Quantum Control for Error Correction using Mother Tee Optimization

Wael Korani

and Malek Mouhoub

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

https://orcid.org/0000-0002-1419-1149

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

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

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

(t) ≤ 2.5 GHz.

The Hamilton for z CC-TS is represented as the

-dimensional block diagonal matrix (Ghosh et al.,

2013).

H(t)

H(t) =

∑

j=1

⊗z

(diag



0,ε

(t),2ε

(t) −η, 3ε

(t) −η



)

z−1

∑

j=1

⊗(z−1)

⊗X

j+1

+ Y

⊗Y

j+1

) (1)







0 1 0 0

1 0

√

2 0

√

2 0

√

0 0

√

3 0







(2)

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

556







0 −1 0 0

1 0 −

√

2 0

√

2 0 −

√

0 0

√

3 0







(3)

where I is represented the identity matrix and each

block works on H

⊗z

, where H

⊗z

is denoted the

Hilbert space. The sum of all B with (z −1) copies

of A is the P

⊗z

operator, where B is represented the

Kronecker products. In addition, the coupling X

the generalized Pauli operators, and

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:

(t) = O

H(t)O

†

, (4)

where O

†

is denoted the operator that truncate

H(t).

(t) is calculated over θ, where θ is denoted the

gate time. The unitary operator is calculated as fol-

lows:

U (Θ) = T exp



(t)



(5)

= T exp



−2πi

(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:

(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)) =





†

target

(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

. 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

- - - - -

+ 2

FPCTs

FCTs

LPCTs

−1

+ 3

No parents

Agents are arranged in descending order

based on their ﬁtness value

+ 1

Feeders

Non-feeders

FPCT s

−4

FCT s

= 3

LPCTs

−4

−1

Fr =

+ 1

NFr =

−1

Figure 1: MTO Topology.

The TMT has two searching levels: At the ﬁrst level,

it updates its position randomly as follows:

k+1

) = P

) + δR(d) (9)

R(d) =

√

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.

k+1

) = P

) + ∆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

k+1

) = P

) +

n−1

∑

i=1

n −i + 1

) −P

)), (13)

where P

) is denoted the current position of any

member in the group, P

) is denoted the current

position of a feeder agent, and P

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):

k+1

) = P

) +

n−1

∑

i=n−N

n −i + 1

) −P

)).

(14)

The LPCTs group updates its agents’ positions as

follows (Korani et al., 2019):

k+1

) = P

) +

−N

∑

i=n−N

n −i + 1

) −P

)).

(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

(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

,d, K

,Cl,and El

: The population size (AFSs)

: The position of the active food sources

d: The dimension of the problem

: 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

,. ..,P

)

Evaluate the ﬁtness value of T agents (S

.. .S

)

Sort solutions in descending order based on the ﬁtness and store them in S

S = Sort(S

.. .S

)

The sorted positions with the same rank of S stored in array A

A = (P

.. .P

)

loop

for k

= 1 to K

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

.. .S

)

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

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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