# Two-qubit quantum gates with minimal pulse sequences

Ignacio R. Sola  corresponding author<EMAIL_ADDRESS>Departamento de Quimica
Fisica, Universidad Complutense, 28040 Madrid, Spain  Seokmin Shin  School of
Chemistry, Seoul National University, 08826 Seoul, Republic of Korea  Bo Y.
Chang  corresponding author<EMAIL_ADDRESS>School of Chemistry, Seoul
National University, 08826 Seoul, Republic of Korea  Departamento de Quimica
Fisica, Universidad Complutense, 28040 Madrid, Spain

###### Abstract

Working with trapped atoms at close distance to each other, we show that one
can implement entangling gates based on non-independent qubits using a single
pulse per qubit, or a single structured pulse. The optimal parameters depend
on approximate solutions of Diophantine equations, causing the fidelity to
never be exactly perfect, even under ideal conditions, although the errors can
be made arbitrarily smaller at the cost of stronger fields. We fully
characterize the mechanism by which the gates operate, and show that the main
source of error in realistic implementations comes from fluctuations in the
peak intensity, which especially damages the fidelity of the gates that use
stronger fields. Working with two-pulse sequences, instead of one, enables the
use of a plethora of mechanisms and a broad range of optimal parameters to
choose from, to achieve high-fidelity gates.

##  I Introduction

Most quantum control protocols rely on complex pulse sequences or pulse
structures in the time domain. We show in this work that, for ordered systems
with a high degree of control in their spatial structure, it is possible to
use the simplest pulse sequences and achieve the same level of control acting
on the spatial degrees of freedom, adding some complexity in the spatial
domain.

Quantum computers are the paramount systems where one needs a maximum degree
of control over their spatial and time domain properties to minimize the
effects of decoherence, and to synchronize the different interference effects
that are involved in the speed-up properties of quantum algorithms [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Atoms trapped by optical tweezers [14, 15,
16, 17, 18], using highly excited Rydberg states for dipole-blockaded
interactions [19, 20, 21, 22, 23], are one of the promising platforms for
quantum computing, due to their extended coherence times [13], strong and
long-range interactions [13], scalability [14, 24], and addressability [25,
26, 27, 23, 28]. This adaptability makes Rydberg atoms a versatile resource
for implementing multi-particle entanglement [29, 30, 31, 32, 6, 33, 34, 35,
36, 37, 38], simple quantum circuits [39, 40, 33, 41, 42, 43, 44, 45, 46, 27,
47, 48, 49, 50, 51] and even quantum gates across different quantum computing
platforms [52, 53, 54, 55, 56].

Current technology allows to control the position and spatial organization of
the atoms in atomic traps with great precision, and this property has been
extensively used for quantum simulations and to prepare different entangled
states [57, 58]. Most quantum circuits, however, have relied on the use of
independent qubits, which for homogeneous qubits impose large interatomic
distances and hence operate with weak dipole blockades, leading to slow two-
qubit gates. Several C-PHASE [39, 48, 59] and C-NOT [40] gate proposals
reported implementation times in the microsecond.

Since the ancillary states are highly excited (although long-lived) Rydberg
states, speeding-up the processes has obvious advantages, as it drastically
reduces the effect of decoherence. But this typically requires working with
closer, and hence non-independent, qubits, which brings an additional level of
control in the atomic positions and the spatial profiles of the laser beams,
for which we proposed a novel spatio-temporal control framework [60, 61, 62].
It turns out that by addressing both qubits at the same time using structured
light and controlling the amplitude of the fields at the location of each
qubit, one can extend the well-known scheme proposed by Jacksch et al. [39]
with minimal changes, but working in the nanosecond regime, at least under
ideal conditions [60]. The scheme, called the SOP (symmetrically orthogonal
protocol) prepared a coherent dark state to transition the population through
Rydberg states, isolating the effects of odd and even pulses in the pulse
sequence, which added to the effect of the dipole blockade [60]. But by
breaking the symmetry of the system with apparent disorder and fully
controlling the spatial profile of the lasers, we showed that a multitude of
schemes could implement the CZ gate with higher fidelity, in 2-qubit [61] and
N-qubit systems [62].

Alternatively, there have been recent promising results addressing two or
three qubits in symmetric arrangements of the atoms, which correspond to a
very specific scenario from our setup of possible arrangements. Here, the
control is enhanced by phase modulation of the pulses [51, 63], so all the
pulse complexity lies again in the time-domain.

It is possible to classify the optimal control protocols obtained by numerical
algorithms and to analyze the correlations among subsets of control
parameters. In particular, we found highly constraint optimal parameters in
protocols that use two-pulse sequences [61]. In this work, we focus on the
minimal pulse sequences, where all the control practically depends only on the
spatial domain. In particular, we find that for non-independent qubits, there
are solutions that require a single pulse, which depends on approximate
solutions of Diophantine equations. By scrutinizing the nature of two-pulse
sequences, we determine the set of possible protocols and analyze the working
principles behind their dynamics. In this work, we also propose a different
physical realization of the non-independent qubit gates, using superposed
Gaussian beams, and provide an analysis of the role of the fluctuation and
noise in the different control parameters on the robustness of the protocols.

##  II Setup

###  II.1 Dynamics

We study here gate protocols based on non-independent qubits, that operate
with pulses that interact with both qubits (or more than one qubit in the
general setup) at the same time. Then one must control both the temporal
features of the pulse sequence (pulse areas, frequencies, relative phases) as
well as the spatial properties of the pulse beams.

An example is the SOP scheme [60], where one applies a sequence of three
structured pulses, using hybrid modes of light (e.g. superposition of TEM
modes), with different amplitudes at the qubit sites:
$\Omega_{k}(\vec{r}_{A},t)=a_{k}\mu_{0r}E_{k}(t)/\hbar=a_{k}\Omega_{k}(t)$,
$\Omega_{k}(\vec{r}_{B},t)=b_{k}\mu_{0r}E_{k}(t)/\hbar=a_{k}\Omega_{k}(t)$.
The first pulse has a large amplitude on qubit $A$, $a_{1}$, and a smaller
amplitude on qubit $B$, $b_{1}$. The second one reverts the role, but with a
phase shift in one amplitude: $a_{2}=-b_{1}$, and $b_{2}=a_{1}$. Finally, the
third pulse is a replica of the first one. The role of the $a$ and $b$
coefficients can be obviously interchanged. Arranging the factors that
participate on the local amplitudes (henceforth called geometrical factors) as
components of vectors ${\bf e}_{k}$ (henceforth structural vectors), then we
observe that ${\bf e}_{1}{\bf e}_{2}=0$ and ${\bf e}_{1}{\bf e}_{3}=1$. The
geometrical factors can be partially incorporated into the Franck-Condon
factors $\mu_{0r}$, so one can assume, without loss of generality, that
$a_{k}$ and $b_{k}$ are normalized to unity ($|{\bf
e}_{k}|=\sqrt{a_{k}^{2}+b_{k}^{2}}=1$).

For atoms a short distance apart, the dipole blockade forbids that more than
one Rydberg state can be populated during the laser action. In the simplest
model that describes the two-qubit gate [61], the system is described by $8$
states: the computational basis and ancillary states with Rydberg excitations,
as the pulse frequencies are chosen to be in resonance with the
$|0\rangle\rightarrow|r\rangle$ transition111The scheme performs similarly
using pulses in resonance from the $|1\rangle$ state to the Rydberg state..
The Hamiltonian is block-diagonal for each computational basis ${\sf
H}_{k}^{V}\oplus{\sf H}_{k}^{A}\oplus{\sf H}_{k}^{B}\oplus{\sf H}^{D}$, where

${\sf H}_{k}^{V}=-\frac{1}{2}\Omega_{k}(t)\left(a_{k}|00\rangle\langle
r0|+b_{k}|00\rangle\langle 0r|+\mathrm{h.c.}\right)$

is the Hamiltonian of a $3$-level subsystem in $V$ configuration, acting in
the subspace of $\\{|00\rangle,|r0\rangle,|0r\rangle\\}$ states, ${\sf
H}_{k}^{A}=-\frac{1}{2}a_{k}\Omega_{k}(t)\left(|01\rangle\langle
r1|+\mathrm{h.c.}\right)$ and ${\sf
H}_{k}^{B}=-\frac{1}{2}b_{k}\Omega_{k}(t)\left(|10\rangle\langle
1r|+\mathrm{h.c.}\right)$ are two-level Hamiltonians acting in the subspace of
$\\{|01\rangle,|r1\rangle\\}$ and $\\{|10\rangle,|1r\rangle\\}$ respectively.
We will refer generally to any of these subsystems with the superscript $S$
($S=V,A,B$). Finally, ${\sf H}^{D}=0|11\rangle\langle 11|$ is the Hamiltonian
acting on the double-excited qubit state $|11\rangle$, decoupled from any
field.

Using temporally non-overlapping pulses, the propagator for the time evolution
is the time-ordered product of the evolution operators for each pulse, ${\sf
U}^{S}=\prod_{k=0}^{N_{p}-1}U^{S}_{N_{p}-k}$, which is analytical. For the $V$
subsystem,

$U^{V}_{k}=\left(\begin{array}[]{ccc}\cos\theta^{V}_{k}&ia_{k}\sin\theta^{V}_{k}&ib_{k}\sin\theta^{V}_{k}\\\
ia_{k}\sin\theta^{V}_{k}&a_{k}^{2}\cos\theta^{V}_{k}+b_{k}^{2}&a_{k}b_{k}\left[\cos\theta^{V}_{k}-1\right]\\\
ib_{k}\sin\theta^{V}_{k}&a_{k}b_{k}\left[\cos\theta^{V}_{k}-1\right]&b_{k}^{2}\cos\theta^{V}_{k}+a_{k}^{2}\end{array}\right)$
(1)

where the mixing angle

$\theta^{V}_{k}=\frac{1}{2}\int_{-\infty}^{\infty}\Omega_{k}(t)dt=\frac{1}{2}A_{k}$

is half the pulse area. For the two-level subsystems $A$ and $B$, we can use
the same expression for the relevant states with $a_{k}=1,b_{k}=0$, for
$U_{k}^{A}$, and vice versa for $U_{k}^{B}$. However, the mixing angles depend
on the local coupling: $\theta^{A}_{k}=a_{k}A_{k}/2$ and
$\theta_{k}^{B}=b_{k}A_{k}/2$. We will refer to the generalized pulse areas,
$2\theta^{S}_{k}$, as GPA.

The SOP uses spatially orthogonal vectors such that the state of the system
after the first pulse acting on $|00\rangle$, is a dark state of the
Hamiltonian for the second pulse ${\sf H}_{2}^{V}$, so the second pulse does
not affect this state. In this way, the SOP works similarly to the JP, but
with non-independent qubits. In this work, we will study families of schemes
that can operate with even fewer pulses, although they typically require the
same (or larger) accumulated pulse area, $A_{T}=\sum_{k}|A_{k}|$. In the
following section, we propose a possible scheme to control the structural
factors over a wide range of values (including negative factors) by using
superposed laser beams.

###  II.2 Implementation

Figure 1: Scheme showing two possible implementations of the spatiotemporal
control of trapped qubits, located at the $\times$ positions, using phase-
locked Gaussian beams TEM00 centered at each qubit,
$\Omega_{a},\Omega_{b},\Omega_{c}$ (second row) or structured pulses
$\Omega_{k}$ whose spatial profile at the qubits coincides with the linear
combination of TEM00 (third row). The sequence of operations that governs the
temporal evolution of the state depends on the pulse sequence, shown in the
first row.

The spatial control is encoded in ${\bf e}_{k}$ and can be achieved by
different means. In [60] we proposed the use of hybrid modes of light. A
possible generalization for spatially non-orthogonal pulses in any
configuration, may require more complex structured light [65, 66, 67], such as
those sketched in Fig.1 (second row). A simpler laboratory implementation,
shown in the third row, can be achieved using a superposition of overlapping
phase-locked Gaussian modes [68, Chen_APE2012] centered at different qubits ,
instead of a single field, for each pulse in the sequence. In the simplest
setup, we will consider just two qubits separated at a distance $R$, shone by
two lasers at each step $k$ of the sequence, ${\Omega}_{ak}(\vec{r},t)$ and
${\Omega}_{bk}(\vec{r},t)$, each focused on a qubit, but with waistbeams that
span both.

We want the lasers to act with spatial coefficient $a_{k}$ at qubit $a$ and
$b_{k}$ at qubit $b$. If the beams are Gaussian (but any form is valid), and
both lasers have the same time-dependence given by the function of time
$f(t)$, the sum of both gives the local field at its peak,
${\Omega}_{k}(\vec{r}_{a},t_{0k})={\Omega}_{ak}(\vec{r}_{a},t_{0k})+{\Omega}_{bk}(\vec{r}_{a},t_{0k})=\left[\widetilde{\Omega}_{ak}+\theta\widetilde{\Omega}_{bk}\right]=a_{k}\widetilde{\Omega}_{0k}$,
where Rabi frequencies with tilde represent their values at peak amplitude,
and $\theta=e^{-\alpha R^{2}}$ ($\alpha$ measures the beam’s waist). Here we
have assumed that the spatial profile of the lasers is the same for all the
pulses in the sequence, as will be the case in most laboratory
implementations. Correspondingly,
${\Omega}_{k}(\vec{r}_{b},t_{0k})={\Omega}_{ak}(\vec{r}_{b},t_{0k})+{\Omega}_{bk}(\vec{r}_{b},t_{0k})=\left[\theta\widetilde{\Omega}_{ak}+\widetilde{\Omega}_{bk}\right]=b_{k}\widetilde{\Omega}_{0k}$.

The geometrical factors can be arranged as a column (row) vector $\vec{e}_{k}$
with components $a_{k}$, $b_{k}$. In addition, we can define the column vector
of field components $\vec{\cal
E}_{k}=\left(\widetilde{\Omega}_{ak},\widetilde{\Omega}_{bk}\right)$, and the
spatial overlap matrix

${\sf S}=\left(\begin{array}[]{cc}1&\theta\\\ \theta&1\end{array}\right)$ (2)

such that $\widetilde{\Omega}_{0k}\vec{e}_{k}={\sf S}\vec{\cal E}_{k}$ and
$\vec{\cal E}_{k}=\widetilde{\Omega}_{0k}{\sf S}^{-1}\vec{e}_{k}$,

$\left(\begin{array}[]{c}\widetilde{\Omega}_{ak}\\\
\widetilde{\Omega}_{bk}\end{array}\right)=\frac{\widetilde{\Omega}_{0k}}{1-\theta^{2}}\left(\begin{array}[]{cc}1&-\theta\\\
-\theta&1\end{array}\right)\left(\begin{array}[]{c}a_{k}\\\
b_{k}\end{array}\right)$ (3)

which gives

$\frac{\widetilde{\Omega}_{bk}}{\widetilde{\Omega}_{ak}}=\frac{x_{k}-\theta}{1-\theta
x_{k}}$ (4)

where $x_{k}=b_{k}/a_{k}$ is the ratio of the geometrical factors. Whenever
$\theta>x_{k}$, assuming $x_{k}\leq 1$, the ratio is negative. This can be
achieved by controlling the relative phase between the pulses. For $x_{k}\leq
1$,
$\left|\widetilde{\Omega}_{bk}/\widetilde{\Omega}_{ak}\right|<\left|b_{k}/a_{k}\right|$.
Under certain conditions, it is possible (and it might be more economic) to
use a single field with more complex spatial structure, such as structured
light, instead of a superposition of overlapping Gaussian pulses. Finally, for
two-qubit of few qubit systems, and positive relative ratios, it is possible
to perform the operation with a single broad pulse, controlling the relative
positions of the atoms with respect to the pulse waistbeam. In the symmetric
arrangement, the pulse should be focused at mid distance between the atoms.

Using superposed Gaussian beams, it is always possible to extend this
procedure to more than $2$ qubits, controlling the geometrical factors by
controlling the ratio of the peak amplitudes of the fields (as well as the
pulse phases). In the general case, one needs to define a different
$\theta_{ab}$ for each pair of qubits. The matrix ${\sf S}$ is always
invertible, as long as $\theta_{ab}\neq 1$, which would imply that two qubits
occupy the same space.

In fact, one can use the superposition of Gaussian pulses as a technique to
remove the effect of one pulse over an unwanted qubit, if we want to work with
independent qubits even when $\alpha\sim\\!R^{-2}$. In this case, the goal is
to make $x_{k}=0$, for which
$\widetilde{\Omega}_{bk}=-\theta\widetilde{\Omega}_{ak}$ and hence
${\Omega}_{k}(\vec{r}_{b},t_{0k})=0$.

##  III Single pulse protocols

Figure 2: Map of the fidelity for the CZ gate as a function of the pulse area
and the ratio between the geometrical factors, for protocols based on a single
pulse acting simultaneously on both qubits. In dashed lines, we show the
protocols for which the action of the laser is minimal in the qubit $b$. The
peaks appear at approximate solutions of a Diophantine equation.

One of the advantages of working with non-independent qubits is that it is
possible to use shorter pulse sequences. In principle, there are enough
control knobs to implement an entangling gate with a single pulse sequence.

For the CZ gate, we use the unconventional (but equivalent) gate definition,
where the amplitudes in each computational state, except the $|11\rangle$,
experience a $\pi$ shift at the end of the gate. We calculate the fidelity as

$F=\frac{1}{16}\left(-U^{A}_{11}-U^{B}_{11}-U^{V}_{11}+1\right)^{2}$ (5)

where every term $U^{S}_{11}$ is the first matrix element of Eq.(1). For every
subsystem, $S$ of states coupled by the radiation, starting from the different
computational states, one must then achieve $\cos\left(\theta^{S}\right)=-1$.
These probability amplitudes correspond to so-called $0$-loop processes [61],
where the amplitude stays solely on the computational basis by the end of the
pulse. For a single pulse dynamics, only $0$-loops can realize the gate.
However, it is very simple to prove that $0$-loops can never be exactly
achieved for the $3$ subsystems with a single pulse, so the gate mechanism
cannot yield perfect fidelities even in the absence of noise or perturbations.
The proof is simple to sketch.

###### Proof.

Let the system have $2$ qubits. For perfect fidelity, the following conditions
must be satisfied:

$\displaystyle\cos(\sqrt{a^{2}+b^{2}}A/2)=-1\rightarrow
A=(4l-2)\pi,l\in\mathbb{Z}$ $\displaystyle\cos(aA/2)=-1\rightarrow
aA=(4l^{\prime}-2)\pi,l^{\prime}\in\mathbb{Z}$
$\displaystyle\cos(bA/2)=-1\rightarrow
bA=(4l^{\prime\prime}-2)\pi,l^{\prime\prime}\in\mathbb{Z}$ (6)

where we used normalized structural factors. It is not possible to fulfill all
the required conditions of Eq.(6) at the same time: Calling $p=4l-2$,
$n=4l^{\prime}-2$, $m=4l^{\prime\prime}-2$, squaring the argument of the third
condition, and comparing with the first two conditions, we obtain the relation
between the integers $m,n,p$: $m^{2}+n^{2}=p^{2}$. Equations like this that
require integer solutions are generically called Diophantine equations. They
have an infinite number of solutions. However, it can be easily shown that the
solutions cannot be constrained such that all $m,n,p$ are of the form
$2,6,10,\ldots 4l-2$. For, let $p>n\geq m$,
$m^{2}=p^{2}-n^{2}=(p+n)(p-n)=16(l+l^{\prime}-1)(l-l^{\prime})$, while by
directly squaring,
$m^{2}=16\left(l^{\prime\prime}\right)^{2}-16l^{\prime\prime}+4$. Dividing
both sides by $16$ we have
$\left(l^{\prime\prime}\right)^{2}-l^{\prime\prime}+0.25=(l+l^{\prime}-1)(l-l^{\prime})$.
The left-hand side cannot be integer, while the right-hand side must be
integer. ∎

It can be shown that the same restrictions apply to all $2$-qubit entangling
gates. This issue becomes more pronounced as the number of qubits increases.
For instance, with $3$ qubits, we have $3$ $V$ subsystems and $3$ two-level
systems where the previous Diophantine approximate solutions must hold, in
addition to a tripod system, which adds another equation like
$m^{2}+n^{2}+p^{2}=q^{2}$, that does not hold solutions for $m,n,p,q$ integers
of the type $2,6,\ldots,4l-2$ or similar.

However, while it is not possible to achieve perfect fidelity, the Equations
(6) can be in principle fulfilled up to any desired accuracy. For instance, in
the CZ gate, $14^{2}\approx 10^{2}+10^{2}$ with a relative error of
approximately $4/200\approx 2$%, so that an approximate solution exists using
equal structural factors in the qubits ($a=b=1/\sqrt{2}$) and a pulse area of
$A\sim 14\pi$, which leads to a fidelity $F=0.992$. In Fig.2 we show a map of
the fidelity of the gate as a function of the pulse area $A$ and the ratio of
the geometrical factors, $x=b/a$. Because the role of the geometrical factors
is equivalent (the fidelity is the same for $x$ and $x^{-1}$), we only show
the map for $x\leq 1$. The density of high-fidelity protocols increases for
small $x$ (alternatively, $x\gg 1$). The simplest solutions involve $bA=2\pi$.
For large $A$ and small $b$, $\sqrt{1-b^{2}}\approx 1$ and $aA\approx A$. This
gives the series of solutions shown by the white dotted line in Fig.2, where
$bA=2\pi$, from which

$bA=\frac{x}{\sqrt{1+x^{2}}}A=2\pi\longrightarrow
A=2\pi\frac{\sqrt{1+x^{2}}}{x}$ (7)

A similar equation must be satisfied by $aA$. Dividing both, we obtain the
values of $x$ at which the fidelity is maximized,

$x_{\mathrm{op}}=\frac{b}{a}=\frac{bA}{aA}=\frac{4l^{\prime\prime}+2}{4l^{\prime}+2}$
(8)

For the smallest possible local area in qubit $b$, $bA=2\pi$
($l^{\prime\prime}=0$), $x_{\mathrm{op}}$ lie in the sequence of inverse odd
numbers, $x_{\mathrm{op}}=1/(2l^{\prime}+1)$. To fully optimize the gate, the
contribution of the $3$ terms $U^{A}_{11},U^{B}_{11},U^{V}_{11}$ must be
maximized, for which the optimal pulse area must be slightly corrected as the
average between the value expected from Eq.(7) with $x_{\mathrm{op}}$, and the
value of the area that maximizes the $U^{V}_{11}$ term,

$A_{\mathrm{op}}=\left(2l+1+\sqrt{\left(2l^{\prime}+1\right)^{2}+\left(2l^{\prime\prime}+1\right)^{2}}\right)\pi$
(9)

where $l\geq l^{\prime}\geq l^{\prime\prime}\in\mathbb{Z}$. The protocol with
smallest possible area ($l,l^{\prime},l^{\prime\prime}=0,0,0$) is achieved
with $A_{\mathrm{op}}=2.4\pi$ at $x_{\mathrm{op}}=1$ giving a relatively low
fidelity of $F=0.804$. The second maxima, at $A=6.17\pi$ with $x=1/3$, gives
already a fidelity $F=0.968$. For very large integers, the relative error can
be as small as desired by increasing the pulse area, properly adjusting the
ratio of the geometrical factors following Eq.(8) and the area with Eq.(9).
Some results are obtained in Fig.7. However, as discussed in Sec.V, taking
into account the effect of fluctuations in the parameters due to shot-to-shot
noise, can shift the maximum fidelities to the lower pulse area protocols.

##  IV Two-pulse protocols

Figure 3: The fidelity for protocols based on two-pulse sequences, as a
function of the pulse areas, inherit the properties of
$-\cos(\theta_{1})-\cos(\theta_{2})$ (left) and
$-\cos(\theta_{1}\pm\theta_{2})$ (center and right). We represent the cosine
scaled and shifted as $(-\cos x+1)/2$ so that its range is between $0$ and
$1$, like the fidelity.

For two-pulse sequences, the time-evolution operator for the $A$ and $B$
subsystems has two terms

$U^{S^{\prime}}_{11}=\cos\left(\alpha_{2}A_{2}/2\right)\cos\left(\alpha_{1}A_{1}/2\right)-\sin\left(\alpha_{2}A_{2}/2\right)\sin\left(\alpha_{1}A_{1}/2\right)$
(10)

where $S^{\prime}=A,B$, $\alpha=a,b$, and the subscript refers to the pulse
order. The first term is responsible for a gate mechanism based on a $0$-loop,
as in single-pulse sequences. The second-term accounts for another mechanism
that prepares the gate, the so-called one-loop, where the first pulse excites
the population to the Rydberg state and the second pulse takes the population
back to the computational basis. For this to happen, the GPA must be an odd
multiple of $\pi$. In the $V$ subsystem, the second term is scaled by the
product of the geometrical factors,
$U^{V}_{11}=\cos\left(A_{2}/2\right)\cos\left(A_{1}/2\right)-{\bf e}_{2}{\bf
e}_{1}\sin\left(A_{2}/2\right)\sin\left(A_{1}/2\right)$ (${\bf e}_{2}{\bf
e}_{1}=a_{1}a_{2}+b_{1}b_{2}$). For each subsystem, it is in principle
possible to have gate mechanisms that behave as 0-loops, a 1-loops, or
superpositions of both. However, because of the ${\bf e}_{2}{\bf e}_{1}$
factor, $U^{V}_{11}$ can only be close to $-1$ if it follows a 0-loop, unless
${\bf e}_{2}{\bf e}_{1}=\pm 1$, that is, if the structural vectors are aligned
or anti-aligned, constraining the parameters to be $x_{2}=\pm x_{1}$.

Figure 4: Fidelity of the gate for two-pulse protocols as a function of
$x_{2}$ and $A_{2}$. In (a) we choose $A_{1}=6\pi$ and $x_{1}=1/3$, which are
parameters that allow for a high-fidelity of the gate even in the absence of
the second pulse, based on a 0-loop mechanism for all subsystems. In (b),
$A_{1}=4\pi$ and $x_{1}=1/4$, so that the gate follows a 1-loop mechanism for
the B subsystem. The maps are very similar to those of single-pulse sequences
but with displaced areas and ratios of the geometrical factors.

We will first analyze 0-loop protocols, which are a natural extension of
single-pulse-based mechanisms. For 0-loop protocols in the $V$ subsystem,
$\cos\left(A_{2}/2\right)\cos\left(A_{1}/2\right)=-1$, which force
$A_{1}=(4l+2)\pi$ and $A_{2}=4m\pi$ ($l,m\in\mathbb{Z}$) or vice versa,
forming the checkered pattern of the map of protocols as a function of the
pulse areas [see Fig.3], which was found in Sola et al.[61] using optimization
algorithms.

In Fig.4(a) we show the fidelity map as a function of $A_{2}$ and $x_{2}$,
after choosing $A_{1}=6\pi$ and $x_{1}=1/3$, which are valid parameters in a
single-pulse protocol. Hence, $A_{2}=0$ is always a possible solution. In
addition, all areas of the form $A_{2}=4m$ ($m\in\mathbb{Z}$) provide high-
fidelity gates. As the choice of $x_{1}$ forces the A and B subsystems to
follow a 0-loop mechanism (since $\cos\theta^{S^{\prime}}_{1}=-1$), then
$x_{\mathrm{op}}=4m^{\prime\prime}/4m^{\prime}$. Obvious solutions of the
corresponding Diophantine equations show up at every $m^{\prime}$ for
$m^{\prime\prime}=0$ (since then $m^{2}=(m^{\prime})^{2}$ exactly), but also,
e.g. at $m=5,m^{\prime}=4,m^{\prime\prime}=3$, for which $x_{op}=0.75$, etc.

In Fig.4(b) we choose $A_{1}=4\pi$ and $x_{1}=1/4$. Solutions exist for all
areas of the second pulse of the form $A_{2}=(4m+2)\pi$. Now
$b_{1}A_{1}\approx\pi$, $a_{1}A_{1}\approx 4\pi$, so the first pulse opens a
1-loop mechanism for the B subsystem, and a 0-loop mechanism for the A
subsystem. Then the sequence of fidelity peaks must occur at
$x_{op}=(2m^{\prime\prime}+1)/(4m^{\prime}+2)$ (for all
$m^{\prime},m^{\prime\prime}\in\mathbb{Z}$. For the smallest possible
$m^{\prime\prime}=0$, $b_{2}A_{2}=x_{2}A_{2}/\sqrt{1+x_{2}^{2}}\approx\pi$ and
hence $A_{2}=\pi\sqrt{1+x_{2}^{2}}/x_{2}$. This is the dotted line shown in
Fig.4(b) for which high-fidelity peaks show up at
$x_{op}=1/2,1/6,1/10,\ldots,1/(4m^{\prime}+2)$.

It is important to note, however, that any superposition of mechanisms can
occur in the A and B subsystems. As long as the pulse areas $A_{1}$ and
$A_{2}$ alternate as $(4l+2)\pi$ and $4m\pi$ or vice versa, it is always
possible to find high-fidelity protocols for any $x_{1}$, because from
Eq.(10), $U^{S}_{11}=\cos\left(\theta_{1}\pm\theta_{2}\right)$ ($\alpha=A,B$),
with $\theta_{k}=\alpha_{k}A_{1}/2$. The minus sign inside the cosine applies
when the ratios ($x_{1}$ and $x_{2}$) or areas ($A_{1}$ and $A_{2}$) change
signs. There will be always values of $x_{1}$, $x_{2}$ (or more precisely, of
$a_{1}A_{1}+a_{2}A_{2}$ and $b_{1}A_{1}+b_{2}A_{2}$), for which
$U^{S}_{11}=-1$ for the choice of pulse areas $A_{1},A_{2}$ that make
$U^{V}_{11}=-1$. Depending on $x_{1}$ and $x_{2}$, the A and B subsystems
belong to a continuous range of mechanisms, from 0-loops to 1-loops, passing
through any combination.

Figure 5: Fidelity map as a function of the pulse areas $A_{1}$ and $A_{2}$
for (a) aligned structural vectors (with $x_{1}=x_{2}=1/5$), (b) anti-aligned
structural vectors ($x_{1}=-x_{2}=1/5$), and (c) orthogonal structural vectors
(with $x_{1}=-1/x_{2}=1/5$).

Can this realization of every possible mechanism include the V subsystem?
Indeed, if the structural vectors are aligned or anti-aligned, ${\bf
e}_{1}=\pm{\bf e}_{2}$, for which $x_{2}=\pm x_{1}$, then the three terms
$U^{S}_{11}$ ($S=A,B,V$) behave as Eq.(10), which can be written as
$\cos\left(\theta^{S}\right)$, with $\theta^{V}=(A_{1}\pm A_{2})/2$,
$\theta^{S^{\prime}}=(\alpha_{1}A_{1}+\alpha_{2}A_{2})/2$. These are exactly
the same equations as in the single-pulse sequence, except that now the
argument depends on the sum of pulse areas,

$A_{T}=A_{1}\pm A_{2}=(4n+2)\pi,\,\,\,n\in\mathbb{Z}$ (11)

where the plus sign applies for aligned vectors and the minus, for anti-
aligned vectors. So every combination of pulse areas that sums $(4n+2)\pi$ can
generate a high-fidelity gate, where the mechanism can be any superposition of
0-loops and 1-loops for all the different subsystems.

Figure 6: Fidelity map as a function of the second pulse area $A_{2}$ and the
ratio of geometrical factors $x$ for aligned (a) $x_{2}=x_{1}$, and (b) anti-
aligned ($x_{2}=-x_{1}$) structural vectors. For the figure, we choose
$A_{1}=7\pi$.

In Fig.5 we show the fidelity map as a function of the pulse areas $A_{1}$ and
$A_{2}$ for $x=x_{1}=x_{2}=1/5$ (left) and $x=x_{1}=-x_{2}=1/5$ (center).
There are high-fidelity straps for pulse areas that sum $(4n+2)\pi$, but not
for all values of $n$. The actual maximum fidelity observed and its location
depends on the choice of $x$. The direction of the straps depends on whether
the vectors are aligned or anti-aligned. These patterns inherit the properties
of $-\cos(\theta_{1}\pm\theta_{2})$ shown in Fig.3. In Fig.6 we show the
fidelity map as a function of $x$ and $A_{2}$, where we fixed $A_{1}=7\pi$,
for both aligned (a) and anti-aligned (b) vectors. As observed,
$A_{2}=(4n-5)\pi$. The solution that appears at $x=0.2$ corresponds to
$A_{2}=3\pi$ (the sum of areas equals $10\pi$). Allowing $x$ to change, one
can typically find high-fidelity protocols for any possible valid $n$, and
hence for any $A_{2}$.222 In spite of the fact that one can find protocols
that maximize the fidelity for any type of mechanism, using optimal control
algorithms in ref [61] we only found protocols for which the $V$ subsystem is
a 0-loop. An intriguing question is why is this so. We believe that the reason
is rooted in the peak fidelities that can be achieved by the different
mechanisms under moderate pulse areas. Because the fidelity of protocols based
on aligned structural vectors behaves as in single-pulse protocols, one needs
$A_{T}\geq 14\pi$ (for $l^{\prime}=l$, $l^{\prime\prime}=0$) to achieve $F\geq
0.99$, while $A_{T}\geq 38\pi$ (for $l^{\prime}=l$, $l^{\prime\prime}=0$) to
achieve $F\geq 0.999$, which was the threshold chosen in [61] to analyze the
chosen optimal protocols. Numerically, one finds many protocols with higher
fidelity and lower pulse areas when the pulse areas are correlated as
$A_{1}=(4l+2)\pi$ and $A_{2}=4m\pi$ or vice versa, imposing a 0-loop in the
$V$ subsystem. Only when one analyzes the set of optimal protocols with lower
fidelities one observes protocols with mechanisms that do not use the 0-loop
for the $V$ subsystem.

Finally, it is even possible to find optimal protocols where the structural
vectors are orthogonal, ${\bf e}_{1}{\bf e}_{2}=0$. They imply a superposition
of the aligned and anti-aligned vectors, for which the fidelity map looks like
the pattern observed in Fig.5(right). The fidelity peaks form now a rotated
lattice. The peaks are a distance of $4\pi$ apart, and the angle of the
lattice depends on the choice of $x$. These are the solutions explored in the
so-called SOP (symmetrical orthogonal protocol), shown in reference [60].

##  V Evaluating the effects of noise

To analyze in detail all the effects of noise on the proposed schemes, one
needs to better define the setup of the system, choosing very concrete
parameters for the lasers and atomic traps, which is outside the scope of this
work. Our analytical approach follows from an approximate Hamiltonian from
which we can obtain the time-evolution operator, so we cannot incorporate the
sources of noise at the level of the dynamical description. From the physical
point of view, the schemes shown here operate using the Rydberg blockade, so
one can expect a similar sensitivity to the fluctuation of the laser
frequency, the spontaneous emission, and the thermal motion of the atoms, as
reported elsewhere. [70] However, because the atoms are much closer, the
dipole blockade is much larger and the pulses much shorter (operating, in
principle in tens of nanoseconds) and much more intense, the phase-induced
detunings or changes in population due to spontaneous decays, which are the
main sources of errors in microsecond experiments, become almost negligible in
our setup. Mainly shot-to-shot fluctuations, rather than decoherence, will
have some impact on the fidelities.

Herein, we develop a simple model to evaluate the impact of fluctuations in
the pulse energy (hence pulse areas) and geometrical factors on the fidelity
for the CZ gate in two-qubit systems, using two partially overlapping pulse
beams centered at each qubit.

The impact of amplitude fluctuations over the pulse areas is direct. For a
pulse with intensity $I_{0}=c\epsilon_{0}^{2}$, given that the area is
$A_{0}=\mu\epsilon_{0}S_{0}/\hbar$, where $S_{0}$ is a shape factor,
neglecting fluctuations in the pulse duration (or rather, subsuming the effect
on the peak intensity fluctuation), the relative error in the pulse areas is

$\delta A_{0}\equiv\Delta A_{0}/A_{0}=\Delta I_{0}/2I_{0}$ (12)

Using stabilized microsecond pulses, $\delta I_{0}$ can be estimated as
$\sim\\!3$% or smaller.

Fluctuations in the geometrical factors depend both on fluctuations in the
laser amplitudes as well as on the thermal motion of the atoms. For the
parameter $b_{k}$ obtained by a superposition of beams

$b_{k}=\frac{\epsilon_{bk}}{\Omega_{0k}}+\theta\frac{\epsilon_{ak}}{\Omega_{0k}}$
(13)

where $\Omega_{0k}=\sqrt{\epsilon_{bk}^{2}+\epsilon_{ak}^{2}}$, we separate
the dependence on $\epsilon$ from the dependence on $R$ through
$\theta=\exp(-\alpha R^{2})$, as $\Delta b_{k}^{2}=(\Delta
b_{k}^{\prime})^{2}+(\Delta b_{k}^{\prime\prime})^{2}$, where

$(\Delta b_{k}^{\prime})^{2}=\left(\frac{\partial
b_{k}}{\partial\epsilon_{ak}}\right)^{2}(\Delta\epsilon_{ak})^{2}+\left(\frac{\partial
b_{k}}{\partial\epsilon_{bk}}\right)^{2}(\Delta\epsilon_{bk})^{2}\ .$

Assuming that the relative errors in the fields are similar,
$\delta\epsilon_{ak}=\delta\epsilon_{bk}=\delta I_{0}/2$,

$\displaystyle(\Delta b_{k}^{\prime})^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left\\{\frac{\epsilon_{ak}^{2}\theta^{2}+\epsilon_{bk}^{2}}{\Omega_{0k}^{2}}+\left(\frac{b_{k}}{\Omega_{0k}}\right)^{2}\frac{\epsilon_{ak}^{4}+\epsilon_{bk}^{4}}{\Omega_{0k}^{2}}\right\\}(\delta
I_{0})^{2}$ $\displaystyle\leq$ $\displaystyle\frac{1}{2}b_{k}^{2}(\delta
I_{0})^{2}$

On the other hand,

$(\Delta b_{k}^{\prime\prime})^{2}=\left(\frac{\partial
b_{k}}{\partial\theta}\right)^{2}\left(\frac{d\theta}{dR}\right)^{2}(\Delta
R)^{2}=\left(\frac{\epsilon_{ak}}{\Omega_{0k}}\right)^{2}\left(2\alpha
R\theta\right)^{2}(\Delta R)^{2}\ .$

Since $2\alpha R^{2}\theta\sim 1$, $(\Delta
b_{k}^{\prime\prime})^{2}\sim\theta^{2}(\delta R)^{2}$, from which

$\Delta b_{k}^{2}\sim\frac{1}{2}b_{k}^{2}(\delta I_{0})^{2}+\theta^{2}(\delta
R)^{2}$ (14)

Equally, for the $a_{k}$ term, we have

$\Delta a_{k}^{2}\sim\frac{1}{2}a_{k}^{2}(\delta
I_{0})^{2}+x^{2}\theta^{2}(\delta R)^{2}$ (15)

so

$\delta x^{2}_{k}=\delta b_{k}^{2}+\delta a_{k}^{2}=(\delta
I_{0})^{2}+\left(2+\frac{1}{x_{k}^{2}(x_{k}^{2}+1)}\right)\theta^{2}(\delta
R)^{2}$ (16)

where we observe that $\Delta x_{k}$ depends on $x_{k}^{-1}$ (because $\delta
x_{k}$ depends on $x_{k}^{-2}$), so we expect the error to be larger for
protocols that work with small $x_{k}$. This is why the unwanted presence of a
second qubit can damage the fidelity of a scheme based on independent qubits.
These detrimental effects can be somehow reduced in the SOP.

To evaluate the error in $\delta R$, we use a simple estimation assuming a
diffusion model for the dispersion of the atoms, $\Delta R\sim\sqrt{2Dt_{g}}$,
where $t_{g}$ is the gate duration and $D$ the diffusion coefficient. In [70],
working with atoms separated $5\mu$m and using gates that operate in
$\sim\\!5\mu$s at $\sim\\!22~{}\mu$K, the authors evaluate $\Delta R$ as
$\sim\\!50\,$nm. If we assume that our gates operate under similar conditions
(e.g. temperature) but $25$ times faster, that would imply $\Delta
R\sim\\!10$nm, for a relative error of $\delta R\sim\\!1$% when the atoms are
approximately $1\mu$m apart, although our approximations may underestimate the
error during the measuring of the gate’s state. To evaluate the effect of the
temperature, we will assume a linear dependence with the mean square
displacement, as in Brownian motion, or for classical and quantum oscillators
under certain limits [71].

Figure 7: Fidelity for single-pulse protocols with different $l=l^{\prime}$
and $l^{\prime\prime}=0$ for different levels of noise in the parameters. The
dotted lines show the errors induced by thermal fluctuations in the positions
of the atoms with relative standard deviations $\delta R=0.02,0.01$ (lower and
higher curves). Solid lines show the errors induced by fluctuations in the
peak intensities of the pulse, with relative standard deviations $\delta
I=0.03,0.015$ (lower and higher curves). The squares are the results in the
absence of fluctuations, and circles with error bars show the results for the
fidelity in the presence of both noise sources, with $\delta R=0.02$, $\delta
I=0.03$. The error bars show the standard deviation in the fidelity, across a
distribution of $1000$ samples.

We use Eqs.(12) and (16) to evaluate a distribution of parameters $A$ and $x$
following the noise statistics. We also include a distribution in the absolute
phase of the lasers with $\Delta\phi=0.01\pi$. Using a sample of $1000$
different parameters, we evaluate the average fidelity and the standard
deviation for several single-pulse optimal protocols (with
$l^{\prime\prime}=0$ and $l=l^{\prime}$) with different noise contributions.
In Fig.7 we show the fidelities in the absence of fluctuations (squares) and
the average fidelities with $\delta I=0.03$, $\delta R=0.01$ (labelled as
“standard”), which are the errors reported in [70]. The error bars show the
standard deviation in the fidelity, which, for $l^{\prime}=6$ reaches
$\sigma=0.17$. The results reveal that fidelity is severely affected for
protocols that use large $l^{\prime}$ (and $l$ and hence $A$), which correlate
to protocols that operate with larger Rabi frequencies and smaller ratios of
the geometrical factors.

The effect of fluctuations in the laser amplitudes (solid lines) is much
stronger than those on the atomic positions (dotted lines). Although the
relative error both in $A$ and $x$ is linearly proportional to the relative
error in the pulse intensities, the required precision in the intensities
should increase for protocols that use stronger fields, as a small error in
$A$ can easily shift the GPA from an odd multiple to an even multiple of $\pi$
(and vice versa), totally changing the excitation mechanism. For intensity
fluctuations of $\sim\\!3$% only the lowest area protocols ($A\leq 10\pi$)
survive with fidelity errors smaller than $5$%. It is really necessary to
reduce the laser fluctuations to one-half of this value or lower ($1$% in the
yellow line) to reduce the errors to less than $2$% in protocols with
$A=14\pi$. In Fig.7, labelled as “ultra”, we also show the results using noise
statistics currently available [31] with state-of-the-art laser stabilization
($\delta I\sim 0.007$, $\Delta\phi\leq 0.01\pi$) and sideband cooling
($T=10~{}\mu$K), which show that errors in fidelity can, in principle, be
reduced to less than $1\%$. In fact, all the errors in such conditions depends
on $\delta I$, as the same results would be obtained at $T=1$ mK.

##  VI Conclusions

In this work, we have studied minimal pulse sequences that implement the CZ
gate on two adjacent and non-independent qubits with high fidelity, where the
number of pulses used per qubit can be as small as one. Indeed, using
structured light, in principle one can implement the gate with a single pulse.
We have proposed a possible implementation using superposed Gaussian beams,
and we have analyzed the role of parameter fluctuations induced by shot-to-
shot noise.

Ultimately, the optimal parameters must be approximate solutions of
Diophantine equations, imposing strict conditions on the pulse areas and
overlaps of the pulses. While perfect fidelities can never be achieved even
under ideal conditions, the errors can be made as small as desired using
intense pulses. The use of two-pulse sequences looses the restrictions on the
values of the parameters that optimize the gate. One finds that a continuum of
mechanisms, described in terms of quantum pathways, can be used for its
implementation, although strong correlations in the areas of the pulses of the
form $A_{1}=(4l+2)\pi$, $A_{2}=4m\pi$ ($l,m\in\mathbb{Z}$) or vice versa, are
typically found in optimal protocols.

By implementing the qubits in atoms trapped at a short distance of each other
(thereby boosting the dipole blockade), the goal is to speed up the gates to
the nanosecond time-scale. We found that intensity fluctuations have a much
stronger impact on the fidelity of the gates than the thermal motion of the
atoms, mainly in protocols that use large pulse areas and hence, assuming
short pulses, strong fields. Our preliminary analysis reveals that the
stabilization of the lasers that allows to reduce the relative errors in the
pulse intensities below $1\%$, may be necessary for the laboratory
implementations of these protocols. On the other hand, the experiments can be
performed at typical ultracold temperatures of $\sim 10$ $\mu$K.

While an in-depth analysis of all protocols can only be made for small pulse
sequences, we expect that the use of protocols with several pulses with
similar total accumulated Rabi frequency, but smaller peak intensities, can
result in higher fidelities and more robust gates.

## Acknowledgements

This research was supported by the Quantum Computing Technology Development
Program (NRF-2020M3E4A1079793). IRS thanks the BK21 program (Global Visiting
Fellow) for the stay during which this project started and the support from
MINECO PID2021-122796NB-I00. SS acknowledges support from the Center for
Electron Transfer funded by the Korean government(MSIT)(NRF-2021R1A5A1030054)

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11institutetext: São Paulo State University - UNESP, Bauru, Brasil 11email:
{mateus.roder, clayton.pereira, leandro.passos, luiz.felix,
<EMAIL_ADDRESS>
https://www.fc.unesp.br/ 22institutetext: Recogna Laboratory -
www.recogna.tech

# A Layer-Wise Information Reinforcement Approach to Improve Learning in Deep
Belief Networks

Mateus Roder 1122 0000-0002-3112-5290 Leandro A. Passos 1122
0000-0003-3529-3109 Luiz Carlos Felix Ribeiro 1122 0000-0003-1265-0273 Clayton
Pereira 1122 0000-0002-0427-4880 João Paulo Papa
1122 0000-0002-6494-7514

###### Abstract

With the advent of deep learning, the number of works proposing new methods or
improving existent ones has grown exponentially in the last years. In this
scenario, “very deep” models were emerging, once they were expected to extract
more intrinsic and abstract features while supporting a better performance.
However, such models suffer from the gradient vanishing problem, i.e.,
backpropagation values become too close to zero in their shallower layers,
ultimately causing learning to stagnate. Such an issue was overcome in the
context of convolution neural networks by creating “shortcut connections”
between layers, in a so-called deep residual learning framework. Nonetheless,
a very popular deep learning technique called Deep Belief Network still
suffers from gradient vanishing when dealing with discriminative tasks.
Therefore, this paper proposes the Residual Deep Belief Network, which
considers the information reinforcement layer-by-layer to improve the feature
extraction and knowledge retaining, that support better discriminative
performance. Experiments conducted over three public datasets demonstrate its
robustness concerning the task of binary image classification.

###### Keywords:

Deep Belief Networks Residual Networks Restricted Boltzmann Machines.

##  1 Introduction

Machine learning-based approaches have been massively studied and applied to
daily tasks in the last decades, mostly due to the remarkable accomplishments
achieved by deep learning models. Despite the success attained by these
techniques, they still suffer from a well-known drawback regarding the
backpropagation-based learning procedure: the vanishing gradient. This kind of
problem becomes more prominent on deeper models since the gradient vanishes
and is not propagated adequately to former layers, thus, preventing a proper
parameter update.

To tackle such an issue, He et al. [4] proposed the ResNet, a framework where
the layers learn residual functions concerning the layer inputs, instead of
learning unreferenced functions. In short, the idea is mapping a set of
stacked layers to a residual map, which comprises a combination of the set
input and output and then mapping it back to the desired underlying mapping.

The model achieved fast popularity, being applied in a wide range of
applications, such as traffic surveillance [7], medicine [12, 8], and action
recognition [2], to cite a few. Moreover, many works proposed different
approaches using the idea of residual functions. Lin et al. [11], for
instance, proposed the RetinaNet, a pyramidal-shaped network that employs
residual stages to deal with one-shot small object detection over unbalanced
datasets. Meanwhile, Szegedy et al. [17] proposed the Inception-ResNet for
object recognition. Later, Santos et al. [16] proposed the Cascade Residual
Convolutional Neural Network for video segmentation.

In the context of deep neural networks, there exist another class of methods
that are composed of Restricted Boltzmann Machines (RBMs) [6], a stochastic
approach represented by a bipartite graph whose training is given by the
minimization of the energy between a visible and a latent layer. Among these
methods, Deep Belief Networks (DBNs) [5] and Deep Boltzmann Machines [15, 13]
achieved a considerable popularity in the last years due the satisfactory
results over a wide variety of applications [14, 3, 18].

However, as far as we are concerned, no work addressed the concept of
reinforcing the feature extraction over those models in a layer-by-layer
fashion. Therefore, the main contributions of this paper are twofold: (i) to
propose the Residual Deep Belief Network (Res-DBN), a novel approach that
combines each layer input and output to reinforce the information conveyed
through it, and (ii) to support the literature concerning both DBNs and
residual-based models.

The remainder of this paper is presented as follows: Section 2 introduces the
main concepts regarding RBMs and DBNs, while Section 3 proposes the Residual
Deep Belief Network. Further, Section 4 describes the methodology and datasets
employed in this work. Finally, Sections 5 and 6 provide the experimental
results and conclusions, respectively.

##  2 Theoretical Background

This section introduces a brief theoretical background regarding Restricted
Boltzmann Machines and Deep Belief Networks.

###  2.1 Restricted Boltzmann Machines

Restricted Boltzmann Machine stands for a stochastic physics-inspired
computational model capable of learning data distribution intrinsic patterns.
The process is represented as a bipartite graph where the data composes a
visible input-like layer $\bm{v}$, and a latent $n$-dimensional vector
$\bm{h}$, composed of a set of hidden neurons whose the model tries to map
such inputs onto. The model’s training procedure dwells on the minimization of
the system’s energy, given as follows:

$E(\bm{v},\bm{h})=-\sum_{i=1}^{m}b_{i}v_{i}-\sum_{j=1}^{n}c_{j}h_{j}-\sum_{i=1}^{m}\sum_{j=1}^{n}w_{ij}v_{i}h_{j},$
(1)

where $m$ and $n$ stand for the dimensions of the visible and hidden layers,
respectively, while $b$ and $c$ denote their respective bias vectors, further,
W corresponds to the weight matrix connecting both layers, in which $w_{ij}$
stands for the connection between visible unit $i$ and the $j$ hidden one.
Notice the model is restricted, thus implying no connection is allowed among
the same layer neurons.

Ideally, the model was supposed to be solved by computing the joint
probability of the visible and hidden neurons in an analytic fashion. However,
such an approach is intractable since it requires the partition function
calculation, i.e., computing every possible configuration of the system.
Therefore, Hinton proposed the Contrastive Divergence (CD) [6], an alternative
method to estimate the conditional probabilities of the visible and hidden
neurons using Gibbs sampling over a Monte Carlo Markov Chain (MCMC). Hence,
the probabilities of both input and hidden units are computed as follows:

$p(h_{j}=1|\bm{v})=\sigma\left(c_{j}+\sum_{i=1}^{m}w_{ij}v_{i}\right),$ (2)

and

$p(v_{i}=1|\bm{h})=\sigma\left(b_{i}+\sum_{j=1}^{n}w_{ij}h_{j}\right),$ (3)

where $\sigma$ stands for the logistic-sigmoid function.

###  2.2 Deep Belief Networks

Conceptually, Deep Belief Networks are graph-based generative models composed
of a visible and a set of hidden layers connected by weight matrices, with no
connection between neurons in the same layer. In practice, the model comprises
a set of stacked RBMs whose hidden layers greedily feeds the subsequent RBM
visible layer. Finally, a softmax layer is attached at the top of the model,
and the weights are fine-tuned using backpropagation for classification
purposes. Figure 1 depicts the model. Notice that $\textbf{W}^{(l)}$,
$l\in[1,L]$, stands for the weight matrix at layer $l$, where $L$ denotes the
number of hidden layers. Moreover, $\bm{v}$ stands for the visible layer, as
well as $\bm{h}^{(l)}$ represents the $l^{th}$ hidden layer.

Figure 1: DBN architecture with two hidden layers for classification purposes.

##  3 Information Reinforcement in DBNs

In this section, we present the proposed approach concerning the residual
reinforcement layer-by-layer in Deep Belief Networks, from now on called Res-
DBN. Since such a network is a hybrid model between sigmoid belief networks
and binary RBMs [5], it is important to highlight some “tricks” to make use of
the information provided layer-by-layer.

As aforementioned, DBNs can be viewed as hybrid networks that model the data’s
prior distribution in a layer-by-layer fashion to improve the lower bound from
model distribution. Such a fact motivated us to make use of the information
learned in each stack of RBM for reinforcement since the greedy-layer pre-
training uses the activation of latent binary variables as the input of the
next visible layer. Generally speaking, such activation is defined by Eq. 2,
and its pre-activation vector, $\bm{a}^{(l)}$, as follows:

$a^{(l)}_{j}=c^{(l)}_{j}+\sum_{i=1}^{m}w^{(l)}_{ij}x^{(l-1)}_{i},$ (4)

where, $c^{(l)}_{j}$ stands for the bias from hidden layer $l$, $m$ is the
number of units present on the previous layer, $w^{(l)}_{ij}$ represents the
weight matrix for layer $l$, and $x^{(l-1)}_{i}$ stands for the input data
from layer $l-1$, where $x^{0}_{i}=v_{i}$.

Therefore, it is possible to use the “reinforcement pre-activation” vector,
denoted as $\hat{\bm{a}}^{(l)}$, from layer $l$, $\bm{\forall}$ $l>1$. Since
the standard RBM output of post-activation (provided by Eq. 2) is in $[0,1]$
interval, it is necessary to limit the reinforcement term of the proposed
approach as follows:

$\hat{\bm{a}}^{(l)}=\dfrac{\delta(\bm{a}^{(l-1)})}{max\\{\delta(a^{(l-1)}_{j})\\}},$
(5)

where, $\delta$ stands for the Rectifier111$\delta(z)=max(0,z)$. function,
while $max$ returns the maximum value from the $\delta$ output vector for
normalization purposes. Then, the new input data and the information
aggregation for layer $l$ is defined by adding the values obtained from Eq. 5
to the post-activation, i.e., applying $\sigma(\bm{a}^{(l-1)})$, as follows:

$x^{(l-1)}_{i}=\sigma(a^{(l-1)}_{j})+\hat{a}^{(l)}_{j},$ (6)

where $x^{(l-1)}_{i}$ stands for the new input data to layer $l$,
$\bm{\forall}$ $l>1$, and its normalized and vectorized form can be obtained
as follows:

$\bm{x}^{(l-1)}=\dfrac{\bm{x}^{(l-1)}}{max\\{x^{(l-1)}_{i}\\}}.$ (7)

It is important to highlight that, in Eq. 5, we only use the positive pre-
activations to retrieve and propagate the signal that is meaningful for
neurons excitation, i.e., values greater than 0, which generates a probability
of more than $50\%$ after applying sigmoid activation.

Figure 2: Res-DBN architecture with 3 hidden layers.

The Figure 2 depicts the Res-DBN architecture, with hidden layers connected by
the weights $\textbf{W}^{(l)}$. The dashed connections stand for the
reinforcement approach, with the information aggregation occuring as covered
by the Eqs. 4 to 7, from a generic hidden layer to the next one
($\bm{h}^{(1)}\rightarrow\bm{h}^{(2)}$, for instance).

##  4 Methodology

In this section, we present details regarding the datasets employed in our
experiments, as well as the experimental setup applied for this paper.

###  4.1 Datasets

Three well-known image datasets were employed throughout the experiments:

  * $\bullet$

MNIST222http://yann.lecun.com/exdb/mnist [10]: set of $28\times 28$ binary
images of handwritten digits (0-9), i.e., 10 classes. The original version
contains a training set with $60,000$ images from digits ‘0’-‘9’, as well as a
test set with $10,000$ images.

  * $\bullet$

Fashion-MNIST333https://github.com/zalandoresearch/fashion-mnist [20]: set of
$28\times 28$ binary images of clothing objects. The original version contains
a training set with $60,000$ images from $10$ distinct objects (t-shirt,
trouser, pullover, dress, coat, sandal, shirt, sneaker, bag, and ankle boot),
and a test set with $10,000$ images.

  * $\bullet$

Kuzushiji-MNIST444https://github.com/rois-codh/kmnist [1]: set of $28\times
28$ binary images of hiragana characters. The original version contains a
training set with $60,000$ images from $10$ previously selected hiragana
characters, and a test set with $10,000$ images.

###  4.2 Experimental Setup

Concerning the experiments, we employed the concepts mentioned in Section 3,
considering two main phases: (i) the DBN pre-training and (ii) the
discriminative fine-tuning. Regarding the former, it is important to highlight
that the information reinforcement is performed during the greedy layer-wise
process, in which the hidden layers ($l={1,2,\dots,L}$) receive the positive
“residual” information. Such a process takes into account a mini-batch of size
$128$, a learning rate of $0.1$, $50$ epochs for the bottommost RBM
convergence, and $25$ epochs for the intermediate and top layers
convergence555Such a value is half of the initial one to evaluate Res-DBN
earlier convergence..

Moreover, regarding the classification phase, a softmax layer was attached at
the top of the model after the DBN pre-training, performing the fine-tuning
process for $20$ epochs through backpropagation using the well-known ADAM [9]
optimizer. The process employed a learning rate of $10^{-3}$ for all layers.
Furthermore, it was performed $15$ independent executions for each model to
provide statistical analysis. To assess the robustness of the proposed
approach, we employed seven different DBN architectures changing the number of
hidden neurons and layers, as denoted in Table 1.

Model | Res-DBN |  | DBN
---|---|---|---
(a) |  $i$:500:500:10 |  |  $i$:500:500:10
(b) |  $i$:500:500:500:10 |  |  $i$:500:500:500:10
(c) |  $i$:500:500:500:500:10 |  |  $i$:500:500:500:500:10
(d) |  $i$:1000:1000:10 |  |  $i$:1000:1000:10
(e) |  $i$:1000:1000:1000:10 |  |  $i$:1000:1000:1000:10
(f) |  $i$:1000:1000:1000:1000:10 |  |  $i$:1000:1000:1000:1000:10
(g) |  $i$:2000:2000:2000:2000:10 |  |  $i$:2000:2000:2000:2000:10

Table 1: Different setups, where $i$ stands for the number of neurons on the
input layer.

##  5 Experiments

In this Section, we present the experimental results concerning seven distinct
DBN architectures, i.e., (a), (b), (c), (d), (e), (f) and (g), over the
aforementioned datasets. Table 2 provides the average accuracies and standard
deviations for each configuration on $15$ trials, where the proposed approach
is compared against the standard DBN formulation in each dataset for each
configuration. Further, results in bold represent the best values according to
the statistical Wilcoxon signed-rank test [19] with significance $p\leq 0.05$
concerning each model configuration. On the other hand, underlined values
represent the best results overall models regarding each dataset, without a
statistical difference, i.e., results similar to the best one achieved.

Experiment | MNIST |  |  | Fashion MNIST |  |  | Kuzushiji MNIST
---|---|---|---|---|---|---|---
Res-DBN |  | DBN |  |  | Res-DBN |  | DBN |  |  | Res-DBN |  | DBN
(a) | $\bm{97.39\pm 0.08}$ |  | $97.23\pm 0.09$ |  |  | $81.13\pm 0.33$ |  | $\bm{81.52\pm 0.27}$ |  |  | $\bm{86.49\pm 0.18}$ |  | $84.78\pm 0.29$
(b) | $\bm{97.61\pm 0.07}$ |  | $97.44\pm 0.11$ |  |  | $81.49\pm 0.50$ |  | $81.41\pm 0.57$ |  |  | $\bm{87.75\pm 0.20}$ |  | $85.81\pm 0.18$
(c) | $97.59\pm 0.10$ |  | $97.57\pm 0.09$ |  |  | $81.66\pm 0.33$ |  | $81.51\pm 0.60$ |  |  | $\bm{88.21\pm 0.18}$ |  | $86.97\pm 0.30$
(d) | $\bm{97.66\pm 0.10}$ |  | $97.40\pm 0.10$ |  |  | $81.55\pm 0.35$ |  | $81.15\pm 0.64$ |  |  | $\bm{87.67\pm 0.19}$ |  | $86.24\pm 0.21$
(e) | $\underline{\bm{97.85\pm 0.06}}$ |  | $97.48\pm 0.12$ |  |  | $\bm{82.05\pm 0.48}$ |  | $81.59\pm 0.51$ |  |  | $\bm{88.95\pm 0.16}$ |  | $87.57\pm 0.20$
(f) | $\underline{97.80\pm 0.37}$ |  | $97.68\pm 0.29$ |  |  | $82.16\pm 0.50$ |  | $82.19\pm 0.46$ |  |  | $\underline{\bm{89.63\pm 0.23}}$ |  | $88.81\pm 0.40$
(g) | $\underline{\bm{97.88\pm 0.19}}$ |  | $97.51\pm 0.30$ |  |  | $\underline{82.73\pm 0.53}$ |  | $\underline{82.63\pm 0.36}$ |  |  | $\underline{\bm{89.45\pm 0.78}}$ |  | $88.70\pm 0.60$

Table 2: Experimental results on different datasets.

Regarding the original MNIST dataset, the preeminence of the proposed model
over the standard version of the RBM is evident, since the best results were
obtained exclusively by Res-DBN and, from these, five out of seven scenarios
presented statistical significance. Such a behavior is stressed in the
Kuzushiji MNIST dataset, where the best results were obtained solely by the
Res-DBN over every possible configuration. The results’ similarity between
these datasets is somehow expected since both are composed of handwritten
digits or letters.

The Fashion MNIST dataset presents the single experimental scenario, i.e.,
model (a), where the proposed model was outperformed by the traditional DBN,
although by a small margin. In all other cases Res-DBN presented results
superior or equal to the traditional formulation, which favors the Res-DBN use
over the DBNs.

Finally, one can observe the best results overall were obtained using a more
complex model, i.e., with a higher number of layers and neurons, as denoted by
the underlined values. Additionally, the proposed model outperformed or at
least is equivalent, to the standard DBN in virtually all scenarios, except
one concerning the Fashion-MNIST dataset.

###  5.1 Training Evaluation

Figures 3, 4, and 5 depict the models’ learning curves over the test sets
regarding MNIST, Fashion MNIST, and Kuzushiji MNIST, respectively. In Figure
3, one can observe that Res-DBN(e) converged faster than the remaining
approaches, obtained reasonably good results after seven iterations. At the
end of the process, Res-DBN(f) and (g) boosted and outperformed Res-DBN(e), as
well as any of standard DBN approaches, depicted as dashed lines.

Figure 3: Accuracy on MNIST test set.

Regarding Fashion MNIST, it can be observed in Figure 4 that Res-DBN(e) was
once again the fastest technique to converge, obtaining acceptable results
after five iterations. However, after iteration number five, all models seem
to overfit, explaining the performance decrease observed over the testing
samples. Finally, after $14$ iterations, the results start increasing once
again, being Res-DBN(g) the most accurate technique after $20$ iterations.

Figure 4: Accuracy on Fashion MNIST test set.

Finally, the Kuzushiji learning curve, depicted in Figure 5, displays a
behavior silimiar to the MNIST dataset. Moreover, it shows that Res-DBN
provided better results than its traditional variant in all cases right from
the beginning of the training. In some cases with a margin greater than $2\%$,
showing a promissing improvement.

Figure 5: Accuracy on Kuzushiji MNIST test set.

##  6 Conclusions

In this paper, we proposed a novel approach based on reinforcing DBN’s layer-
by-layer feature extraction in a residual fashion, the so-called Residual Deep
Belief Network. Experiments conducted over three public datasets confirm the
sturdiness of the model. Moreover, it is important to highlight faster
convergence achieved by Res-DBN in front of DBN, once half of the epochs were
employed for pre-training hidden layers, and the results outperformed the
latter model.

Regarding future work, we intend to investigate the model in the video domain,
applying it to classification and recognition tasks, as well as to propose a
similar approach regarding Deep Boltzmann Machines.

## Acknowledgments

The authors are grateful to FAPESP grants #2013/07375-0, #2014/12236-1,
#2017/25908-6, #2019/07825-1, and #2019/07665-4, as well as CNPq grants
#307066/2017-7, and #427968/2018-6.

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