\text{(ext)}}(x)J^{\text{(ext)}}(x)+\lambda\delta\mathcal{H}_{ij},\\\
&\delta\mathcal{H}_{ij}=J^{(\text{ext})}(x_{i})\cdot
S^{A}_{\text{imp},i}+J^{(\text{ext})}(x_{j})\cdot
S^{A}_{\text{imp},j},\end{split}$ (43)

where $\mathcal{H}$ is the Hamiltonian for the original system, and $\lambda$
is the coupling constant. The corresponding current operator is determined by

$\begin{split}&I^{(\text{ext})}=\frac{\mathrm{d}N^{(\text{ext})}}{\mathrm{d}t}=-\mathrm{i}[N^{(\text{ext})},\mathcal{H}_{\text{tot}}],\\\
&N^{(\text{ext})}=\int^{x_{0}}\mathrm{d}x~{}J^{\text{(ext)}}(x).\end{split}$
(44)

Using the chiral anomaly

$[J^{\text{(ext)}}(x),J^{\text{(ext)}}(x^{\prime})]=\frac{\mathrm{i}}{2\pi}\delta^{\prime}(x-x^{\prime}),$
(45)

the current operator can be expressed as

$\begin{split}I^{(\text{ext})}(x_{0})=J^{(\text{ext})}(x_{0})+\frac{\lambda}{2\pi}&\left[\delta(x-x_{i})S^{A}_{\text{imp},i}\right.\\\
&\left.+\delta(x-x_{j})S^{A}_{\text{imp},j}\right].\end{split}$ (46)

Then the conductance is related by the Kubo formula to the correlation
function of this particular spin component placed at impurity site $i$ and
$j$. The part of the finite frequency conductance that depends on the fusion
channel of the anyons is

$\delta
G_{ij}(\omega)\propto\lambda^{2}\omega^{\frac{2k}{2+k}}\mathcal{F}_{k}(a_{ij})\cos(\omega
x_{ij}),$ (47)

where $x_{ij}=x_{i}-x_{j}$. Hence by measuring the conductance at finite
frequency, one can effectively measure the bi-anyonic fusion channel. This
idea lies at the foundation of our proposal for the topological quantum
computation discussed in detail in Sec. VI. Here we have only showed that
$G_{ij}(\omega)$ will take different values for different fusion channels,
without pinning down the exact values. This is enough from a pragmatic
viewpoint - the association of experimentally measured $G_{ij}(\omega)$ with
the specific fusion channel will be individually determined through the
calibration process in actual implementations.

##  VI Topological Quantum Computing Protocol

Currently, there are four main different approaches to the quantum computing
protocols: the quantum circuit model Nielsen and Chuang (2010), the
measurement-based quantum computation Raussendorf and Briegel (2001);
Raussendorf et al. (2003), the adiabatic quantum computation Farhi et al.
(2000); Das and Chakrabarti (2008), and the topological quantum computation
Kitaev (2003); Nayak et al. (2008); Pachos (2012); Lahtinen and Pachos (2017).
Compared with the other approaches, the topological quantum computation has
the advantage of building the fault-tolerance already on the level of
hardware, utilizing the anyonic ground state manifold with topological
degeneracy Kitaev (2006). Conventionally, the topological quantum computation
is performed by adiabatic braiding of anyons, which is prone to uncontrollable
error channels such as accidental braiding with unwanted anyons and thermal
excitation of unaccounted anyons. However, these flaws can be mitigated by
combining the topological quantum computation with measurement-only quantum
computation Bonderson et al. (2008a), where the adiabatic braiding of anyons
is replaced by repeated projective measurements of the bi-anyonic fusion
channels of anyons. Such a protocol is particularly suitable for the quantum
computer based on the chiral Kondo chain model considered here, since the
anyons are localized at the impurity sites, which are neither mobile nor
excitable.

In the previous sections, we have shown that the chiral Kondo chain model with
$Sp(2k)$ symmetry possesses non-Abelian anyons in its ground state. For $k=2$,
we have Ising anyons, and for $k=3$, we have Fibonacci anyons. In the
following discussion we will focus on the case $k=3$ with Fibonacci anyons
which support universal quantum computations Nayak et al. (2008). As the first
step we will propose the microarchitecture of the quantum computer based on
the 3-channel chiral Kondo chain, and then consider the choice of materials
for its realization. Finally we will discuss the sources and effects of errors
on the proposed quantum computer.

###  VI.1 Microarchitecture of the Quantum Computer

Let us summarize our results. The Fibonacci anyons created by coupling the
superconducting islands to three (pseudo)spin degenerate chiral channels are
localized on the superconducting islands. At criticality, these Fibonacci
anyons are decoupled from the rest of the system. We suggest to use these
immobile anyons for topological quantum computation through the measurement-
only quantum computing protocol Bonderson et al. (2008a). The essential idea
is that the physical transport of an anyon can be replaced by the quantum
teleportation of the information encoded in the ground state, and such quantum
teleportation can be realized by repeated projective measurements of the bi-
anyonic fusion channel among two given anyons and an extra auxiliary anyon. To
promote the quantum teleportation between two anyons into the braiding of
them, we require a further extra auxiliary anyon. The corresponding details
are reviewed in Appendix C. Consequently, the realization of the braiding
requires a total number of four anyons, two of which are auxiliary. If the
measurements of the bi-anyonic fusion channel can be done nondemolitionally,
as we argue below, the equivalent adiabatic braiding can be performed in a
perfect way, without accidental braiding of unwanted anyons and without
thermal excitation of unaccounted anyons. In our proposal, all anyons are
localized at the impurity sites, they are neither mobile nor excitable. The
measurements of the bi-anyonic fusion channel are done through local selective
coupling to an extra single channel of chiral current, which is non-
topological. However, as will be discussed in Sec. VI.3, such a measurement is
an irrelevant perturbation, which cannot alter the type of anyons the host
system supports. Consequently, such a measurement is indeed nondemolitional.

The suggested microarchitecture of the quantum computer is schematically shown
in Fig. 5. The grey slabs are the superconducting islands that localize anyons
intended as the computational units, while the blue slabs are the
superconducting islands that localize anyons intended as the auxiliary units.
The Fibonacci anyons are created by coupling to the (pseudo)spin-degenerate
chiral modes. There are three chiral modes, and the topological material
responsible for each of them are shown in orange, green and red in the figure.
The measurements of the bi-anyonic fusion channels are controlled by the
switches coupling to yet another chiral mode, shown as the solid black line in
the figure, and the measurement apparatus is shown in blue in the figure. The
microarchitecture is design in a way such that it can be scaled up two-
dimensionally, although it is essentially equivalent to a one-dimensional
array of superconducting islands coupled by the same set of chiral modes.

Figure 5: The illustration of the microarchitecture of the quantum computer.
The detailed description can be found in the main text.

On top of the microarchitecture shown in Fig. 5, the measurement of the fusion
channel of two arbitrary anyons can be performed by closing the corresponding
switches and decoding the signal. This is possible due to the fact that the
signal has features that depend on the fusion channel, as indicated by Eq.
(35) and (47). As a result, we need a calibration process performed on every
two-unit combination of the microarchitecture components, such that the
decoding algorithm for interpreting the signal can be determined. Afterwards,
the braiding of two consecutive computational anyons can be performed by the
non-deterministic sequence of repeated measurements reviewed in Appendix C.
The braidings of the Fibonacci anyons can then be used to efficiently
approximate arbitrary unitary quantum gates Nayak et al. (2008); Kitaev
(1997); Dawson and Nielsen (2006), as reviewed in Appendix B. The design of
the implementation of such approximations up to arbitrarily given precision is
called quantum compiling Maronese et al. (2022), and there is a well-
established protocol for Fibonacci anyons Rouabah et al. (2021). In
conclusion, the quantum computation can be performed on the quantum computer
as designed in Fig. 5 by programming the switches. Since the switches are
simply classical bits, they can be controlled by a classical computer. This
consists of a two-layer design of the quantum computer, where the quantum
algorithms are executed by manipulating the classical bits, although in a non-
deterministic way depending on the quantum output throughout the process.

###  VI.2 Choice of Materials

It appears that the main challenge here will be to find a proper material that
provides the two-fold degenerate chiral modes. We are aware of two classes of
suitable materials, as discussed below.

Firstly, there are topological insulators doped with magnetic ions. The theory
suggests Bi2Te3, Bi2Se3, and Sb2Te3 doped with Cr Wang et al. (2013), and the
experiment also suggests MnBi2Te4 Ge et al. (2020). The low-energy bands of
these materials consist of a bonding and an antibonding state of $p_{z}$
orbitals, labeled by $\ket{P2^{-}_{z},\sigma}$ and $\ket{P1^{+}_{z},\sigma}$,
respectively. The generic form of the effective Hamiltonian describing these
four bands is

$\begin{split}&\mathcal{H}_{3D}=\epsilon(k)I\otimes I+k_{x}\sigma_{x}\otimes
I+k_{y}\sigma_{y}\otimes\tau_{z}\\\
&\quad\quad+Ak_{z}\sigma_{y}\otimes\tau_{y}+M(k)\sigma_{z}\otimes
I-\Delta\sigma_{z}\otimes\tau_{z},\\\
&M(k)=B_{0}+B_{1}k_{z}^{2}+B_{2}(k_{x}^{2}+k_{y}^{2}),\\\
&\epsilon(k)=D_{0}+D_{1}k_{z}^{2}+D_{2}(k_{x}^{2}+k_{y}^{2}),\end{split}$ (48)

where $A,B_{0},B_{1},B_{2},D_{0},D_{1},D_{2},\Delta$ are all constant
coefficients. Here $\sigma$ matrices act within the states of the same parity
and $\tau$ matrices act on parity indices. According to Ref. Wang et al.,
2013, at sufficiently large $\Delta$ which originates from doping with
magnetic Cr, one will have $C=2$ Chern insulator. In this case the edge states
have the same parity, for instance these may be only
$\ket{P2^{-}_{z},\uparrow}$ and $\ket{P1^{+}_{z},\downarrow}$. It is important
for us that these states are not spin polarized and hence states with both
spins can tunnel into the quantum dot. $C=2$ state was observed in MnBi2Te4Ge
et al. (2020) in high magnetic field polarizing the spins on Mn. The presence
of such magnetic field may be a problem for the superconducting island.

The second class includes the chiral charge-density-wave (CDW) state of kagome
metals AV3Sb5 (A= K, Rb, Cs). Some of these materials undergo a phase
transition to a 3Q CDW phase with complex bond order hosting loop currents and
plaquette fluxes Feng et al. (2021); Yu et al. (2021). The evidence for such a
phase comes from the experiments detecting time reversal symmetry breaking
without creation of magnetic moments Mielke et al. (2022); Khasanov et al.
(2022) ascribed to loop currents Lin and Nandkishore (2022); Christensen et
al. (2022). Such state will have spin degenerate chiral edge modes. In the
natural state, however, the chemical potential lies slightly above the van
Hove singularity and as a consequence the CDW state is not insulating. Hence
gating will be necessary to enforce the requirements of our model.

###  VI.3 Analysis of Errors

In general, there are two types of errors that should be dealt with carefully
in quantum computations: the random error due to coupling to the environment
and the unitary error due to imprecision in applying quantum gates.
Topological quantum computations using non-Abelian anyons provide hardware-
level protection against both types of errors Nayak et al. (2008). However,
there are still two extra error channels that hinder the actual fault
tolerance. One is the wandering of anyons and the other is thermally excited
anyonic pairs. They can cause braiding with unwanted anyons and appearance of
unaccounted anyons. These difficulties are avoided in the quantum computing
platform based on the chiral Kondo chain. As discussed previously, the
localization of the anyons on the superconducting islands and the decoupling
of the anyons from the rest of the system make the quantum component of the
two-layer design inherently error-free, since the anyons are immobile and non-
excitable. On the other hand, the classical component itself can be made
fault-tolerant following the well-established protocols for the classical
computers. As a result, the remaining error channel lies in the measurement
process, where the computing system is coupled to a measurement circuit (the
black circuit with switches shown in Fig. 5). For each step of the measurement
process, we selectively close two switches and read the signal to interpret
the resulting fusion channel of the two selected anyons. In such a step, the
computing system is coupled to the measurement circuit locally, as shown in
Eq. (43). The scaling dimension of the local perturbation
$\delta\mathcal{H}_{ij}$ in Eq. (43) is larger than 1, thus being irrelevant.
This can be understood alternatively via Zamolodchikov’s $c$-theorem
Zamolodchikov (1986). According to Eq. (6), if we add another chiral edge to a
single superconducting island, we will increase the central charge. Here, the
chiral edge is added locally to two of the superconducting islands, so the
locally enhanced central charge will eventually flow back to the unperturbed
value on the global scale. As a result, the effect of the measurement circuit
on the computing system is only perturbative, which cannot alter the type of
anyons the host system supports, although the associated perturbation itself
is non-topological. This leaves us with only the readout error associated with
the measurement apparatus (the blue circuit shown in Fig. 5), which can be
easily handled by performing projective measurements on the same two anyons
repeatedly.

At last we have to mention another possible source of errors: perturbations
lifting the degeneracy between the quantum dots energy levels. This was
already discussed in Sec. II, but is worth repeating. As we have argued, we
should be concerned only with perturbations which couple to the anyons and
hence may lift the ground state degeneracy. The most likely perturbation shown
in Eq. (7), which is diagonal in the dot indices, does not even couple to the
symplectic spins [sela]. Moreover, the symplectic symmetry greatly weakens the
influence of those perturbations which do couple to the symplectic spins
acting as effective magnetic fields. For the $SU(2)$ symmetry such
perturbations would present a serious challenge, but for the symplectic
symmetry where anyons are weakly coupled to the bulk, they do not. Indeed,
according to Eq. (28) the “magnetic” field $H_{j}$ couples to the anyon zero
mode at impurity site $j$ as

$\delta\mathcal{H}=H_{j}\Gamma_{j}J_{-1}\Phi_{2}(x_{j}).$ (49)

This perturbation is strongly irrelevant and as long as one manages to keep it
much smaller than the Kondo temperature, its influence will be small [sela].
The above discussion suggests that the chiral Kondo chain model with
symplectic symmetry may indeed present a considerable advantage over previous
proposals with unitary symmetry.

##  VII Summary and Outlook

In this paper, we have briefly reviewed the symplectic Kondo effect and
extended it to a chiral Kondo chain model. We have showed that under the
condition of perfect symplectic symmetry the chiral Kondo chain model is
integrable for an arbitrary number of impurities, and possesses non-Abelian
anyons in its ground state. These non-Abelian anyons are localized on the
impurity sites and decoupled from the rest of the system at criticality. We
have further analyzed various perturbations around integrability and concluded
that they do not change the low energy behaviors. The symplectic symmetry is
responsible for several distinct features of the model, favorable for quantum
information applications. It relieves the strict requirement of the $SU(2)$
Kondo effect for the fine tuning of the energy levels and exchange
interactions of different channels. Both the anyon contribution to the
impurity spin in Eq. (28) and the coupling between the anyon and the bulk have
higher scaling dimensions than those for the unitary symmetry, which
drastically reduces the influence of perturbations at criticality. We believe
that these features make the chiral Kondo chain with symplectic symmetry a
favorable candidate for implementation of the measurement-only topological
quantum computing protocol. The essential ingredient of this protocol is the
projective measurement of the bi-anyonic fusion channel. We have shown that
such measurements can be executed by locally coupling an extra single channel
of chiral current to the two anyons and measuring the corresponding
conductance at finite frequency. Besides, we have shown that the measurements
act as irrelevant perturbations to the host system, which cannot alter the
type of anyons the host system supports. We have proposed a two-layer
microarchitecture of the quantum computer based on the chiral Kondo chain
model with symplectic symmetry, where the quantum algorithms can be executed
by manipulating the classical switches in a non-deterministic way. We have
also discussed the sources and effects of errors on the proposed quantum
computer, and argued that the design is intrinsically fault-tolerant, and the
remaining error channels can be handled efficiently. The main challenge to our
proposal is to find a proper material that provides the required two-fold
degenerate chiral modes. We have considered two potential candidate materials,
with discussion of difficulties associated with their implementations
respectively. We look forward to collaborations with experimentalists to push
forward the realization of our proposal.

##  VIII Acknowlegements

AMT is grateful for D. Goldhaber-Gordon for valuable advice concerning the
experimental realization of the suggested model, to J. von Delft and to A.
Pasupathy and Q. Li for interest to the work. AMT and EJK are grateful for E.
Sela and M. Lotem for sharing their preliminary numerical results of the
suggested model. AMT and TR are supported by Office of Basic Energy Sciences,
Material Sciences and Engineering Division, U.S. Department of Energy (DOE)
under Contract No. DE-SC0012704.

##  Appendix A Correlation of $Sp_{1}(2k)$ Primary Fields

The singular behavior of the correlation functions of $Sp_{1}(2k)$ primary
fields at short distance can be determined by a powerful method called the
boundary conformal field theory (boundary CFT) Cardy (1984, 1989); Cardy and
Lewellen (1991). This method expresses the effect of the impurity spins as
conformally invariant boundary conditions on the chiral fermions, and further
trades the boundary CFT for a bulk CFT with boundary states. There is a
bijection between the primary fields in the bulk CFT and the conformally
invariant boundary conditions, such that boundary states can be constructed by
fusing the bulk primary fields (here the spin of the chiral fermions) with a
reference boundary state (here the impurity spin). This has been used to
tackle the problem of a single impurity spin, known as the fusion ansatz
Affleck (1990); Affleck and Ludwig (1991b, c); Affleck and Ludwig (1992),
which we now review.

We here denote the boundary condition using upper case $A$, and the conformal
tower of the primary field $\Phi_{a}$ using lower case $a$. The boundary state
$\ket{A}$ is then expanded in terms of the so-called Ishibashi state $\ket{a}$
constructed from the conformal tower $a$ Ishibashi (1989), and the expansion
coefficients $\langle\langle a|A\rangle$ fulfill the following Cardy
conditions Cardy (2004):

$\sum_{b}S^{a}_{b}n^{b}_{AB}=\langle A|a\rangle\rangle\langle\langle
a|B\rangle,$ (50)

where $S^{a}_{b}$ is the modular $S$-matrix of the bulk CFT Philippe Francesco
(2012), and $n^{b}_{AB}$ is the multiplicity of the conformal tower $b$ under
the boundary conditions $A$ and $B$. Utilizing the Verlinde formula for the
modular $S$-matrix Verlinde (1988), the Cardy conditions can be solved by the
following fusion ansatz

$\langle\langle a|B\rangle=\langle\langle
a|A\rangle\frac{S_{c}^{a}}{S_{0}^{a}},\quad
n^{a}_{AB}=\sum_{b}N_{bc}^{a}n^{b}_{AA},$ (51)

where the boundary state $\ket{B}$ is obtained by fusing the boundary state
$\ket{A}$ with the primary field $\Phi_{c}$, $0$ denotes the vacuum, and the
coefficients $N^{a}_{bc}$ are those appearing in the fusion rule for the bulk
primary fields:

$\Phi_{a}\times\Phi_{b}=\sum_{c}N_{ab}^{c}\Phi_{c}.$ (52)

Using the fusion ansatz in Eq. (51), the correlation between the primary
fields can be evaluated Cardy and Lewellen (1991):

$\begin{split}\left\langle\Phi_{a}\left(z_{1}\right)\Phi_{a}\left(z_{2}\right)\right\rangle&=\frac{1}{\left(z_{1}-z_{2}\right)^{2h_{a}}}\frac{\langle\langle
a|B\rangle}{\langle\langle 0|B\rangle}\\\
&=\frac{1}{\left(z_{1}-z_{2}\right)^{2h_{a}}}\frac{S^{a}_{c}/S^{a}_{0}}{S^{0}_{c}/S^{0}_{0}},\end{split}$
(53)

where $h_{a}$ is the scaling dimension of the primary field $\Phi_{a}$, the
impurity spin is placed at the origin, the complex coordinate is defined as
$z\equiv\tau-\mathrm{i}x$, and the correlation is evaluated at
$\operatorname{Im}z_{1}<0$ and $\operatorname{Im}z_{2}>0$ around the impurity
spin. Eq. (53) can be easily evaluated as long as the modular $S$-matrix for
the primary fields is known.

The above described fusion ansatz can be readily applied to the case of a
single impurity spin. For the case of multiple impurity spins, a generalized
multi-fusion ansatz is proposed under the condition that each impurity spin
affects the chiral fermions independently Lopes et al. (2020); Gabay et al.
(2022):

$n_{FM}^{a}=\sum_{b}N_{bc}^{a}n_{FM-1}^{b},$ (54)

where $F$ refers to the free boundary condition on the left, $M$ refers to the
boundary condition of $M$ impurity spins on the right, and $M$ is obtained by
fusing $(M-1)$ with the primary field $\Phi_{c}$. Iterating Eq. (54), and
using the analogy with the fusion category theory, an effective primary field
$\Phi_{c_{\text{eff}}}$ and an effective boundary $B_{\text{eff}}$ can be
defined to represent the collective effect of the $M$ impurity spins:

$\begin{split}&n_{FM}^{a}=\sum_{c_{\text{eff}}}\operatorname{dim}\left[V_{c\ldots
c}^{c_{\text{eff}}}\right]n_{FB_{\text{eff}}}^{a},\\\ &\underbrace{c\times
c\times\ldots\times
c}_{M}=\sum_{c_{\text{eff}}}\operatorname{dim}\left[V_{c\ldots
c}^{c_{\mathrm{eff}}}\right]c_{\mathrm{eff}},\\\
&n_{FB_{\mathrm{eff}}}^{a}=\sum_{b}N_{bc_{\mathrm{eff}}}^{a}n_{FF}^{b},\end{split}$
(55)

where $\operatorname{dim}\left[V_{c\ldots c}^{c_{\text{eff}}}\right]$ is the
dimension of the Hilbert space obtained by $(M-1)$ fusions of $c$ into
$c_{\text{eff}}$. Consequently, the correlation between the primary fields
$\Phi_{a}$ can be evaluated as

$\begin{split}\left\langle\Phi_{a}\left(z_{1}\right)\Phi_{a}\left(z_{2}\right)\right\rangle&=\frac{1}{\left(z_{1}-z_{2}\right)^{2h_{a}}}\frac{\left\langle\langle
a|B_{\mathrm{eff}}\right\rangle}{\left\langle\langle
0|B_{\mathrm{eff}}\right\rangle}\\\
&=\frac{1}{\left(z_{1}-z_{2}\right)^{2h_{a}}}\frac{S_{c_{\mathrm{eff}}}^{a}/S_{0}^{a}}{S_{c_{\mathrm{eff}}}^{0}/S_{0}^{0}},\end{split}$
(56)

where $z_{1}$ sits to the left of the $M$ impurity spins, and $z_{2}$ sits to
the right of the $M$ impurity spins. In case of the $Sp_{1}(2k)$ theory, the
modular-$S$ matrix is Kac and Wakimoto (1988)

$S_{a}^{a^{\prime}}=\sqrt{\frac{2}{2+k}}\sin\left[\frac{\pi(a+1)\left(a^{\prime}+1\right)}{2+k}\right].$
(57)

As a result, the correlation function in Eq. (56) encodes the information of
the total fusion channel of $M$ anyons localized at the impurity sites,
through the correspondence between the primaries and the anyons. Such a
correspondence can be checked by looking at the fusion rules for the
$Sp_{1}(2k)$ primaries shown in Eq. (21), and the results for the two relevant
cases $k=2$ and $k=3$ are shown in Eq. (31) and (33) respectively.

##  Appendix B Fibonacci Anyons

In addition to the vacuum $\mathbf{1}$, there is a single type of anyon
$\tau$. The fusion rules are

$\tau\times\mathbf{1}=\mathbf{1}\times\tau=\tau,\quad\tau\times\tau=\mathbf{1}+\tau.$
(58)

The quantum dimension is

$d_{\tau}=\frac{1+\sqrt{5}}{2}.$ (59)

The non-Abelian nature of the Fibonacci anyons is contained in its nontrivial
fusion matrix:

$F_{\tau}^{\tau\tau\tau}=\begin{pmatrix}d^{-1}_{\tau}&d^{-1/2}_{\tau}\\\
d^{-1/2}_{\tau}&-d^{-1}_{\tau}\end{pmatrix}.$ (60)

The nontrivial braiding matrices for the Fibonacci anyons are

$R_{\mathbf{1}}^{\tau\tau}=e^{\mathrm{i}4\pi/5},\quad
R_{\tau}^{\tau\tau}=e^{-\mathrm{i}3\pi/5}.$ (61)

For implementation of two-level gates, we need three anyons Nayak et al.
(2008). The three anyon basis can be chosen as shown in the upper panel of
Fig. 6, and the braiding operators are shown in the lower panel of Fig. 6.

Figure 6: The computational basis and the braiding operators using three
anyons. The states $\ket{0}$ and $\ket{1}$ constitute two levels of the qubit,
while the state $\ket{N}$ is a non-computational auxiliary state.

The two braiding operators can be represented in terms of the three anyons
basis as Nayak et al. (2008)

$\begin{split}&\rho\left(\sigma_{1}\right)\begin{pmatrix}|0\rangle\\\
|1\rangle\\\ |N\rangle\end{pmatrix}=\begin{pmatrix}e^{-4\pi i/5}&0&0\\\
0&e^{3\pi i/5}&0\\\ 0&0&e^{3\pi i/5}\end{pmatrix}\begin{pmatrix}|0\rangle\\\
|1\rangle\\\ |N\rangle\end{pmatrix},\\\
&\rho\left(\sigma_{2}\right)\begin{pmatrix}|0\rangle\\\ |1\rangle\\\
|N\rangle\end{pmatrix}=\begin{pmatrix}-\frac{e^{-\pi
i/5}}{d_{\tau}}&-\frac{ie^{-i\pi/10}}{\sqrt{d_{\tau}}}&0\\\
-\frac{ie^{-i\pi/10}}{\sqrt{d_{\tau}}}&-\frac{1}{d_{\tau}}&0\\\ 0&0&e^{3\pi
i/5}\end{pmatrix}\begin{pmatrix}|0\rangle\\\ |1\rangle\\\
|N\rangle\end{pmatrix}.\end{split}$ (62)

It is easy to see that the states $\ket{0}$ and $\ket{1}$ constitute the
computational basis, while the state $\ket{N}$ stands alone as an non-
computational auxiliary state. According to the Solovay-Kitaev theorem Kitaev
(1997); Dawson and Nielsen (2006), any unitary quantum gate can be
approximated by the two braiding operators up to arbitrary precision
efficiently. For example, the Hadamard gate can be approximated as Rouabah et
al. (2021):

$\begin{split}W_{H}&=\sigma_{1}^{+4}\sigma_{2}^{-2}\sigma_{1}^{+2}\sigma_{2}^{-2}\sigma_{1}^{+2}\sigma_{2}^{+2}\sigma_{1}^{-2}\sigma_{2}^{+4}\sigma_{1}^{+2}\sigma_{2}^{-2}\sigma_{1}^{-2}\sigma_{2}^{+2}\sigma_{1}^{+2}\\\
&=-\frac{i}{\sqrt{2}}\left(\begin{array}[]{cc}1.0040+0.0056i&0.9959-0.0048i\\\
0.9959+0.0048i&-1.0040+0.0056i\end{array}\right),\end{split}$ (63)

where the extra phase is not a problem, since eventually a projective
measurement will be performed.

##  Appendix C Measurement-Only Topological Quantum Computation

We review the basic ingredients of the measurement-only topological quantum
computation proposed in Ref. Bonderson et al., 2008a in terms of Fibonacci
anyons, adopting the diagrammatic notations used by Kitaev Kitaev (2006);
Bonderson et al. (2008a, b). It includes the main components listed below.

Orthogonal projective measurement of fusion channel
$c\in\\{\mathbf{1},\tau\\}$ of a state $\ket{\psi}$:

$\begin{split}&\operatorname{Prob}(c)=\left\langle\psi\left|\Pi_{c}\right|\psi\right\rangle,\quad|\psi\rangle\mapsto\frac{\Pi_{c}|\psi\rangle}{\sqrt{\left\langle\psi\left|\Pi_{c}\right|\psi\right\rangle}},\\\
&\Pi_{a}\Pi_{b}=\delta_{ab}\Pi_{a},\end{split}$

whose diagrammatic representation is

Conventionally, the line for the vacuum $c=\mathbf{1}$ is omitted for better
visualization. In our proposal, the set of orthogonal projective measurements
is realized by measuring the bi-anyonic fusion channel through the correlation
function of the impurity spins, see Sec. V.

The information-encoded state $\psi(\tau,\cdots)$

where only the anyon on which we want to perform the operation is shown.

Maximally entangled Bell state $\ket{\tau,\tau;\mathbf{1}}$

where the line for the vacuum is conventionally omitted.

Firstly, let us consider the quantum teleportation using three anyons, which
is diagrammatically represented as

where $\varkappa$ is a pure phase. In this way, the information-encoded in
anyon 3 with others of $\psi$ is transported to anyon 1 with others of $\psi$,
using anyon 2 as an auxiliary anyon. However, the measurement of anyons 2 and
3 may not be in the vacuum channel, this is where the repeated measurements
come in. We denote the fusion outcome of anyons 1 and 2 as $e_{i}$, and the
one of anyons 2 and 3 as $f_{i}$. Imagine we start with $e_{1}=\mathbf{1}$,
and we measure anyons 2 and 3 with outcome $f_{1}$. If $f_{1}=\mathbf{1}$,
then we are done; otherwise we measure anyons 1 and 2 with outcome $e_{2}$.
This gives us a sequence of outcomes $(e_{1},f_{1};e_{2},f_{2};\ldots)$. The
average number of attempts until $f_{n}=\mathbf{1}$ is $\langle
n\rangle\leqslant d^{2}_{\tau}<3$, and the probability of needing $n>N$
attempts is

$\operatorname{Prob}\left(f_{1},\ldots,f_{N}\neq\bm{1}\right)\leq\left(1-d_{\tau}^{-2}\right)^{N}\approx
0.618^{N}$

which is exponentially suppressed. Such a non-deterministic operation is
dubbed as a forced measurement, and it is diagrammatically represented as

Next, let us consider promoting the quantum teleportation into braiding. This
can be done through the forced measurement using two auxiliary anyons, which
is diagrammatically represented as

Consequently, we can use a total number of four anyons to realize a binary
braiding.

## References

  * Kitaev (2003) A. Kitaev, Annals of Physics 303, 2 (2003), ISSN 0003-4916, URL https://www.sciencedirect.com/science/article/pii/S0003491602000180.
  * Nayak et al. (2008) C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008), URL https://link.aps.org/doi/10.1103/RevModPhys.80.1083.
  * Pachos (2012) J. Pachos, _Introduction to Topological Quantum Computation_ , Introduction to Topological Quantum Computation (Cambridge University Press, 2012), ISBN 9781107005044\.
  * Lahtinen and Pachos (2017) V. Lahtinen and J. K. Pachos, SciPost Phys. 3, 021 (2017), URL https://scipost.org/10.21468/SciPostPhys.3.3.021.
  * Kitaev (2001) A. Y. Kitaev, Physics-Uspekhi 44, 131 (2001), URL https://dx.doi.org/10.1070/1063-7869/44/10S/S29.
  * Sarma et al. (2015) S. D. Sarma, M. Freedman, and C. Nayak, npj Quantum Information 1, 15001 (2015), URL https://doi.org/10.1038/npjqi.2015.1.
  * Potok et al. (2007) R. M. Potok, I. G. Rau, H. Shtrikman, Y. Oreg, and D. Goldhaber-Gordon, Nature 446, 167 (2007), URL https://doi.org/10.1038/nature05556.
  * Keller et al. (2015) A. J. Keller, L. Peeters, C. P. Moca, I. Weymann, D. Mahalu, V. Umansky, G. Zaránd, and D. Goldhaber-Gordon, Nature 526, 237 (2015), URL https://doi.org/10.1038/nature15261.
  * Karki et al. (2023) D. B. Karki, E. Boulat, W. Pouse, D. Goldhaber-Gordon, A. K. Mitchell, and C. Mora, Phys. Rev. Lett. 130, 146201 (2023), URL https://link.aps.org/doi/10.1103/PhysRevLett.130.146201.
  * Lopes et al. (2020) P. L. S. Lopes, I. Affleck, and E. Sela, Phys. Rev. B 101, 085141 (2020), URL https://link.aps.org/doi/10.1103/PhysRevB.101.085141.
  * Gabay et al. (2022) D. Gabay, C. Han, P. L. S. Lopes, I. Affleck, and E. Sela, Phys. Rev. B 105, 035151 (2022), URL https://link.aps.org/doi/10.1103/PhysRevB.105.035151.
  * Lotem et al. (2022) M. Lotem, E. Sela, and M. Goldstein, Phys. Rev. Lett. 129, 227703 (2022), URL https://link.aps.org/doi/10.1103/PhysRevLett.129.227703.
  * Komijani (2020) Y. Komijani, Phys. Rev. B 101, 235131 (2020), URL https://link.aps.org/doi/10.1103/PhysRevB.101.235131.
  * Bonderson et al. (2008a) P. Bonderson, M. Freedman, and C. Nayak, Phys. Rev. Lett. 101, 010501 (2008a), URL https://link.aps.org/doi/10.1103/PhysRevLett.101.010501.
  * Li et al. (2023) G. Li, E. J. König, and J. I. Väyrynen, Phys. Rev. B 107, L201401 (2023), URL https://link.aps.org/doi/10.1103/PhysRevB.107.L201401.
  * König and Tsvelik (2023) E. J. König and A. M. Tsvelik, Annals of Physics p. 169231 (2023), ISSN 0003-4916, URL https://www.sciencedirect.com/science/article/pii/S0003491623000167.
  * Flint et al. (2008) R. Flint, M. Dzero, and P. Coleman, Nature Physics 4, 643 (2008).
  * Verstegen (1991) D. Verstegen, Communications in Mathematical Physics 137, 567 (1991), URL https://doi.org/10.1007/BF02100278.
  * Kimura (2021) T. Kimura, Journal of the Physical Society of Japan 90, 024708 (2021), eprint https://doi.org/10.7566/JPSJ.90.024708, URL https://doi.org/10.7566/JPSJ.90.024708.
  * (20) Through private communication with E. Sela we learned about preliminary numerical renormalization group data for the chiral Kondo chain model with symplectic symmetry by M. Lotem, E. Sela et al., which suggests that both the level disorder in Eq. (7) and the Kondo anisotropy in Eq. (34) are relevant perturbations. Our conclusion, however, is that the level disorder does not directly couple to the impurity spins, and the Kondo anisotropy is exactly marginal being equivalent to a coupling of chiral currents. As is stated in Sec. IV.2, a particularly simple argument favoring our conclusion can be given for the $k=2$ case.
  * Lin et al. (1998) H.-H. Lin, L. Balents, and M. P. A. Fisher, Phys. Rev. B 58, 1794 (1998), URL https://link.aps.org/doi/10.1103/PhysRevB.58.1794.
  * Konik et al. (2002) R. M. Konik, H. Saleur, and A. W. W. Ludwig, Phys. Rev. B 66, 075105 (2002), URL https://link.aps.org/doi/10.1103/PhysRevB.66.075105.
  * Andrei (1980) N. Andrei, Phys. Rev. Lett. 45, 379 (1980), URL https://link.aps.org/doi/10.1103/PhysRevLett.45.379.
  * Vigman (1980) P. Vigman, Soviet Journal of Experimental and Theoretical Physics Letters 31, 364 (1980).
  * Tsvelick and Wiegmann (1983) A. Tsvelick and P. Wiegmann, Advances in Physics 32, 453 (1983), eprint https://doi.org/10.1080/00018738300101581, URL https://doi.org/10.1080/00018738300101581.
  * Andrei et al. (1983) N. Andrei, K. Furuya, and J. H. Lowenstein, Rev. Mod. Phys. 55, 331 (1983), URL https://link.aps.org/doi/10.1103/RevModPhys.55.331.
  * Andrei and Lowenstein (1979) N. Andrei and J. H. Lowenstein, Phys. Rev. Lett. 43, 1689 (1979), URL https://link.aps.org/doi/10.1103/PhysRevLett.43.1698.
  * Ogievetsky et al. (1987) E. Ogievetsky, N. Reshetikhin, and P. Wiegmann, Nuclear Physics B 280, 45 (1987), ISSN 0550-3213, URL https://www.sciencedirect.com/science/article/pii/0550321387901386.
  * Berg et al. (1978) B. Berg, M. Karowski, P. Weisz, and V. Kurak, Nuclear Physics B 134, 125 (1978), ISSN 0550-3213, URL https://www.sciencedirect.com/science/article/pii/0550321378904893.
  * Reshetikhin (1985) N. Y. Reshetikhin, Theoretical and Mathematical Physics 63, 555 (1985), URL https://doi.org/10.1007/BF01017501.
  * Jimbo (1986) M. Jimbo, in _Field Theory, Quantum Gravity and Strings_ , edited by H. J. de Vega and N. Sánchez (Springer Berlin Heidelberg, Berlin, Heidelberg, 1986), pp. 335–361.
  * Reshetikhin and Weigmann (1987) N. Reshetikhin and P. Weigmann, Physics Letters B 189, 125 (1987), ISSN 0370-2693, URL https://www.sciencedirect.com/science/article/pii/0370269387912822.
  * Lotem et al. (2023) M. Lotem, E. Sela, and M. Goldstein, Phys. Rev. B 107, 155417 (2023), URL https://link.aps.org/doi/10.1103/PhysRevB.107.155417.
  * Affleck and Ludwig (1991a) I. Affleck and A. W. W. Ludwig, Phys. Rev. Lett. 67, 161 (1991a), URL https://link.aps.org/doi/10.1103/PhysRevLett.67.161.
  * Liberman et al. (2021) A. Liberman, A. K. Mitchell, I. Affleck, and E. Sela, Phys. Rev. B 103, 195131 (2021), URL https://link.aps.org/doi/10.1103/PhysRevB.103.195131.
  * Mitchell et al. (2021) A. K. Mitchell, A. Liberman, E. Sela, and I. Affleck, Phys. Rev. Lett. 126, 147702 (2021), URL https://link.aps.org/doi/10.1103/PhysRevLett.126.147702.
  * Bouwknegt and Schoutens (1999) P. Bouwknegt and K. Schoutens, Nuclear Physics B 547, 501 (1999), ISSN 0550-3213, URL https://www.sciencedirect.com/science/article/pii/S0550321399000954.
  * Walton (1990) M. A. Walton, Nuclear Physics B 340, 777 (1990), ISSN 0550-3213, URL https://www.sciencedirect.com/science/article/pii/055032139090470X.
  * Cummins (1991) C. J. Cummins, Journal of Physics A: Mathematical and General 24, 391 (1991), URL https://dx.doi.org/10.1088/0305-4470/24/2/012.
  * Affleck (1990) I. Affleck, Nuclear Physics B 336, 517 (1990), ISSN 0550-3213, URL https://www.sciencedirect.com/science/article/pii/055032139090440O.
  * Affleck and Ludwig (1991b) I. Affleck and A. W. Ludwig, Nuclear Physics B 352, 849 (1991b), ISSN 0550-3213, URL https://www.sciencedirect.com/science/article/pii/055032139190109B.
  * Witten (1984) E. Witten, Communications in Mathematical Physics 92, 455 (1984), URL https://doi.org/10.1007/BF01215276.
  * James et al. (2018) A. J. A. James, R. M. Konik, P. Lecheminant, N. J. Robinson, and A. M. Tsvelik, Reports on Progress in Physics 81, 046002 (2018), URL https://dx.doi.org/10.1088/1361-6633/aa91ea.
  * Trebst et al. (2008) S. Trebst, M. Troyer, Z. Wang, and A. W. W. Ludwig, Progress of Theoretical Physics Supplement 176, 384 (2008), ISSN 0375-9687, URL https://doi.org/10.1143/PTPS.176.384.
  * Génetay Johansen and Simula (2021) E. Génetay Johansen and T. Simula, PRX Quantum 2, 010334 (2021), URL https://link.aps.org/doi/10.1103/PRXQuantum.2.010334.
  * (46) We note that $\Phi_{1}$ and $\Phi_{2}$ are primary fields, while $\Gamma$ is the Fibonacci anyon. They share the same fusion rules but they are not identified with each other.
  * Nielsen and Chuang (2010) M. A. Nielsen and I. L. Chuang, _Quantum Computation and Quantum Information: 10th Anniversary Edition_ (Cambridge University Press, 2010).
  * Raussendorf and Briegel (2001) R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001), URL https://link.aps.org/doi/10.1103/PhysRevLett.86.5188.
  * Raussendorf et al. (2003) R. Raussendorf, D. E. Browne, and H. J. Briegel, Phys. Rev. A 68, 022312 (2003), URL https://link.aps.org/doi/10.1103/PhysRevA.68.022312.
  * Farhi et al. (2000) E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, _Quantum computation by adiabatic evolution_ (2000), eprint arXiv:quant-ph/0001106.
  * Das and Chakrabarti (2008) A. Das and B. K. Chakrabarti, Rev. Mod. Phys. 80, 1061 (2008), URL https://link.aps.org/doi/10.1103/RevModPhys.80.1061.
  * Kitaev (2006) A. Kitaev, Annals of Physics 321, 2 (2006), ISSN 0003-4916, january Special Issue, URL https://www.sciencedirect.com/science/article/pii/S0003491605002381.
  * Kitaev (1997) A. Y. Kitaev, Russian Mathematical Surveys 52, 1191 (1997), URL https://dx.doi.org/10.1070/RM1997v052n06ABEH002155.
  * Dawson and Nielsen (2006) C. M. Dawson and M. A. Nielsen, Quantum Info. Comput. 6, 81–95 (2006), ISSN 1533-7146.
  * Maronese et al. (2022) M. Maronese, L. Moro, L. Rocutto, and E. Prati, _Quantum Compiling_ (Springer International Publishing, Cham, 2022), pp. 39–74, ISBN 978-3-030-89746-8, URL https://doi.org/10.1007/978-3-030-89746-8_2.
  * Rouabah et al. (2021) M. T. Rouabah, N. E. Belaloui, and A. Tounsi, Journal of Physics: Conference Series 1766, 012029 (2021), URL https://dx.doi.org/10.1088/1742-6596/1766/1/012029.
  * Wang et al. (2013) J. Wang, B. Lian, H. Zhang, Y. Xu, and S.-C. Zhang, Phys. Rev. Lett. 111, 136801 (2013), URL https://link.aps.org/doi/10.1103/PhysRevLett.111.136801.
  * Ge et al. (2020) J. Ge, Y. Liu, J. Li, H. Li, T. Luo, Y. Wu, Y. Xu, and J. Wang, National Science Review 7, 1280 (2020), ISSN 2095-5138, eprint https://academic.oup.com/nsr/article-pdf/7/8/1280/38881797/nwaa089.pdf, URL https://doi.org/10.1093/nsr/nwaa089.
  * Feng et al. (2021) X. Feng, K. Jiang, Z. Wang, and J. Hu, Science Bulletin 66, 1384 (2021), ISSN 2095-9273, URL https://www.sciencedirect.com/science/article/pii/S2095927321003224.
  * Yu et al. (2021) L. Yu, C. Wang, Y. Zhang, M. Sander, S. Ni, Z. Lu, S. Ma, Z. Wang, Z. Zhao, H. Chen, et al., _Evidence of a hidden flux phase in the topological kagome metal CsV 3Sb5_ (2021), eprint arXiv:2107.10714.
  * Mielke et al. (2022) C. Mielke, D. Das, J. X. Yin, H. Liu, R. Gupta, Y. X. Jiang, M. Medarde, X. Wu, H. C. Lei, J. Chang, et al., Nature 602, 245 (2022), URL https://doi.org/10.1038/s41586-021-04327-z.
  * Khasanov et al. (2022) R. Khasanov, D. Das, R. Gupta, C. Mielke, M. Elender, Q. Yin, Z. Tu, C. Gong, H. Lei, E. T. Ritz, et al., Phys. Rev. Res. 4, 023244 (2022), URL https://link.aps.org/doi/10.1103/PhysRevResearch.4.023244.
  * Lin and Nandkishore (2022) Y.-P. Lin and R. M. Nandkishore, Phys. Rev. B 106, L060507 (2022), URL https://link.aps.org/doi/10.1103/PhysRevB.106.L060507.
  * Christensen et al. (2022) M. H. Christensen, T. Birol, B. M. Andersen, and R. M. Fernandes, Phys. Rev. B 106, 144504 (2022), URL https://link.aps.org/doi/10.1103/PhysRevB.106.144504.
  * Zamolodchikov (1986) A. B. Zamolodchikov, JETP Lett. 43, 730 (1986).
  * Cardy (1984) J. L. Cardy, Nuclear Physics B 240, 514 (1984), ISSN 0550-3213, URL https://www.sciencedirect.com/science/article/pii/0550321384902414.
  * Cardy (1989) J. L. Cardy, Nuclear Physics B 324, 581 (1989), ISSN 0550-3213, URL https://www.sciencedirect.com/science/article/pii/055032138990521X.
  * Cardy and Lewellen (1991) J. L. Cardy and D. C. Lewellen, Physics Letters B 259, 274 (1991), ISSN 0370-2693, URL https://www.sciencedirect.com/science/article/pii/037026939190828E.
  * Affleck and Ludwig (1991c) I. Affleck and A. W. Ludwig, Nuclear Physics B 360, 641 (1991c), ISSN 0550-3213, URL https://www.sciencedirect.com/science/article/pii/055032139190419X.
  * Affleck and Ludwig (1992) I. Affleck and A. W. W. Ludwig, Phys. Rev. Lett. 68, 1046 (1992), URL https://link.aps.org/doi/10.1103/PhysRevLett.68.1046.
  * Ishibashi (1989) N. Ishibashi, Modern Physics Letters A 04, 251 (1989), URL https://doi.org/10.1142/S0217732389000320.
  * Cardy (2004) J. Cardy (2004), URL https://arxiv.org/abs/hep-th/0411189.
  * Philippe Francesco (2012) D. S. Philippe Francesco, Pierre Mathieu, _Conformal Field Theory_ (Springer New York, NY, 2012).
  * Verlinde (1988) E. Verlinde, Nuclear Physics B 300, 360 (1988), ISSN 0550-3213, URL https://www.sciencedirect.com/science/article/pii/0550321388906037.
  * Kac and Wakimoto (1988) V. G. Kac and M. Wakimoto, Advances in Mathematics 70, 156 (1988), ISSN 0001-8708, URL https://www.sciencedirect.com/science/article/pii/0001870888900552.
  * Bonderson et al. (2008b) P. Bonderson, K. Shtengel, and J. Slingerland, Annals of Physics 323, 2709 (2008b), ISSN 0003-4916, URL https://www.sciencedirect.com/science/article/pii/S0003491608000171.


Then $\\{\lambda_{t,0}(\boldsymbol{\theta})\\}_{t=p_{0}+1}^{\infty}$ is also
stationary for any $\boldsymbol{\theta}$ and ergodic for
$\boldsymbol{\theta}\in\mathcal{V}(\boldsymbol{\theta}_{0})$. Since
$\lambda_{t,0}$ does not depend on its lagged values, (35) holds with
$\widetilde{C}_{T}(u,\boldsymbol{v},\widehat{\boldsymbol{\theta}})$ replaced
by ${C}_{T}(u,\boldsymbol{v},\widehat{\boldsymbol{\theta}})$ for

${C}_{T}(u,\boldsymbol{v},\boldsymbol{\theta})=\frac{1}{T}\sum_{t=2}^{T}\left[u^{Y_{t}}-g(u,\lambda_{t,0}(\boldsymbol{\theta}))\right]\alpha\bigl{(}{\boldsymbol{Z}}_{t,0},\boldsymbol{v}\bigr{)}.$

The Taylor expansion gives

$C_{T}(u,\boldsymbol{v},\widehat{\boldsymbol{\theta}})=C_{T}(u,\boldsymbol{v},{\boldsymbol{\theta}}_{0})+\frac{\partial
C_{T}(u,\boldsymbol{v},{\boldsymbol{\theta}})}{\partial\boldsymbol{\theta}}|_{\boldsymbol{\theta}=\boldsymbol{\theta}^{*}}(\widehat{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0}),$

where $\boldsymbol{\theta}^{*}$ is a point between $\boldsymbol{\theta}_{0}$
and $\widehat{\boldsymbol{\theta}}$. Here
$C_{T}(u,\boldsymbol{v},{\boldsymbol{\theta}}_{0})$ is a sum of stationary and
ergodic variables such that

$\displaystyle C_{T}(u,\boldsymbol{v},{\boldsymbol{\theta}}_{0})$
$\displaystyle\stackrel{{\scriptstyle\mathsf{P}}}{{\rightarrow}}\mathbb{E}\,\left[u^{Y_{1}}-g(u,\lambda_{1,0}(\boldsymbol{\theta}_{0}))\right]\alpha\bigl{(}{\boldsymbol{Z}}_{1,0},\boldsymbol{v}\bigr{)}$
$\displaystyle=\mathbb{E}\,\left\\{\left[g(u,\lambda_{1,A})-g(u,\lambda_{1,0}(\boldsymbol{\theta}_{0}))\right]\alpha({\boldsymbol{Z}}_{1,0},\boldsymbol{v})\right\\}$

as $T\to\infty$ and the convergence holds uniformly in $u$ and
$\boldsymbol{v}$. Furthermore,

$\displaystyle\frac{\partial
C_{T}(u,\boldsymbol{v},{\boldsymbol{\theta}})}{\partial\boldsymbol{\theta}}=-\frac{1}{T}\sum_{t=2}^{T}\frac{\partial
g(u,z)}{\partial
z}|_{z=\lambda_{t,0}(\boldsymbol{\theta})}\frac{\partial\lambda_{t,0}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\alpha(\boldsymbol{Z}_{t,,0}\boldsymbol{v}),$

where

$\frac{\partial\lambda_{t,0}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}=\frac{\partial
h_{0}(\boldsymbol{Y}_{t-1:t-p_{0}},\boldsymbol{X}_{t-1};\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}.$

It follows from (17) and from the properties of $h_{0}$ that
$\bigl{|}{\partial
C_{T}(u,\boldsymbol{v},{\boldsymbol{\theta}})}/{\partial\boldsymbol{\theta}}|_{\boldsymbol{\theta}=\boldsymbol{\theta}^{*}}\bigr{|}=O_{P}(1)$
uniformly in $u\in[0,1]$ and $\boldsymbol{v}\in{\rm I\kern-1.99997ptR}^{d}$.
The assertion then follows from (18). ∎

For the proof of Theorem 4.5 let $h_{0}$ be the linear function from (19) and
$\boldsymbol{\Theta}$ the corresponding parametric space. For
$\boldsymbol{\theta}\in\boldsymbol{\Theta}$ define recursively for $t\geq 2$

$\widetilde{\lambda}_{t,0}(\boldsymbol{\theta})=\omega+\alpha
Y_{t-1}+\beta\widetilde{\lambda}_{t-1,0}(\boldsymbol{\theta})+\boldsymbol{\gamma}^{\top}\boldsymbol{X}_{t-1}$
(36)

for some initial $\widetilde{\lambda}_{1,0}=l_{1}$. Let

$\widetilde{\boldsymbol{Z}}_{t,0}(\boldsymbol{\theta})=(Y_{t-1},\widetilde{\lambda}_{t,0}(\boldsymbol{\theta}),\boldsymbol{X}_{t-1}^{\top})^{\top}.$

######  Lemma A.6.

Let $\\{Y_{t}\\}$ and $\\{\boldsymbol{X}_{t}\\}$ be stationary and ergodic
sequences, $l_{1}\geq 0$, and $\widetilde{\lambda}_{t,0}(\boldsymbol{\theta})$
be defined by (36) for $\boldsymbol{\theta}\in\boldsymbol{\Theta}_{0}$, where
$\boldsymbol{\Theta}_{0}$ is a compact subset of $\boldsymbol{\Theta}$. Then

$\widetilde{\lambda}_{t,0}(\boldsymbol{\theta})=\omega\frac{1-\beta^{t-1}}{1-\beta}+\sum_{j=0}^{t-2}\beta^{j}[\alpha
Y_{t-1-j}+\boldsymbol{\gamma}^{\top}\boldsymbol{X}_{t-1-j}]+\beta^{t-1}l_{1}.$

Furthermore, the derivatives of
$\widetilde{\lambda}_{t,0}(\boldsymbol{\theta})$ are

$\displaystyle\frac{\partial\widetilde{\lambda}_{t,0}(\boldsymbol{\theta})}{\partial\omega}$
$\displaystyle=\frac{1-\beta^{t-1}}{1-\beta},$
$\displaystyle\frac{\partial\widetilde{\lambda}_{t,0}(\boldsymbol{\theta})}{\partial\alpha}$
$\displaystyle=Y_{t-1}+\beta\frac{\partial\widetilde{\lambda}_{t-1,0}(\boldsymbol{\theta})}{\partial\alpha}=\sum_{j=0}^{t-2}\beta^{j}Y_{t-1-j},$
$\displaystyle\frac{\partial\widetilde{\lambda}_{t,0}(\boldsymbol{\theta})}{\partial\boldsymbol{\gamma}}$
$\displaystyle=\boldsymbol{X}_{t-1}+\beta\frac{\partial\widetilde{\lambda}_{t-1,0}(\boldsymbol{\theta})}{\partial\boldsymbol{\gamma}}=\sum_{j=0}^{t-2}\beta^{j}\boldsymbol{X}_{t-1-j},$
$\displaystyle\frac{\partial\widetilde{\lambda}_{t,0}(\boldsymbol{\theta})}{\partial\beta}$
$\displaystyle=\widetilde{\lambda}_{t-1,0}(\boldsymbol{\theta})+\beta\frac{\partial\widetilde{\lambda}_{t-1,0}(\boldsymbol{\theta})}{\partial\beta}=\sum_{j=0}^{t-2}\beta^{j}\widetilde{\lambda}_{t-1-j,0}(\boldsymbol{\theta}),$

and it holds that

$\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}_{0}}\frac{1}{T}\sum_{t=2}^{T}\widetilde{\lambda}_{t,0}(\boldsymbol{\theta})=O_{\mathsf{P}}(1).$
(37)

and

$\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}_{0}}\frac{1}{T}\sum_{t=2}^{T}\frac{\partial\widetilde{\lambda}_{t,0}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}=O_{\mathsf{P}}(1).$
(38)

###### Proof.

The expressions for $\widetilde{\lambda}_{t,0}(\boldsymbol{\theta})$ and its
derivatives follow from direct computations. To prove (37) observe that

$\frac{1}{T}\sum_{t=2}^{T}\widetilde{\lambda}_{t,0}(\boldsymbol{\theta})=\frac{\omega}{1-\beta}-\frac{\omega}{1-\beta}\frac{1}{T}\sum_{t=2}^{T}\beta^{t-1}+\frac{1}{T}\sum_{t=2}^{T}\sum_{j=0}^{t-2}\beta^{j}[\alpha
Y_{t-1-j}+\boldsymbol{\gamma}^{\top}\boldsymbol{X}_{t-1-j}]+l_{1}\frac{1}{T}\sum_{t=2}^{T}\beta^{t-1}.$

Clearly then,

$\frac{\omega}{1-\beta}\frac{1}{T}\sum_{t=2}^{T}\beta^{t-1}\to 0\quad\text{
and }\quad l_{1}\frac{1}{T}\sum_{t=2}^{T}\beta^{t-1}\to 0$

as $T\to\infty$. Furthermore, notice that

$\frac{1}{T}\sum_{t=2}^{T}\sum_{j=0}^{t-2}\beta^{j}\alpha
Y_{t-1-j}=\alpha\frac{1}{T}\sum_{k=1}^{T-1}Y_{k}\sum_{j=0}^{T-k-1}\beta^{j}=\alpha\frac{1}{T}\sum_{k=1}^{T-1}Y_{k}\frac{1-\beta^{T-k}}{1-\beta}\leq\frac{\alpha}{1-\beta}\frac{1}{T}\sum_{k=1}^{T}Y_{k}.$

The term with $\boldsymbol{\gamma}^{\top}\boldsymbol{X}_{t-1-j}$ can be
treated analogously. Since $\boldsymbol{\Theta}_{0}$ is compact and
$\\{Y_{t}\\}$ and $\\{\boldsymbol{X}_{t}\\}$ are ergodic, we get (37). The
previous also implies that (38) holds for the derivatives with respect to
$\omega$, $\alpha$ and $\boldsymbol{\gamma}$ respectively, so it suffices to
show the claim for the derivative with respect to $\beta$. To this end observe
that,

$\frac{1}{T}\sum_{t=2}^{T}\frac{\partial\widetilde{\lambda}_{t,0}(\boldsymbol{\theta})}{\partial\beta}=\frac{1}{T}\sum_{t=2}^{T}\sum_{j=0}^{t-2}\beta^{j}\widetilde{\lambda}_{t-1-j,0}(\boldsymbol{\theta})\leq\frac{1}{1-\beta}\frac{1}{T}\sum_{k=1}^{T-1}\widetilde{\lambda}_{k,0}(\boldsymbol{\theta}),$

where we used the same arguments as before. Thus, (38) follows from (37).

∎

######  Lemma A.7.

Let $\\{Y_{t}\\}$ and $\\{\boldsymbol{X}_{t}\\}$ satisfy $\mathcal{A}2$. Let
$\boldsymbol{\Theta}_{0}$ be a compact subset of $\boldsymbol{\Theta}$. For
$\boldsymbol{\theta}\in\boldsymbol{\Theta}_{0}$ define

${\lambda}_{t,0}(\boldsymbol{\theta})=\frac{\omega}{1-\beta}+\sum_{j=0}^{\infty}\beta^{j}[\alpha
Y_{t-1-j}+\boldsymbol{\gamma}^{\top}\boldsymbol{X}_{t-1-j}].$

Then ${\lambda}_{t,0}(\boldsymbol{\theta})$ is finite with probability 1 and
the sequence $\\{{\lambda}_{t,0}(\boldsymbol{\theta})\\}$ is stationary and
ergodic. Moreover, there exists $\rho\in(0,1)$ such that

$\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}_{0}}|\widetilde{\lambda}_{t,0}(\boldsymbol{\theta})-\lambda_{t,0}(\boldsymbol{\theta})|\leq
W_{t}\rho^{t},$

where $W_{t}$ is a positive random variable such that
$\mathbb{E}\,W_{t}^{\kappa}<\infty$ for some $\kappa>0$.

###### Proof.

The finiteness follows together with the ergodicity and stationarity by
classical arguments, see (Brockwell and Davis,, 2006, Proposition 3.1.1). The
inequality follows from the fact that

$\lambda_{t,0}(\boldsymbol{\theta})=\omega+\alpha
Y_{t-1}+\boldsymbol{\gamma}^{\top}\boldsymbol{X}_{t-1}+\beta\lambda_{t-1,0}(\boldsymbol{\theta})$

and by comparing this with (36) which yields

$|{\lambda}_{t,0}(\boldsymbol{\theta})-\widetilde{\lambda}_{t,0}(\boldsymbol{\theta})|\leq\beta^{t-1}|\lambda_{1,0}-l_{1}|,$

where $l_{1}$ is the initial value introduced after (36), and by the fact that
$\boldsymbol{\Theta}_{0}$ is a compact set. ∎

Proof of Theorem 4.5. The test statistic $\Delta_{T,W}$ is of the form

$\frac{1}{T}\Delta_{T,W}=\int_{0}^{1}\int_{{\rm
I\kern-1.39998ptR}^{d}}|\widetilde{C}(u,\boldsymbol{v},\widehat{\boldsymbol{\theta}})|^{2}w(u)\omega(\boldsymbol{v})\mathrm{d}\boldsymbol{v}\mathrm{d}u+o_{\mathsf{P}}(1)$

for

$\widetilde{C}_{T}(u,\boldsymbol{v},\boldsymbol{\theta})=\frac{1}{T}\sum_{t=2}^{T}\left[u^{Y_{t}}-g(u,\widetilde{\lambda}_{t,0}(\boldsymbol{\theta}))\right]\alpha\bigl{(}\widetilde{\boldsymbol{Z}}_{t,0}(\boldsymbol{\theta}),\boldsymbol{v}\bigr{)}.$

The Taylor expansion of
$\widetilde{C}(u,\boldsymbol{v},\widehat{\boldsymbol{\theta}})$ in
$\boldsymbol{\theta}_{0}$ gives

$\widetilde{C}(u,\boldsymbol{v},\widehat{\boldsymbol{\theta}})=\widetilde{C}(u,\boldsymbol{v},{\boldsymbol{\theta}}_{0})+\frac{\partial\widetilde{C}(u,\boldsymbol{v},{\boldsymbol{\theta}})}{\partial\boldsymbol{\theta}}|_{\boldsymbol{\theta}=\boldsymbol{\theta}^{*}}(\widehat{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0}),$

where $\boldsymbol{\theta}^{*}$ is a point between $\boldsymbol{\theta}_{0}$
and $\widehat{\boldsymbol{\theta}}$. Then

$\displaystyle\left|\frac{1}{T}\frac{\partial\widetilde{C}(u,\boldsymbol{v},{\boldsymbol{\theta}})}{\partial\boldsymbol{\theta}}|_{\boldsymbol{\theta}=\boldsymbol{\theta}^{*}}\right|\leq
2M_{1}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}_{0}}\frac{1}{T}\sum_{t=2}^{T}\frac{\partial\widetilde{\lambda}_{t,0}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}+4|v_{2}|\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}_{0}}\frac{1}{T}\sum_{t=2}^{T}\frac{\partial\widetilde{\lambda}_{t-1,0}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}=O_{\mathsf{P}}(1)$

and thus,

$\widetilde{C}(u,\boldsymbol{v},\widehat{\boldsymbol{\theta}})=\widetilde{C}(u,\boldsymbol{v},{\boldsymbol{\theta}}_{0})+R_{1,T}(u,\boldsymbol{v}),$

where $|R_{1,T}(u,v)|\leq\|\boldsymbol{v}\|o_{P}(1)$. Furthermore,
$\widetilde{C}(u,\boldsymbol{v},{\boldsymbol{\theta}}_{0})=A_{1}(u,\boldsymbol{v})+A_{2}(u,\boldsymbol{v}),$
where

$\displaystyle A_{1}(u,\boldsymbol{v})$
$\displaystyle=\frac{1}{T}\sum_{t=2}^{T}\left[u^{Y_{t}}-g(u,\lambda_{t,A})\right]\alpha\bigl{(}\widetilde{\boldsymbol{Z}}_{t,0}(\boldsymbol{\theta}_{0}),\boldsymbol{v}\bigr{)},$
$\displaystyle A_{2}(u,\boldsymbol{v})$
$\displaystyle=\frac{1}{T}\sum_{t=2}^{T}\left[g(u,\lambda_{t,A})-g(u,\lambda_{t,0}(\boldsymbol{\theta}_{0}))\right]\alpha\bigl{(}\widetilde{\boldsymbol{Z}}_{t,0}(\boldsymbol{\theta}_{0}),\boldsymbol{v}\bigr{)}.$

Then $T\cdot A_{1}(u,\boldsymbol{v})$ is a sum of martingale differences,
which are uniformly bounded for $u\in[0,1]$ and $\boldsymbol{v}\in{\rm
I\kern-1.99997ptR}^{d}$, so they have a finite uniformly bounded variance.
Therefore, it follows from the property of martingale differences that
$\mathbb{E}\,A_{1}(u,\boldsymbol{v})=0$ and
$\mathsf{Var}A_{1}(u,\boldsymbol{v})\to 0$, which shows that
$A_{1}(u,\boldsymbol{v})\stackrel{{\scriptstyle\mathsf{P}}}{{\rightarrow}}0$
uniformly in $(u,\boldsymbol{v}^{\top})^{\top}\in[0,1]\times{\rm
I\kern-1.99997ptR}^{d}$. On the other hand write,

$A_{2}(u,\boldsymbol{v})=Q_{T}(u,\boldsymbol{v})+R_{2,T}(u,\boldsymbol{v}),$

where

$Q_{T}(u,\boldsymbol{v})=\frac{1}{T}\sum_{t=2}^{T}\left[g(u,\lambda_{t,A})-g(u,\lambda_{t,0}(\boldsymbol{\theta}_{0}))\right]\alpha\bigl{(}{\boldsymbol{Z}}_{t,0}(\boldsymbol{\theta}_{0}),\boldsymbol{v}\bigr{)}$

and

$\displaystyle R_{2,T}(u,\boldsymbol{v})$
$\displaystyle=\frac{1}{T}\sum_{t=2}^{T}\left[u^{Y_{t}}-g(u,\lambda_{t,0}(\boldsymbol{\theta}_{0}))\right]\left[\alpha\bigl{(}\widetilde{\boldsymbol{Z}}_{t,0}(\boldsymbol{\theta}_{0}),\boldsymbol{v}\bigr{)}-\alpha\bigl{(}{\boldsymbol{Z}}_{t,0}(\boldsymbol{\theta}_{0}),\boldsymbol{v}\bigr{)}\right]$
$\displaystyle+\frac{1}{T}\sum_{t=2}^{T}\left[g(u,\lambda_{t,0}(\boldsymbol{\theta}_{0}))-g(u,\widetilde{\lambda}_{t,0})(\boldsymbol{\theta}_{0}))\right]\alpha\bigl{(}\widetilde{\boldsymbol{Z}}_{t,0}(\boldsymbol{\theta}_{0}),\boldsymbol{v}\bigr{)},$

where we used the identity

$\sum_{i}(a_{i}-\widetilde{b}_{i})\widetilde{c}_{i}=\sum_{i}(a_{i}-b_{i})c_{i}+\sum_{i}(a_{i}-b_{i})(\widetilde{c}_{i}-c_{i})+\sum_{i}(b_{i}-\widetilde{b}_{i})\widetilde{c}_{i}.$

Now it follows from (17) that $g(\cdot,z)$ is $M_{1}$-Lipschitz in $z$, and
the function $\alpha(\cdot,\cdot)$ is a sum of cosine and sine terms of a
scalar product, and that
$\boldsymbol{v}^{\top}\widetilde{\boldsymbol{Z}}_{t,0}(\boldsymbol{\theta})-\boldsymbol{v}^{\top}{\boldsymbol{Z}}_{t,0}(\boldsymbol{\theta})=v_{2}\bigl{[}\widetilde{\lambda}_{t,0}(\boldsymbol{\theta})-{\lambda}_{t,0}(\boldsymbol{\theta})\bigr{]}.$
Hence,

$|R_{2,T}(u,\boldsymbol{v})|\leq(4\|\boldsymbol{v}\|+2M_{1})\frac{1}{T}\sum_{t=2}^{T}\big{|}\widetilde{\lambda}_{t,0}(\boldsymbol{\theta})-{\lambda}_{t,0}(\boldsymbol{\theta})\bigr{|}$

and Lemma A.7 together with Lemma A.1 imply that
$|R_{2,T}(u,\boldsymbol{v})|\leq\|\boldsymbol{v}\|o_{P}(1)$. Furthermore,
$\\{\lambda_{t,0}(\boldsymbol{\theta}_{0})\\}$ and
$\\{{\boldsymbol{Z}}_{t,0}(\boldsymbol{\theta}_{0})\\}$ are sequences of
stationary and ergodic random variables and hence

$Q_{T}(u,\boldsymbol{v})\stackrel{{\scriptstyle\mathsf{P}}}{{\rightarrow}}\mathbb{E}\,\left[g(u,\lambda_{1,A})-g(u,\lambda_{1,0}(\boldsymbol{\theta}_{0}))\right]\alpha\bigl{(}{\boldsymbol{Z}}_{1,0}(\boldsymbol{\theta}_{0}),\boldsymbol{v}\bigr{)}=\zeta(u,\boldsymbol{v}),$

which concludes the proof of the assertion. ∎