Quantum Control for Error Correction using Mother Tee Optimization
Wael Korani
a
and Malek Mouhoub
b
Department of Computer Science, University of Regina, Regina, Saskatchewan, Canada
Keywords:
Mother Tree Optimization, Continuous Optimization, Quantum Computing, Artificial Intelligence.
Abstract:
Quantum Control Problem (QCP) for Error Correction (EC) is a significant issue that helps in producing an
efficient quantum computer. The QCP for EC can be tackled using Stochastic Local Search (SLS) methods.
However, these techniques might produce low quality results for large dimensional quantum systems. Lately,
Nature-Inspired (NI) algorithms including different variants of Particle Swarm Optimization (PSO) and Def-
erential Evolution (DE) were implemented in several studies to tackle the QCP for EC, but the results were
not promising. In this paper, we propose a quantum model that is built on our NI algorithm, called Mother
Tree Optimization for QCP (MTO-QCP), to overcome the stagnation issue that the other methods suffer from.
In order to assess the performance of MTO-QCP, we conducted several preliminary experiments to adjust our
MTO parameters. In this regard, our MTO-QCP achieves high-fidelity (> 99.99%) for a Single-Shot (SS)
three-qubit gate control at gate operation time of 26 ns. This recommended fidelity is an acceptable threshold
fidelity for fault-tolerant Quantum Computing (QC) problems.
1 INTRODUCTION
Quantum mechanics research has been significantly
growing as a corner stone in computation in different
branches of science (Caves, 2013) such as, metrol-
ogy (Giovannetti et al., 2011), secure communica-
tion (Scarani et al., 2009), femtosecond lasers (Assion
et al., 1998), and nuclear magnetic reassurance (Hop-
kins et al., 2003). The heart of these applications is
quantum dynamics, which is a desired state or oper-
ation that could be reached by controlling a quantum
system.
Control theory is a process of guiding the system’s
dynamics to a desired state or to optimize the dynam-
ics of a given objective function, as in QCP for EC.
This theory is built on a mathematical model for a
physical system. Quantum control as a part of con-
trol theory is a process of producing a feasible solu-
tion of control parameters for a given control problem
to produce an efficient system. It is known that the
quantum control process is a very complicated pro-
cess, because the dynamics of quantum system is of-
ten nonlinear and/or noncontinuous. Analytical solu-
tion for these kind of models is very sophisticated and
needs much computations and resources. An alterna-
tive solution is reinforcement learning that facilitates
a
https://orcid.org/0000-0002-1419-1149
b
https://orcid.org/0000-0001-7381-1064
representing a physical system in an explicit mathe-
matical model (Sutton et al., 1992). In the last couple
of decades, several attempts were conducted to use
optimization methods for quantum states problems to
achieve efficient systems.
QC is growing significantly due to the availability
of real quantum computers (QCos) that allow users to
perform quantum experiments. QC helps in speeding
up the computations using quantum bit (qubit) strings
and quantum gates (QGs). QCos have been used in
different quantum applications using specific num-
ber of qubits. For instance, in (Sisodia et al., 2017),
discrimination of orthogonal entangled states has a
significant role in quantum information processing.
A SS Toffoli (controlled-controlled-NOT (CCNOT))
gate is recommended for quantum information pro-
cessing and other classical quantum application (Za-
hedinejad et al., 2015). The noise of the output (fi-
delity) is one of the common challenges in the perfor-
mance of quantum applications (Linke et al., 2017).
In order to achieve a scalable QC, a number of
high fidelity QGs is required to construct a quan-
tum circuit (CCr) (Nielsen and Chuang, 2002). The
fragility of the state of a QCos is compensated using
quantum EC (Cory et al., 1998). In the past, several
experiments were conducted on single and two-qubit
operations, but the quantum EC code requires at least
three-qubit using the Toffoli gate (Cory et al., 1998).
In (Zahedinejad et al., 2016), the fidelity was calcu-
554
Korani, W. and Mouhoub, M.
Quantum Control for Error Correction using Mother Tee Optimization.
DOI: 10.5220/0010194905540561
In Proceedings of the 13th International Conference on Agents and Artificial Intelligence (ICAART 2021) - Volume 2, pages 554-561
ISBN: 978-989-758-484-8
Copyright
c
2021 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
lated using a decomposition method that used lower
order qubit gates to build a multi-qubit gates. How-
ever, this method is not recommended, as it takes long
operation times.
The rest of the paper is structured as follows. Sec-
tion 2 lists some related studies and our motivation to
propose a quantum model. Section 3 discusses quan-
tum control and quantum gate design model. Sec-
tion 4 gives a brief introduction about our recently
proposed MTO. Section 5 explains the optimization
quantum control that our model will solve. Section 6
shows our proposed model combining our MTO algo-
rithm and QCP for EC. Section 7 reports on our ex-
periments. Finally, section 8 lists concluding remarks
and ideas for future work.
2 RELATED WORK
NI techniques as a part of the approximate class
of algorithms received more attention in the last
of decades for solving different hard optimization
problems including QCPs. While exact optimiza-
tion techniques require a prior knowledge about sys-
tem dynamics such as Bayesian and Markovian feed-
back (Wiseman et al., 2002), NI techniques solve
QCPs as a black box (i.e. the systems characteristics
are unknown). NI methods were successfully imple-
mented to find a desired state for QCPs. In (Srinivas
and Deb, 1994), Particle Swarm Optimization (PSO),
Differential Evolution (DE) and Genetic Algorithms
(GAs) were implemented to solve quantum control
schemes. However, these algorithms could not find a
satisfactory solution when the number of control pa-
rameters increases.
Optimization methods have been successfully im-
plemented to tackle difficult and sophisticated non-
linear or/and non-continuous optimization problems
including quantum problems. Several attempts were
conducted to tackle the QCP for EC. In (Fletcher,
2013), Quasi-Newton (QN) based on the Broyden-
Fletcher-Goldfarb-Shanno (BFGS) approximation of
Hessian is implemented in a model to tackle QCP
for EC. The Nelder-Mead simplex (NMS) method
that is a direct method and does not require deriva-
tives (Olsson and Nelson, 1975), Particle Swarm Op-
timization (PSO) that is built on the movement of
birds (Clerc and Kennedy, 2002), and Differential
Evolution (DE) that is built on Darwinian princi-
ple (Zahedinejad et al., 2014) were applied to tackle
QCPs. In (Ghosh et al., 2013), the authors intro-
duced a quantum-control procedure to produce an op-
timal pulse for a Toffoli gate. Quasi-Newton, sim-
plex methods, variants of PSO, Greedy algorithms as
well as DE techniques, all failed to solve the QCP for
EC. Indeed, it has been reported in (Zahedinejad et al.,
2016) that all listed algorithms (QN, NMS, PSO, and
DE) fail to generate satisfactory fidelity to QCP for
EC higher than 99.3% (with the best fidelity, 99.31%,
obtained by DE). However, the recommended level of
fidelity to QCP for EC is 99.99% (Ghosh et al., 2013).
However, the previous implemented optimization
techniques suffer from several drawbacks that make
them inappropriate to tackle a challenging problem
such as QCP. The simplex technique may optimize
a problem without computing its derivatives, which
usually consume much computing power. Although
the simplex technique is simple and robust in tackling
small dimension optimization problems, it might eas-
ily fail for large dimension problems (B
˝
urmen et al.,
2006) and (Kelley, 1999).
GAs and DE algorithms should not be used to
solve a given problem where there is a traditional
exact method that can solve it, because these algo-
rithms cannot be better with less computational ef-
fort (Schwefel, 2000). In (Back, 1996), Back con-
cluded that these algorithms are not appropriate meth-
ods to solve strongly convex problems. In (Leung
et al., 1997), Leung et al. concluded that GAs suffer
from premature convergence. The authors suggested
that increasing the population size plays an impor-
tant role to help overcome this problem. In (Hrstka
and Ku
ˇ
cerov
´
a, 2004), DE may offer easier conver-
gence by increasing the number of parents and reduc-
ing the scaling factor; however, DE, may then suffer
from high computational time. (Chen et al., 2004)and
(Shi and Eberhart, 2001), PSO variants and other al-
gorithms are growing fast, and they offer an alterna-
tive way for tackling complex problems, but they have
some limitations: parameter tuning, stagnation, and
time critical applications
In 2015, the QCP for EC was solved using a
proposed model that is built on DE variant and de-
composition multi-gates concept. In (Zahedinejad
et al., 2015), the result shows that the proposed model
achieved fast high-fidelity Toffoli gate. In 2016 (Za-
hedinejad et al., 2016), the same authors implemented
other fast three-qubit gates that outperforms the de-
composition of qubit gates to lower order (i.e, one-
quibt or/and two-qubit). In (Palittapongarnpim et al.,
2017), the authors developed an optimization variant
of DE to solve the QCP, where PSO and other evo-
lutionary algorithms failed to achieve satisfactory re-
sults.
However, in (Zahedinejad et al., 2016), the au-
thors proposed a variant of DE called SuSSADE to
solve the QCP for EC and achieved high fidelity.
The proposed variant SuSSADE was not well de-
Quantum Control for Error Correction using Mother Tee Optimization
555
scribed and how the author implemented to solve
QCP. The lack of description of SuSSADE and the
drawbacks of other algorithms motivate us to propose
a quantum control model that is built on our NI algo-
rithm called Mother Tree Optimization (MTO) (Ko-
rani et al., 2019) algorithm, and using SS multi-
QGs (Lanyon et al., 2009). The MTO has the capa-
bility to escape from local solutions using our fixed
off-spring topology, and it is appropriate to solve high
dimension problems compared to other PSO vari-
ants (Korani et al., 2019) and (Korani and Mouhoub,
2020). Here, a SS is represented a continuous time
evolution that is also controlled to realize the QG. The
decomposition of multi-QGs to lower order of qubit
gates affects the performance and makes it slower.
Thus, we use fast SS multi-QG as a basic component
for achieving high fidelity, because it works signifi-
cantly faster than the decomposition technique. In our
model, the optimization problem is constructed based
on Unitary Evaluation (UE) function that includes a
Hamiltonian description of the quantum physical sys-
tem, and it delivers high fidelity QG.
Our model, based on our MTO, is implemented
to tackle the QCP for EC. MTO is a NI optimiza-
tion technique that is simple and can be implemented
and programmed in a few lines of code. MTO is ap-
plied on one instance (three-qubit gate control using a
gate operation time of 26 ns) as an example, because
QCP experiments take long time that might reach a
month. MTO achieves promising results that reaches
a high fidelity value (> 99.99%). We expect that
our quantum model can be implemented for differ-
ent number of qubit gate control at different operat-
ing gate times to achieve the optimal parameters for
piecewise-constant pulse.
3 QUANTUM CONTROL
The framework of quantum control procedure appli-
cations, that its dynamics are governed by quantum
mechanics, is called quantum control (Dong and Pe-
tersen, 2010). The QG design is an example of quan-
tum control procedure that uses the control theory
to apply logic gates on different quantum bits (Za-
hedinejad et al., 2014). This QG design is used in QC
to speed up computation using quantum bit (qubit).
The mathematical model of QG design is discussed
in this section.
3.1 Quantum Gate Design Model
QC is built on quantum control theory to perform
computation in an efficient way, and it helps in speed-
ing up the computation of different operations such
as factorization (Shor, 1999) and searching process
in database (Grover, 1996). The interaction between
quantum system and environment in QC implemen-
tation may introduce noise called errors. The resul-
tant errors will reduce the efficiency and invalidate
the benefits of using quantum resources (Shor, 1996).
Therefore, achieving protected quantum information
could be obtained by decreasing the error rate to a
specific threshold. Our aim is to propose a quantum
model to increase the system fidelity to a certain level
as shown in our case study for QCP of EC.
Qubit is the basic unit in QC, and is used to per-
form computations. This qubit has characteristics dif-
ferent than the classical bit. It may exist in any su-
perposition state, unlike the classical bit that has only
two states (0 or 1). In decomposition case, QC op-
erations can be reduced to a set of lower order qubit
gates (Barenco et al., 1995). However, the decom-
position process will lead to increase the processing
time and decrease the fidelity. Toffoli gates that work
on more than two qubits is one of the recommended
ways to increase system efficiency and fidelity. The
Toffoli gate is a controlled/controlled-not gate work-
ing on three-qubits.
We implement the same mathematical model as
in (Ghosh et al., 2013) and (Spiteri et al., 2018). We
consider z coupled super conducting artificial atoms
with parameters appropriate for a transmon system
(TS). In addition, TS can also be capacitively coupled
(CC). The TS includes individual transmon that has
specific number of energy states (m), and j [z] :=
[1,2,...,z] are denoted their positions. The CC be-
tween transmons prodcues an interaction between all
adjacent transmons in XY plane. The produced cou-
pling strength is evaluated to be g = 30MHz. The ap-
plied pulse in ideal gate has a shifted frequency that
is located in range of 2.5 GHz ε
j
(t) 2.5 GHz.
The Hamilton for z CC-TS is represented as the
m
z
-dimensional block diagonal matrix (Ghosh et al.,
2013).
H(t)
h
:=
ˆ
H(t) =
z
j=1
P
z
I
(diag
0,ε
k
(t),2ε
k
(t) η, 3ε
k
(t) η
0
)
+
g
2
z1
j=1
P
(z1)
I
(X
j
X
j+1
+ Y
j
Y
j+1
) (1)
X
j
=
0 1 0 0
1 0
2 0
0
2 0
3
0 0
3 0
j
(2)
ICAART 2021 - 13th International Conference on Agents and Artificial Intelligence
556
Y
j
i
=
0 1 0 0
1 0
2 0
0
2 0
3
0 0
3 0
j
(3)
where I is represented the identity matrix and each
block works on H
z
4
, where H
z
4
is denoted the
Hilbert space. The sum of all B with (z 1) copies
of A is the P
z
I
operator, where B is represented the
Kronecker products. In addition, the coupling X
j
is
the generalized Pauli operators, and
Y
j
i
is a simulation
for uniform coupling, giving that there is no relation-
ship between coupling operators and k (Ghosh et al.,
2013).
Equation 1 is reduced to subspace due to the
block-diagonal property of Hamiltonian:
ˆ
H
p
(t) = O
z
ˆ
H(t)O
z
, (4)
where O
z
is denoted the operator that truncate
ˆ
H(t).
ˆ
H
p
(t) is calculated over θ, where θ is denoted the
gate time. The unitary operator is calculated as fol-
lows:
U (Θ) = T exp
Z
Θ
0
dt
H
p
(t)
i~
(5)
= T exp
2πi
Z
Θ
0
dt
ˆ
H
p
(t)
,
where T operator is denoted to the time-ordering.
U (Θ) of the most three excitations are used to define
Transmon states, where is denoted U (Θ) computa-
tion space. The projection is calculated as follows:
U
cs
(Θ) = PU (Θ)P
, (6)
The projected unitary is an operator that is used to
calculate the fidelity of the system. The fidelity is
evaluated by measuring the difference between the ac-
tual result and target gate U
target
. The fidelity of the
system is calculated as follows:
F (Θ, n; ε(t)) =
1
2
n
tr
U
target
U
cs
(Θ)
[0,1] (7)
and it represents the QG operation for given shifted
frequencies. The value of desired fidelity is (>
99.99%).
F (Θ, n; ε(t)) 99.99% (8)
4 MOTHER TREE
OPTIMIZATION (MTO)
ALGORITHM
MTO and its variant, MTO-CL, are built on a co-
operative system of Douglas fir trees (Korani et al.,
2019). The basic idea of the MTO algorithm is built
on our agent communication topology called Fixed-
Offspring (FO) (Korani et al., 2019). The number of
agents in the population is represented by group of
Active Food Sources (AFSs). The size of the popula-
tion is denoted as N
T
. Figure 1 shows that the pop-
ulation is divided into four different partitions. Posi-
tion of each agent is updated based on the group that
agent belongs to (Korani et al., 2019). Top Mother
Tree (TMT) updates its position randomly as shown
in Equation 9, 10, 11, and 12 (Korani et al., 2019).
Figure 1 shows the Partially Connected Trees (PCTs)
and Fully Connected Trees (FCTs). In addition, the
PCTs is divided into FPCTs and LPCTs as shown in
Figure 1.
TMT
1
st
2
nd
- - - - -
- - - - -
N
T
2
N
T
2
+ 2
N
T
FPCTs
FCTs
LPCTs
N
T
2
1
N
T
2
+ 3
No parents
Agents are arranged in descending order
based on their fitness value
N
T
2
+ 1
Feeders
Non-feeders
N
FPCT s
=
N
T
4
2
N
FCT s
= 3
N
LPCTs
=
N
T
4
2
N
os
=
N
r
2
1
N
Fr =
N
T
2
+ 1
N
NFr =
N
T
2
1
Figure 1: MTO Topology.
The TMT has two searching levels: At the first level,
it updates its position randomly as follows:
P
1
(x
k+1
) = P
1
(x
k
) + δR(d) (9)
R(d) =
R
R ·R
|
(10)
~
R = 2 (round (rand (d,1)) 1) rand (d,1) (11)
where δ is the root signal step size, R(d) is a random
vector, and d is the dimension of the problem (in our
case 26*3). At the second level, the TMT updates its
position randomly with smaller step size as follows.
P
1
(x
k+1
) = P
1
(x
k
) + R(d) (12)
where is the Mycorrhizal Fungi Network (MFN).
Values of δ and are adapted based on several pre-
liminary experiments.
The FPCTs group updates its agents’ positions
as follows:
Quantum Control for Error Correction using Mother Tee Optimization
557
P
n
(x
k+1
) = P
n
(x
k
) +
n1
i=1
1
n i + 1
(P
i
(x
k
) P
n
(x
k
)), (13)
where P
n
(x
k
) is denoted the current position of any
member in the group, P
i
(x
k
) is denoted the current
position of a feeder agent, and P
n
(x
k+1
) is denoted
the updated position of the current member (Korani
et al., 2019). The FCTs group updates its members’
positions as follows (Korani et al., 2019):
P
n
(x
k+1
) = P
n
(x
k
) +
n1
i=nN
os
1
n i + 1
(P
i
(x
k
) P
n
(x
k
)).
(14)
The LPCTs group updates its agents’ positions as
follows (Korani et al., 2019):
P
n
(x
k+1
) = P
n
(x
k
) +
N
T
N
os
i=nN
os
1
n i + 1
(P
i
(x
k
) P
n
(x
k
)).
(15)
Algorithm 1 is the pseudo-code of the MTO method.
More details about this algorithm can be found in (Ko-
rani et al., 2019).
5 THE QUANTUM CONTROL
FOR ERROR CORRECTION
PROBLEM
The QCP for EC is formulated as an optimization
problem that could be tackled using our MTO algo-
rithm. Our objective is to find a solution for a three-
qubit optimization problem. In addition, our opti-
mization problem should meet a significant constraint
such that the fidelity meets a threshold (99.99%). The
transmon-shifted frequencies are considered control
pulses. These pulses are discretized using piecewise-
constant. More formally, our quantum optimization
problem is formalized as follows:
find ε(t),t [0, Θ], (16a)
subject 2.5 ε
k
(t) 2.5 ,n = 1, 2, . . . , k,
(16b)
F (Θ, n; ε(t)) 99.99%. (16c)
The frequency components in Equation 16b are repre-
sented as piecewise-constant step function that has a
time step duration = 1 ns. The gate time is an integer
number of t for a given simulation. Equation 16c
has nθ /t degrees of freedom, so that increasing the
gate time θ or decreasing the time step t, results in
increasing the probability of finding solution. The
degree of freedom represents the piecewise-constant
values.
Algorithm 1: MTO (Korani et al., 2019).
Require: : N
T
,P
T
,d, K
rs
,Cl,and El
N
T
: The population size (AFSs)
P
T
: The position of the active food sources
d: The dimension of the problem
K
rs
: The number of kin recognition signals
Cl: The number of climate change events (0 for MTO)
El: The elimination percentage
Distribute T agents uniformly over the search space (P
1
,. ..,P
T
)
Evaluate the fitness value of T agents (S
1
.. .S
T
)
Sort solutions in descending order based on the fitness and store them in S
S = Sort(S
1
.. .S
T
)
The sorted positions with the same rank of S stored in array A
A = (P
1
.. .P
T
)
loop
for k
rs
= 1 to K
rs
do
Use equations (9)–(15) to update the position of each agent in A
Evaluate the fitness of the updated positions
Sort solutions in descending order and store them in S
Update A
end for
if Cl = 0 then
BREAK;
else
Select the best agents in S ((1 - El) S)
Store the best selected position in Abest
Distort Abest (mulitply by random vector)
Distort(Abest) = Abest R(d)
Remove the rest of the population (El)S
Generate agents equal to the the number of removed agents
Cl = Cl 1
end if
end loop (Cl > 0)
S = Sort(S
1
.. .S
T
)
Global Solution = Min(S)
return Global Solution.
We implement our mathematical model for QCP for
EC to design SS high fidelity QGs (Toffoli gates). In
our model, the quantum dynamics is represented us-
ing Hamiltonian evolution. However, the complexity
of our quantum optimization problem becomes very
difficult with increasing the number of transmons n.
In addition, there are some limitation of the selected
θ and t to meet the practical requirements of quan-
tum computers.
6 PROPOSED MODEL
Our proposed model includes two major parts: the
optimization component (MTO) and the mathemati-
cal model for QCP for EC. The optimizer is shown on
ICAART 2021 - 13th International Conference on Agents and Artificial Intelligence
558
Unitary Evaluation (UE)
UE = (data)
PE = f (UE)
Projection Evaluation (PE)
Fidelity (fd)
f d = f (PE)
Recieve fidelity
and update the
update the fitnes
value
Update the position
of each particle
MTOCL
f d
data = (T, n)
Figure 2: SS high-fidelity three-qubit Toffoli gate.
the left hand side while the quantum model is located
on the right hand side, as shown in Figure 2. The QCP
for EC represents SS high-fidelity three-qubit Toffoli
gate.
The data exchange between our optimizer (MTO)
and our quantum model (QCP for EC) runs for a pre-
defined number of iterations until stopping criteria is
reached. In the first iteration, the optimizer generates
random agents (equal to the population size), and each
agent has a vector of all required parameters’ values
to the quantum model, so that each agent in the popu-
lation represents a solution for our quantum problem.
The number of parameters is equal to (θ n), where θ
is the gate operation time (in our case 26 ns), and n is
the number of transmonsn (in our case 3 qubit). In the
quantum model side, the computations is performed
in several stages. In the first stage, the UE function
receives the (θ n) (26*3) values for each agent in the
population from our MTO optimizer and computes
the most three excitation. In the second stage, the
output of UE function feeds the Projection Evalua-
tion (PE) function to calculates the projection of these
excitations as explained in Section 3. Finally, the fi-
delity function receives the output of the PE function
and compares to the target gate U
target
. The fidelity is
calculated for each agent and then the results are sent
back to the MTO optimizer as updated fitness values
for their associated agents. In the optimizer side, in
the next iteration, each agent in the population will
update its position (quantum model solution) accord-
ing to its updated fitness value (the received fidelity).
The system keeps running and improves the fidelity
record until it reaches the stopping criteria.
7 EXPERIMENTATION
We have two objectives in this experiments: find a
solution for our proposed quantum model and evalu-
ate the performance of our MTO algorithm in solv-
ing hard optimization problem. Our experiment is
designed to achieve high output fidelity of our pro-
posed model (QCP for EC) using our recently pro-
posed MTO method. In addition, we test the reliabil-
ity, efficiency, and validity of using MTO optimizer as
a recently proposed NI algorithm in solving QCP for
EC optimization problem. In (Korani et al., 2019), the
authors recommended two different population sizes
40 or 60. In our experiments, we set the population
size to be 60 active food sources, because it achieves
better results than using other population sizes. In this
regard, when we increase the population size the fi-
delity improves more quickly. The QCP for EC is
considered one of the recent challenging optimization
problems that may consume much resources and time
that may reach to months. We set a limit of 10
7
it-
erations as the stopping criterion, which could takes
months to finish. The time of the function evalua-
tion totally depends on the computing resources that
the user might use. In our experiment, our proposed
model has been implemented using Python language
and all experiments have been performed on a Mac-
Book Pro with 2.3 GHz Intel Core i9 and 16GB 2400
MHz DDR4.
The performance of MTO is evaluated in tackling
QCP for EC with respect to other well-known opti-
mization algorithms. In (Ghosh et al., 2013), Ghosh
et. al. have implemented and evaluated other ba-
sic optimization algorithms to solve the same quatum
optimization problem, and the results are reported in
Table 2. We compare these results to those we ob-
tained for MTO. The authors in (Ghosh et al., 2013)
conducted experiments comparing the performance of
different optimization methods to their proposed DE
variant. The parameters of MTO are adjusted based
on extensive preliminary experiments that took long
time as shown in Table 1.
Table 1: Parameter Settings.
Algorithm Parameters settings
MTO Root signal δ = 50.0
or MFN signal = 40.03
MTO-CL Small deviation φ = 7.0
Population size = 60
CL = 10
EL = 20%
Our quantum control model proposed in Section 6 is
solved using MTO optimizer. Pulses are generated
using MTO for a simulated SS high-fidelity three-
Quantum Control for Error Correction using Mother Tee Optimization
559
Figure 3: Optimal pulses for designing a Toffoli gate.
qubit Toffoli gate as described in Figure 2. In our
experiment, the gate time is adjusted to 26 ns, and
the model produces feasible solution in each itera-
tion. At the end of the experiments, the optimal solu-
tion is recorded, which represents optimal pulses for
our quantum model. The optimal pulses (piecewise-
constant pulses) is depicted in Figure 3 for designing
a Toffoli gate that achieves high fidelity (>99.994%).
The dots circles in the figure denote the learning pa-
rameters for MTO. Our results show that our MTO
optimizer could be considered as an alternative solu-
tion for QCP for EC compared to that results reported
in (Ghosh et al., 2013) using different optimization al-
gorithms: Quasi-Newton, Simplex, and DE as shown
in Table 2. The main issue of using MTO in solving
the QCP for EC is that it takes long time to produce
the desired solution. In addition, solving the QCP for
EC requires a steady system that could work for long
time without interruption.
Table 2: The fidelity of 3-qubit Toffoli gate using different
optimization algorithms (Ghosh et al., 2013).
θ = 26 and n = 3
Optimization method Fidelity
Quasi-Newton 0.9912
Simplex 0.9221
DE 0.9931
SuSSADE 0.9999
MTO or MTOCL 0.99994
8 CONCLUSION AND FUTURE
WORKS
We have designed a quantum model for solving QCP
for EC using our recently proposed nature-inspired al-
gorithm called MTO. Our model has been evaluated
to solve one quantum instance for 3-qubit and gate
time 26 ns. The results show the capability of MTO
optimizer to tackle the QCP for EC and achieve the
desired fidelity. MTO can be used as an alternative
method for solving different quantum instances. We
expect that MTO could be used for solving higher
qubit order 4-qubit and 5-qubit of QCP of EC. In-
deed, in the near future, we plan to implement more
instances for 3-qubit and 4-qubit QCP for EC using
more computing resources. The number of qubit n
will be set to increase the difficulty of the problem.
The frequency will be tuned until we get the best fre-
quency along with gate operation time θ to produce
the desired high fidelity.
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