# Quantum walk state transfer on a hypercube

M. Štefaňák and S. Skoupý  Department of Physics, Faculty of Nuclear Sciences
and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115
19 Praha 1 - Staré Město, Czech Republic<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate state transfer on a hypercube by means of a quantum walk where
the sender and the receiver vertices are marked by a weighted loops. First, we
analyze search for a single marked vertex, which can be used for state
transfer between arbitrary vertices by switching the weighted loop from the
sender to the receiver after one run-time. Next, state transfer between
antipodal vertices is considered. We show that one can tune the weight of the
loop to achieve state transfer with high fidelity in shorter run-time in
comparison to the state transfer with a switch. Finally, we investigate state
transfer between vertices of arbitrary distance. It is shown that when the
distance between the sender and the receiver is at least 2, the results
derived for the antipodes are well applicable. If the sender and the receiver
are direct neighbours the evolution follows a slightly different course.
Nevertheless, state transfer with high fidelity is achieved in the same run-
time.

##  1 Introduction

One of the fruitful application of quantum walks [1, 2, 3] is the task of
state transfer [4] between two vertices of a graph. Consider the particle
initially localized on the sender vertex. Our goal is to transfer it with high
probability to the receiver vertex by a quantum walk evolution. There are two
basic approaches to this problem. If we have control over all sites we can
globally design the dynamics such that the walker is transferred between a
selected pair of vertices at a certain time. For the continuous time
evolution, where the dynamics is governed by the Schroedinger equation with a
given hamiltonian, the problem was investigated to large detail in the context
of spin chains [5, 6, 7, 8, 9, 10], discrete quantum networks of various
topologies [11, 12, 13, 14, 15] and continuous time quantum walks [16, 17, 18,
19, 20]. In the discrete time case this approach was studied on various graphs
such as circle [21, 22], regular graphs [23] or more general networks [24].
Recently, discrete time quantum walks with evolution operators defined by a
product of two reflections [25] were also applied for state transfer. The
paper [26] focused on walks with the Grover coin, while [27], [28]
investigated a more general setting of coins constructed from weighted tail-
arc incidence matrices, finding several classes of graphs supporting state
transfer.

The second approach to state transfer utilizes quantum search [29, 30, 31, 32,
33], where the dynamics at the sender and receiver vertices is altered
locally. In search we evolve the system from the equal weight superposition to
a state localized on the marked vertex. For state transfer the walk starts
localized on the sender vertex and evolves through the equal weight
superposition to a state localized at the receiver vertex. One possibility is
to mark both vertices that want to communicate at the same time. In the
discrete time case this method was proposed for state transfer on lattices
[34] and further analyzed on various types of finite graphs, e. g. on cycles
and their variants [16, 35], star and complete graph with loops [36], complete
bipartite graph [37], circulant graphs [38] or butterfly network [39]. In the
continuous time evolution this approach was investigated on Erdos-Renyi graphs
[33]. Nevertheless, running the search algorithm with two marked vertices does
not necessarily guarantee state transfer with fidelity close to 1. However, if
the search success probability is high, one can use the switch approach [40,
41]. Here we initially mark only the sender vertex and evolve the search
algorithm for one period to reach a state close to the equal weight
superposition, then switch the marking from the sender to the receiver vertex
and evolve the search algorithm for one more period.

In comparison with the first method the modification of search for state
transfer has the advantage that in principle any two pairs of vertices can
establish communication without the need to adjust the dynamics on the rest of
the graph. They don’t even have to know each others location, only some global
property of the graph e.g. number of vertices which influences the run-time.

In the present paper we consider state transfer on a hypercube by a discrete
time quantum walk where the sender and the receiver vertices are marked with a
weighted loop. First, we investigate the search for a single marked vertex,
which is then directly utilized to perform state transfer with a switch. We
follow the approach of [29] and reduce the problem to a walk on a finite line
with position dependent coins. The difference to the original search algorithm
is that the resulting approximate invariant subspace is three-dimensional, as
the loop effectively adds an eigenvector corresponding to an eigenvalue 1. The
optimal weight resulting in high success probability is determined to be
$n/2^{n}$, which is in accordance with previous studies [42, 43, 44, 45, 46,
47].

Next, we focus on the state transfer between marked vertices which are
antipodal. In such a case, the problem still can be reduced to a finite line,
however, the approximate invariant subspace has dimension 5. Apart from an
eigenvalue 1 there are two pairs of complex conjugated eigenvalues. We show
that state transfer with high fidelity can be achieved by tuning the weight
such that the phases of the eigenvalues become harmonic. The resulting quantum
walk performs state transfer faster than the approach with the switch.

Finally, we consider state transfer between marked vertices at arbitrary
distance. Here we mostly rely on numerical simulations, nevertheless, we
determine an exact 1-eigenvector of the evolution operator relevant for the
dynamics of state transfer. When the sender and the receiver vertices are not
connected by an edge the evolution of the system is very close to the case of
the antipodal vertices. The case of state transfer between direct neighbours
is bit distinct. We show that the system evolves approximately in a three-
dimensional subspace. Nevertheless, state transfer with high fidelity is
achieved for the same parameters, i.e. weight of the loop and the number of
steps, as before.

The rest of the paper is organized as follows: we set the notation and
investigate the search for a single marked vertex and state transfer with
switch in Section 2. In Section 3 we analyze in detail the state transfer
between antipodal vertices. The investigation is extended to vertices of
arbitrary distance in Section 4. Finally, we conclude and present an outlook
in Section 5.

##  2 Search on a hypercube with a weighted loop and state transfer with a
switch

Let us begin with the description of an unperturbed Grover walk on an
$n$-dimensional hypercube. We adopt the notation from the seminal paper [29]
on the quantum walk search on a hypercube. The graph has $2^{n}$ vertices
which are labeled by $n$-bit strings $\vec{x}=x_{1}\ldots x_{n}$, where
$x_{i}=0,1$. To each vertex we assign a basis vector $|\vec{x}\rangle$ of the
position space

${\mathcal{H}}_{P}={\rm Span}\left\\{|\vec{x}\rangle|\vec{x}=x_{1}\ldots
x_{n},\ x_{i}=0,1\right\\}.$

Each vertex of the hypercube has degree $n$, the neighbouring vertices have
Hamming distance 1, i.e. they differ in a single bit. Since hypercube is
regular, we can introduce the coin space

${\mathcal{H}}_{C}={\rm Span}\left\\{|d\rangle|d=1,\ldots n\right\\},$

in the same way at every vertex. The Hilbert space of the quantum walk on the
hypercube is then

${\mathcal{H}}={\mathcal{H}}_{P}\otimes{\mathcal{H}}_{C}.$

The unitary evolution operator of the walk is a product of the shift and the
coin operator

$U=S\cdot C.$

The shift is given by

$S=\sum_{d=1}^{n}\sum_{\vec{x}}|\vec{x}\oplus\vec{e}_{d},d\rangle\langle\vec{x},d|,$

where $\vec{e}_{d}$ is a basis vector in the direction $d$, i.e.
$\vec{e}_{d}=e^{d}_{1}\ldots e^{d}_{n}$ with $e^{d}_{j}=\delta_{j,d}$, and
$\oplus$ denotes addition modulo 2. For the unperturbed walk the coin operator
is homogeneous across the graph, i.e.

$C=I_{P}\otimes G$

where $I_{P}$ is the identity in the position space and $G$ is the Grover
diffusion operator

$G=2|s_{C}\rangle\langle
s_{C}|-I_{C},\quad|s_{C}\rangle=\frac{1}{\sqrt{n}}\sum_{d=1}^{n}|d\rangle.$

Turning to the quantum search, we have to perturb the walk. We attach a loop
to the marked vertex (but only at the marked vertex) and consider a different
coin. The graph is no longer regular, since the marked vertex which can be
labeled without loss of generality as $\vec{0}$, has a degree $n+1$. The
Hilbert space of the perturbed walk is now given by a direct sum

${\mathcal{H}}^{\prime}=\bigoplus_{\vec{x}}{\mathcal{H}}_{\vec{x}}$

of local Hilbert spaces

$\displaystyle\vec{x}\neq\vec{0}:\quad$
$\displaystyle\quad{\mathcal{H}}_{\vec{x}}={\rm
Span}\left\\{|\vec{x},d\rangle|d=1,\ldots n\right\\},$
$\displaystyle\vec{x}=\vec{0}:\quad$
$\displaystyle\quad{\mathcal{H}}_{\vec{0}}={\rm
Span}\left\\{|\vec{0},d\rangle|d=0,1,\ldots n\right\\}.$

The evolution operator is modified into

$U^{\prime}=S^{\prime}\cdot C^{\prime},$

where the shift is given by

$S^{\prime}=\sum_{d=1}^{n}\sum_{\vec{x}}|\vec{x}\oplus\vec{e}_{d},d\rangle\langle\vec{x},d|+|\vec{0},0\rangle\langle\vec{0},0|.$

For the coin we keep the Grover diffusion operator $G$ on the non-marked
vertices and alter it at the marked vertex $\vec{0}$ to $-G^{\prime}$

$C^{\prime}=(I_{P}-|\vec{0}\rangle\langle\vec{0}|)\otimes
G-|\vec{0}\rangle\langle\vec{0}|\otimes G^{\prime}.$

Here $G^{\prime}$ denotes the modified Grover coin with a loop of weight $l$

$G^{\prime}=2|s_{l}\rangle\langle
s_{l}|-I_{0},\quad|s_{l}\rangle=\frac{1}{\sqrt{n+l}}\left(\sqrt{l}|0\rangle+\sum_{d=1}^{n}|d\rangle\right).$

For the initial state of the search, we consider the equal weight
superposition of all basis states except of the state of the loop, i.e.

$|\psi_{0}\rangle=\frac{1}{\sqrt{n2^{n}}}\sum_{d=1}^{n}\sum_{\vec{x}}|\vec{x},d\rangle.$

In the same way as in [29], the problem can be reduced to the walk on a finite
line with a non-homogeneous coin. Define the following $2n+1$ orthonormal
basis states

$\displaystyle|x,R\rangle$ $\displaystyle=\frac{1}{\sqrt{(n-x){n\choose
x}}}\sum_{|\vec{x}|=x}\sum_{x_{d}=0}|\vec{x},d\rangle,\quad x=0,\ldots n-1,$
$\displaystyle|x,L\rangle$ $\displaystyle=\frac{1}{\sqrt{x{n\choose
x}}}\sum_{|\vec{x}|=x}\sum\limits_{x_{d}=1}^{n}|\vec{x},d\rangle,\quad
x=1,\ldots n,$ $\displaystyle|0,\circlearrowleft\rangle$
$\displaystyle=|\vec{0},0\rangle.$

The unperturbed shift operator acts in this basis as

$S=\sum_{x=0}^{n-1}\left(|x,R\rangle\langle x+1,L|+|x+1,L\rangle\langle
x,R|\right)$

while for the perturbed shift we have to add the loop at 0

$S^{\prime}=S+|0,\circlearrowleft\rangle\langle 0,\circlearrowleft|.$

The coins $C$ and $C^{\prime}$ acts locally on each position $x$, and their
actions differ only at the marked vertex 0

$\displaystyle C$ $\displaystyle=\sum_{x=0}^{n}|x\rangle\langle x|\otimes
C_{x},$ $\displaystyle C^{\prime}$ $\displaystyle=|0\rangle\langle 0|\otimes
C_{0}^{\prime}+\sum_{x=1}^{n}|x\rangle\langle x|\otimes C_{x}.$

The local coin operators for $x\neq 0,n$ are in the
$\left\\{|R\rangle,|L\rangle\right\\}$ basis given by

$C_{x}=\begin{pmatrix}\cos\theta_{x}&\sin\theta_{x}\\\
\sin\theta_{x}&-\cos\theta_{x}\end{pmatrix},$

where the angle $\theta_{x}$ is determined by

$\quad\cos\theta_{x}=1-\frac{2x}{n},\quad\sin\theta_{x}=\frac{2}{n}\sqrt{x(n-x)}$

For $x=n$ the coin space is one-dimensional and the coin reduces to $C_{n}=1$,
and the same holds for $x=0$ in the unpertubed case. For the perturbed case,
in contrast to [29], the coin space at the marked vertex $x=0$ is two-
dimensional, and in the basis
$\left\\{|R\rangle,|\circlearrowleft\rangle\right\\}$ the local coin operator
reads

$C_{0}^{\prime}=\begin{pmatrix}-\frac{n-l}{n+l}&-\frac{2\sqrt{nl}}{n+l}\\\
-\frac{2\sqrt{nl}}{n+l}&\frac{n-l}{n+l}\end{pmatrix}.$

The initial state of the search is in the basis
$\left\\{|x,R\rangle,|x,L\rangle,|0,\circlearrowleft\rangle\right\\}$
expressed as

$\displaystyle|\psi_{0}\rangle=$ $\displaystyle\
\frac{1}{2^{\frac{n}{2}}}(|0,R\rangle+|n,L\rangle)+\frac{1}{2^{\frac{n}{2}}}\sum_{x=1}^{n-1}\left(\sqrt{{n-1\choose
x-1}}|x,L\rangle+\sqrt{{n-1\choose x}}|x,R\rangle\right).$

Let us show that the search algorithm evolves approximately in a three-
dimensional subspace spanned by
$\\{|\psi_{0}\rangle,|0\circlearrowleft\rangle,|\psi_{1}\rangle\\}$,
where111Similar state was employed in the analysis of [29], but here we modify
it slightly - we omit the last term in the summation, which simplifies the
calculations for the state transfer.

$\displaystyle|\psi_{1}\rangle$
$\displaystyle=\frac{1}{c}\sum_{x=0}^{n/2-2}\frac{1}{\sqrt{2{n-1\choose
x}}}\left(|x,R\rangle-|x+1,L\rangle\right),\quad
c=\sqrt{\sum_{x=0}^{n/2-2}\frac{1}{{n-1\choose x}}}.$

Note that with the estimate

$\sum_{x=0}^{n/2-2}\frac{1}{{n-1\choose x}}\approx 1+\frac{1}{n-1},$

we can approximate the normalization of $|\psi_{1}\rangle$ by

$\frac{1}{c}\approx\sqrt{1-\frac{1}{n}}\approx 1-\frac{1}{2n}.$ (1)

First, by direct calculation one can show that

$|\alpha_{1}\rangle=\sqrt{\frac{l2^{n}}{n+l2^{n}}}|\psi_{0}\rangle-\sqrt{\frac{n}{n+l2^{n}}}|0,\circlearrowleft\rangle,$

is an exact eigenvector of $U^{\prime}$ with eigenvalue 1. Second, for small
values of the weight $l$ the states

$\displaystyle|\alpha_{2}\rangle$
$\displaystyle=\sqrt{\frac{n}{n+l2^{n}}}|\psi_{0}\rangle+\sqrt{\frac{l2^{n}}{n+l2^{n}}}|0,\circlearrowleft\rangle,$
$\displaystyle|\alpha_{3}\rangle$ $\displaystyle=|\psi_{1}\rangle,$

are approximate eigenvectors. We find that the action of $U^{\prime}$ on
$|\alpha_{2,3}\rangle$ is given by

$\displaystyle U^{\prime}|\alpha_{2}\rangle=$ $\displaystyle\
|\alpha_{2}\rangle-\frac{2}{n+l}\sqrt{l+\frac{n}{2^{n}}}\left(\sqrt{l}|0,\circlearrowleft\rangle+\sqrt{n}|1,L\rangle\right),$
$\displaystyle U^{\prime}|\alpha_{3}\rangle=$ $\displaystyle\
|\alpha_{3}\rangle-\frac{\sqrt{2l}}{c(n+l)}\left(\sqrt{n}|0,\circlearrowleft\rangle-\sqrt{l}|1,L\rangle\right)-$
$\displaystyle-\frac{2}{c\sqrt{2{n-1\choose\lfloor\frac{n}{2}\rfloor-2}}}\left(\frac{\lfloor\frac{n}{2}\rfloor-1}{n}|\lfloor\frac{n}{2}\rfloor-2,R\rangle+\frac{\sqrt{(\lfloor\frac{n}{2}\rfloor-1)(n-\lfloor\frac{n}{2}\rfloor+1)}}{n}|\lfloor\frac{n}{2}\rfloor,L\rangle\right),$

from which we determine the matrix elements

$\displaystyle\langle\alpha_{2}|U^{\prime}|\alpha_{2}\rangle$
$\displaystyle=1-2\frac{l+\frac{n}{2^{n}}}{n+l},$
$\displaystyle\langle\alpha_{3}|U^{\prime}|\alpha_{3}\rangle$
$\displaystyle=1-\frac{l}{c^{2}(n+l)}-\frac{\lfloor\frac{n}{2}\rfloor-1}{c^{2}n{n-1\choose\lfloor\frac{n}{2}\rfloor-2}},$
$\displaystyle\langle\alpha_{3}|U^{\prime}|\alpha_{2}\rangle$
$\displaystyle=-\langle\alpha_{2}|U^{\prime}|\alpha_{3}\rangle=\frac{1}{c}\frac{\sqrt{2n(l+\frac{n}{2^{n}})}}{n+l}.$

In the basis
$\left\\{|\alpha_{1}\rangle,|\alpha_{2}\rangle,|\alpha_{3}\rangle\right\\}$
the evolution operator of the search $U^{\prime}$ can be approximated by a
matrix

$U^{\prime}_{ef}=\begin{pmatrix}1&0&0\\\ 0&\cos{\omega}&\sin\omega\\\
0&-\sin\omega&\cos\omega\end{pmatrix}$

with the angle $\omega$ given by

$\omega\approx\sin\omega=\langle\alpha_{3}|U^{\prime}|\alpha_{2}\rangle\approx\frac{(2n-1)\sqrt{l+\frac{n}{2^{n}}}}{\sqrt{2n}(l+n)},$

where we have used the approximation of the normalization (1). Hence, the
evolution keeps the state $|\alpha_{1}\rangle$ unchanged and acts as an
approximate rotation in the $|\alpha_{2}\rangle$, $|\alpha_{3}\rangle$ plane
by an angle $\omega$.

Let us now estimate the success probability for a given $l$. In terms of
$|\alpha_{j}\rangle$, the initial state and the target state (loop at the
marked vertex) are given by

$\displaystyle|\psi_{0}\rangle$
$\displaystyle=\sqrt{\frac{l2^{n}}{n+l2^{n}}}|\alpha_{1}\rangle+\sqrt{\frac{n}{n+l2^{n}}}|\alpha_{2}\rangle,$
$\displaystyle|0,\circlearrowleft\rangle$
$\displaystyle=-\sqrt{\frac{n}{n+l2^{n}}}|\alpha_{1}\rangle+\sqrt{\frac{l2^{n}}{n+l2^{n}}}|\alpha_{2}\rangle.$

Maximal success probability is achieved if we rotate from $|\alpha_{2}\rangle$
to $-|\alpha_{2}\rangle$, i.e. if we reach a state

$|\psi(T)\rangle\approx\sqrt{\frac{l2^{n}}{n+l2^{n}}}|\alpha_{1}\rangle-\sqrt{\frac{n}{n+l2^{n}}}|\alpha_{2}\rangle.$

In this state, we find the walker in the loop with probability

$P(l)=|\langle
0,\circlearrowleft|\psi(T)\rangle|^{2}\approx\frac{4nl2^{n}}{(n+l2^{n})^{2}}.$
(2)

This function has a maximum equal to one for $l=\frac{n}{2^{n}}$, confirming
that the optimal weight can be chosen as $l=\frac{d}{N}$ for a $d-$regular
graph even if we consider the loop only at the marked vertex. Note that for
$l=\frac{n}{2^{n}}$ the vectors $|\alpha_{1,2}\rangle$ simplify into

$\displaystyle|\alpha_{1}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|\psi_{0}\rangle-|0,\circlearrowleft\rangle\right),$
$\displaystyle|\alpha_{2}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|\psi_{0}\rangle+|0,\circlearrowleft\rangle\right).$

Figure 1: Success probability of the search algorithm on the hypercube of
dimension 10 as a function of the weight $l$. Black curve is given by (2),
gray circles correspond to numerical simulation.

Turning to the run-time of the search algorithm, to rotate from
$|\alpha_{2}\rangle$ to $-|\alpha_{2}\rangle$ the number of steps has to be
taken approximately as

$T_{1}\approx\frac{\pi}{\omega}\approx\frac{\pi\sqrt{2n}(l+n)}{(2n-1)\sqrt{l+\frac{n}{2^{n}}}}.$

For the optimal weight, we find

$T_{1}\approx\frac{n\pi(1+2^{-n})}{2n-1}2^{\frac{n}{2}}\sim\frac{\pi}{2}2^{\frac{n}{2}}=O(\sqrt{N}).$

We can now utilize the derived results for search to perform state transfer
between any two vertices using the switch approach [40, 41]. We begin at the
sender vertex $\vec{s}$ in the state $|\vec{s},0\rangle$. Note that at this
stage the weighted loop is considered only on the sender vertex. The system is
evolved for $T_{1}$ steps, reaching close to the equal-weight superposition
state $|\psi_{0}\rangle$. Then we switch the marked vertex from the sender to
the receiver $\vec{r}$, and evolve for $T_{1}$ more steps. With high fidelity
the particle will be found in the loop at the receiver vertex, i.e. in the
state $|\vec{r},0\rangle$. The total number of steps for state transfer with
switch is then given by

$T_{2}=2T_{1}\approx\frac{2n\pi(1+2^{-n})}{2n-1}2^{\frac{n}{2}}\sim\pi
2^{\frac{n}{2}}.$ (3)

##  3 State transfer between the antipodal vertices

Let us consider the state transfer between the antipodal vertices (without
loss of generality $\vec{0}$ and $\vec{1}=1\ldots 1$). Due to the symmetry of
this configuration the problem can be again reduced to a finite line of length
$n$, with loops on both ends, i.e. in comparison with the search we have one
additional basis state $|n,\circlearrowleft\rangle=|\vec{1},0\rangle$. The
evolution operator is given by

$U^{\prime\prime}=C^{\prime\prime}\cdot S^{\prime\prime},$

where the shift reads

$S^{\prime\prime}=S+|0,\circlearrowleft\rangle\langle
0,\circlearrowleft|+|n,\circlearrowleft\rangle\langle n,\circlearrowleft|,$

and the coin operator is decomposed as

$C^{\prime\prime}=\left(|0\rangle\langle 0|+|n\rangle\langle n|\right)\otimes
C_{0}^{\prime}+\sum_{x=1}^{n-1}|x\rangle\langle x|\otimes C_{x}.$

The initial state of the state transfer algorithm is the loop at the sender
vertex $0$, i.e.

$|\psi(0)\rangle=|0,\circlearrowleft\rangle.$

We show that the algorithm evolves approximately in a five dimensional
subspace spanned by
$\\{|\psi_{0}\rangle,|0,\circlearrowleft\rangle,|n,\circlearrowleft\rangle,|\psi_{1}\rangle,|\psi_{2}\rangle\\}$,
where we have denoted

$|\psi_{2}\rangle=\frac{1}{c}\sum_{x=0}^{n/2-2}\frac{1}{\sqrt{2{n-1\choose
x}}}\left(|n-x,L\rangle-|n-x-1,R\rangle\right).$

First, by direct calculation one can show that

$|\beta_{1}\rangle=\sqrt{\frac{l2^{n}}{l2^{n}+2n}}|\psi_{0}\rangle-\sqrt{\frac{n}{l2^{n}+2n}}(|0,\circlearrowleft\rangle+|n,\circlearrowleft\rangle),$

is an exact eigenvector of $U^{\prime\prime}$ with eigenvalue 1. Next,
consider the following four orthonormal states

$\displaystyle|\beta_{2}\rangle$
$\displaystyle=\sqrt{\frac{2n}{l2^{n}+2n}}|\psi_{0}\rangle+\sqrt{\frac{l2^{n-1}}{l2^{n}+2n}}(|0,\circlearrowleft\rangle+|n,\circlearrowleft\rangle),$
$\displaystyle|\beta_{3}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|\psi_{1}\rangle-|\psi_{2}\rangle\right),$
$\displaystyle|\beta_{4}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|0,\circlearrowleft\rangle-|n,\circlearrowleft\rangle\right),$
$\displaystyle|\beta_{5}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|\psi_{1}\rangle+|\psi_{2}\rangle\right).$
(4)

The action of the evolution operator on these vectors is given by

$\displaystyle U^{\prime\prime}|\beta_{2}\rangle=$
$\displaystyle|\beta_{2}\rangle-\frac{\sqrt{2l(l+n\
2^{1-n})}}{n+l}(|0,\circlearrowleft\rangle+|n,\circlearrowleft\rangle)-\frac{\sqrt{2n(l+n\
2^{1-n})}}{n+l}(|1,L\rangle+|n-1,R\rangle),$ $\displaystyle
U^{\prime\prime}|\beta_{3}\rangle=$
$\displaystyle|\beta_{3}\rangle-\frac{\sqrt{ln}}{c(n+l)}(|0,\circlearrowleft\rangle+|n,\circlearrowleft\rangle)+\frac{l}{c(n+l)}(|1,L\rangle+|n-1,R\rangle)-$
$\displaystyle-\frac{1}{c\sqrt{{n-1\choose\lfloor\frac{n}{2}\rfloor-2}}}\left(\frac{\lfloor\frac{n}{2}\rfloor-1}{n}(|\lfloor\frac{n}{2}\rfloor-2,R\rangle+|\lfloor\frac{n+1}{2}\rfloor+2,L\rangle)+\right.$
$\displaystyle\left.+\frac{\sqrt{(\lfloor\frac{n}{2}\rfloor-1)(n-\lfloor\frac{n}{2}\rfloor+1)}}{n}(|\lfloor\frac{n}{2}\rfloor,L\rangle+|\lfloor\frac{n+1}{2}\rfloor,R\rangle)\right),$
$\displaystyle U^{\prime\prime}|\beta_{4}\rangle=$
$\displaystyle\frac{n-l}{n+l}|\beta_{4}\rangle-\frac{\sqrt{2ln}}{l+n}(|1,L\rangle-|n-1,R\rangle),$
$\displaystyle U^{\prime\prime}|\beta_{5}\rangle=$
$\displaystyle|\beta_{5}\rangle-\frac{\sqrt{ln}}{c(n+l)}(|0,\circlearrowleft\rangle-|n,\circlearrowleft\rangle)+\frac{l}{c(n+l)}(|1,L\rangle-|n-1,R\rangle)-$
$\displaystyle-\frac{1}{c\sqrt{{n-1\choose\lfloor\frac{n}{2}\rfloor-2}}}\left(\frac{\lfloor\frac{n}{2}\rfloor-1}{n}(|\lfloor\frac{n}{2}\rfloor-2,R\rangle-|\lfloor\frac{n+1}{2}\rfloor+2,L\rangle)+\right.$
$\displaystyle\left.+\frac{\sqrt{(\lfloor\frac{n}{2}\rfloor-1)(n-\lfloor\frac{n}{2}\rfloor+1)}}{n}(|\lfloor\frac{n}{2}\rfloor,L\rangle-|\lfloor\frac{n+1}{2}\rfloor,R\rangle)\right).$

From this we find that the diagonal matrix elements of $U^{\prime\prime}$ read

$\displaystyle\langle\beta_{2}|U^{\prime\prime}|\beta_{2}\rangle=$
$\displaystyle 1-\frac{2\left(l+n\ 2^{1-n}\right)}{n+l},$
$\displaystyle\langle\beta_{3}|U^{\prime\prime}|\beta_{3}\rangle=$
$\displaystyle\langle\beta_{5}|U^{\prime\prime}|\beta_{5}\rangle=1-\frac{l}{c^{2}(n+l)}-\frac{\lfloor\frac{n}{2}\rfloor-1}{c^{2}n{n-1\choose\lfloor\frac{n}{2}\rfloor-2}},$
$\displaystyle\langle\beta_{4}|U^{\prime\prime}|\beta_{4}\rangle=$
$\displaystyle 1-\frac{2l}{n+l}.$

The only non-zero off-diagonal matrix elements are

$\displaystyle\langle\beta_{3}|U^{\prime\prime}|\beta_{2}\rangle$
$\displaystyle=-\langle\beta_{2}|U^{\prime\prime}|\beta_{3}\rangle=\frac{\sqrt{2n\left(l+n2^{1-n}\right)}}{c(n+l)},$
$\displaystyle\langle\beta_{5}|U^{\prime\prime}|\beta_{4}\rangle$
$\displaystyle=-\langle\beta_{4}|U^{\prime\prime}|\beta_{5}\rangle=\frac{\sqrt{2ln}}{c(n+l)}.$

Hence, for large $n$ and small $l$ the evolution operator in the approximate
invariant subspace is close to the unitary operator

$U^{\prime\prime}_{ef}=\begin{pmatrix}1&0&0&0&0\\\
0&\cos\omega_{1}&-\sin\omega_{1}&0&0\\\ 0&\sin\omega_{1}&\cos\omega_{1}&0&0\\\
0&0&0&\cos\omega_{2}&-\sin\omega_{2}\\\
0&0&0&\sin\omega_{2}&\cos\omega_{2}\end{pmatrix},$

where the angles $\omega_{j}$ are given by

$\displaystyle\omega_{1}$
$\displaystyle\approx\frac{\sqrt{2n\left(l+n2^{1-n}\right)}}{c(n+l)},\quad\omega_{2}\approx\frac{\sqrt{2ln}}{c(n+l)}.$

To achieve state transfer with high fidelity we can tune the weight $l$ to
make the frequencies harmonic. In particular, for

$l=\frac{2}{3}\frac{n}{2^{n}},$ (5)

we find (with the approximation (1))

$\omega_{2}=\frac{\omega_{1}}{2}\approx\frac{2n-1}{\sqrt{3}n\left(1+\frac{2}{3}2^{-n}\right)}2^{-\frac{n}{2}}\sim\frac{2}{\sqrt{3}}2^{-\frac{n}{2}}.$
(6)

Note that for this value of $l$ the basis states $|\beta_{1,2}\rangle$ reduce
to

$\displaystyle|\beta_{1}\rangle$
$\displaystyle=\frac{1}{2}|\psi_{0}\rangle-\sqrt{\frac{3}{8}}(|0,\circlearrowleft\rangle+|n,\circlearrowleft\rangle),$
$\displaystyle|\beta_{2}\rangle$
$\displaystyle=\frac{\sqrt{3}}{2}|\psi_{0}\rangle+\sqrt{\frac{1}{8}}(|0,\circlearrowleft\rangle+|n,\circlearrowleft\rangle).$
(7)

The eigenvectors and eigenvalues of $U^{\prime\prime}_{ef}$ are then given by

$\displaystyle|\omega_{1}^{(\pm)}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|\beta_{2}\rangle\mp
i|\beta_{3}\rangle\right),\quad\lambda_{1}^{(\pm)}=e^{\pm 2i\omega_{2}},$
$\displaystyle|\omega_{2}^{(\pm)}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|\beta_{4}\rangle\mp
i|\beta_{5}\rangle\right),\quad\lambda_{2}^{(\pm)}=e^{\pm i\omega_{2}}.$

The initial and the final state (loops at the sender and the receiver vertex)
of the STA can be decomposed as

$\displaystyle|0,\circlearrowleft\rangle=$
$\displaystyle-\sqrt{\frac{3}{8}}|\beta_{1}\rangle+\frac{1}{\sqrt{8}}|\beta_{2}\rangle+\frac{1}{\sqrt{2}}|\beta_{4}\rangle$
$\displaystyle=$
$\displaystyle-\sqrt{\frac{3}{8}}|\beta_{1}\rangle+\frac{1}{4}(|\omega_{1}^{(+)}\rangle+|\omega_{1}^{(-)}\rangle)+\frac{1}{2}(|\omega_{2}^{(+)}\rangle+|\omega_{2}^{(-)}\rangle),$
$\displaystyle|n,\circlearrowleft\rangle=$
$\displaystyle-\sqrt{\frac{3}{8}}|\beta_{1}\rangle+\frac{1}{\sqrt{8}}|\beta_{2}\rangle-\frac{1}{\sqrt{2}}|\beta_{4}\rangle$
$\displaystyle=$
$\displaystyle-\sqrt{\frac{3}{8}}|\beta_{1}\rangle+\frac{1}{4}(|\omega_{1}^{(+)}\rangle+|\omega_{1}^{(-)}\rangle)-\frac{1}{2}(|\omega_{2}^{(+)}\rangle+|\omega_{2}^{(-)}\rangle).$

The time evolution of the walk is then given by

$\displaystyle|\psi(t)\rangle=$
$\displaystyle-\sqrt{\frac{3}{8}}|\beta_{1}\rangle+\frac{1}{4}(e^{i2\omega_{2}t}|\omega_{1}^{(+)}\rangle+e^{-i2\omega_{2}t}|\omega_{1}^{(-)}\rangle)+\frac{1}{2}(e^{i\omega_{2}t}|\omega_{2}^{(+)}\rangle+e^{-i\omega_{2}t}|\omega_{2}^{(-)}\rangle).$

Hence, the fidelity of state transfer into the loop at the receiver vertex at
time $t$ is equal to

${\cal F}(t)=|\langle
n,\circlearrowleft|\psi(t)\rangle|^{2}=\frac{1}{64}\left(3+\cos\left(2\omega_{2}t\right)-4\cos\left(\omega_{2}t\right)\right)^{2},$
(8)

which reaches unity for

$T_{3}=\frac{\pi}{\omega_{2}}\approx\frac{\pi\sqrt{3}n\left(1+\frac{2}{3}2^{-n}\right)}{2n-1}2^{-\frac{n}{2}}\sim\frac{\pi\sqrt{3}}{2}2^{\frac{n}{2}}.$
(9)

In comparison to state transfer with a switch (3) the run-time is faster by a
factor $\frac{\sqrt{3}}{2}$.

For illustration, we display in Figure 2 the fidelity of state transfer
between the most distant vertices on a hypercube of dimension 10.

Figure 2: Fidelity of state transfer between the antipodal vertices on a
hypercube of dimension 10. Black curve correspond to (8) (with an exact value
of $c$), gray circles are obtained from numerical simulation.

We point out that the Grover walk on the hypercube without marked vertices is
capable of state transfer to the antipodal vertex in polynomial time in $n$
[48]. However, this works only for the antipodes. As we illustrate with the
numerical investigation in the following section, marking sender and receiver
vertices with a weighted loop allows for state transfer at arbitrary distance.

##  4 State transfer between vertices of arbitrary distance

When the sender and the receiver vertices are not antipodes the reduction to a
line is not possible anymore. For this reason, we turn to numerical
simulations. Our investigations indicate that state transfer with high
fidelity can be achieved for arbitrary distance $d$ with the same weight of
the loops (5) and time of measurement (9) as for the state transfer to the
antipodal vertex ($d=n$). Because of the symmetries of the hypercube it is
sufficient to consider one particular vertex of a distance $d$ from the
sender. Without loss of generality we consider the sender at the vertex
$\vec{0}$ and the receiver at $\vec{r}$, where the Hamming weight of the
$n$-bit string $\vec{r}$ is $d$. The initial and the target vectors of the
state transfer algorithm are the loop at the sender and the receiver vertices,
i.e. $|\vec{0},0\rangle$ and $|\vec{r},0\rangle$.

In Figure 3 we display the course of fidelity of state transfer for a
hypercube of dimension $n=10$ as a function of the number of steps $t$. We
consider two choices of distance between the sender and the receiver vertex -
$d=5$ depicted by the gray circles and $d=1$ corresponding to the blue
triangles. For $d=5$, and in fact for all $d\neq 1$, the fidelity closely
follows the curve (8) derived for the state transfer to the antipode. However,
in the case of state transfer between the direct neighbours the curve is less
steep and the peak is visibly wider. Nevertheless, the maximal fidelity is
reached at approximately the same time given by (9). The plot indicates that
the algorithm behaves in a different way for $d=1$ and $d\geq 2$.

Figure 3: Fidelity of state transfer on a hypercube of dimension 10. We
consider the receiver vertex at distance 5 (gray circles) and 1 (blue
triangles) from the sender vertex. For $d=5$ the curve follows the result for
the antipode (8). For $d=1$ the peak of the fidelity is visibly wider and the
course can be approximated by (12).

Let us first consider the case $d\geq 2$, where the evolution is close to the
case of the state transfer to the antipode. In Figures 4 and 5 we focus on the
fidelity of state transfer. For Figure 4 we fix the dimension of the hypercube
($n=10$) and display the fidelity of state transfer in dependence on the
distance $d$. Fidelity reaches values around 0.997 in all cases except for
$d=2$, where it is close to 0.994. In Figure 5 we consider the minimal
fidelity for the state transfer on hypercubes of dimensions ranging from $n=5$
to $n=12$. There is a clear increasing trend of the minimal fidelity.

Figure 4: Fidelity of state transfer at time $T_{3}$ (9) in dependence on the
distance between the sender and the receiver. We consider a hypercube of
dimension 10. Figure 5: Minimal fidelity of state transfer at time $T_{3}$ (9)
for a hypercube of dimension $n$.

Let us now investigate the relevant eigenvalues and eigenvectors of the
evolution operator. In analogy with (4) and (7) we denote the following
orthogonal vectors

$\displaystyle|\gamma_{1}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}|\psi_{0}\rangle-\sqrt{\frac{3}{8}}(|\vec{0},0\rangle+|\vec{r},0\rangle),$
(10) $\displaystyle|\gamma_{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{2}|\psi_{0}\rangle+\sqrt{\frac{1}{8}}(|\vec{0},0\rangle+|\vec{r},0\rangle),$
$\displaystyle|\gamma_{4}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|\vec{0},0\rangle-|\vec{r},0\rangle).$

Direct calculation shows that $|\gamma_{1}\rangle$ is an exact eigenvector of
$U^{\prime\prime}$ with eigenvalue 1 irrespective of $d$, i.e. including
$d=1$. In fact, $|\gamma_{1}\rangle$ is an eigenstate of both the shift and
the coin operator. To complete the approximate invariant subspace we turn to
the numerical evaluation. Two pairs of complex conjugated eigenvalues
$\lambda_{j}^{(\pm)}=e^{\pm i\omega_{j}}$, $j=1,2$ with the smallest non-zero
eigenfrequencies $\omega_{j}$ are determined numerically, and the
corresponding eigenvectors are found. The upper plot in Figure 6 shows the
eigenfrequencies for the hypercube of dimension 10, confirming that for all
distances $d=2,\ldots,n$ they are close to the ideal theoretical values
determined by (6) for the state transfer to the antipodal vertex. The lower
plot shows the ratio $\omega_{1}/\omega_{2}$. We see that the frequencies are
close to a 2:1 resonance. In Figure 7 we consider the overlap of the
corresponding numerically determined eigenvectors with the states
$|\gamma_{2}\rangle$ and $|\gamma_{4}\rangle$ for a hypercube of dimension
$n=8$. The scalar products are close to the ideal value $1/\sqrt{2}$ derived
for the state transfer to the antipode. We conclude that the analytical
results derived in Section 3 are well applicable to state transfer between
vertices of arbitrary distance $d\geq 2$.

Figure 6: Upper plot shows numerically evaluated relevant eigenfrequencies for
a hypercube of dimension $n=10$ depending on the distance between the sender
and the receiver $d$. Circles correspond to the smaller angle $\omega_{2}$,
triangles to $\omega_{1}$. Dashed lines highlight the values (6) given from
the state transfer to the antipode. The lower plot shows the ratio
$\omega_{1}/\omega_{2}$ for different distances. Figure 7: Overlap between the
relevant eigenvectors of the evolution operator and the vectors
$|\gamma_{2}\rangle$ and $|\gamma_{4}\rangle$, respectively. Black circles
correspond to $\langle\omega_{1}^{(\pm)}|\gamma_{2}\rangle$, blue triangles to
$\langle\omega_{2}^{(\pm)}|\gamma_{4}\rangle$. Dashed red line marks
$1/\sqrt{2}$ which is the ideal theoretical value of the scalar products.
Hypercube of dimension 8 is considered.

Turning to the case of sender and receiver being directly connected by an
edge, we find that the distinction with the case of $d\geq 2$ is twofold.
First, the projection of the initial and the target state onto
$\ker(U^{\prime\prime}-1)$ is larger than what holds for an arbitrary distance
(10). Let us denote by $d_{s}$ and $d_{r}$ the direction to the sender and
receiver, respectively, so that e.g. $|\vec{0},d_{r}\rangle$ corresponds to
the directed edge from the sender vertex $\vec{0}$ to the receiver vertex
$\vec{r}$. A rather tedious calculation reveals that the projection is given
by the following normalized eigenvector (note that $|\vec{0},d_{r}\rangle$ and
$|\vec{r},d_{s}\rangle$ are not orthogonal to $|\psi_{0}\rangle$)

$|\gamma_{0}\rangle=\frac{x}{\sqrt{2}}(|\vec{0},0\rangle+|\vec{r},0\rangle)+y|\psi_{0}\rangle+\frac{z}{\sqrt{2}}(|\vec{0},d_{r}\rangle+|\vec{r},d_{s}\rangle),$
(11)

where the coefficients $x,y,z$ read

$\displaystyle x$ $\displaystyle=\sqrt{\frac{3a}{b}},\quad a=2^{n}+2n-4,\quad
b=3\cdot 2^{n}+8n-12-2^{2-n},$ $\displaystyle y$
$\displaystyle=-\frac{2(n-1)}{\sqrt{ab}},\quad
z=-\frac{\sqrt{n2^{n+1}}(1-2^{1-n})}{\sqrt{ab}}.$

For large $n$, we find that $x$ approaches unity while $y$ and $z$ vanish
according to

$\displaystyle x=1-O(2^{-n}),\quad y=O(n2^{-n}),\quad z=O(\sqrt{n2^{-n}}).$

Therefore, for large $n$ the vector

$|\tilde{\gamma_{0}}\rangle=\frac{1}{\sqrt{2}}(|\vec{0},0\rangle+|\vec{r},0\rangle),$

is an approximate eigenvector corresponding to 1.

Second distinction is in the relevant part of the spectrum of the evolution
operator. Indeed, numerical evaluation indicates that the pair
$\lambda_{1}^{(\pm)}$ is no longer present in the spectrum of
$U^{\prime\prime}$. On the other, for the remaining pair $\lambda_{2}^{(\pm)}$
the angle $\omega_{2}$ is close to the formula (6), see Figure 8. Hence, the
approximate invariant subspace is only three-dimensional, consisting of
$|\gamma_{0}\rangle$ and $|\omega_{2}^{(\pm)}\rangle$. The initial and the
target vectors of the state transfer algorithm are decomposed into this basis
as

$\displaystyle|\vec{0},0\rangle=$
$\displaystyle\frac{x}{\sqrt{2}}|\gamma_{0}\rangle+\frac{1}{2}(|\omega_{2}^{(+)}\rangle+|\omega_{2}^{(-)}\rangle)+|\delta_{s}\rangle,$
$\displaystyle|\vec{r},0\rangle=$
$\displaystyle\frac{x}{\sqrt{2}}|\gamma_{0}\rangle-\frac{1}{2}(|\omega_{2}^{(+)}\rangle+|\omega_{2}^{(-)}\rangle)+|\delta_{r}\rangle,$

where $|\delta_{r,s}\rangle$ are some residual non-normalized vectors, which
vanish for large $n$ as $x$ tends to 1. The state of the system evolves
according to

$|\psi(t)\rangle=\frac{1}{\sqrt{2}}|\tilde{\gamma_{0}}\rangle+\frac{1}{2}(e^{-i\omega_{2}t}|\omega_{2}^{(+)}\rangle+e^{i\omega_{2}t}|\omega_{2}^{(-)}\rangle)+|\delta_{s}(t)\rangle.$

Neglecting the residual vectors, the fidelity of state transfer at time $t$
then follows the curve

${\cal
F}(t)=|\langle\vec{r},0|\psi(t)\rangle|^{2}=\frac{1}{4}(x^{2}-\cos(\omega_{2}t))^{2}.$
(12)

At time $T_{3}$ (9) fidelity reaches the value

${\cal F}=\frac{1}{4}(1+x^{2})^{2}.$ (13)

In Figure 9 we compare this result with the fidelity at time $T_{3}$ obtained
from numerical simulation. For large $n$, where we can replace
$|\gamma_{0}\rangle$ with $|\tilde{\gamma_{0}}\rangle$ and $x$ with 1, we find
that the fidelity at time $t$ reads

${\cal F}(t)=\sin^{4}{\left(\frac{\omega_{2}t}{2}\right)},$

which reaches unity for number of steps equal to $T_{3}$. This explains the
broader peak in Figure 3 in comparison with the case $d=5$.

Figure 8: Comparison of numerically evaluated relevant eigenvalues
$\lambda_{2}^{(\pm)}$ with the analytical value for the case of state transfer
between the neighbouring vertices. Black dots correspond to the phase of
$\lambda_{2}^{(+)}$ for different dimensions of the hypercube $n$. The red
curve is given by (6). Figure 9: Fidelity of state transfer to a neighbouring
vertex at time $T_{3}$ for a given dimension of the hypercube $n$. Black dots
are obtained from numerical simulations. The red curve is taken from the
approximation (13).

##  5 Conclusions

Two approaches to state transfer on a hypercube were investigated in detail.
The first one utilizes quantum walk search, where we evolve from the loop at
the sender vertex to the initial state of the search algorithm, switch the
marked vertex from the sender to the receiver, and evolve to the loop at the
receiver vertex. It was shown that high fidelity of state transfer is achieved
for the weight given by the ratio of the degree of the vertex to the total
number of vertices. The run-time of the state transfer with a switch is then
twice the number of steps of the search. In the second approach we have
considered marking both the sender and the receiver vertices simultaneously.
The case of antipodal vertices was investigated analytically, and we have
shown that the optimal weight is 2/3 of the weight for the search. This
results in a run-time faster by a factor of $\sqrt{3}/2$ when compared to the
state transfer with a switch. Utilizing numerical simulations we have
illustrated that the second approach works well for arbitrary distance between
the sender and the receiver, although the evolution differs for the case of
direct neighbours. This confirms that on a hypercube network any two parties
can establish communication without the need to know each others position and
using only local operations and global properties of the network (dimension of
the hypercube).

When the sender and the receiver are marked simultaneously the evolution is
determined by two pairs of complex conjugated eigenvalues. State transfer with
high fidelity was achieved by tuning the weight of the loops such that the two
relevant frequencies became harmonic. It is an open question if such an effect
can be demonstrated on different graphs. In addition, the ratio of the optimal
weights for the state transfer and search, was shown to be 2/3 for the
hypercube. One can ask if this is specific for a hypercube or it applies to a
broader set of graphs.

Both authors are grateful for financial support from RVO 14000 and ”Centre for
Advanced Applied Sciences”, Registry No. CZ.02.1.01/0.0/0.0/16 019/0000778,
supported by the Operational Programme Research, Development and Education,
co-financed by the European Structural and Investment Funds. MŠ acknowledges
the financial support from Czech Grant Agency project number GAČR 23-07169S.
SS is grateful for financial support from SGS22/181/OHK4/3T/14. We would like
to express our gratitude to prof. Igor Jex for stimulating discussions and
support over the years. Live long and prosper!

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# EXPLORING SELF-SUPERVISED REPRESENTATION LEARNING FOR LOW-RESOURCE MEDICAL
IMAGE ANALYSIS

###### Abstract

The success of self-supervised learning (SSL) has mostly been attributed to
the availability of unlabeled yet large-scale datasets. However, in a
specialized domain such as medical imaging which is a lot different from
natural images, the assumption of data availability is unrealistic and
impractical, as the data itself is scanty and found in small databases,
collected for specific prognosis tasks. To this end, we seek to investigate
the applicability of self-supervised learning algorithms on small-scale
medical imaging datasets. In particular, we evaluate $4$ state-of-the-art SSL
methods on three publicly accessible _small_ medical imaging datasets. Our
investigation reveals that in-domain low-resource SSL pre-training can yield
competitive performance to transfer learning from large-scale datasets (such
as ImageNet). Furthermore, we extensively analyse our empirical findings to
provide valuable insights that can motivate for further research towards
circumventing the need for pre-training on a large image corpus. To the best
of our knowledge, this is the first attempt to holistically explore self-
supervision on low-resource medical datasets. Source codes are available at:
https://github.com/soumitri2001/SmallDataSSL

**footnotetext: denotes equal contribution.

Index Terms—  Self-supervised learning, low-resource, medical image analysis,
contrastive learning, non-contrastive

##  1 Introduction

With its label efficiency and the ability to learn robust representations,
self-supervised learning (SSL) [1, 2, 3, 4, 5] has shown promising performance
across various visual understanding tasks. The typical paradigm for SSL has
been to perform pre-training on a large-scale dataset (such as ImageNet [6]),
followed by application to the downstream task with in-domain data. This,
however, poses a fundamental challenge – there is often a dearth of available
data for pre-training in a domain that is very different from commonly
available ones. One such field is medical image analysis, which comprises
disease detection [7], classification [8] and segmentation [9] from medical
scan images. Understandably, medical images vary greatly, not only across
anatomical regions, but also in their capturing mechanisms. Moreover, due to
privacy issues (and otherwise), medical data is hardly available on a large
scale; they are mostly collected in small-scale databases private to a
geographical location or laboratory. This is more prominent for specialized
prognosis cases (such as histopathological scans for cancer detection). The
need for in-domain training thereby arises in such cases. One might argue that
supervised in-domain learning is a possible solution; however, it must be
noted that deep learning models are highly prone to overfitting when it comes
to small-scale datasets. SSL techniques, on the other hand, can learn very
robust features without the need for labels, making them immune to such
problems [1, 10]. Furthermore, medical image annotation is a cumbersome
process requiring a lot of effort from skilled clinicians, further motivating
the use of self-supervision for medical imaging tasks [11, 12, 13].

While SSL has been leveraged to large-scale medical datasets [14, 8, 15],
there has been no prior work that specifically focuses on self-supervision for
smaller medical datasets. In fact, even in the broader vision community, only
a couple of works [16, 17] to date have approached self-supervision in low-
resource settings. The work by Wallace et. al. [16], which evaluated some of
the pretext-task-based SSL algorithms [18, 19, 20] over a wide range of visual
domains, was one of the earliest. However, with recent (non-)contrastive
approaches achieving state-of-the-art performance on vision tasks, mere
evaluation of classical pretext tasks alone is rather outdated.

To this end, we explore the applicability of SSL algorithms on small medical
imaging datasets. Specifically, our study encompasses both contrastive (CL)
and non-contrastive (NCL) learning algorithms in literature, and uses datasets
[21, 22, 23] that have been proposed for clinical purposes and not for mere
benchmarking. We further analyse the findings with suitable ablations to
present insights and trends across datasets, which can open up new research
directions. Last but not the least, the empirical results of our paper can be
directly referred to as a benchmark by other researchers working on similar
lines. Our study is motivated by the recent work of Cao et. al. [17], which
approached SSL in a scaled-down fashion from both model and data aspects.
However, it should be noted that we deal entirely with medical image data and
our study includes recent non-contrastive SSL variants as well, which have
been overlooked in [17].

Summing up, the salient contributions of our work are:

  1. 1.

We extensively evaluate $4$ recent state-of-the-art SSL algorithms on three
real-world small-scale medical image classification datasets.

  2. 2.

We analyse our findings across various ablations and experimental setups to
present several insights and noteworthy trends, which ought to pave the way
for future studies.

  3. 3.

To the best of our knowledge, this is one of the first attempts to explore
both data and label efficiency together in the context of medical image
analysis.

Fig. 1: Generic representation of self-supervised learning.

##  2 Materials and Methods

###  2.1 Self-Supervised Learning Algorithms

In this paper, we consider two families of SSL algorithms, namely contrastive
[1, 24] and non-contrastive [3, 2] learning, which have yielded state-of-the-
art performance at various vision tasks, surpassing traditional SSL works such
as inpainting [18] or jigsaw puzzle [19].

Contrastive Learning: These algorithms [1, 25, 4] aim at pulling the
representations of similar samples i.e. positive pairs closer while
simultaneously pushing away dissimilar samples (i.e. negative pairs). In its
typical setup [1, 25], two augmented views of the same image $x_{i}$ are
passed through a shared CNN encoder and its cascaded projector head, with the
representations obtained being $z_{i}^{(1)}$ and $z_{i}^{(2)}$. For each
augmented representation $z_{i}^{(m)}$, the contrastive loss objective sets
the other corresponding augmentation $z_{i}^{n}$ as the positive label
$(m{\neq}n)$. Having thus reduced to a classification problem, the objective
is formulated as a cross-entropy loss function with respect to $z_{i}^{(m)}$,
also known as the NT-Xent loss [26]. The overall loss would be a sum over all
the individual losses, given by Eq. 1, with temperature parameter $\tau$.

$\mathbf{L}_{CL}=-\sum\limits_{\begin{subarray}{c}m\in\\{1,2\\}\\\
i\in\\{1...B\\}\end{subarray}}\log\frac{\exp(z_{i}^{(1)}\cdot
z_{i}^{(2)}/\tau)}{\exp(z_{i}^{(1)}\cdot z_{i}^{(2)}/\tau)+\sum_{j\neq
i}\exp(z_{i}^{(m)}\cdot z_{j}^{(p)}/\tau)}$

(1)

In this work, we have chosen SimCLR [1] and DCL [4] as the representative
contrastive SSL methods, which have shown state-of-the-art performance across
various high-resource vision tasks. While SimCLR is a straightforward
implementation of the NT-Xent contrastive loss (Eq. 1), DCL removes the
negative-positive coupling multiplier term [4] and formulates a decoupled
version of the loss.

Non-Contrastive Learning: Contrastive SSL methods require large mini-batch
sizes for increased negative pairs (SimCLR [1]) and/or large memory banks
(MoCo [25]). Such a scaling in batch size and number of overall samples leads
to smaller generalization gaps which boost performance [27]. However, such
requirements make Contrastive SSL resource-inefficient. To alleviate this, NCL
methods such as SimSiam [2] and VICReg [3] have been introduced in recent
literature. These methods do not require any negative pairs and therefore
perform well even with smaller mini-batch sizes. In particular, they attempt
to minimize redundancy (information collapse) between components of the
encoded representations generated from two views of the input. VICReg does so
by an information maximization paradigm involving a variance-preservation term
[3] for the embedded components. On the other hand, SimSiam [2] minimizes the
negative cosine similarity of one view with the projector output of the other.
It calculates a symmetric loss while applying a stop gradient to the non-
projected view as follows:

$\displaystyle\mathbf{L}_{SimSiam}=\frac{1}{2}{sim}(p_{1},stgrad(z_{2}))+\frac{1}{2}{sim}(p_{2},stgrad(z_{1}))$
(2)

Here, $sim(\cdot)$ is the negative cosine similarity function given by Eq, 3,
while $stgrad(\cdot)$ refers to gradient stopping operation.

$sim(u,v)=-\frac{u}{\|u\|_{2}}\cdot\frac{v}{\|v\|_{2}}$ (3)

In our study, we have adopted VICReg and SimSiam as the representative NCL
algorithms.

###  2.2 Datasets

Three publicly accessible small-scale medical imaging datasets [21, 22, 23]
have been used in this study. Note that all of these datasets were proposed
keeping clinical use in mind, which ensures the practicality of our
investigation.

Breast: The BreaKHis400x [21] is a breast histopathological dataset comprising
microscopic biopsy images of 588 benign and 1232 malignant breast tumors at
400x magnification.

Colorectal: This is a class-balanced histopathological dataset comprising 5000
tissue images evenly divided among 8 classes [22], proposed for the detection
of colorectal cancer.

Pneumonia: This is a lung X-ray dataset by Kermany et. al. [23] comprising
5856 images, out of which 1583 are “Normal” while 4273 belong to class
“Pneumonia”.

Each of the datasets was randomly split into train, validation and test sets
in the ratios 70%, 10% and 20% respectively. Some representative sample images
are shown in Figure 2.

Fig. 2: Sample images from each of the datasets used.

##  3 EXPERIMENTAL SETUP

###  3.1 Implementation details

Pre-training: We used a randomly initialized ResNet50 encoder, a standard
architecture used for SSL training [1, 4]. For SimCLR and DCL, (i.e.
contrastive), a $2$-layer MLP projector head with $128$-dimensional output [1]
was cascaded with the encoder. For SimSiam [2]and VICReg [3], we followed
their original configurations – $2$-layer MLP projector/expander heads with
outputs of $512$ and $4096$ dimensions respectively. Batch sizes of $16$, $32$
and $64$ were considered, learning rate of $0.01$, SGD optimizer and training
for $250$ epochs.

Downstream: We used linear evaluation, wherein we froze the pre-trained
encoder and trained a linear classifier on top of the frozen representations.
Following the literature [2, 4], we attached a $2$-layered MLP classifier and
trained it for $100$ epochs using Adam optimizer with a StepLR scheduler
$(\gamma=0.5)$ decaying over episodes of $25$ epochs. The best individual
hyperparameter settings obtained from careful tuning have been provided in
Table 1.

Method Dataset | Breast | Colorectal | Pneumonia
---|---|---|---
DCL | 16(0.01) | 16(0.01) | 64(0.5)
SimCLR | 16(0.01) | 16(0.005) | 64(0.1)
SimSiam | 16(0.01) | 16(0.01) | 16(0.01)
VICReg | 16(0.005) | 16(0.005) | 16(0.01)

Table 1: Optimal hyperparameters for linear evaluation. The values have been
reported as $BatchSize(LearningRate)$.

All of the models were implemented in PyTorch [28] and the experiments were
conducted on a 16GB Nvidia Tesla V100 GPU. We shall release our codes upon
acceptance.

###  3.2 Evaluation

Protocol: We used two types of evaluation – (1) using a KNN classifier
directly on the frozen pre-trained representations and (2) linear probing. The
purpose of KNN evaluation is to evaluate the robustness of the SSL features
without any further training/fine-tuning, whereas linear probing follows
supervised training of an MLP classifier on top of the frozen backbone. For
KNN, the value of $k$ was set to 200, following recent literature [4].

Metrics: $Accuracy$ and weighted $F1$-$score$, the two most commonly used
classification metrics, have been used in this study to evaluate the
performance of SSL algorithms on the mentioned small-scale medical imaging
datasets.

##  4 Results and Analysis

We report the performances of the SSL algorithms on the three medical image
datasets using the metrics described above. The best results of KNN-based
evaluation are found at a batch size of $64$, which have been tabulated in
Table 2.

Method Dataset | Breast | Colorectal | Pneumonia
---|---|---|---
DCL [4] | 0.837 / 0.829 | 0.796 / 0.795 | 0.771 / 0.759
SimCLR [1] | 0.835 / 0.828 | 0.792 / 0.790 | 0.766 / 0.748
SimSiam [2] | 0.842 / 0.836 | 0.746 / 0.739 | 0.715 / 0.676
VICReg [3] | 0.838 / 0.831 | 0.784 / 0.780 | 0.787 / 0.773

Table 2: KNN evaluation results. Scores have been reported as
$accuracy/F1$-$score$.

As for the linear evaluation setup, the best hyperparameter settings have been
reported in Table 1, based on which the linear evaluation results are
tabulated in Table 3. For reference, we also experimented with a linear-like
transfer learning setup where the frozen backbone comprises supervised
ImageNet [6] trained representations (denoted by ImgNet+TL).

Method Dataset | Breast | Colorectal | Pneumonia
---|---|---|---
DCL [4] | 0.939 / 0.939 | 0.921 / 0.921 | 0.790 / 0.773
SimCLR [1] | 0.925 / 0.924 | 0.874 / 0.874 | 0.761 / 0.734
SimSiam [2] | 0.908 / 0.907 | 0.898 / 0.898 | 0.776 / 0.754
VICReg [3] | 0.939 / 0.939 | 0.915 / 0.915 | 0.817 / 0.806
ImgNet+TL | 0.916 / 0.915 | 0.911 / 0.910 | 0.798 / 0.775

Table 3: Linear evaluation results. Scores have been reported as
$accuracy/F1$-$score$. Fig. 3: KNN evaluation $Accuracy$ (%) vs Batch Size for
4 different methods across all 3 datasets.

From the empirical results obtained, we now further analyze their implications
along various axes of comparison.

Contrastive vs Non-contrastive: An important point of investigation would be
how the two SSL paradigms evaluated in this paper, namely CL and NCL, perform
at medical image classification under low data setting. From the KNN
evaluations in Table 2, we observe that the NCL variants consistently perform
at par with the CL approaches, outperforming the latter in Breast and
Pneumonia datasets. The trend is repeated in the linear probing results (Table
3), where VICReg outperforms both DCL and SimCLR on the above mentioned
datasets, while being very competitive to DCL on the Colorectal dataset. We
attribute this to the fact that NCL learning algorithms can work effectively
with fewer number of data samples, since there is no explicit need for a
higher number of samples, as in the case of CL methods [1, 10]. Thus, given a
fixed batch-size that is not sufficiently large, NCL methods would yield
superior results compared to CL ones.

Effect of batch size: It is well-known that SSL approaches are influenced by
the batch size taken. For CL, higher batch size ensures increase in negative
samples [10, 4], while in NCL methods, feature normalization is done along the
batch dimension [3]. To investigate these trends for small-scale medical data,
we ablated the batch size across $16$, $32$ and $64$ and plotted the KNN
accuracy scores in Figure 3. Note that batch size of $128$ was infeasible due
to resource constraints.

CL methods are expected to show improvement with increase in batch size [1,
10] due to the rise in negative samples. Surprisingly, we do not see such a
generalised trend for either of SimCLR or DCL. However, from Figure 3, it is
evident that VICReg shows improvement with increase in batch sizes, which is
in accordance with the findings of the original paper [3]. This can be
attributed to the feature normalization operation in its loss function, where
a larger batch size would ensure less noisy gradients, thereby improving
generalisability. On the other hand, for SimSiam, the peak is usually at an
intermediate value, asserting its effectiveness under lower batch size
configurations [2]. This conveys that non-contrastive SSL approaches are
better suited for low-resource medical imaging compared to CL methods.

Effect of class imbalance, SSL vs Supervised: SSL methods have been found to
be intrinsically robust to dataset imbalance [29]. This is more appropriately
shown by the consistently high $F1$-scores that keep parity with the accuracy
values for imbalanced datasets (Breast and Pneumonia). In fact, we note that
for Pneumonia, there is a large difference between $accuracy$ and $F1$-$score$
for the supervised setup $(\simeq 0.023)$ as compared to the top-peforming SSL
algorithm $(\simeq 0.010)$. Table 3 also reveals that linear probing of self-
supervised representations outperforms transfer learning of frozen supervised
ImageNet [6] features. Qualitatively, the $t$-SNE plots in Figure 4 clearly
depict the skewed distribution of the supervised model compared to a more
uniform distribution obtained from SSL. Thus, it is logical to infer that
class imbalance is handled more effectively by self-supervision, even under
low-resource settings.

Fig. 4: $t$-SNE comparison of top-performing SSL model vs supervised setting
across all 3 datasets.

##  5 CONCLUSION

In this paper, for the first time, we explored the performance of SSL on
small-scale medical imaging datasets. We holistically evaluated four such
methods across three real-world biomedical datasets. From our findings, it
seems fair to conclude that SSL methods, especially non-contrastive ones, can
be effectively leveraged to low-resource medical imaging data. However, we
also note that certain dataset-specific nuances alter the expected SSL trends,
which requires further research. Tackling such biases can potentially lead to
more robust and data-efficient SSL algorithms.

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