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

# Quantum State Transfer in Interacting, Multiple-Excitation Systems

Alexander Yue  Department of Physics and Astronomy, University of California,
Davis, CA 95616, USA  Rubem Mondaini  Department of Physics, University of
Houston, Houston, Texas 77004, USA  Qiujiang Guo  School of Physics, ZJU-
Hangzhou Global Scientific and Technological Innovation Center, and Zhejiang
Key Laboratory of Micro-nano Quantum Chips and Quantum Control,
Zhejiang University, Hangzhou, China  Richard T. Scalettar  Department of
Physics and Astronomy, University of California, Davis, CA 95616, USA

###### Abstract

Quantum state transfer (QST) describes the coherent passage of quantum
information from one node in a network to another. Experiments on QST span a
diverse set of platforms and currently report transport across up to tens of
nodes in times of several hundred nanoseconds with fidelities that can
approach 90% or more. Theoretical studies examine both the lossless time
evolution associated with a given (Hermitian) lattice Hamiltonian and methods
based on the master equation that allows for losses. In this paper, we
describe Monte Carlo techniques which enable the discovery of a Hamiltonian
that gives high-fidelity QST. We benchmark our approach in geometries
appropriate to coupled optical cavity-emitter arrays and discuss connections
to condensed matter Hamiltonians of localized orbitals coupled to conduction
bands. The resulting Jaynes-Cummings-Hubbard and periodic Anderson models can,
in principle, be engineered in appropriate hardware to give efficient QST.

##  I Introduction

Given a Hamiltonian $\hat{\cal H}$, the time evolution of a quantum mechanical
system from an initial wave function $|\psi(0)\rangle$ is governed by the
unitary map $|\psi(t)\rangle=\hat{\cal U}(t)|\psi(0)\rangle=e^{-{\rm
i}\hat{\cal H}t}|\psi(0)\rangle$ [1]. One expects, generically, that wave
functions spread with time since their initial state encodes uncertainty in
momentum as well as position. The most simple example is that of a free
particle in one dimension where, if $|\psi(0)\rangle$ is spatially localized
with uncertainty $(\Delta x)(0)^{2}=\sigma_{x}^{2}$, the subsequent
development will be such that $(\Delta
x)(t)^{2}=\sigma_{x}^{2}\big{[}\,1+\hbar t/2m\sigma_{x}^{2}\,\big{]}$. At long
times $t$ the position uncertainty is proportional to $\sqrt{t}$, reflecting
the analogy between the diffusion and the (imaginary time) Schrödinger
equations. As a consequence, achieving “high-fidelity Quantum State Transfer,”
(QST) [2] in which an excitation is transported from one particular spatial
location to another without spreading, requires special engineering. Solving
this problem for appropriate hardware is essential to a host of potential
applications in quantum computing [3, 4, 5].

At the experimental level, QST is being explored in a variety of realizations:
from atom-photon coupling in optical cavities [6], Yb atoms connected by
optical fibers [7], optical cavities connected by waveguides [8], a chain of
superconducting qubits with tunable nearest-neighbor couplings [9], and the
implementation of phonon-mediated QST [10]. Theoretical studies of QST
encompass a similar goal – to determine a Hamiltonian $\hat{\cal H}$ that
evades the usual wave packet spreading. This program inverts the more usual
quantum mechanical problem of solving for the time evolution operator
$\hat{\cal U}$ of a given $\hat{\cal H}$, to the determination of an
appropriate $\hat{\cal H}$ which leads to a desired $\hat{\cal U}$.

One illustration of this process is the prescription of Christandl et al. [11]
who showed that a one-dimensional hopping Hamiltonian with link-dependent
hybridization $J_{i}=J_{0}\sqrt{i\,(N-i)}$ exhibits perfect QST of a state
initially localized at site $i=1$ to arrival with unit probability to site
$i=N$ at a time $t=\pi/(2J_{0})$. Here $\\{\,J_{i}\,\\}$ for
$i=1,2,\ldots,(N-1)$ are the hoppings from site $i$ to site $i+1$ and $N$ is
the chain length. Many existing schemes for QST [12, 13, 14], like that of
Christandl, focus on single excitations (or multiple, non-interacting
excitations) in one dimension. However, experiments have begun to probe many-
body effects in QST [15, 16], as well as higher dimensions [17]. Naturally,
the theoretical determination of a Hamiltonian that incorporates correlations
between several particles is a challenging task. In this paper, we describe a
general Monte Carlo method to compute Hamiltonians which result in high-
fidelity QST.

Our paper is outlined as follows: Section II describes the class of model
Hamiltonians that define the space over which our Monte Carlo searches and the
method itself. Here we emphasize an analogy between coupled cavity arrays and
the periodic Anderson model (PAM) of condensed matter physics. Section III
then provides some initial illustrations of how our approach works,
demonstrating that it can reproduce solutions to the single excitation systems
discussed in Ref. [11]. It also benchmarks how the number of Monte Carlo steps
scales with target fidelity and system chain length.

Section IV turns to the main focus of the paper – the unsolved problem of
achieving high-fidelity QST with multiple interacting excitations. Here, in
the Jaynes-Cummings-Hubbard (JCH) model, the mixed bosonic and fermionic
statistics of photons and emitters, respectively, induce many-body effects. In
the case of the PAM, it is the on-site repulsion that introduces correlations,
and one must constrain the Monte Carlo search to prevent the simulation from
finding trivial, non-interacting solutions. As we advance, Sec. V considers
what happens when we try to achieve high fidelity QST between multiple pairs
of initial and final states The Christandl prescription [11] automatically
accomplishes perfect QST between all pairs representing transitions between
states located symmetrically about the chain center, as might be inferred from
the symmetry of the $\\{\,J_{i}\,\\}$.. Lastly, Sec. VI summarizes our results
and describes the outlook for related calculations.

##  II Models and Methodology

In this section, we will describe the systems we consider and define their
Hamiltonians. We will also characterize the computational methods we use to
tune coupling rates to achieve high-fidelity QST.

###  II.1 Models

The Jaynes-Cummings-Hubbard model [19] describes the coupling of photons and
atom-like emitters in a lattice of optical cavities. Photons can move between
nearest-neighbor cavities, and within each cavity, they can be absorbed or
emitted by atoms in that cavity. In this paper, we focus on one-dimensional
lattices and on systems in which every cavity contains exactly one emitter.
With these constraints, the JCH Hamiltonian is

$\displaystyle\hat{\cal H}_{\textnormal{JCH}}$
$\displaystyle=\sum\limits_{i=1}^{N}\Omega_{i}^{\phantom{\dagger}}\hat{a}_{i}^{\dagger}\hat{a}_{i}^{\phantom{\dagger}}+\sum\limits_{i=1}^{N-1}J_{i}^{\phantom{\dagger}}\big{(}\,\hat{a}_{i+1}^{\dagger}\hat{a}_{i}^{\phantom{\dagger}}+\hat{a}_{i}^{\dagger}\hat{a}_{i+1}^{\phantom{\dagger}}\,\big{)}$
$\displaystyle+\sum\limits_{i=1}^{N}\omega_{i}^{\phantom{\dagger}}\hat{\sigma}_{i}^{+}\hat{\sigma}_{i}^{-}+g_{i}\big{(}\,\hat{a}_{i}^{\dagger}\hat{\sigma}_{i}^{-}+\hat{\sigma}_{i}^{+}\hat{a}_{i}^{\phantom{\dagger}}\,\big{)}\
.$ (1)

We define the length of the system, $N$, as the number of cavities.
$\hat{a}_{i}^{\dagger}\big{(}\hat{a}_{i}^{\phantom{\dagger}}\big{)}$ are
photon creation (annihilation) operators in cavity $i$ and
$\hat{\sigma}_{i}^{+}\big{(}\hat{\sigma}_{i}^{-}\big{)}$ are excitation (de-
excitation) operators for the emitter in cavity $i$. The model is
parameterized by cavity energies $\Omega_{i}$, inter-cavity photon hopping
rates $J_{i}$, emitter energy levels $\omega_{i}$ and photon-emitter coupling
rates $g_{i}$. The geometry of such a system can be seen in Figure 1 (top).
Initial studies of time evolution in the JCH model in the single-excitation
sector have recently been presented in [20]. We will consider both situations
in which the couplings are constrained to be symmetric about the chain center
and ones in which they vary freely. It is important to note that we treat
$J_{i},g_{i}$ as experimentally tunable parameters, but assume that
$\Omega_{i},\omega_{i}$ remain fixed. As might be expected, this choice can be
important depending on whether the states between which the QST is occurring
obey the same symmetry.

To solve the JCH model (II.1), and the associated time evolution operator
$\hat{\cal U}$, one can employ exact diagonalization. Since $\hat{\cal H}_{\rm
JCH}$ conserves the total number of excitations, $N_{\rm
exc}\equiv\sum_{i}\langle\big{(}\hat{a}_{i}^{\dagger}\hat{a}_{i}^{\phantom{\dagger}}+\hat{\sigma}_{i}^{+}\hat{\sigma}_{i}^{-}\big{)}\rangle$,
the matrix for $\hat{\cal H}_{\rm JCH}$ is block diagonal, and we can consider
each excitation number subspace individually. For the $N=4$ geometry
illustrated in Fig. 1(top), the Hilbert space dimension is ${\cal D}=8$ when
$N_{\rm exc}=1$, since the single excitation can be a photon in any of the
cavities or an excited atom. ${\cal D}=32$ when $N_{\rm exc}=2$ since there
are four states with two photons in a single cavity and $\frac{8\cdot
7}{2}=28$ states with either two photons in different cavities, two excited
atoms, or one photon and one excited atom. Similar enumerations can be made
for other lattice sizes $N$ and numbers of excitations $N_{\rm exc}$. The
dimension ${\cal D}$ grows very rapidly with $N$ and $N_{\rm exc}$.

In one dimension, there are close analogies between quantum spin-$1/2$ models
and fermionic tight-binding models as emphasized by the Jordan-Wigner
transformation [21]. Indeed, the XX spin-$1/2$ Hamiltonian in the sector of
total spin $S_{z}=-N/2+1$ (one up spin in a background of down spins), and the
boson and fermion tight binding models in the one-particle sector, have
identical single-particle eigenvalues. These connections persist in higher
numbers of excitations, with the only distinction being in the allowed
occupations of the single-particle levels.

However, in the JCH model, the fact that photon occupations can exceed one,
but the emitters are strictly two-level systems negates these connections by
mixing spin-1/2 and bosonic operators. An important consequence is that the
eigenvalues of the JCH model in the two-excitation sector are not simply
combinations of the single excitation eigenvalues. In this way, the JCH model,
despite appearing “quadratic” in its operators, poses a non-trivial many-body
problem in the same way as the periodic Anderson model below, where explicit
quartic interaction terms are present. A more detailed discussion of the many-
particle eigenvalue structure of bosonic, fermions, and “mixed” operator
Hamiltonians is presented in Appendix A.

Figure 1:  Top: Geometry of the one-dimensional Jaynes-Cummings-Hubbard Model
with $N=4$ cavities each containing one emitter. Photons can “hop” between
adjacent cavities (squares) $i$ and $i+1$ via the coupling rates $J_{i}$.
Within cavity $i$ a photon can be absorbed and excite emitter $i+N$ (circles)
via the coupling rates $g_{i}$. The same process can be reversed, traveling
from emitter to cavity via $g_{i}$. The numbers indicate our convention for
labeling the states in the Hilbert space, i.e. the rows of our matrix in the
single excitation sector. Bottom: Geometry of the Periodic Anderson Model with
$N=4$ sites. Spin up and spin down electrons can “hop” between adjacent
conduction orbitals $i$ and $i+1$ via the hybridizations $t_{i}$ and between a
conduction orbital $i$ and localized orbital $i+N$ via the hybridizations
$v_{i}$, but not between adjacent localized orbitals. When two electrons
occupy the same conduction or localized orbital, they have an on-site
repulsion, $U^{c},U^{d}$, respectively.

The central structure of the JCH model, the coupling of one “mobile”
collection of quantum degrees of freedom to a set of localized modes is
reminiscent of the Kondo [22] and periodic Anderson models [23, 24] used to
describe conduction electrons interacting with local magnetic orbitals. The
Hamiltonian of the PAM is

$\displaystyle\hat{\cal H}_{\textnormal{PAM}}$
$\displaystyle=-\sum_{i,\sigma}t_{i}^{\phantom{\dagger}}\big{(}\,\hat{c}_{i+1,\sigma}^{\dagger}\hat{c}_{i\sigma}^{\phantom{\dagger}}+\hat{c}_{i\sigma}^{\dagger}\hat{c}_{i+1,\sigma}^{\phantom{\dagger}}\,\big{)}+$
(2) $\displaystyle\sum_{i,\sigma}v_{i}^{\phantom{\dagger}}\big{(}\,$
$\displaystyle\hat{d}_{i\sigma}^{\dagger}\hat{c}_{i\sigma}^{\phantom{\dagger}}+\hat{c}_{i\sigma}^{\dagger}\hat{d}_{i\sigma}^{\phantom{\dagger}}\,\big{)}+\sum\limits_{i=1}^{N}U^{c}\,\hat{n}^{c}_{i\uparrow}\hat{n}^{c}_{i\downarrow}+U^{d}\,\hat{n}^{d}_{i\uparrow}\hat{n}^{d}_{i\downarrow}\
.$

Similar to the JCH model, the length of the system, $N$, is the number of
conduction sites.
$\hat{c}_{i\sigma}^{\dagger}\big{(}\hat{c}_{i\sigma}^{\phantom{\dagger}}\big{)}$
are conduction electron creation (annihilation) operators at site $i$ and spin
$\sigma$,
$\hat{d}_{i\sigma}^{\dagger}\big{(}\hat{d}_{i\sigma}^{\phantom{\dagger}}\big{)}$
are creation (annihilation) operators of localized electrons at site $i$ and
spin $\sigma$, and
$\hat{n}^{c}_{i\uparrow},\hat{n}^{c}_{i\downarrow},\hat{n}^{d}_{i\uparrow},\hat{n}^{d}_{i\downarrow}$
are the associated number operators.

The model is parameterized by conduction electron hopping $t_{i}$, inter-
orbital hybridizations $v_{i}$, and on-site repulsions $U^{c}$ and $U^{d}$.
The geometry of the PAM is illustrated in Fig. 1 (bottom). The structure in
the top and bottom panels emphasizes the geometric similarities between the
JCH model and the PAM but obscures the fact that the PAM has two spin species
of fermionic conduction electrons, whereas the JCH model has a single species
of mobile bosons. Similarly, the localized orbital accommodates spin-up and
spin-down electrons, unlike the single excitation of the atoms in the JCH.
These differences become manifest when $N_{\rm exc}>1$.

The brute force methodology for solving $\hat{\cal H}_{\rm PAM}$ is identical
to that for $\hat{\cal H}_{\rm JCH}$: choose a basis, construct the matrix,
and diagonalize. The occupation number states for the PAM are, of course,
different from those for the JCH, due to the difference in the statistics of
the operators and the presence of an additional spin degree of freedom in the
PAM.

In the absence of the interaction terms, QST in the PAM with multiple
excitations is the same as for a single excitation – a consequence of the fact
that the many-particle eigenenergies are simple sums of the single particle
excitations when $U^{c}=U^{d}=0$ and all the degrees of freedom share
fermionic statistics. As noted above, this is not the case for the mixed
character of the degrees of freedom in the JCH model. The effect of
interactions on multi-excitation QST in the Hubbard model has been considered
in [25].

###  II.2 Methods

A high-level description of Monte Carlo methods emphasizes their flexibility:
given a collection of degrees of freedom $\\{\,\phi_{i}\,\\}$ and a
probability distribution ${\cal P}\big{(}\\{\,\phi_{i}\,\\}\big{)}$ one
defines an appropriate transition matrix ${\cal
T}\big{(}\\{\,\phi_{i}\,\\}\rightarrow\\{\,\phi_{i}\,\\}^{\prime}\big{)}$ so
that, as ${\cal T}$ is repeatedly invoked to generate a sequence of
configurations, the degrees of freedom are generated with the desired
probability This is done by requiring ${\cal T}$ to obey “detailed balance”
via, for example, the heat bath or Metropolis algorithms. This ensures that
${\cal P}\big{(}\\{\,\phi_{i}\,\\}\big{)}$ is an eigenvector of ${\cal T}$ of
largest eigenvalue. It is not necessary to normalize ${\cal P}$, since only
the ratio ${\cal P}\big{(}\\{\,\phi_{i}\,\\}\,\big{/}\,{\cal
P}\big{(}\\{\,\phi_{i}\,\\}^{\prime}\big{)}$ is needed. .

To accomplish efficient QST, our procedure is then as follows. We define a
target time evolution operator $\hat{\cal U}^{\,\bf t}$ which, having chosen a
basis for our Hilbert space, has matrix elements ${\cal U}^{\,\bf t}_{ab}$
with $a,b$ labeling the basis vectors (rows and columns of $\hat{\cal U}$).
$a$ and $b$ run from $1$ to the dimension ${\cal D}$ of the Hilbert space.
This time evolution operator is a function of the parameters of the
Hamiltonian and of the time, and encodes the desired perfect QST. If we want
to go from a specific basis state $|\alpha\rangle$ (e.g. one which represents
excitations initially localized on particular sites) to another specific state
$|\beta\rangle$ with excitations at some final locations) we require column
$\alpha$ of the matrix for $\hat{\cal U}$ to consist entirely of zeros except
for row $\beta$.

In order to have our Monte Carlo enforce these matrix elements we define an
“action” that quantifies the difference between the matrix of our current time
evolution operator and the target:

$\displaystyle{\cal
S}\equiv\frac{1}{T}\sum_{a,b}{\vphantom{\sum}}^{\prime}\,\Big{(}{\,\cal
U}^{\,\bf t}_{ab}-{\cal U}_{ab}\,\Big{)}^{2}.$ (3)

Here the prime on the sum emphasizes that it runs only over those matrix
elements (columns of ${\cal U}$) which are targeted. If we want to specify
perfect QST for only one pair of initial and final states, the sum in Eq. 3 is
only over one column. Adding requirements for perfect QST between additional
pairs expands the number of terms in the sum for ${\cal S}$. This will be
discussed in Sec. V.

The Monte Carlo then proceeds by making changes to each of the parameters in
the Hamiltonian in succession, and accepting those changes with probability
given by the “heat bath” prescription: $p=e^{-\Delta{\cal
S}}\,/\,\big{(}1+e^{-\Delta{\cal S}}\big{)}$. Here $\Delta{\cal S}$ is the
change in the action induced when the Hamiltonian parameter is altered.
Equation 3 contains a “temperature” $T$. To achieve perfect QST, one desires
to converge to ${\cal S}=0$ so that the time evolution operator exactly
matches the target. This would eventually occur if we set $T=0$ to suppress
all fluctuations about the target. The purpose of making $T$ finite is to
prevent getting caught in metastable states in the course of optimizing the
Hamiltonian to perfect QST, and also to allow large Monte Carlo moves to be
accepted early in the simulation. An effective annealing schedule [27] is to
start at a high temperature $T_{\rm max}$ and then reduce $T$ logarithmically
in steps towards a small but finite $T_{\rm min}$. A procedure which is found
empirically to work reasonably well is to start at the hoppings $J_{i}=1$ and
reduce the temperature from $T_{\rm max}\sim 100$ to $T_{\rm min}\sim 0.1$.

However, this primitive annealing can be improved. The “dual annealing”
method, also called “generalized simulated annealing,” [28, 29] provides
better convergence (a factor of 2-3) [30] and hence is used in the results
which follow. The “dual annealing” method uses the same main accepting and
rejecting protocol as the primitive Monte Carlo but includes several
variations and additional procedures, as described below. We have implemented
some of these changes into our Monte Carlo algorithms to speed up convergence.

The most potent change is that instead of using a uniform distribution to
generate random perturbations, it uses the Cauchy-Lorentz distribution, which
was shown to speed up convergence in [31]. Another key modification is that
while the temperature globally decreases over many iterations, it is
occasionally “re-annealed,” sending the temperature higher for a brief period
and increasing the rate of temperature decline. This puts the optimization
algorithm into alternating periods of frequent, larger changes, and
infrequent, smaller changes. The two frequencies of the temperature annealing
process is where the method “dual annealing” gets its namesake. An important
caveat to this annealing schedule is that the best position is saved before
each re-annealing of the temperature so that if no better solution is found
during the high-activity period, then the program can return to this last
solution and avoid getting sent off track.

We conclude this section by commenting on the computational cost and the
explored phase space. Regarding cost, we must re-compute the matrix for
$\hat{\cal U}$ after each change in a Hamiltonian parameter. This involves
diagonalizing and exponentiating the Hamiltonian, operations that scale with
the cube of the Hilbert space dimension ${\cal D}$. As noted earlier, ${\cal
D}$ is a rapidly increasing function of $N$ and $N_{\rm exc}$. For the JCH
model with $N=8$ and $N_{\rm exc}=2$ we have ${\cal D}=128$, so a sweep
through all 16 parameters $\\{\,J_{i},g_{i}\,\\}$ lattice takes ${\cal
O}\big{(}16\cdot(128)^{3}\big{)}\sim 10^{7}$ operations. Doing $10^{4}$ sweeps
would then represent ${\cal O}(10^{11})$ operations and take tens of minutes
on a several GHz processor workstation. Thus, the time grows rapidly with $N$
and $N_{\rm exc}$. However, the system sizes accessible to the theory are
still of the same order as those being explored experimentally at present
[32].

Regarding the phase space, the JCH and PAM have $2N-1$ and $2N+1$ parameters,
respectively, which are adjusted to optimize for high-fidelity QST. To
transfer each initial and final state pair, there are ${\cal D}$ constraints
involved in matching ${\cal U}$ to ${\cal U}^{\,\bf t}$. We can see that as
$N$ and $N_{\rm exc}$ grow, the number of constraints begins vastly to exceed
the number of adjustable parameters. Thus there is no a priori guarantee that
a good solution will be found. One of our chief messages is that, despite
having (many) more equations than unknowns, achieving good QST is still
feasible.

Figure 2:  The distribution of function calls used to reach target levels of
fidelity for one excitation systems of various sizes. For each number of
cavities, our Monte Carlo methods were run $200$ times with different seeds
and the numbers of function calls used at the benchmarks of $0.95$ (blue),
$0.98$ (red), and $0.99$ (green) fidelity were recorded. In the plot, each box
represents the range between the $25$th percentile and the $75$th percentile
of function calls. The black bar marks the median value and the black diamond
marks the mean. Note that the $y$ axis is on a log scale. The scaling to reach
a fixed fidelity is exponential in $N$. Each line is fit to $A_{f}e^{m_{f}N}$.
Here $A_{1}=272,A_{2}=994,A_{3}=2368$.

##  III QST in the Single-Excitation Sector

Here we will demonstrate our methods on a problem with known solutions – QST
in the single-excitation sector of a 1D coupled cavity array. In this case,
perfect QST can be found reliably by utilizing Monte Carlo with an action that
targets “special eigenvalues” found in [20]. However, in the next section, we
will see that optimizing the time-evolution matrix is a much more versatile
method.

When working in these systems, we define our basis to include the states with
a single excitation, either a photon in one of the cavities or an excitation
of one of the emitters. We define our Monte Carlo’s error as the value in our
time-evolution matrix representing the probability that a particle will
transfer between the leftmost cavity and the rightmost cavity at a time
$t_{p}$.

We observe that these Monte Carlo methods can reliably reach any high fidelity
between these two states given enough Monte Carlo iterations. To benchmark our
methods, we ran independent optimizations many times with different random
seeds and recorded the number of function calls (i.e., the attempt to update a
single Hamiltonian parameter, recomputing the corresponding unitary ${\cal
U}$) required to converge to various fidelity levels for transfer between the
ends of the top row of sites [see Fig. 1(a)]. These results are displayed in
Figure 2.

We observe that the number of necessary function calls increases as we demand
higher fidelity or increase the system’s complexity (distance of propagation).
The correlation between the number of cavities and function calls appears
roughly linear on the log scale, implying an exponential relationship. There
also appears to be a small parity bias favoring systems with an odd number of
cavities. As is typical with stochastic optimization methods, the variance is
quite large. There is no guarantee that a target fidelity will be reached
under any number of function calls. Note that the computational difficulty in
solving larger systems sizes increases both because the optimization requires
more function calls, but also because each function call becomes more
expensive, as adding more states creates a larger matrix that must be
diagonalized to obtain ${\cal U}$.

Our method is not limited to QST between the top leftmost and rightmost sites.
We can target transfer between any pair of states by specifying a different
target time-evolution operator. Aside from transfers between states that
violate a conservation or symmetry law, there are many pairings of states to
choose from. In Fig. 3, we exhaustively demonstrate that all single target
transfers can be achieved in a length $N=3$ system with ${\cal D}=6$ total
states.

Figure 3:  An $N=3$ coupled cavity array with a single excitation. The six
possible states are arranged along the $x$ and $y$ axes. Each square
represents a transfer between two states. The value of each square is the
maximum fidelity achieved after $3000$ function calls when optimizing for that
transfer specifically. Left: Each optimization has coupling where symmetry is
enforced. Right: A repeat of this process without enforcing symmetry on the
couplings.

From the left side of Fig. 3 we observe that the optimization for each state
transfer converged to either $0.5$ or to $1$. We note that every value on the
anti-diagonal has converged to $1$, meaning we can transfer any state to its
“mirror” state with perfect fidelity. Some other elements can only reach $0.5$
due to our symmetry constraints. On the right side without such constraints we
see all optimizations reach almost $1$. As we primarily focus on symmetric
transfers, we will typically apply symmetry constraints to halve the number of
optimization parameters.

##  IV Multiple Excitations

We now move on to the main focus of the paper: QST with multiple, interacting
excitations. This is a difficult task – when adding a second excitation in
either the JCH or PAM models the basis size is roughly squared, but the number
of tunable hoppings stays the same. Thus it is no longer feasible to require
perfect state transfer for all system states. Instead we must focus on a
smaller number of state pairings and refined methods.

###  IV.1 Jaynes-Cummings-Hubbard Model

In the JCH model, the obvious first guess to finding high fidelity QST in the
multiple excitation sector is to use solutions that worked with one
excitation. However, as shown in Ref. [20], the behavior of a two-excitation
system bears little resemblance to that of a one-excitation system with the
same coupling values and therefore the one-excitation solution cannot be used.

Another approach that comes to mind is to apply the eigenvalue targeted
optimization method that we used to find perfect QST in the one-excitation JCH
model in [20]. But this requires foreknowledge of the eigenvalues that lead to
perfect QST, which we do not have in the two-excitation system. In addition,
the eigenvalue patterns in the one excitation system cannot be reached in the
two-excitation case, due to the many differences in their respective formats
discussed in Appendix A.

We therefore turn to our Monte Carlo methods that target the time-evolution
operator. In the JCH system we will transfer from the state with two
(photonic) excitations in the top leftmost cavity to the state with two
excitations in the top rightmost cavity to serve as an example of our method.
Just as described in Sec. III, we compute the time evolution matrix at time
$t_{p}$ and optimize for our desired values.

While the two-excitation JCH system is significantly harder to optimize than
the one-excitation JCH system due to its increased basis size and complexity,
our Monte Carlo methods can still achieve high-fidelity QST given enough
iterations. For example, in Fig. 4, we highlight a solution in an $N=8$ JCH
model that achieves a fidelity of $0.994$.

Note that the transfer time $t_{p}$ can be engineered to any desired value by
scaling the coupling values. As found in Ref. [11], and is clear from
dimensional analysis, multiplying each coupling value by a constant $J_{0}$
scales the transfer time by $J_{0}^{-1}$.

Coupling Values
$J_{1},J_{7}$ $J_{2},J_{6}$ $J_{3},J_{5}$ $J_{4}$ 926.29 5124.13 4273.83
9241.97 $g_{1},g_{8}$ $g_{2},g_{7}$ $g_{3},g_{6}$ $g_{4},g_{5}$ 15384.66
16447.19 16654.13 2055.53

Figure 4:  Demonstration of high-fidelity transfer between two states in the
JCH model with two excitations in $N=8$ cavities. We consider transfer
beginning with two photons co-located in the leftmost cavity and transferred
to the rightmost cavity. In the figure, we plot the fidelity of this transfer
over time. The peak fidelity is $0.994$. In the top left inset, note that the
system oscillates rapidly between having both particles in one cavity and
having one particle in the cavity and one in its associated emitter. The
period of these oscillations is approximately $10^{5}$ times smaller than the
principal oscillation’s period. To illustrate this principal oscillation, the
main plot displays the sum of the probability of these two states. This sum is
close to a sine wave, with small disordered oscillations visible in the lower
right inset.

It is interesting to note that the evolution of this system has several
differences from its one-excitation and cavity-only counterparts. The fidelity
over time graph for cavity-only perfect QST was characterized by a single sine
wave [11], and the single excitation JCH model appeared to be a combination of
two similar frequencies. However, the solution in Figure 4 contains an
extremely rapid oscillation between cavity and emitter that is superimposed
upon the main oscillatory behavior. Physically, this means that getting both
excitations across the chain is almost guaranteed, but determining whether the
emitter is excited requires extremely high precision. In future research, it
may be beneficial to create a method that finds solutions without rapid
oscillations.

The transfer time is also significantly larger than in the cavity-only and
single-excitation systems relative to the scale of the coupling values. In
both of those systems, the transfer time was inversely proportional to the
scale of all the hopping rates. As some of the coupling rates in the two-
excitation system are significantly larger than in the cavity-only and single-
excitation solutions, one might expect the transfer time to be shorter, but
instead, it is nearly two orders of magnitude larger. Since the two-excitation
system has a significantly larger basis size, the longer transfer time may be
because more frequencies must coincide simultaneously to get perfect QST.
However, the higher coupling values explain the rapid oscillations between the
end cavity and emitter.

We can benchmark the number of function calls required to reach high-fidelity
transfer in the two-excitation case as well. This is recorded in Fig. 5 using
the same methods as the convergence plot for the one-excitation case in Fig.
2. While the exponential dependence on the number of cavities persists, the
prefactor $m_{f}$ in the exponential is typically much larger, rendering much
more costly annealing. This indicates that the landscape of solutions that
minimize ${\cal S}$ is shallower and that the number of solutions that reach
the target fidelity is reduced – this is characteristic of increasingly
quantum chaotic systems.

Figure 5:  Distribution of function calls needed to reach target levels of
fidelity for two-excitation systems. In each system, we target the transfer of
two particles localized in the top left cavity transferring to the top right
cavity. Our Monte Carlo methods were tested in the same manner as in Fig. 2.
Each optimization run was given a maximum of $400,000\cdot(\text{number of
independent variables})$ function calls before termination. The percentage of
runs that reached the benchmark within the maximum function calls is annotated
below each bar when it is not $100\%$ Note that the $y$ axis is on a log
scale. Each line is fit to $A_{f}e^{m_{f}N}$. Here
$A_{1}=657,A_{2}=1161,A_{3}=610$.

###  IV.2 Periodic Anderson Model

We now move on to the Periodic Anderson Model. In this model, adding a second
excitation (of opposite spin) will introduce correlations due to the
interaction constants $U^{c}$, $U^{d}$ in the Hamiltonian. Without such
interactions, the system is identical to two separate one-excitation systems,
and the solution is unchanged.

In Ref. [25], it was found that in the single-band Hubbard model, near-perfect
QST could be achieved using the single excitation solution and certain optimal
values of the interaction $U$. This prompted us to search for similar values
in the PAM. We begin with the solved coupling values for the single-excitation
system and manually search for special $U^{c},U^{d}$ values that exhibit high-
fidelity QST with two excitations. We applied this method for an $N=3$ length
system, but we see that even for such a small case, we do not find nontrivial
$U^{c},U^{d}$ that allow a one-excitation solution to work with two
interacting excitations. In a sweep of both $U^{c}$ and $U^{d}$ from $0$ to
three times the average coupling value, the highest nontrivial solution had
fidelity $0.229$ when transferring one up and one down electron together
across the top chain and $0.817$ when swapping the positions of a top and down
electron placed at opposite ends of the top chain. Thus, it is clear that we
must find entirely new solutions.

We move on to applying our time-evolution matrix targeting Monte Carlo methods
to the PAM systems. These methods are implemented in a similar way as the JCH
system; we vary our coupling values, create and diagonalize the Hamiltonian,
and compute the error from the time evolution matrix. We find that our
prescription can still find high fidelity QST in the PAM system, as
demonstrated in Fig. 6. In this solution, we achieve fidelity of $0.996$.

The behavior of this system over time appears to follow a sinusoidal path with
an additional oscillation of higher frequency. In contrast to the JCH model,
we see much lower amplitude rapid oscillations in the transfer fidelity. This
oscillation is between the target state with both fermions in the top right
site and a equal superposition of the two states with one fermion in the top
right site and one in the rightmost offshoot site (conduction and localized
orbitals). A benefit to this is that any measurements will be less impacted by
imprecision at the measurement time. The transfer time is of a similar order
of magnitude to the two-excitation JCH solution.

One difficulty we encounter when searching for solutions in the PAM system is
that we must avoid the “trivial solution.” When both $U^{c}=U^{d}=0$ the
system is effectively interactionless and the one-excitation solution
reemerges. To counteract this, we placed bounds on how small the interaction
coupling values can be. In our algorithms, we discouraged the trivial solution
by increasing the calculated error of proposed coupling values if either of
the on-site interactions were less than $0.1$ times the average strength of
the coupling values.

In terms of convergence, although our optimization method is the same as for
the JCH model, between a PAM system and a JCH system of equal length, the PAM
system will be more computationally expensive to optimize, assuming the two
excitations have opposite spin. In that case, the excitations in the PAM model
are distinguishable, and the basis will be roughly twice as large, meaning
eight times more difficult to diagonalize. In addition, the PAM has two more
independent variables that must be optimized, namely the interaction constants
$U^{c}$, $U^{d}$.

Ultimately we conclude that we can find high-fidelity solutions for the PAM
with multiple interacting particles. We found high-fidelity transfers for
multiple pairs of states in PAM systems with two electrons and up to $10$
total sites. This further shows the efficacy of this method in optimizing QST
in a large variety of challenging models.

Coupling and interaction values
$t_{1},t_{3}$ $t_{2}$ $v_{1},v_{4}$ $v_{2},v_{3}$ $U^{c}$ $U^{d}$ 419.87
1188.11 478.51 1292.36 13683.6 4014.55

Figure 6:  High-fidelity transfer between two states in the $N=4$ PAM with one
spin-up fermion and one spin-down fermion. We used our Monte Carlo algorithms
to target transfer between the state corresponding to both fermions localized
in the top left site and the state corresponding to both in the top right
site. This transfer is challenging as the on-site repulsion components of the
Hamiltonian tend to discourage the particles from localizing in the same site.
It is important to note that although the optimization algorithm can tune
these repulsion constants, the final result does not minimize these values.
The system reaches a peak fidelity of $0.996$.

##  V Multiple Transfers

In this section, we will target Hamiltonians which achieve good QST between
several different pairs of initial and final states, with multiple
excitations. As we have detailed in our methods section, we can select any
number of matrix elements to be optimized in our Monte Carlo. However, the
difficulty of finding a solution increases with the number of constraints. We
reliably found high-fidelity QST when targeting only one pair of states in
Sec. IV, but on the other extreme, our searches indicate that there is no good
solution whereby every state is reflected across the center (although such a
“complete” solution is possible in the single-excitation sector [20]). We will
now investigate how many sets of states we can transfer with one single set of
coupling values.

To test the impact of adding constraints to our Monte Carlo algorithms, we run
a series of optimizations on an $N=4$ JCH system with $N_{\rm exc}=2$. First,
we create a list of all desired pairs of states to transfer between by pairing
all states with their mirror image states (ignoring self-symmetric ones).
Then, we run many independent optimizations for different numbers of target
transfers. On each run with $N_{T}$ target transfers, we randomly selected
$N_{T}$ pairs of states from our set. When calculating the final error we
divide by $N_{T}$ so that the error is one minus the average fidelity of all
transfers. The results of these optimizations are shown in Fig. 7.

As expected, adding more target transfers (more constraints), utilizing the
same set of couplings, results in lower average fidelity. The decrease from
$N_{T}=1$ to $N_{T}=2$ is the most dramatic, meaning we cannot guarantee very
high fidelity for anything more than one target transfer. However, even with
six transfers, we can achieve an average fidelity of eighty percent after just
$2\cdot 10^{6}$ function calls, which is promising. In real-world
applications, the fidelity may be higher than seen here–we have chosen random
transfers for this assessment, but the fidelity is typically much higher when
the target transfers are similar to each other.

Figure 7:  Average convergence of fidelity for different numbers of target
transfers $({N_{T}})$ in an $N=4$ JCH system. The $x$ axis is the number of
function calls. The average error along the $y$ axis is one minus the average
transfer fidelity, averaged over one hundred independent optimizations. Note
that these average errors are far above the lowest errors we can find given
enough iterations. Indeed, the best average fidelities of our $100$ runs for
$N_{T}$ transfers one through six were
$(1.000,0.994,0.991,0.986,0.976,0.972)$, respectively. So while it is the case
that an optimization with fewer target transfers usually converges more
rapidly, we emphasize that we can eventually achieve high fidelity in the
systems with many target transfers.

##  VI Conclusions

We have demonstrated the efficacy of a novel Monte Carlo/dual annealing method
at finding high-fidelity QST in systems with multiple, interacting
excitations. To optimize the fidelity between two states in a system, we used
tailored Monte Carlo algorithms that optimize our coupling values to produce
the time-evolution matrix corresponding to our transfer. In addition, we have
benchmarked the convergence in the JCH model with and without multiple
excitations to quantify the difficulty that interacting excitations have on
engineering good QST. Lastly, we gauged the limits of our optimizations by
solving for increasing numbers of state transfers in the same system.

We note that one can also perform a Monte Carlo in the Hamiltonian parameters
to minimize the “infidelity”, $1-|\langle\psi(t_{\rm QST}|\psi_{\rm
target}\rangle|^{2}$, that is, the difference between the wave function at
some QST transfer time and a target wavefunction [17]. This latter approach
has some potential advantages in scaling with the Hilbert space dimension,
since one can use Lanczos-type algorithms to compute the action of ${\cal U}$
on the initial wave function rather than constructing the full matrix of
${\cal U}$. Exploration of the most efficient optimization procedure is an
interesting area for future work.

Initial time evolution studies in the Jaynes-Cummings-Hubbard model in the
single-excitation sector are described in [20]. There, a different approach to
determining the Hamiltonian was used, one in which minimization of the action
altered the hoppings such that a set of target eigenvalues was achieved. The
method proposed here is much more powerful for two reasons. Most importantly,
while the eigenvalues that will result in good QST are known in certain
special cases [11], they are not known generally. The approach described here
eschews the need for preknowledge of the eigenvalues and focuses only on the
actual objective, namely QST. It also has the flexibility to search for
different types of evolution (multiple pairs of initial and final states).
Meanwhile, the practicality of the two methods is comparable. Both involve
multiple diagonalizations of a ${\cal D}$ dimensional matrix, and both involve
tuning ${\cal O}(N)$ free parameters in order to achieve ${\cal O}({\cal D})$
constraints (a column of $\hat{\cal U}$ in the approach described here, the
eigenvalues of $\hat{\cal H}$ in the approach of Ref. [20]).

In recent years, high-fidelity state transfer has been a central objective of
the quantum information science community, explored experimentally in coupled
atoms and photons [33, 34] using microwave photons [35], quantum dots [36] and
with superconducting qubits [37, 17].

In particular, experimental realizations of the JCH model have progressed in
various platforms [38, 32]. As quantum computers increase in scale, the high-
fidelity transfer of states will be critical to achieving an interconnected
quantum network. Our Monte Carlo approach provides a path for optimizing
couplings, which is practical for networks of the same size as those currently
constructed. The work reported here was performed on a small collection of
standard workstations, allowing the possibility of substantially larger
systems using more powerful hardware, even though the computational cost
scales exponentially.

A reasonable extension that follows from the versatility of the methods we
outlined in this paper is to apply these methods to other models, for example,
the Kondo and Hubbard Hamiltonians. Another interesting extension is the
application of the methods developed here to higher dimensions. Initial work
with 2D superconducting qubit arrays has been undertaken in Ref. [17]. The
results reported there point to the generality of using Monte Carlo to
engineer high-fidelity QST between desired states in higher dimension.

Acknowledgements A.Y. and R.T.S. were supported by the grant DE‐SC0014671
funded by the U.S. Department of Energy, Office of Science. R.M. acknowledges
support from the NSFC Grants No. NSAF-U2230402, 12111530010, 12222401, and No.
11974039. A.Y. conducted the simulations. All authors participated in the
ideas behind the research, the analysis of the data, and the writing of the
manuscript.

##  Appendix A Eigenvalues of bosonic, fermionic, and mixed statistics

Prior investigations of Monte Carlo optimization of QST [20] were centered
around the eigenvalues of the Hamiltonian. This numerical work generalized the
discussion of [11], which found that integer spacing between the ratios of
successive eigenvalues in a one-excitation system’s Hamiltonian was both a
necessary and sufficient condition for that system to exhibit perfect QST.

While we have emphasized in this paper that targeting the time-evolution
operator $\hat{\cal U}$ via Monte Carlo appears to be the more flexible and
general method for achieving high-fidelity QST, in this Appendix, for
additional insight, we explore two features of the eigenvalue distribution for
two-excitation QST: (i) a comparison of the ratios of the eigenvalues after
the Monte Carlo has optimized ${\cal U}$ to initial random couplings; and (ii)
the relation between two excitation and single excitation eigenvalues for the
JCH model.

###  A.1 Eigenvalue properties after optimization of $\hat{\cal U}$

One indication that eigenvalues still remain pertinent to QST in systems with
multiple, interacting excitations is an analysis of the ratios between
successive eigenvalues for optimized Hamiltonians and for random ones. As
found by [11], the ratios of spaces between eigenvalues is critical for QST in
the one-excitation sector. We examine a large sample of independently
optimized coupling values and determine if the ratios between successive
eigenvalues follow a different distribution than when the coupling values are
chosen randomly. Indeed, we noticed a distinct difference between the two
distributions—this ratio is close to $1$ significantly more often in systems
of optimized couplings than in systems of random couplings. A deeper analysis
of systems optimized for QST using random matrix theory is done in [17].

Coupling Values

$J_{1},J_{5}$ | $J_{2},J_{4}$ | $J_{3}$
---|---|---
20.531 | 5.309 | 16.228
$g_{1},g_{6}$ | $g_{2},6_{5}$ | $g_{3},6_{4}$
1.023 | 1.544 | 3.645

Figure 8:  This figure highlights the pattern of eigenvalues in our six-length
JCH model solved for many transfers. In blue, we plot the $12$ eigenvalues for
our optimized Hamiltonian with one excitation. Only the nonnegative
eigenvalues are displayed in the plot, but each eigenvalue comes with a
negative counterpart. Degenerate eigenvalues are plotted together and labelled
with their multiplicity. In red, we plot the $78$ linear combinations of two
of the $12$ base eigenvalues. In green, we plot the $72$ eigenvalues of the
system with two excitations. Note that the green two-excitation eigenvalues
overlap extremely closely with the red combinations of the base eigenvalues.
This pattern is not typically observed for systems that do not exhibit QST.

Another interesting pattern emerges in certain small systems. In our
optimizations for these systems we find some some rare solutions with many
high fidelity transfers. For instance, in a $6$ length system we found a
system that has $9$ unique simultaneous high fidelity transfers (those being
the states where excitations in the cavities are transferred to their
horizontally mirrored states). In such a system, we would expect that any
patterns in the eigenvalues that relate to QST would be more apparent.

###  A.2 Multi-excitation and single-excitation eigenvalues relationship

It is well known that Hamiltonians which are quadratic in the creation and
destruction operators have a many-particle spectrum that is obtained trivially
from the single-particle one, $E^{(1)}_{\alpha}$. If the operators are
fermionic, the two-particle eigenvalues are
$\\{E^{(2)}\\}=E^{(1)}_{\alpha}+E^{(1)}_{\beta}$, with $\alpha>\beta$, while
if the operators are bosonic the same relation holds with $\alpha\geq\beta$.
The fact that $\alpha<\beta$ is not counted separately from $\alpha>\beta$
reflects the indistinguishabilty of the quantum particles, while the Pauli
exclusion principle removes $\alpha=\beta$ for fermions. In considering the
spectrum of the JCH model, which is quadratic but has operators of mixed
character (bosonic photons and fermionic emitters), it is natural to ask if
the collection of two excitation eigenvalues $\\{E^{(2)}\\}$ nevertheless have
any relationship to $E^{(1)}_{\alpha}$. (It is clear the eigenvalues can never
be precisely given from single excitation combinations, as the “blocking” of
excitations in emitters reduces the number of possible states, and hence the
number of eigenvalues is different.)

To address this question, we first optimized $\hat{\cal U}$ for high-fidelity
QST for an $N=6$ JCH system with two excitations, and then computed the
resulting eigenvalues. Next, we computed the eigenvalues for the same system
and the same coupling values but with one excitation. In this case, we indeed
see a high overlap of eigenvalues between the multiple excitation system and
combinations of the base system eigenvalues. We found that this attribute of
high overlap of eigenvalues is descriptive but not prescriptive of high
fidelity QST as there exist many systems with this quality do not exhibit high
fidelity QST..

Despite the fact that the Hamiltonian appears to be quadratic, the operators
from which it is built do not all share a common set of (bosonic or fermionic)
commutation relations. Thus the many particle eigenenergies and eigenfunctions
cannot be built in a simple way from the one particle sector, as is the case
with a purely bosonic or fermion quadratic Hamiltonian.

While we have presented data here for the JCH model, the application of these
eigenvalue analyses on the Periodic Anderson Model is a interesting subject
for further study. In the PAM, interactions are introduced through the on-site
interactions $U^{c},U^{d}$. This form of interactions has the benefit of not
changing the basis size, which keeps the number of eigenvalues unaltered.
However, the effect of the interaction constants $U^{c},U^{d}$ on the
eigenvalues appears to be significantly more complex than in the JCH model,
and no patterns associated with high fidelity QST have been observed during
optimization.

## References

  * [1] Here and throughout this paper we set $\hbar=1$.
  * Cirac  _et al._ [1997] J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, Quantum state transfer and entanglement distribution among distant nodes in a quantum network, Physical Review Letters 78, 3221 (1997).
  * DiVincenzo [2000] D. P. DiVincenzo, The physical implementation of quantum computation, Fortschritte der Physik: Progress of Physics 48, 771 (2000).
  * Beals  _et al._ [2013] R. Beals, S. Brierley, O. Gray, A. W. Harrow, S. Kutin, N. Linden, D. Shepherd, and M. Stather, Efficient distributed quantum computing, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, 20120686 (2013).
  * Preskill [2018] J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  * Stute  _et al._ [2013] A. Stute, B. Casabone, B. Brandstätter, K. Friebe, T. Northup, and R. Blatt, Quantum-state transfer from an ion to a photon, Nature photonics 7, 219 (2013).
  * Olmschenk  _et al._ [2009] S. Olmschenk, D. Matsukevich, P. Maunz, D. Hayes, L.-M. Duan, and C. Monroe, Quantum teleportation between distant matter qubits, Science 323, 486 (2009).
  * Vogell  _et al._ [2017] B. Vogell, B. Vermersch, T. E. Northup, B. P. Lanyon, and C. A. Muschik, Deterministic quantum state transfer between remote qubits in cavities, Quantum Science and Technology 2, 045003 (2017).
  * Li  _et al._ [2018] X. Li, Y. Ma, J. Han, T. Chen, Y. Xu, W. Cai, H. Wang, Y. Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, Perfect quantum state transfer in a superconducting qubit chain with parametrically tunable couplings, Phys. Rev. Appl. 10, 054009 (2018).
  * Bienfait  _et al._ [2019a] A. Bienfait, K. J. Satzinger, Y. Zhong, H.-S. Chang, M.-H. Chou, C. R. Conner, É. Dumur, J. Grebel, G. A. Peairs, R. G. Povey, _et al._ , Phonon-mediated quantum state transfer and remote qubit entanglement, Science 364, 368 (2019a).
  * Christandl  _et al._ [2004] M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, Perfect state transfer in quantum spin networks, Phys. Rev. Lett. 92, 187902 (2004).
  * Kay [2006] A. Kay, Perfect state transfer: Beyond nearest-neighbor couplings, Phys. Rev. A 73, 032306 (2006).
  * Kay [2010] A. Kay, Perfect, efficient, state transfer and its application as a constructive tool, International Journal of Quantum Information 08, 641 (2010).
  * Keele and Kay [2022] C. Keele and A. Kay, Combating the effects of disorder in quantum state transfer, Phys. Rev. A 105, 032612 (2022).
  * Kandel  _et al._ [2021a] Y. P. Kandel, H. Qiao, S. Fallahi, G. C. Gardner, M. J. Manfra, and J. M. Nichol, Adiabatic quantum state transfer in a semiconductor quantum-dot spin chain, Nature Communications 12, 1 (2021a).
  * van Diepen  _et al._ [2021] C. J. van Diepen, T.-K. Hsiao, U. Mukhopadhyay, C. Reichl, W. Wegscheider, and L. M. K. Vandersypen, Electron cascade for distant spin readout, Nature Communications 12, 77 (2021).
  * Xiang  _et al._ [2024] L. Xiang, J. Chen, Z. Zhu, Z. Song, Z. Bao, X. Zhu, F. Jin, K. Wang, S. Xu, Y. Zou, H. Li, Z. Wang, C. Song, A. Yue, J. Partridge, Q. Guo, R. Mondaini, H. Wang, and R. T. Scalettar, Enhanced quantum state transfer: Circumventing quantum chaotic behavior (2024), arXiv preprint, arXiv:2402.00936 [quant-ph] .
  * [18] The Christandl prescription [11] automatically accomplishes perfect QST between all pairs representing transitions between states located symmetrically about the chain center, as might be inferred from the symmetry of the $\\{\,J_{i}\,\\}$.
  * Angelakis  _et al._ [2007] D. G. Angelakis, M. F. Santos, and S. Bose, Photon-blockade-induced Mott transitions and ${XY}$ spin models in coupled cavity arrays, Phys. Rev. A 76, 031805 (2007).
  * Baum  _et al._ [2022] E. Baum, A. Broman, T. Clarke, N. C. Costa, J. Mucciaccio, A. Yue, Y. Zhang, V. Norman, J. Patton, M. Radulaski, and R. T. Scalettar, Effect of emitters on quantum state transfer in coupled cavity arrays, Phys. Rev. B 105, 195429 (2022).
  * Jordan and Wigner [1928] P. Jordan and E. Wigner, Über das Paulische Äquivalenzverbot, Zeitschrift für Physik 47, 631–651 (1928).
  * [22] F. B. Anders, The Kondo effect, Correlated Electrons: From Models to Materials .
  * Sinjukow and Nolting [2002] P. Sinjukow and W. Nolting, Exact mapping of periodic Anderson model to Kondo lattice model, Physical Review B 65, 212303 (2002).
  * Vekić  _et al._ [1995] M. Vekić, J. Cannon, D. Scalapino, R. Scalettar, and R. Sugar, Competition between antiferromagnetic order and spin-liquid behavior in the two-dimensional periodic Anderson model at half filling, Phys. Rev. Lett. 74, 2367 (1995).
  * Yang  _et al._ [2006] S. Yang, Z. Song, and C. P. Sun, Quantum state swapping via a qubit network with Hubbard interactions, Phys. Rev. B 73, 195122 (2006).
  * [26] This is done by requiring ${\cal T}$ to obey “detailed balance” via, for example, the heat bath or Metropolis algorithms. This ensures that ${\cal P}\big{(}\\{\,\phi_{i}\,\\}\big{)}$ is an eigenvector of ${\cal T}$ of largest eigenvalue. It is not necessary to normalize ${\cal P}$, since only the ratio ${\cal P}\big{(}\\{\,\phi_{i}\,\\}\,\big{/}\,{\cal P}\big{(}\\{\,\phi_{i}\,\\}^{\prime}\big{)}$ is needed.
  * Kirkpatrick  _et al._ [1983] S. Kirkpatrick, C. D. Gelatt Jr, and M. P. Vecchi, Optimization by simulated annealing, Science 220, 671 (1983).
  * Xiang  _et al._ [2013] Y. Xiang, S. Gubian, B. Suomela, and J. Hoeng, Generalized Simulated Annealing for Global Optimization: The GenSA Package, The R Journal 5, 13 (2013).
  * Pedamallu and Ozdamar [2008] C. S. Pedamallu and L. Ozdamar, Investigating a hybrid simulated annealing and local search algorithm for constrained optimization, European Journal of Operational Research 185, 1230 (2008).
  * Mullen [2014] K. M. Mullen, Continuous global optimization in R, Journal of Statistical Software 60, 1–45 (2014).
  * Tsallis and Stariolo [1996] C. Tsallis and D. A. Stariolo, Generalized simulated annealing, Physica A: Statistical Mechanics and its Applications 233, 395 (1996).
  * Li  _et al._ [2022] B.-W. Li, Q.-X. Mei, Y.-K. Wu, M.-L. Cai, Y. Wang, L. Yao, Z.-C. Zhou, and L.-M. Duan, Observation of non-Markovian spin dynamics in a Jaynes-Cummings-Hubbard model using a trapped-ion quantum simulator, Phys. Rev. Lett. 129, 140501 (2022).
  * Matsukevich and Kuzmich [2004] D. Matsukevich and A. Kuzmich, Quantum state transfer between matter and light, Science 306, 663 (2004).
  * Majumdar  _et al._ [2012] A. Majumdar, A. Rundquist, M. Bajcsy, V. D. Dasika, S. R. Bank, and J. Vučković, Design and analysis of photonic crystal coupled cavity arrays for quantum simulation, Phys. Rev. B 86, 195312 (2012).
  * Kurpiers  _et al._ [2018] P. Kurpiers, P. Magnard, T. Walter, B. Royer, M. Pechal, J. Heinsoo, Y. Salathé, A. Akin, S. Storz, J.-C. Besse, _et al._ , Deterministic quantum state transfer and remote entanglement using microwave photons, Nature 558, 264 (2018).
  * Kandel  _et al._ [2021b] Y. P. Kandel, H. Qiao, S. Fallahi, G. C. Gardner, M. J. Manfra, and J. M. Nichol, Adiabatic quantum state transfer in a semiconductor quantum-dot spin chain, Nature Communications 12, 2156 (2021b).
  * Bienfait  _et al._ [2019b] A. Bienfait, K. J. Satzinger, Y. Zhong, H.-S. Chang, M.-H. Chou, C. R. Conner, É. Dumur, J. Grebel, G. A. Peairs, R. G. Povey, _et al._ , Phonon-mediated quantum state transfer and remote qubit entanglement, Science 364, 368 (2019b).
  * Toyoda  _et al._ [2013] K. Toyoda, Y. Matsuno, A. Noguchi, S. Haze, and S. Urabe, Experimental realization of a quantum phase transition of polaritonic excitations, Phys. Rev. Lett. 111, 160501 (2013).
  * [39] We found that this attribute of high overlap of eigenvalues is descriptive but not prescriptive of high fidelity QST as there exist many systems with this quality do not exhibit high fidelity QST.