1] Instituto de Física Enrique Gaviola, Universidad Nacional de Córdoba ,
CONICET, Facultad de Matemática, Astronomía, Física y Computación, Av. Medina
Allende s/n, Ciudad Universitaria, CP:X5000HUA Córdoba, Argentina.

2] Instituto de Modelado e Innovación Tecnológica (CONICET-UNNE) and Facultad
de Ciencias Exactas, Naturales y Agrimensura, Universidad Nacional del
Nordeste, Avenida Libertad 5400, W3404AAS Corrientes, Argentina.

# Pretty good quantum state transfer on isotropic and anisotropic Heisenberg
spin chains with tailored site dependent exchange couplings

Pablo Serra  Alejandro Ferrón  Omar Osenda  [  [

###### Abstract

Using a global optimization algorithm we obtain spin chains with site-
dependent exchange coefficients which allow almost perfect quantum state
transfer between the extremes of the chains without any further time-dependent
external control. We consider chains with isotropic and anisotropic Heisenberg
Hamiltonian with up to 100 spins. The method allow us to choose the arrival of
the transferred state by changing the range available to the exchange coupling
strengths. We consider short transferred times, in particular shorter than
those achievable with known time-dependent control schemes. The chains
obtained with the optimization method show some remarkable and interesting
traits as, for instance, the scaling of the magnitude of the exchange
couplings needed to achieve near perfect state transfer with the length of the
chain and the arrival time. This scaling makes it possible to decide if the
chain with site-dependent coefficients can be implemented in an actual system
according to the range of interactions allowed in it. We compare the
robustness of the transmission protocol against static disorder on the
exchange coefficients using different figures of merit, which allow us to
conclude that the isotropic Heisenberg chain is the best option when compared
with anisotropic chains. This comparison is relevant since the method allow us
to start with chains that all achieve near perfect quantum state transfer,
regime that was not available previously in these chains without time-
dependent external control.

##  1 Introduction

The transfer of quantum state along communication channels has been under
study for almost two decades now as a task of primordial importance to Quantum
Information Processing [1]. The cornerstone of the subject on quantum spin
chains was laid by Bose [2, 3], who introduced the figure of merit to analyze
the quality of the resulting transfer processes.

The most simple transfer protocol on a spin chain can be described rather
easily: a given state is prepared at one extreme of the chain and after some
time another state, the transferred one, is retrieved at the other extreme.
The transfer is better for larger values of the fidelity, and perfect if the
state retrieved is the same that the one that was prepared, which corresponds
to the maximum value allowed to the fidelity.

The dynamical behaviour between the preparation and retrieval times can be
controlled but, in many cases, it is preferred that the time evolution of the
quantum state is governed only by the time-independent spin chain Hamiltonian,
which depends only on the exchange couplings between the spins on the chain.
The unforced dynamics is preferred under the rationale that the precise tuning
of the control fields applied to the quantum spin chain add other sources of
noise that spoil the state transfer process [4, 5, 6, 7, 8]. However, the
strength of the exchange couplings can be modulated to improve the quality of
the transfer process, resulting in a number of strategies and dynamical
regimes [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]

The addition of noise sources is problematic enough but it is not the only
drawback in actual implementations, since the strength of the coupling between
the spin chain and the control fields and the frequency spectrum available are
also causes for concern [14].

Despite what has been said above, much progress has been made in the field of
controlled state transfer, from the study of the most basic aspects of
controllability [15, 16, 18, 17], to obtaining particular states in different
types of chains [23, 19, 21, 22]. There is a host of physical systems where
some quantum state transfer process can be implemented [24, 25, 26, 27, 28,
29, 30, 31]. The strategies to control a given chain are also multiple,
varying from using only one actuator [18, 19, 20], two actuators [33] or even
controlling almost all (or all) the spins on the chain [34, 35, 36].

On the other hand, perfect state transfer without other interactions than the
exchange couplings between the spins of the chain can only be achieved in
chains with the so called XX Hamiltonian [5]. The Heisenberg Hamiltonian only
admits what has been termed pretty good [38, 37, 39, 40] or near perfect state
transfer, i.e. choosing adequate values for the exchange couplings strength it
is possible to obtain fidelity values differentially close to unity. This
scenario can be rigorously proved for chains whose length is of the form
$N=2^{k}$, where $k$ is some natural number. It is interesting that these
results can be proved using linear algebra theorems that provide insights
about the Hamiltonian spectrum.

Regrettably, the theorems that assure that the pretty good state transfer
regime exists do not provide neither the order of magnitude of the couplings
nor the transmission time at which to look for them. Taking this into account,
the optimization of the fidelity as a function of the exchange couplings
strength and the transfer time can offer a different approach to the problem.
There has been attempts in this direction, for instance in Ref. [41] machine
learning is employed to obtain the exchange couplings of the XX Hamiltonian
that is associated to perfect state transfer. We are interested in calculate a
set of parameters of the Hamiltonian that places us in the pretty good
transfer regime. That is, we do not require the best set, but a very good set
of parameters. The problem will be written as the finding of a set of
parameters that produce a (very good local) maximum in a cost function.

These class of problems, where we look for good maximum of an arbitrary cost
function, are suitable to be treated by stochastic optimization methods. In
particular, in this paper we use a method developed by one of us, which has
been shown to be very efficient in optimizing functions in parameter space of
dimensions even greater than those of interest in this study: the pivot method
(PM) for global optimization [42, 43]. The pivot method is a stochastic method
to search for the global minimum of a multiple-minimum function defined on a
high-dimensional phase space. To find the global minimum of a high-dimensional
corrugated landscape is, in general, a NP-hard problem, and therefore there is
not a stochastic method that can assure that the global minimum is reached.
However, the pivot method has shown to be an efficient method to find a very
good minimum, that actually is what we need in the search for good quantum
state transfer in chain-spin systems described in this paper.

The paper is organized as follows, In Section 2 the model Hamiltonians to be
considered are introduced together with the transfer protocol and the figures
of merit that quantify the quality of the transfer process, in particular the
relationship between the fidelity of transmission and the transferred
population, which is the probability that a single excitation is transferred
from one end of the chain to the other. In this Section we also describe how
the optimization algorithm must be implemented to find the site-dependent
exchange couplings that maximize the transferred population for a chosen
transferral time, at the same time we introduced all the notation related to
the numerical procedure.

In Section 3 the transfer properties of isotropic Heisenberg spin chains with
modulated exchange coupling coefficients are thoroughly analyzed. Particular
emphasis is placed on obtaining chains that transmit states with high fidelity
for different transfer (or arrival) times. It is shown that the magnitude of
the largest exchange coefficient of a modulated chain depends on the arrival
time and an approximate formula that fits the dependency is proposed. It is
studied how the transmission of a chain whose exchange coefficients have been
optimized is affected by the presence of static disorder. It is shown that the
transferred population shows scaling properties as a function of time when
comparing the transfer of chains, of fixed length, but whose coefficients have
been optimized for different arrival times. We also present a brief discussion
about the quality of the transfer protocol if the initial state can not be
prepared exactly like the protocol prescribes. Once the relationship between
the chain length and the maximum value of the exchange coefficients needed to
achieve near perfect quantum state transfer at a given arrival time is
established, we discuss in which kind of physical system it could be possible
to implement chains like the ones treated in the Section.

The analysis described in the paragraph above is repeated in Section 4 for
anisotropic Heisenberg spin chains. The conclusions and the discussion of our
results are presented in Section 5 where we discuss the different dynamical
behaviours that appear in the disordered spin chains.

##  2 Model Hamiltonians, transfer protocol and Methods

The XXZ Hamiltonian of a quantum spin chain with only nearest-neighbor
interactions can be written as

$H=-\sum_{i=1}^{N-1}J_{i}\left(\sigma_{i}^{x}\sigma_{i+1}^{x}+\sigma_{i}^{y}\sigma_{i+1}^{y}+\Delta\sigma_{i}^{z}\sigma_{i+1}^{z}\right),$
(1)

where $N$ is the number of spins, or the length of the chain, the $\sigma$’s
are the Pauli matrices, the $J_{i}$’s are the exchange coupling strengths,
$J_{i}>0,\forall i$, and $\Delta$ is an anisotropy parameter. The Heisenberg
Hamiltonian correspond to the isotropic case, $\Delta=1$ and $\Delta=0$ is
known as the XX Hamiltonian. Note that the Hamiltonian in Eq. (1) is
adimensional, besides we take $\hbar=1$.

The Hamiltonian in Eq. (1) commutes with the total magnetization in the
$z-$direction

$\left[H,\sum_{i}\sigma_{i}^{z}\right]=0,$ (2)

so the Hamiltonian can be diagonalized in subspaces with fixed number of
excitations, i.e. in subspaces with a given number of spins up. It is
customary to use the computational basis, where for a single spin
$|0\rangle=|\downarrow\rangle$ and $|1\rangle=|\uparrow\rangle$, so
$|\mathbf{0}\rangle=|0000\ldots 0\rangle$ is the state with zero spins up of
the whole chain, and

$\sum_{i=1}^{N}\sigma_{i}^{z}|\mathbf{0}\rangle=-N|\mathbf{0}\rangle.$ (3)

The $N$ states with a single spin up are denoted as follows

$|\mathbf{1}\rangle=|10\ldots 0\rangle,|\mathbf{2}\rangle=|010\ldots
0\rangle,\ldots,|\mathbf{N}\rangle=|00\ldots 1\rangle.$ (4)

i.e. the state $|\mathbf{j}\rangle$ is the state of the chain with only the
$j$-th spin up. The one-excitation basis plays a fundamental role in the
simplest transfer protocol. In this protocol the initial state of the chain,
$|\Psi(0)\rangle$, is prepared as

$|\Psi(0)\rangle=|\psi(0)\rangle\otimes|\mathbf{0}\rangle_{N-1}$ (5)

where $|\psi(0)\rangle=\alpha|0\rangle+\beta|1\rangle$, is an arbitrary one-
spin pure state, with $\alpha$ and $\beta$ complex constants such that
$|\alpha|^{2}+|\beta|^{2}=1$, and $|\mathbf{0}\rangle_{N-1}$ is the state
without excitations of a chain with $N-1$ spins. The state in Eq. (5) can be
rewritten as

$|\Psi(0)\rangle=\alpha|\mathbf{0}\rangle+\beta|\mathbf{1}\rangle.$ (6)

Using the time evolution operator

$U(t)=\exp{(-iHt)},$ (7)

the state of the chain at time $t$ can be obtained as

$|\Psi(t)\rangle=U(t)|\Psi(0)\rangle=\alpha U(t)|\mathbf{0}\rangle+\beta
U(t)|\mathbf{1}\rangle.$ (8)

It is clear that the state transfer is perfect when for some time $T_{per}$
the state of the chain is given by

$|\Psi(T_{per})\rangle=e^{i\theta}\left(\alpha|\mathbf{0}\rangle+\beta|\mathbf{N}\rangle\right),$
(9)

where $\theta$ is some real arbitrary phase. So, the transfer fidelity as a
function of time is calculated as

$f_{\alpha,\beta}(t)=|\alpha\langle\Psi(t)|\mathbf{0}\rangle+\beta\langle\Psi(t)|\mathbf{N}\rangle|^{2}.$
(10)

The fidelity in Eq. (10) depends on the particular initial state, Eq. (6),
chosen to be transferred. Usually, to quantify the quality of the state
transfer protocol, it is introduced the average of $f_{\alpha,\beta}(t)$ over
all the possible values of $\alpha$ and $\beta$. It can be shown that this
averaged fidelity, $F(t)$, can be written as

$F(t)=\frac{\sqrt{|P(t)|}}{3}+\frac{P(t)}{6}+\frac{1}{2},$ (11)

where

$P(t)=|\langle\mathbf{1}|U(t)|\mathbf{N}\rangle|^{2}$ (12)

is called the transferred population between the first and last sites of the
chain, and we assume that $\cos(\gamma)=1$, where
$\gamma=arg(f_{\alpha,\beta}(t))$, as in Ref. [2]. The pretty good transfer
occurs when

$P(t)=1-\epsilon,$ (13)

where $\epsilon\ll 1$.

The optimization algorithm

The XX Hamiltonian shows perfect state transfer for a number of Exchange
Couplings Distributions (ECDs). For ECD we understand a particular set of
values $\\{J_{i}\\}_{i=1}^{N-1}$ that univocally determines a particular
Hamiltonian for a chain. For example, it is well known that the ECD given by
$J_{i}=\sqrt{i(N-i)}$ determines the XX Hamiltonian that shows perfect state
transfer at time $T_{per}=\pi/2$.

For Heisenberg chains $T_{per}$ it is not known, so the optimization method
can be designed to obtain an ECD for a chosen transfer time.

Our method allow us to choose the transfer time $T$, however we will focus on
the range $T\in\left[\frac{N}{2},5N\right]$. The range chosen allows us to
compare our results for the Heisenberg Hamiltonian with those that are well
known for the XX Hamiltonian. There is not an obvious upper value for the
transfer times to be explored other than the search of fast transfer times. It
is worth to mention that transfer times on the order $10N$ are achievable
controlling the dynamics by applying external magnetic fields to the spins at
the extremes of an unmodulated chain (with all its couplings equal), so
obtaining smaller transfer times for uncontrolled state transfer could make
preferable the task of engineering the couplings strength rather than
implementing the control fields [1].

Since the global optimization procedure always find a good maximum for the
function that is being optimized, in our case the transferred population
$P(T)$, where $T$ is the arrival (or transfer) time, we proceed to look for
the maximum transferred population inside a hypercube $0\leq J_{i}\leq
J_{max}$ containing the possible values for the exchange couplings. The
hypercube is the high-dimensional phase space (a N-dimensional hypercube of
size $J_{max}$) that was mentioned in the Introduction. In the context of
optimization methods the function that is maximized or minimized is usually
called cost function.

Figure 1: The cartoon depicts the ingredients of the procedure designed to
find EC distributions that can be used to achieve near perfect QST. a) A
centro-symmetric spin chain with five spins and only two different exchange
couplings $J_{1}$ and $J_{2}$. b) For the five spin chain a ECD that optimize
the cost function for an arrival time $T$ is obtained using the pivot method .
The ECD is denoted by $\\{J^{(k)}\\}_{T}=(J^{(k)}_{1},J^{(k)}_{2})$, which is
a point in the hypercube with side length $J_{max}^{(k)}$. c) The smaller
hypercube with side length $J_{max}^{(1)}$ and the second one smaller, with
side length $J_{max}^{(2)}=J_{max}^{(1)}+\delta$, are depicted using light
blue and blue squares. The corresponding ECD that optimize the cost function
for each hypercube is depicted as a starry dot inside the hypercube.

There is not a known relationship between the maximum value necessary for the
couplings $J_{i}$ to achieve near perfect quantum state transfer at a given
time $T$, for this reason it is useful to consider a succession of values
$J^{(k)}_{max}$. For each one of this values of the hypercube side length the
pivot method determines an ECD denoted as $\\{J_{k}\\}_{T}$ where $T$ is the
transfer time. The succession of lengths of the hypercube side is given by
$J_{max}^{(k)}=J_{max}^{(1)}+(k-1)\delta J$, for $k=1,2,3,\ldots$ and $\delta
J=0.5$. As the rigorous results about the existence of the pretty good
transfer scenario are proved for centro-symmetric ECDs, we impose this
restriction to the exchange couplings, $J_{N-i}=J_{i}\;;\;i=1,\cdots,[N/2]$.
The cartoon in Fig. 1 depicts the main ingredients of the procedure to find
the exchange coupling distributions that will lead to a chain that can attain
the near perfect QST. On the other hand, it is clear that the optimization of
the transferred population in a particular hypercube with side length
$J_{max}^{(k)}$ provides a optimized value for the transferred population,that
we denote as $P^{(k)}$, and that depends on the side length.

Figure 2: The exchange coupling distribution $\\{J^{(k)}\\}_{T}$ vs the length
of the hypercube side $J_{max}$ for a system of $N=30$ spins. As explained in
the text, the ECDs corresponding to a system with $N=30$ have $15$ different
exchange couplings, which are shown in the figure using colored lines and
symbols. The ECDs were obtained using the pivot method for arrival time $T=2N$
. Each set of points of a given colour corresponds to a the values of a given
exchange coupling, for instance the black dots correspond to $J_{1}$, the red
ones to $J_{2}$ and so on. The figure clearly shows that the particular values
of each exchange coupling strength $J_{i}$ depend on $J_{max}$. The black
dashed line correspond to $J=J_{max}$ and is used a guide to the eye. Note
that for small values of $J_{max}$ the exchange couplings strengths are all
close to $J_{max}$, while for large values of $J_{max}$ the pivot method does
not find values of $J_{j}$ that further optimize the population transfer and
each exchange coupling $J_{i}$ reaches a plateau.

Before proceeding with the analysis of the results that can be obtained using
the pivot method, we want stress that there is not a simple way to choose the
actual value of $\delta J$. From what is known for the Heisenberg and XX
models it is a fair assumption that, for models within the $XX$ universality,
the pretty good transfer scenario could be reached with exchange couplings
bounded by $N/2$, since this is the value that correspond to the maximal
exchange coupling for the perfect state transfer case in the XX model.
Nevertheless, the perfect state transfer happens for transfer times that are
independent of the chain length so it can be also expected that the bound on
the exchange couplings are a decreasing function of the transferral time (this
is confirmed by the results that are shown latter in the paper). So $\delta
J=0.5$ is a compromise between the different scales that we thought could be
present when dealing with models with different values of $\Delta$ and arrival
times. In this sense, the value chosen is not optimal, but allow us to search
for optimal values of the exchange coupling distribution increasing the volume
of the hypercube and starting the search from a “good” initial ECD.

The workings of the pivot method are as follows. Starting from an initial set
of points in the phase space on which the cost function depends, the method
explores this space until an optimal value is found [42, 43]. In our case each
point is an ECD that determines a Hamiltonian. Using the spectrum and
eigenvalues of the Hamiltonian the transferred population $P(t)$ is determined
for a transfer time $T$ using Eq. (12). The pivot method finds the largest
possible value of the transferred population, $P^{(k)}(T)$, inside an
hypercube of side $J^{(k)}_{max}$.

Fig. 2 shows the typical results that can be found from a run of the procedure
described in the paragraphs above. The Hamiltonian considered is the isotropic
one. The figure shows the values of the fifteen different EC values in
$\\{J^{(k)}\\}_{T}$, for a chain with $N=30$ spins as functions of the length
of the hypercube side and for a transfer time $T=2N$. The value of a
particular coupling changes in a random-walk way when the length of the
hypercube size is increased, eventually reaching a large plateau.

The data shown in Fig. 2 was obtained starting from a ECD with
$J_{i}^{(1)}=1,\forall i$. We run the PM algorithm to find the largest
possible value for the transferred population $P^{(2)}$ corresponding to an
ECD such that $J_{i}^{(2)}\leq J_{max}^{(2)}$. Then, the length of the
hypercube side was increased to $J_{max}=J_{max}^{(3)}$ and the PM was run
again. The algorithm was employed up to $J_{max}^{(K)}=N$ for some $K$. As
usual for stochastic global optimization algorithms, we apply to the output
set a deterministic gradient algorithm [44] in order to obtain a better
accuracy. Throughout this work, mainly in the figures, we will refer to a
succession of points $(J_{max}^{(k)},P^{(k)}(T)))$ as the transferred
population as a function of $J_{max}$, as if it were effectively a function.
This is done for brevity and under the understanding that each point of the
succession $P^{(k)}(T)$ must be calculated for an arrival time $T$ and with a
specifically designed distribution of coefficients $J_{max}^{(k)}$. In the
same sense, once an ECD has been designed to achieve a high TP at a given
arrival time $T$ the function in Eq. (12) can be evaluated at any time.

The procedure described in the paragraph above is not the most general but, as
we will show latter, results in ECDs with very good properties, in particular
the procedure is able to find the regime where $P(t)=1-\epsilon$.

##  3 Results: Quantum state transfer: the Heisenberg Hamiltonian

Figure 3: The transferred population for a chain with Heisenberg Hamiltonian
and $N=30$ spins. The data corresponds to arrival times $T=N/2,N,2N$ and $3N$,
which are depicted as black, red, green and blue dots, the lines joining the
dots are used as a guide for the eye. Each point corresponds to a pair
$(J_{max}^{(k)}/N,P^{(k)})$, look at the text for definitions. The inset shows
a zoom of the region where the transferred population exceeds the value
$0.99$.

The corresponding values of the transferred population $P^{(k)}(T)$ obtained
with the ECD shown in Fig. 2 are shown in Fig. 3, which also shows transferred
population corresponding to arrival times $N/2$ and $N$ for a chain length
$N=30$. Each point corresponds to a pair $(J_{max}^{(k)}/N,P^{(k)}(T))$, the
lines are included as a guide for the eye. From the data sets it is clear that
in general the ordering $P^{(k)}(T_{<})<P^{(k)}(T_{>})$ holds, where
$T_{>}>T_{<}$ are two ordered arrival times. However, since $P^{(k)}(T)$ is a
discrete sample of a random-like quantity for some $k$’s the ordering is
reversed. All in all, the data in the figure is compatible with two reasonable
physical traits, i) for a fixed $J_{max}^{(k)}$ larger arrival times results
in larger transferred population, and ii) to reach a given target transferred
population taking larger arrival times ensures that the target is attained for
lower $J_{max}^{(k)}$.

The inset in Fig. 3 shows a detailed view of the region where the transferred
population $P>0.99$. This zoom level allows us to appreciate that the transfer
is of a very high quality and this is due to the excellent properties of the
coefficient distribution found by the optimization method. Another feature
that can be clearly observed in the inset has to do with the decreasing value
of $J_{max}$ for which the transferred population reaches a given preset
value, say $P=0.99$. The inset shows that this value is reached for
$J_{max}\sim 0.8N$ for an arrival time equal to $N/2$, $J_{max}\sim 0.4N$ for
an arrival time equal to $N$, etc. Even at this level the of magnification the
largest values of $P^{(k)}$ attained for the largest values of $J_{max}^{(k)}$
explored can not be fully appreciated, since they are larger than $0.999$.

The results presented so far in the figures correspond to a chain of length
$N=30$, in any case we have checked that the algorithm to find all the
coefficients of a chain works correctly for chains with lengths between 10 and
90 and for various arrival times, mainly $N/2,N,2N,3N,4N$ and $5N$. In the
Supplemental Material [53] we include a table with the 25 different
coefficients of a chain of length $N=50$, for the times mentioned, in addition
to the value of the transferred population that is obtained when a chain is
assembled with these exchange coefficients. The coefficients in the Table will
be used latter on.

From a theoretical point of view, that the regime where $P=1-\epsilon$ can
only be attained using $J_{max}^{(k)}\approx N$ is not an impediment or a
problem. For an actual implementation this trait could render the ECDs
responsible of that regime useless, since changing the strength of the
interaction between the components of a microscopic system on one or two
orders of magnitude (depending on the length of the chain) is extremely
difficult. So, in the rest of this Section we will focus on the properties of
the ECDs obtained with small or moderate values of $J_{max}^{(k)}$ to evaluate
them as alternative distributions that can be experimentally implemented.

One concern about the ECDs obtained for small values of $J_{max}$ is that the
transferred population obtained from them could be unstable under static
perturbations of the couplings, so the transferred population would be easily
ruined regardless of the very good quality of the transfer produced by the
unperturbed distribution.

Figure 4: Color map of the averaged transferred population as a function of
the maximum value of the exchange coupling $J_{max}$ and the static disorder
amplitude. Calculations were performed for a chain with $N=30$ spins and for
different arrival times: (a) $T=N$, (b) $T=2N$ and (c) $T=3N$

The stability of quantum state transfer in spin chains under static
perturbation, or static disorder, i.e. the robustness of the transferred
population when the ECDs is subject to static perturbations has been
extensively studied in the literature. The static disorder is understood as a
consequence of manufacturing errors of the physical system that is modeled by
the spin chain. In theoretical studies the static perturbations are modeled
adding a random term proportional to the coupling strength [7, 8, 45]

$J_{i}\longrightarrow J_{i}(1+a\xi_{i}),$ (14)

where $a$ is the strength of the disorder and $\xi$ is a random variable with
zero mean and unity variance.

When static disorder is added, the transferred population averaged over
realizations of the disorder is introduced as a figure of merit

$\bar{P}=\left\langle P\right\rangle_{\xi},$ (15)

where the average is a simple equally weighted average. This means that for a
chain with $N$ spins a realization of the disorder implies that $N-1$ random
numbers are generated $\\{\xi_{i}^{j}\\}$ and the ECD to be analyzed is
changed as

$J_{i}\longrightarrow J_{i}(1+a\xi_{i}^{j}),$ (16)

where $1\leq j\leq n_{r}$, and $n_{r}$ is the number of realizations to be
taken into account. Once a particular ECD has been generated, Eq. (16), the
corresponding transferred population at arrival time $T$ is calculated,
$P^{j}_{\xi}(T)$, and the averaged TP is calculated as

$\bar{P}(T)=\frac{1}{n_{r}}\sum_{j=1}^{n_{r}}P^{j}_{\xi}(T).$ (17)

Fig. 4 shows, as color maps, the averaged transferred population (TP) as a
function of $J_{ax}^{(k)}$ and the disorder strength for a chain with $N=30$
and three different arrival times. As the color map indicates, there are
segments of the $J$-axis where the TP is particularly unstable, so even for
very small disorder strengths the TP becomes quite poor ($P<0.9$), note the
dark orange and blue fringes that appear in every map shown. Nevertheless, the
segments become sparser and the fringes appear for larger disorder strengths
when the arrival time considered is longer, and, for the three cases depicted
there are always regions where the TP is very high and robust (the brilliant
yellow regions) even for small $J$-values.

For further analyze the properties of ECDs found using the PM that produce
fairly good state transfer ($P\gtrsim 0.95$) it is useful to look back at the
time evolution of the population transfer, Eq. (12). So far we have studied
the values reached by the population transfer at the times for which the ECD
was optimized. Now we want to study the population transfer as a continuous
function of time, Eq. (12). It is clear that the time-dependent functions will
be different depending on the ECD that is considered but, as we will show,
these different functions are closely related.

In this paper we consider that $T=\kappa N$, where $\kappa=1/2,1,2,3,\ldots$.
It is clear that the population transfer obtained with an ECD optimized to
show pretty good transmission at $T$ will show successive peaks at times
$T_{p}=(2p+1)T$, where $p=1,2,3,\ldots$, since the time evolution should be
almost periodic and the time for the state to go back and forth between the
extremes of the chain is $2T$. For perfect state transfer the successive peaks
of the TP are all of the same height, but for pretty good quantum state
transfer the height will not be constant. The TP then, as a function of the
scaled time $t/N$, will show peaks at the scaled arrival times
$T^{sc}=(2p+1)\kappa$.

Figure 5: The transferred population obtained with a Heisenberg chain with
$N=50$ vs the scaled time $t/N$. The TP corresponding to three different
arrival times, $T=\kappa N$, with $\kappa=1/2,2$ and $5$ is shown using black,
red and blue curves, respectively. Note, for instance, that for $\kappa=1/2$
the black curve shows peaks at $t/N=1/2,3/2,5/2,\ldots$ The ECD for each case
was designed to achieve a target population of $0.95$. The particular values
of the coupling coefficients used to assemble the transferring chains can be
found in the Supplemental Material [53].

Fig. 5 shows the time evolution of the TP for three different arrival times.
The corresponding ECDs were designed using the PM algorithm. The TP calculated
using the ECD that is designed to produce a fairly good TP at arrival time
$N/2$, the black curve, shows several peaks at scaled times
$t/N=1/2,3/2,\ldots,(2j+1)/2,\ldots$, for $j\in\mathbb{N}$. The TP calculated
using the ECD that must produced a peak at arrival time $2N$ shows several
peaks at scaled times $2,6,10,\ldots$, while the horizontal scale only allows
for a single peak of the TP calculated using the ECD that corresponds to
$T=5N$. It can be seen that the peaks are broader for larger arrival times.
However the apparent differences in time scale and width of the peaks
corresponding to different arrival times, changing the scaling on the
horizontal axis reveals a more profound relationship between them, as shown in
Fig. 6.

Figure 6: a) The first five peaks of $P(\\{J_{k}\\}_{\kappa N},t)$ vs the
scaled time $t/(\kappa N)$ for $\kappa=1/2,1,2,3,4$ and $5$, are shown using
purple, magenta, black, red, green and blue curves, respectively.. The
excellent superposition between the curves does not allow to distinguish
between them if several peaks are included, because of this a single peak is
shown in b). The particular values of the coupling coefficients used to
assemble the transferring chains can be found in the Supplemental Material
[53].

Calling $P(\\{J_{k}\\}_{T},t)$ the TP calculated using an ECD optimized for an
arrival time $T$, Fig. 6 shows the first five peaks of $P(\\{J_{k}\\}_{T},t)$
vs the scaled time $t/(\kappa N)$, for $\kappa=1/2,1,2,3,4$ and $5$. The
scaling is the cause the curves superposition and reveals the physics involved
in the TP.

Although the ECD for each chain size and arrival time is obtained using a
stochastic method, it is interesting to note that the distributions thus
obtained have common features that allow deciding, according to the strength
of the interactions available in a particular implementation, which ones are
the arrival times compatible with these strengths.

So far, our results show that it is possible to find ECDs such that near
perfect quantum state transfer is achieved for arrival times as short as
$T=N/2$, if the strength of the interaction can be taken large enough. This is
not what happens in actual implementations, where the range of strengths at
disposal are limited by technological or physical reasons. So, it is
reasonable to ask how to proceed when the range of interactions is limited.
The answer, in a nutshell, is that the ECD should be sought at longer arrival
times. In what follows we will show that the ECDs found using the pivot method
show a relatively simple behavior that helps to determine the appropriate
arrival time compatible with the range of available interactions.

We look for the smallest hypercube that contains values of the exchange
coefficients such that the transferred population reaches the target value of
0.98 and we call the corresponding side length value $J_{min}$, which depends
on the length of the chain and the arrival time.

Fig. 7 shows $J_{min}$ for different chain lengths and arrival times. The
values can be fitted, with a very good agreement, using an exponential
function

$J_{fit}=A(N)+B(N)e^{-C(N)\frac{T}{N}}.$ (18)

The data and the fitting show that for fixed target TP the side length is a
rapidly decreasing function of the arrival time. We think that this type of
behavior is rather general if the target population is low enough (0.98, or
lower) and the chain is long enough. The constants $A$, $B$ and $C$ on Eq.
(18) depend on the chain length, although $C(N)$ seems to be independent of
$N$ once the uncertainties in the data are taken into account.

Figure 7: The minimum length of the hypercube side necessary to obtain a
population transfer of 0.98, $J_{min}$, versus the transfer time, for chains
of length (from bottom to top) $N=20,30,40,50,60$ and $70$, the data is shown
using black, red, green, blue, etc, solid dots, respectively. The lines are
included as a guide for the eye

.

Although the results shown in figure 6 tell us that for long arrival times the
intensities of the exchange interactions must all be of the same order, it is
interesting to note that there are different actual physical systems in which
it is possible to tune the particular values of said interactions on ranges
and signs compatible with those studied in the present work. For instance, in
Ref. [48] which deals with the perspectives of exchange coupled quantum dot
spin chains, it is shown that positive exchange interactions can be tuned
between 0 and 125 Mhz. The tunability can be used to implement SWAP operations
between qubits located in adjacent quantum dots or to fix each exchange
interaction coupling to a desired value. In Ref. [49] it is shown that the
sign of the exchange interactions can be chosen and it can be tuned between 0
and 40 $\mu$eV. It is worth noting here that the one-excitation transfer
protocol works in the same way for spin chains with all the exchange
interactions positive or negative. The tunability of the exchange couplings
strengths also has been demonstrated in chains of superconducting qubits see,
for instance [26].

Before changing the model, it is interesting to note some of the consequences
of our results with respect to the “pretty good quantum state transfer"
scenario, since until now the results presented in this Section do not include
the chain lengths where the regime $P=1-\epsilon$ can be proven to exist.

As well as the results shown explicitly in figures , we also obtained results
for chains with even and odd number of spins, in addition to chains with
length equal to powers of two . We have explored the behavior of chains with
lengths between $N=10$ and $N=90$ spins, while comparing the results of chains
of lengths $N=30$ and $N=31$ with $N=32$ we have not found qualitative
differences between the largest transfer values accessible for any pair of
chains that were compared. We also compared the results for chains with $N=60$
and $N=61$ with $N=64$ and, again, there were not discernible qualitative
differences. This study was also made for different transmission times, which
once again, showed that there are no qualitative differences in the largest
values that can be obtained for the population transferred in chains of
similar length, beyond their length being even, odd or a power of two. We
think this presents fairly conclusive evidence that such a scenario is
accessible for spin chains of any length with the Heisenberg Hamiltonian

It is interesting to wonder if the coupling distributions found using the
pivot method allow a robust transmission of states that are initially not pure
due to problems with their preparation. The Heisenberg Hamiltonian commutes
with the total magnetization for any set of values that the ECDs adopt, and
this can be used to estimate the population transfer in the case that the
initial state is an “almost pure" state such as those used. in Nuclear
Magnetic Resonance experiments.

Let us suppose that the state prepared initially can be written as

$\rho_{i}=(1-\eta)|\psi\rangle\langle\psi|\otimes|\mathbf{0}\rangle\langle\mathbf{0}|+\eta\rho_{u},$
(19)

where $\psi$ is the pure state that is intended to be prepared at the first
site of the chain and transmitted with

$1-\eta=\langle\psi|\otimes\langle\mathbf{0}|\rho_{i}|\psi\rangle\otimes|\mathbf{0}\rangle$
(20)

so $\rho_{u}$ accounts for all the undesirable parts of the state that can not
be eliminated because the preparation of the state is not “perfect”, and
$\eta\ll 1$. Then the state after the transmission protocol can be written as

$\rho(t)=U(t)\left((1-\eta)|\psi\rangle\langle\psi|\otimes|\mathbf{0}\rangle\langle\mathbf{0}+\eta\rho_{u}\right)U^{\dagger},$
(21)

and the fidelity with the “expected” state is given by

$F=\langle\psi|\mbox{Tr}_{N-1}(\rho(t))|\psi\rangle.$ (22)

So, the transferred population of an “almost pure” state would be bounded by

$P_{\rho}\sim P(1-\eta),$ (23)

where $P$ is the transfer probability of the one-excitation protocol. In the
paper by Di Franco et al. [46], it is discussed how to obtain perfect transfer
in the event that there is no “initialization of the state to be transmitted",
that is, a series of gates are applied to the ends of the chain, which allows
perfect transmission of a mixed or pure state when the chain has been designed
for perfect transmission in the sub space of an excitation. Our estimate of an
upper bound for the transmission of poorly prepared states is consistent with
the results of Di Franco et al. .

##  4 Results: Quantum state transfer for chains with XXZ Hamiltonian

It has been suggested by Kay [37] that some transfer properties of quantum
spin chains with exchange couplings roughly equal to those of the Heisenberg
chain can be studied using perturbation theory. There are several reasons for
doing that study. In particular, it is of great interest to study how the
transfer properties of the Heisenberg Hamiltonian, with their inherent
difficulties, change towards the properties of the XX Hamiltonian, more direct
to understand and for which several transfer regimes have been identified that
show a direct relationship between the quality and quantity of modulation
exerted on the coefficients and the quality of the transfer obtained by said
modulation.

Figure 8: The transferred population $P$ vs $J_{max}$ for two chains with
$N=50$ and a) $\Delta=0.8$, b) $\Delta=1.2$. The data included in both top
panel correspond to arrival times $N/2$, $N,2N,3N,4N$ and $5N$. The behavior
shown by the transferred population in the top panel shows that, even for the
shorter arrival times, the values of the exchange couplings needed to attain
$P>0.999$ are bounded by $J_{max}\lesssim 3$. In the case $\Delta=1.2$, shown
in the bottom panel, the values of $J_{max}$ needed to attain high quality
transfer, $P>0.99$ can become quite large for short arrival times, in a
similar way to the Heisenberg case, see Fig. 3.

In this Section we apply to anisotropic Hamiltonians with $\Delta\neq 1$, see
Eq. (1), the same methods that we have applied to the Heisenberg Hamiltonian
in Section 3.

Fig. 8 shows the behavior of the TP as a function of the side length of the
hypercube in whose volume the global optimization algorithm looks for the
distribution of couplings that produces the largest possible transfer. The
results shown in panel a) correspond to $\Delta=0.8$, while the results shown
in panel b) correspond to $\Delta=1.2$, respectively.

Comparing the TP shown in the two panels of Fig. 8 it is clear that the case
with $\Delta=0.8$ transfers considerably better and with the difference being
greater for shorter arrival times. It is important to note that in the case of
the figure corresponding to the Heisenberg Hamiltonian case, Fig. 3, the TP
was shown as a function of the length of the hypercube side renormalized by
the length of the chain, while in Fig. 8 it is shown as a function of the
length of the side without any renormalization. The inset in Fig. 8 a) shows
the TP attaining values well above 0.999 for every arrival time considered.

As was said in Section 3, when it comes to qualifying the performance of a
chain built with a given distribution of coefficients, it is not only
necessary to take into account the value of $J_{max}$ that limits the possible
values of the coefficients and the magnitude of the transferred population, in
addition the stability of the TP must be studied in the face of static
disturbances.

Fig. 9 shows the behavior of the averaged TP for three chains with the same
length, $N=51$. The three columns, from left to right, correspond to
Hamiltonians with $\Delta=0.8,1$ and $1.2$. All the necessary ECD were
obtained using the PM. The top row correspond to the shorter arrival time,
$T=\left[\frac{N}{2}\right]$, while the bottom row correspond to the larger
arrival time $T=2\left[\frac{N}{2}\right]$. The figure summarizes quite well
our findings: i) For moderate or small values of the maximum value allowed to
the exchange coefficients, the figure shows values up to $J_{max}=14$, both
anisotropic chains attaining larger TP when the strength of the static
disorder is negligible, but the chain with $\Delta$ larger than unity is very
strongly affected by disorder, making the chain useless for disorder strengths
larger than $0.1$, as can be seen in the two panels in the right column and
shown as a narrow fringe of brilliant yellow and light orange, the two colors
that correspond to a high-quality transfer. ii) Conversely, the TP obtained
using the chain with $\Delta=0.8$ is weakly affected by static disorder,
presenting a TP of a very high quality, as is depicted by the broad fringe of
brilliant yellow and light orange that appear in both panels in the left
column. iii) Finally, the Heisenberg chain is very resistant to static
disturbances, even for a range of intensities of the disorder even greater
than the chain with $\Delta<1$. However, the first type of chain does not
reach as high transmission values as the second one for these values of
$J_{max}$.

The behavior of the TP shown in the right and left columns of Fig. 9 can be,
in practice, translated to any value of $\Delta>1$ and $\Delta<1$,
respectively. Chains whose Hamiltonians have $\Delta<1$ and ECDs optimized
using the PM show excellent quantum state transmission, with very high values
of $P$ and are very resilient against static disorder. Moreover, the ECDs
obtained for short arrival times are bounded by $J_{max}\sim\mathcal{O}(1)$
independently of the chain length, at least for $N<100$ the range of chain
lengths that we have explored. In a sense, it is reasonable to say that chains
with Hamiltonians with $\Delta<1$ are XX-like in as much as their quantum
state transfer properties are involved.

The rapid departure of the behavior of the transferred population when the
Hamiltonian of the chain has $\Delta<1$, from the behavior observed when the
chain obeys the Heisenberg Hamiltonian can be seen in the Fig. 10. The figure
shows $1-P$ as a function of $J_{max}$, in the left panel the data corresponds
to chains with $\Delta\leq 1$, while the right panels shows the data
corresponding to chains with $\Delta\geq 1$. The color of the data points and
curves in both panels have been chosen so that the chains at the same
"distance" from the isotropic case have the same color, so the data for the
chains with $\Delta=0.8$ and 1.2 are shown with black points, for the chains
with $\Delta=0.85$ and 1.15 with red points, and the following with green and
blue dots. The resemblance with “ $\lambda$-point” plot is striking.

The value of $J_{max}$ for which the PT reaches a given target value for
chains with $\Delta>1$ is an increasing function of $1/(1-\Delta)$. To
determine this function, it would be necessary to study much longer chains,
with the consequent expenditure in computing time, anyway for finite chains
there can not be any divergences so, in the worst case scenario $J_{max}$
grows up to values $\mathcal{O}(N)$ for arrival times $\mathcal{O}(N)$.

Figure 9: Color map of the averaged TP as a function of the maximum value of
the exchange coupling and the static disorder amplitude. Results were obtained
for a chain with $N=51$ spins, two different arrival times $T=N/2$ and $T=N$
and three different values of the anisotropy parameter $\Delta=0.8,1.0$ and
$1.2$.

Figure 10: $1-P$ vs $J_{max}$ for chains with $N=61$ and for an arrival time
$T=60$. The transferred population is shown in this way to emphasize the quite
different behavior observed by the transmission in chains with
$\Delta<1$,whose data is shown in left half of the figure, from the one
observed in chains with $\Delta>1$, whose data is shown in the right half of
the figure. Note that the values of the variable $J_{max}$ increase from 1 (in
the middle of the horizontal axis) to 15 (at the left and right extremes of
the horizontal axis) . The same colors are used for data that correspond to
chains with the same $\Delta^{\prime}$, with $\Delta=1\pm\Delta^{\prime}$.
Note that $\Delta^{\prime}=0.2$ is shown with black dots,
$\Delta^{\prime}=0.15$ with red dots and so on.

Even when the averaged transferred population could be very high, it is
logical to ask what the "worst case" scenario is like. This question is
particularly important in experiments in which only a single-shot transmission
attempt is made and there are no error correction algorithms available. To
obtain the worst case scenario, it is necessary to calculate numerous
realizations of the disorder and keep track of the smallest transferred
population.

In the Supplemental Materials we have included a figure comparing the averaged
transferred population and the worst transferred population as functions of
the disorder amplitude and for different chains . We also included a plot
showing the standard deviation of the transferred population. All in all, the
results, although preliminary, show that for moderate disorder amplitude the
Heisenberg isotropic chain is the better option as a transmission channel and
that the statistics of transferred population in quite different between the
different model chains. Other authors also have studied the statistics of the
quantum state transfer, in particular G. Nikolopoulos [52]. In this paper it
is shown that the minimal fidelity is equal to the worst population
transferred for $P>0.5$. The regime of interest is found for values of
$P>0.5$, so the worst case TP is a good qualifier of the transfer process.

##  5 Discussion and Conclusions

If the strength of the couplings is not an issue, which in a microscopic
system does not seem realistic, it is possible to design chains with the
Heisenberg Hamiltonian that transfer an excitation between the ends of the
chain with very good probability at times as short as $N/2$ or $N$. In any
case, if the intensity of the couplings is limited (at most couplings less
than or equal to two), considering arrival times of the order of 4 or $5N$ is
recommended. This last point is of particular importance for possible
practical implementations. In the same sense, if the spin Hamiltonian is an
effective approximation for the low energy spectrum of a given physical
system, it is advisable to look for a physical regime for the original system
in which the resulting model Hamiltonian is anisotropic with $\Delta<1$.

Based on the results shown Fig. 9 about the behavior of the averaged TP we
analyzed the stability of the transmission process for relatively short
arrival times, $N/2$ and $N$. We also obtained results for longer times and
the same chain length $N=50$, which are not consigned to any figure, and the
conclusion drawn in Section 4 holds, that is, chains with $\Delta<1$ transfer
excitations with greater probability than chains with $\Delta=1$ , which in
turn transmits better than those with $\Delta>1$. For times $4N$ or $5N$, the
chains with $\Delta\leq 1$ reach very high values of the transmission
probability ($P>0.9998$), but in those with Heisenberg’s Hamiltonian the
averaged TP decay faster than on the chains with $\Delta<1$ when the intensity
of the disorder is increased . The chains with $\Delta>1$ never reach the
highest levels of quality in transmission than are attained by chains with
$\Delta\leq 1$ and are severely affected by static disorder.

The linear scaling of the transferred population, Fig. 6, is noteworthy for a
number or reasons, i) the ECD which are used to obtain the PT are bounded by a
quantity that changes non-linearly with the arrival time, ii) each ECD is
obtained independently for each time, and iii) the TP is a nonlinear functions
of the couplings. So far we have not reached a conclusion about how to look
for scaling properties in the eigenvalues of the Hamiltonian of the chain and
the coefficients of the eigenvectors that enter into the calculation of the
transferred population.

The results found show us that the ECDs found are relatively robust but, even
in the best case, if there is a static disorder present with a strength larger
than $10\%$, these distributions no longer offer a valid alternative to build
chains of spins that serve for the transfer of quantum states. In the case of
chains with XXZ Hamiltonians with $\Delta>1$ even smaller disorder strengths
render useless the chain as a transmission channel. Some of these features can
be better appreciated in the figure that is included in [53]. So, from the
quantum state transfer point of view the regime of interest can be found for
$a<0.1$. Notwithstanding this, the patterns observed in Fig. 4 and 8, which
can be clearly appreciated for disorder strengths larger than $0.1$, seem to
indicate that there are different dynamical regimes. The propagation of
information can show very different speed bounds in disordered spin chains and
be suppressed for strong enough disorder see, for instance, Ref. [47]. Assert
that the observed patterns are due to this requires more research, which is
being carried out, and depends on the study of the scaling properties of the
matrix elements of the evolution operator associated with the problem.

Fig. 3 shows that, apparently, the system is more robust for smaller transfer
times than for larger ones, at least for some intervals of the variable
$J_{max}$. This is so, in part, because the figure shows the transfer
probability against a single parameter $J_{max}$ while the detailed behavior
of $P$ depends on the whole set of exchange couplings. All the exchange
couplings are bounded by $J_{max}$, but this bound does not give a detailed
information about the distribution of the $J_{i}$. Stochastic optimization
methods can be trapped in a "good" minimum and do not provide information
about the landscape topology of the valley where the “good" minimum lies and
the width of this valley is what determines the robustness of the transfer
process against disorder. All this results in that the exchange coupling
distributions found for short times are more robust against disorder than the
exchange coupling distributions found for longer times, in certain intervals
of the variable $J_{max}$, see Fig. 3 a), b) and c).

The transferred population $P$ is a probability, but the information that it
shows is not enough to quantify large changes in the ECDs. An interesting
quantity to study is the probability of transferring an excitation from a
given site of the chain, $m$, to any other site of the chain, $n$, given by
$P_{mn}(J_{max})$, where $J_{max}$ is the value which was used to determine
the ECD. It is clear that the TP is given by $P_{1,N}$. Studying the
statistical divergence [50] between two probability distributions
$D(\\{P_{nm}(J_{max})\\},\\{P_{nm}(J_{max}^{\prime}\\})$, it is possible to
determine their properties and how susceptible they are to disorder. We use
$\\{P_{nm}(J_{max})\\}$ to denote the set of probabilities with $1\leq m,n\leq
N$ . We have carried out preliminary studies using the Renyi divergence [51],
$D_{1/2}$ , and verified that the divergence reproduces the patterns structure
observed in Fig. 3 and Fig. 6.

Fig. 3 shows vertical regions, for instance in Fig. 3 c) there are three
clearly appreciable regions whose limits are $J_{max}\leq 5$, $5\leq
J_{max}\leq 12$ and $12\leq J_{max}$. These limits correspond, exactly, with
plateaus observed in the coupling coefficient distributions. In the different
regions it can be observed that the susceptibility to disorder is different,
as the color map indicates. For example, again in Fig. 3c), it can be seen
that the distributions of coupling coefficients such that $5\leq J_{max}\leq
12$ are more susceptible to disorder than the distributions at both sides. In
the $5\leq J_{max}\leq 12$ region the propagation of excitations is strongly
suppressed by the increase in the strength of the disorder. In regions where
the system is not very susceptible to disorder, the excitation that is
initially located at one end of the chain spreads over a large portion of the
chain and refocuses at the time of transmission. In areas that are strongly
affected by the disorder, the initial excitation spreads over a few sites in
the chain and the refocusing hardly occurs, which results in a very poor
transferred population. This analysis was carried out calculating the inverse
participation ratio, a quantity that is commonly used to analyze transport
properties in disordered systems and quantum state transmission, see Ref. [6]
and references therein.

The coupling distributions resulting from the application of the PM to chains
with the Heisenberg Hamiltonian show that it is possible to reach a regime
where $P=1-\epsilon$ for chains of any length, not only for chains whose
length is a power of 2, for arrival times of the order of N as long as the
magnitude of the couplings is not bounded. At present we cannot affirm whether
the regime for chains with lengths that are powers of two is the same as for
chains of arbitrary length. We will implement more numerical calculations to
try to elucidate this issue.

Acknowledgments The authors acknowledge partial financial support from CONICET
(PIP11220150100327, PUE22920170100089CO). O.O and P.S. acknowledges partial
financial support from CONICET and SECYT-UNC.

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  * [53] Supplemental Material

Pretty good quantum state transfer on isotropic and anisotropic Heisenberg
spin chains with tailored site dependent exchange couplings

Supplemental Material

## Comparison between the averaged and "worst" transferred populations

Figure 11:  The average transferred population (continuous line) and the
"worst case" in a single experiment (dashed line) versus the strength of the
static disorder. The data shown was obtained for chains of length $N=51$,
$J_{max}=14$ and $N_{r}=10000$ realizations of the static disorder. See the
text for details.

The figure above shows the average transferred population versus the strength
of the static disorder (solid lines) for chains of length $N=51$. The arrival
time is $T=N$. The red line corresponds to a Heisenberg chain, $\Delta=1$,
while the blue and green lines correspond to $\Delta=0.8$ and $1.2$,
respectively, while the brown lines correspond to a spin chain whose dynamics
is governed by the XX Hamiltonian, i.e. $\Delta=0$. . The dashed lines
correspond to the "worst case", that is, the minimum population transfer
obtainable in a single experiment. It can be observed that without static
disorder the $\Delta<1$ chain has a better performance than $\Delta=1$ and
$\Delta>1$, but that for disorder strengths larger than 0.05 the Heisenberg
chain is clearly superior as a transmission channel, being the distance
between the average and the "worst case" always smaller for that chain.

For comparison reasons we have also included results for an XX chain which are
depicted using brown lines, despite that the main interest of the manuscript
are chains with XXZ Hamiltonians. It is interesting to note that, with respect
to the averaged transferred population, the XX TP shows three different
regimes. In the first one its TP is indistinguishable from the TP obtained for
the $\Delta<1$, for disorder amplitudes $a\lesssim 0.05$. For larger values of
the disorder amplitude and up to $a\approx 0.075$ the TP of the XX chain
remains larger or comparable to the Heisenberg case, while for $a>0.075$ the
TP of the Heisenberg chain is always larger than all the other cases.

From the results shown for the averaged TP it could be concluded that, if it
is known a priori which is the expected amplitude of disorder and it is
possible to implement any type of chain, then the most suitable transmission
channels are the XX chain or the Heisenberg chain. Nevertheless, as shown by
the worst case curves, the TP of the XX chain shows by far the worst “worst
case”, see the brown dashed curve in the figure. Moreover, even though that
all the curves were obtained using the same number of disorder realizations,
it is clear from the large fluctuations shown by the worst case for the XX
chain that the statistics of the TP of the XXZ cases is quite different from
that of the XX chain. The large fluctuations of the worst case are not as
surprising since it is not a good statistical estimator as the standard
deviation or the average are. The convergence of the worst case TP to its true
value is quite slower in the XX case when compared with the other cases.

The right panel shows the statistical standard deviation of the transferred
population. The color code is the same as that used in the panel on the left.
Again it can be seen that the dispersion of the values in the case of the
Heisenberg chain is less than for the other two chains. If $P_{i}$ is the
transferred population obtained for the $i-$th realization of the disorder,
the averaged transferred population was introduced in the main text

$P=\left\langle P\right\rangle_{\xi}=\frac{1}{M}\sum_{i=1}^{N_{r}}P_{i},$ (24)

where $N_{r}$ is the number of realizations considered, the minimum
transferred population is given by

$P_{min}=\min_{i}(P_{i}),$ (25)

and the statistical standard deviation of the transferred population can be
calculated as

$\sigma=\frac{1}{N}\sum_{i=1}^{N_{r}}(P_{i}-P)^{2}.$ (26)

Here, it is worth to point out that the standard deviation of the XX chain is
quite close to the anisotropic $\Delta<1$ case.

## Exchange coupling distribution for a Heisenberg Chain with $N=50$

In the following we give a table with the numerical values of all the
quantities used in Figures 5 and 6 of the main text. They correspond to an
Isotropic Heisenberg model with $N=50$ sites , and transfer times
$T=\kappa\,N$, with $\kappa=1/2,\;1,\;2,\;3,\;4,$ and $5$.

$\kappa$ | $1/2$ | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---|---
$P$ | 0.9557 | 0.9531 | 0.9550 | 0.9564 | 0.9530 | 0.9573
$J_{max}$ | 19.0 | 9.5 | 4.5 | 3.5 | 2.5 | 2.0
$J_{1}$ | 0.39944 | 0.20749 | 0.10238 | 0.067054 | 0.051485 | 0.041025
$J_{2}$ | 10.848 | 4.2954 | 1.9585 | 3.3773 | 1.0095 | 0.92646
$J_{3}$ | 9.6847 | 2.9696 | 2.0284 | 0.60075 | 0.93922 | 0.3956
$J_{4}$ | 18.273 | 8.5247 | 4.2234 | 2.7508 | 1.413 | 1.2784
$J_{5}$ | 7.2718 | 7.3141 | 2.4777 | 2.1946 | 1.2422 | 1.7371
$J_{6}$ | 7.4144 | 4.1991 | 2.2276 | 3.4409 | 1.9408 | 1.7506
$J_{7}$ | 12.375 | 6.3237 | 2.3133 | 2.2059 | 1.1415 | 1.7628
$J_{8}$ | 16.283 | 5.2431 | 3.4801 | 2.1465 | 1.1807 | 1.6982
$J_{9}$ | 14.393 | 6.6373 | 3.3664 | 0.92832 | 2.0173 | 0.42639
$J_{10}$ | 8.6577 | 7.2860 | 3.8836 | 1.4471 | 1.5871 | 1.7395
$J_{11}$ | 15.283 | 5.8929 | 2.9566 | 2.2067 | 1.6632 | 1.0526
$J_{12}$ | 15.390 | 7.6673 | 2.7579 | 3.1806 | 1.7769 | 1.9995
$J_{13}$ | 18.846 | 7.1904 | 4.1133 | 3.0605 | 1.9180 | 1.9632
$J_{14}$ | 12.222 | 7.6271 | 4.1504 | 3.3954 | 1.5966 | 1.8553
$J_{15}$ | 17.139 | 7.5777 | 3.8859 | 3.4753 | 1.7934 | 1.9868
$J_{16}$ | 18.045 | 6.8020 | 4.5000 | 3.5000 | 2.1608 | 1.9033
$J_{17}$ | 14.860 | 9.1612 | 3.2385 | 3.5000 | 1.9981 | 1.8423
$J_{18}$ | 18.980 | 6.7268 | 4.5000 | 2.2440 | 2.1971 | 1.7891
$J_{19}$ | 19.000 | 7.9982 | 4.2445 | 3.1035 | 1.6376 | 1.6230
$J_{20}$ | 13.197 | 8.9228 | 3.7993 | 1.9047 | 2.5000 | 1.1124
$J_{21}$ | 19.000 | 7.9298 | 3.9049 | 3.2019 | 1.8421 | 1.6021
$J_{22}$ | 10.844 | 6.5456 | 2.9507 | 1.8041 | 1.8788 | 1.3331
$J_{23}$ | 12.583 | 3.6055 | 3.9359 | 2.8118 | 1.0171 | 0.9925
$J_{24}$ | 13.037 | 5.6579 | 1.4901 | 2.6015 | 0.87489 | 1.3820
$J_{25}$ | 5.645 | 3.7480 | 3.9673 | 1.5246 | 2.3294 | 1.6348
Table 1:  Values of $P$ and $\\{J_{i}\\}$ obtained with the pivot method for
global optimization inside an hypercube of size $J_{max}$ for $N=50$, used to
made Figures 5 and 6 of the paper.