# Optimal Stochastic Resource Allocation for Distributed Quantum Computing

Napat Ngoenriang1, Minrui Xu2, Sucha Supittayapornpong1, Dusit Niyato2, Han
Yu2, Xuemin (Sherman) Shen3
1School of Information Science and Technology, Vidyasirimedhi Institute of
Science and Technology, Thailand
2School of Computer Science and Engineering, Nanyang Technological University,
Singapore
3Department of Electrical and Computer Engineering, University of Waterloo,
Waterloo, ON, Canada

###### Abstract

With the advent of interconnected quantum computers, i.e., distributed quantum
computing (DQC), multiple quantum computers can now collaborate via quantum
networks to perform massively complex computational tasks. However, DQC faces
problems sharing quantum information because it cannot be cloned or duplicated
between quantum computers. Thanks to advanced quantum mechanics, quantum
computers can teleport quantum information across quantum networks. However,
challenges to utilizing efficiently quantum resources, e.g., quantum computers
and quantum channels, arise in DQC due to their capabilities and properties,
such as uncertain qubit fidelity and quantum channel noise. In this paper, we
propose a resource allocation scheme for DQC based on stochastic programming
to minimize the total deployment cost for quantum resources. Essentially, the
two-stage stochastic programming model is formulated to handle the uncertainty
of quantum computing demands, computing power, and fidelity in quantum
networks. The performance evaluation demonstrates the effectiveness and
ability of the proposed scheme to balance the utilization of quantum computers
and on-demand quantum computers while minimizing the overall cost of
provisioning under uncertainty.

###### Index Terms:

Distributed Quantum Computing, Quantum Networks, Resource Allocation,
Stochastic Programming

##  I Introduction

To date, quantum computing is tremendously fast and efficient, being able to
do computations in a matter of seconds which would take decades for older
supercomputers [1]. In 2019, a prototype of Google’s quantum computer was able
to finish the computation and demonstrate the effectiveness of quantum
mechanics [2]. Breakthroughs in quantum computing are essential and influence
various applications, such as artificial intelligence (AI), molecular
modelling, weather forecasting, and drug development [3]. Since the
development of quantum computers is still in its infancy, distributed quantum
computing (DQC) has emerged in significance to solve more complex
computational tasks. In the next quantum computing era, IBM [4] and Google [5]
aim to introduce a practical DQC, which is anticipated in 2025. Although the
advent of DQC has also advanced processing speed for a variety of
heterogeneous tasks, quantum resources are still limited and need to be
managed effectively while performing computations.

The basis for the development of quantum computing is formed based on
properties of quantum mechanics, i.e., superposition, entanglement, and
interference, as shown in Fig. 1. Superposition permits encoding in a mixture
of two states (i.e., qubits). A qubit offers more information storage and
computation choices than binary bits in classical computers. In quantum
mechanics, entanglement describes a correlation between two qubits in which
the values of one qubit may depend on another. To observe the values of
qubits, measurement, often referred to as interference, can be used to
interrupt the processing of quantum computers.

Quantum algorithms enable qubits in quantum computers by utilizing quantum
mechanics. For instance, Shor’s [6] and Grover’s [7] algorithms were developed
to deal with factorization and the search for unstructured data, respectively,
which are highly challenging for classical computers. To tackle these tasks
with quantum computers, at least the order of $10^{6}$ physical qubits are
required [6]. However, a recently developed quantum computer can only contain
tens of qubits [8]. Therefore, it becomes increasingly challenging to handle
and control information in quantum computers due to the limited number of
qubits, the instability of qubits, and the amount of information required for
complex computational tasks. As a result, the concept of DQC has been
presented.

In DQC, quantum teleportation, or the transfer of qubits, is required for
quantum computers to connect and collaborate. In this regard, multiple quantum
computers can work collaboratively to compute a large-scale complex
computational task at quadratic or exponential speed-up. Moreover, most of
common quantum algorithms can benefit from their distributed equivalents. For
example, distributed Shor’s algorithm can reduce computational complexity
compared to original Shor’s algorithm [9]. Additionally, distributed Grover’s
algorithm has a considerably lower query time than Grover’s algorithm [10].
Therefore, distributed quantum algorithms can enhance practical feasibility of
quantum computers in tackling complex computational tasks in practice.

Figure 1: The illustration of system model of distributed quantum computing.

Although DQC and distributed quantum algorithms have evolved and accelerated
to handle complex computational tasks, quantum resources, e.g., quantum
computers and quantum channels, must be optimally allocated to perform
heterogeneous computational tasks. However, the efficient utilization of
quantum resources in DQC is still facing challenges. First, the utilization of
quantum resources depends on the demands of computational tasks, which are not
known precisely at the time of quantum computer deployment. Second, quantum
computers’ availability and computing power may or may not be available to
compute computational tasks. Third, fidelity degradation occurs when
performing quantum teleportation and transferring qubits over quantum
networks, which degrades the efficacy of the entangled qubits. Due to the
uncertainty, allocating quantum resources in DQC may result in both under and
over-deployment of quantum computers. In addition, because of the limitations
of quantum resources, quantum computers can be purchased or outsourced from
other organizations such as Amazon Braket [11] to complete complex
computational tasks. However, the cost of on-demand quantum computers is
relatively more expensive.

To address the aforementioned challenges, in this paper, we propose a resource
allocation scheme based on stochastic programming to minimize the total
deployment cost to compute computational tasks in DQC. The main contributions
of this paper can be summarized as follows:

  * •

We propose an optimal resource allocation scheme to compute computational
tasks in DQC. In particular, the optimal resource allocation are jointly
obtained under the uncertainties of future demands of computational tasks and
instability of quantum characteristics.

  * •

We conduct extensive experiments to demonstrate the importance and
effectiveness of the proposed optimal resource allocation scheme, which
achieves the lowest total deployment cost.

##  II Related Work

Quantum computing was developed to increase the capability of existing
computational resources [12]. One of challenges in quantum computing is
quantum algorithms implemented on quantum computers, for example, Shor’s [6]
and Grover’s [7] algorithms are requiring a massive number of qubits to
execute. For instance, Shor’s algorithm was developed to handle a
factorization problem with around $10^{6}$ qubits, which is too complex for
classical computers [6]. In addition, Grover’s algorithm was presented to
search unordered data by encoding inputs with dimension $N$ as superposition
with $\sqrt{N}$ qubits, which gives a quadratic computing speed-up [7] to
search operations in non-structure databases. These quantum algorithms can be
adopted to outperform existing conventional algorithms with quadratic or
exponential speed-up computations [13].

However, due to challenges in scaling up the number of qubits, quantum
computers are not yet ready to replace traditional computers [14]. Therefore,
DQC was introduced to combine several quantum computers to handle more complex
computational tasks [14, 15]. Most existing works look at distributed quantum
mechanics and algorithms to be the basis in DQC [14, 15]. Similar to quantum
algorithms, distributed Shor’s [9] and Grover’s [10] algorithms were widely
used. The distributed Shor’s algorithm has a computing complexity of
$O((logN)^{2})$, which outperforms the original Shor’s algorithm, i.e.,
$O((logN)^{2}log(logN)log(log(logN)))$, where $N$ is the number to be factored
[9]. Grover’s algorithm incurs expensive query time due to its use of
superpositions as inputs. According to [9], the distributed Grover’s algorithm
was devised to improve the original Grover’s algorithm by reducing the query
times from $\frac{\pi}{4}\sqrt{2^{n}}$ to $\frac{\pi}{4}\sqrt{2^{n-k}}$, where
$n$ is the number of input bits of the original problem, whereas $n-k$ is the
number of input bits for the decomposed subfunctions.

Next, we discuss resource allocation problems in distributed computing,
quantum computing, and DQC, which is the motivation for our work, as follows:

  * 1)

Resource Allocation in Distributed Computing: To enhance the performance of
distributed computing, the authors [16] reviewed existing resource allocation
schemes under dynamic environments in distributed computing with various types
of classical resources, e.g., computing, power, and storage resources. In
particular, the authors in [17] formulated the problem of resource allocation
in cloud computing as a stochastic programming model by considering the
uncertainty of user requirements.

  * 2)

Resource Allocation in Quantum Computing: The authors in [18, 19] proposed and
analyzed the significance of resource allocation problems and adaptive
resource allocation problems, respectively, in quantum cloud computing, using
quantum and cloud characteristics such as execution times, cloud query times,
and circuit compilation times.

  * 3)

Resource Allocation in Distributed Quantum Computing: In [20], the authors
proposed network flow optimization for DQC. The authors used a weighted round-
robin algorithm to pre-compute traffic flows for all the possible paths for
each application and then allocate resources to the applications in the round-
robin, where the maximum net rate of the application is a ratio of the round
size proportional to its weight.

However, all the existing studies overlook the issue of quantum resource
allocation, e.g., quantum computers and channels, in addition to the
uncertainty of quantum computing demands, computational power, and fidelity in
DQC, which directly affects the use of quantum resources in DQC.

##  III System Model

###  III-A System Overview

We consider the system model of a quantum computer operator that provisions a
quantum computing task by using quantum computers as shown in Fig. 1. There
are two options for the quantum computer operator to compute computational
tasks on DQC, i.e., using the deployed quantum computers or using on-demand
quantum computers from other organizations such as Amazon Braket [11]. Let
$\mathcal{J}=\\{1,\dots,j,\dots,J\\}$ be a set of quantum computers. The
number of quantum bits, or qubits (a quantum computation unit), is required to
complete the computational task, denoted by $n$. In addition, $n$ qubits mean
that $n$ classical bits can represent up to $2^{n}$ different possibilities
[12]. In detail, we consider that quantum computer $j$ owns $k_{j}$ qubits.
Alternatively, on-demand quantum computers can be purchased from other
organizations and then connected across the deployed quantum computers to
compute computational tasks. Let $\mathcal{R}=\\{1,\dots,r,\dots,R\\}$
represent a set of on-demand quantum computers.

Figure 2: A procedure of distributed quantum computing of two quantum
computers with quantum teleportation.

###  III-B Quantum Teleportation Across Quantum Networks

Due to the no-cloning theorem in quantum mechanics, qubits cannot be
duplicated or cloned across quantum computers [14]. The concept of quantum
teleportation enables two quantum computers to exchange qubits, as shown in
Fig. 2. The quantum teleportation transfers a pair of qubits executing from
the source to the destination quantum computers to local operations and then
measures the qubits previous to the qubits’ decoherence (i.e., loss of
information in qubits due to the instability of quantum characteristics).
Consequently, multiple quantum computers can connect and collaborate to
complete computational tasks. The quantum computers interconnected via links
are directional with a fixed capacity, indicated by $C_{i,j}>0,$ where $i$ and
$j$ are two connected quantum computers. However, errors may occur while
preserving the entangled qubits between quantum computers. In particular, the
fidelity of the shared Bell pair, indicated by $q_{i,j}$, is a performance
that measures the effectiveness of the entanglement between the desired and
actual states of the quantum teleportation [21]. The range of fidelity is
$[0,1]$, where $1$ refers to the best performance that the maximally entangled
qubits can be reached [20].

###  III-C Uncertainty

We classified the uncertainty in DQC into three types, i.e., i) the demand of
the computational task, ii) the availability and computing power of the
quantum computers, and iii) the fidelity of the entangled qubits. Let
$\Omega=\\{\omega_{1},\dots,\omega_{|\Omega|}\\}$ denote the set of scenarios
that describes the demand of computational tasks, the computing power of
quantum computers, and the fidelity of the Bell pairs, where $|\Omega|$ is the
number of total scenarios. Let $\pi(\omega)$ be the probability that the
scenario $\omega\in\Omega$ is realized, where $\pi(\omega)$ can be calculated
based on historical data [17].

  * i)

The actual demands of computational tasks are unknown at the time of deploying
the quantum computer. Different computational tasks such as minimization,
material science, and machine learning problems may require various qubits
[22]. For example, 58 qubits are required in material science problems [23].
Let $\tilde{n}(\omega)$ be an integer parameter indicating the demand of
computational tasks.

  * ii)

The precise availability and computing power of the quantum computer are
unknown since it could be set aside for other applications or because its
backend might not support all of them [24]. Let $\tilde{k}_{j}(\omega)$ be an
integer parameter indicating the computing power of the quantum computer $j$
in qubits.

  * iii)

Specifically, the fidelity of the shared entangled qubits, also known as the
Bell pair, in DQC is also not known exactly [21]. Let
$\tilde{q}_{i,j}(\omega)$ denote the fidelity of the Bell pair between quantum
computers $i$ and $j$.

For example, $\tilde{n}(\omega)=1,\tilde{k}_{1}(\omega)=2,$ and
$\tilde{q}_{1,2}(\omega)=0.5$ mean that the demand of the computational task
is 1 qubit (or $2^{1}$ in bit), the computing power of the quantum computer
$1$ is 2 qubits, and the fidelity of the Bell pair for quantum teleportation
between the quantum computers $1$ and $2$ is 0.5.

###  III-D Cost

One of the main obstacles to the practical deployment of DQC is the increased
costs of using quantum computers, requiring computing power, and using the
Bell pair for the entangled qubits [25]. Four associated costs of quantum
computing are described. $c^{(dep)}_{j}$ represents the deployment cost of the
quantum computer $j$. $c^{(com)}_{j}$ represents the unit cost of the
computing power of the quantum computer $j$ in qubits. $c^{(Bell)}_{i,j}$
represents the cost of the Bell pair for the shared entangled qubits of the
quantum computers $i$ and $j$. $c^{(ond)}_{r}$ represents the additional cost
associated with the deployment of an on-demand quantum computer $r$.

##  IV Problem Formulation

This section presents the deterministic integer programming and stochastic
integer programming formulations to minimize the total deployment cost for the
quantum computer operator.

###  IV-A Deterministic Integer Programming

If the demand of the computational task, computing power of quantum computers,
and fidelity of the Bell pairs are completely known, quantum computers can
certainly be deployed. Therefore, an on-demand quantum computer is not
necessary. This problem can be formulated as deterministic integer non-linear
programming [26]. A decision variable is listed below.

  * •

$x_{j}$ indicates whether the quantum computer $j$ is reserved to compute the
computational task.

The deterministic formulation is defined as follows:

$\displaystyle\operatorname*{\min_{x_{j}}:}\sum_{j=1}^{J}c^{(dep)}_{j}x_{j}+\sum_{j=1}^{J}c^{(com)}_{j}x_{j}+\sum_{j=1}^{J}c^{(Bell)}_{i,j}x_{i}x_{j},$
(1)

subject to

$\displaystyle\sum_{j=1}^{J}k_{j}x_{j}\geq 2^{n},$ (2) $\displaystyle
C_{i,j}q_{i,j}\geq\min\\{x_{i}k_{i},x_{j}k_{j}\\},$ $\displaystyle\forall
i,j\in\mathcal{J},i\neq j,$ (3) $\displaystyle x_{j}\in\\{0,1\\},$
$\displaystyle\forall j\in\mathcal{J}.$ (4)

The objective in (1) is to minimize the total deployment cost for the quantum
computer operator that depends on the deployment of the quantum computers. The
constraint in (2) enforces that the computing power of the used quantum
computers must meet the demand of the computational task in $n$ qubits
indicating $2^{n}$ bits. The constraint in (3) ensures that the entanglement
of qubits can be successful if the computing power of the used quantum
computers does not exceed the link capacity [20]. The constraint in (4)
indicates that $x_{j}$ is a binary variable.

###  IV-B Stochastic Integer Programming

In reality, the demands of the computational task, computing power, and
fidelity of the Bell pairs cannot be predicted at the time of quantum computer
deployment. Thus, deterministic integer programming is unsuitable. Therefore,
we formulate a two-stage stochastic non-linear integer programming [26]. The
deployment of quantum computers is defined in the first stage, and their
utilization and on-demand quantum computers are defined in the second stage.
The first-stage and second-stage decisions are made before and after observing
the actual demand, computing power of the quantum computers, and the fidelity
of the Bell pairs, respectively. Decision variables are listed as follows:

  * •

$\tilde{x}_{j}(\omega)$ indicates whether the deployed quantum computer $j$ is
used to compute the computational task.

  * •

$y_{r}(\omega)$ indicates whether the on-demand quantum computer $r$ is
deployed to compute the computational task.

The stochastic formulation is defined as follows:

$\displaystyle\operatorname*{\min_{x_{j},\tilde{x}_{j}(\omega),y_{r}(\omega)}:}\sum_{j=1}^{J}c^{(dep)}_{j}x_{j}+\mathbb{E}[\mathcal{Q}(\tilde{x}_{j},y_{r},\omega)],$
(5) $\displaystyle\mathbb{E}$
$\displaystyle[\mathcal{Q}(y_{r},\omega)]=\sum_{\omega\in\Omega}\pi(\omega)\bigg{(}\sum_{j=1}^{J}c^{(com)}_{j}\tilde{x}_{j}(\omega)$
$\displaystyle+\sum_{j=1}^{J}\sum_{i=1}^{J\backslash\\{j\\}}c^{(Bell)}_{i,j}\tilde{x}_{i}(\omega)\tilde{x}_{j}(\omega)+\sum_{r=1}^{R}c^{(ond)}_{r}y_{r}(\omega)\bigg{)},$
(6)

subject to

$\displaystyle\tilde{x}_{j}(\omega)\leq x_{j},$ $\displaystyle\forall
j\in\mathcal{J},\forall\omega\in\Omega,$ (7)
$\displaystyle\sum_{j=1}^{J}\tilde{k}_{j}(\omega)\tilde{x}_{j}(\omega)+\sum_{r=1}^{R}g_{r}y_{r}(\omega)\geq
2^{\tilde{n}(\omega)},$ $\displaystyle\forall\omega\in\Omega,$ (8)
$\displaystyle
C_{i,j}\tilde{q}_{i,j}(\omega)\geq\min\\{\tilde{k}_{i}(\omega)\tilde{x}_{i}(\omega),\tilde{k}_{j}(\omega)\tilde{x}_{j}(\omega)\\},$
$\displaystyle\forall i,j\in\mathcal{J},i\neq j,\forall\omega\in\Omega,$ (9)
$\displaystyle\tilde{x}_{j}(\omega),y_{j}(\omega)\in\\{0,1\\},$
$\displaystyle\forall j\in\mathcal{J},\forall\omega\in\Omega.$ (10)

The objective in (6) is to minimize the total deployment cost for the quantum
computer operator under uncertainty. The constraint in (7) ensures that the
utilized quantum computer for computing the computational task does not exceed
the selected quantum computer. The constraint in (8) enforces that the
computing power of both the utilized and on-demand quantum computers can
complete the computational task requiring $n$ qubits, where $g_{r}$ indicates
the computing power of the on-demand quantum computer. The constraint in (9)
ensures that the computing power of the quantum computers does not exceed the
capability of the link capacity. The constraint in (10) indicates that
$\tilde{x}_{j}(\omega)$ and $y_{j}(\omega)$ are a binary variable.

In addition, the stochastic non-linear programming can be reformulated as a
deterministic non-linear equivalent formulation [26], i.e., mixed-integer non-
linear programming (MINLP), which can be solved by conventional optimization
solver software using synergies with mixed-integer programming and non-linear
programming methods [27].

##  V Performance Evaluation

Figure 3: Cost breakdown under different demands of the computational task.

Figure 4: Cost structure under different computing power of the quantum
computer.

Figure 5: Cost breakdown under different fidelity of the Bell pairs.

Figure 6: Cost breakdown under different on-demand QC costs.

Figure 7: Cost breakdown under different probabilities.

Figure 8: Cost comparison among the proposed, EVF, and random models.

###  V-A Parameter Setting

We consider the system model of DQC, where the quantum computer operator
consists of 10 quantum computers. We set the cost values, measured in
normalized monetary, for the deployment of quantum computers, computing power,
and Bell pairs for the shared entangled qubits to be 5000, 1000, and 450,
respectively, based on [25]. All Bell pairs among quantum computers have
identical costs and qubit capacities, i.e., 257 qubits. The cost and computing
power of on-demand quantum computers are $25000$ and $127$, respectively. In
the stochastic model, we consider two scenarios, i.e., $|\Omega|=2.$ The first
scenario is $\omega_{1}$ in which the demand of the computational task is 10,
the computing power of the quantum computers is 127 qubits, and the fidelity
of the Bell pairs is 1 (i.e., the best performance). The second scenario is
$\omega_{2}$ in which there is no demand for the computational task, all
quantum computers have no available computing power, and the fidelity of the
Bell pairs is 0 (i.e., the worst performance). We assume the default
probability values with $\pi(\omega_{1})$ = 0.8 and $\pi(\omega_{2})=0.2$.

We conduct experiments via GAMS script, which can be solved by MINLP solver
[27]. Some parameters are varied in different experiments.

###  V-B Impact of Demands, Computing Power, and Fidelity

In these experiments, we consider three important factors: i) the demand of
the computational task, ii) the computing power of quantum computers, and iii)
the fidelity of the Bell pairs across quantum computers.

  * i)

We vary the demand of the computational task from 6 to 11 qubits. Fig. 5
demonstrates the cost breakdown. The quantity of deployed quantum computers
also increases as the demand rises. Moreover, when the demand for
computational tasks equals 11, on-demand quantum computers must be deployed to
compute the computational task, as it exceeds the computing power of the
currently deployed quantum computers.

  * ii)

We vary the computing power (i.e., the number of qubits) of quantum computers.
For ease of presentation, all quantum computers are identical. The cost
structure, i.e., the first-stage, second-stage, and total deployment costs, is
shown in Figure 5. As the computing power increases, we see that total
deployment costs are falling. To achieve the lowest total cost, there is a
trade-off between the first-stage cost to use the deployed quantum computers
and the second-stag cost to use on-demand quantum computers.

  * iii)

We vary the fidelity of the entangled qubits in DQC from 0 to 1. To simplify
the evaluation, all the Bell pairs are identical. The results are shown in
Fig. 5. When the fidelity of the Bell pairs is less than 0.5, on-demand
quantum computer deployments are necessary as the entanglement of the qubits
cannot be achieved. The computational tasks can be computed completely when
the fidelity of the entangled qubits is equal to or greater than 0.5.

###  V-C Impact of Cost of On-Demand Quantum Computers

We observe the total deployment cost of the proposed scheme by varying the
cost of on-demand quantum computers. The cost breakdown is shown in Fig. 8.
When the cost of an on-demand quantum computer equals to, or less than 2x,
both the deployed quantum computers and on-demand quantum computers are used.
The cost of using the deployed quantum computer is lower than those of using
on-demand quantum computers when the on-demand quantum computer is more
expensive, i.e., higher than 2x. Moreover, the proposed scheme can adapt the
utilization of quantum computers and on-demand quantum computers to minimize
the total deployment cost despite the variations in on-demand quantum computer
costs.

###  V-D Impact of Probability of Scenarios

We vary the probability of the scenario $\omega_{1}$, which corresponds to
having the demand of the computational task, the computing power, and the
positive performance of the entangled qubits. The cost breakdown is shown in
Fig. 8. We observe that when the probability of the scenario is equal to or
less than 0.2, the on-demand quantum computer is required due to a small
amount of demand. The proposed scheme suggests using the deployed quantum
computers instead of on-demand quantum computers when the probability of the
demand, the computing power, and the fidelity increase.

###  V-E Cost Comparison

We compare the total deployment cost of the proposed stochastic model with the
random model and the Expected Value Formulation (EVF) model. The EVF model
solves the deterministic model with using the average values of the uncertain
parameters. The cost of deploying the on-demand quantum computer varies. The
deployed quantum computer in the first stage is chosen at random in the random
model. Figure 8 compares the total costs of the three models. The proposed
model, as we can see, delivers the lowest total cost. Additionally, the EVF
and random models cannot adapt to the variations in costs of the on-demand
quantum computers, while the proposed scheme can always achieve the most cost-
effective solution.

##  VI Conclusion

In this paper, we have proposed an optimal resource allocation scheme to
efficiently utilize quantum resources in DQC to compute complex computational
tasks at scale. We have formulated a two-stage stochastic programming to
minimize the total deployment cost under uncertainties of the demands of
computational tasks, the computing power of quantum computers, and the quantum
characteristics over quantum networks. The experimental findings demonstrate
that the proposed scheme can minimize the total deployment cost under
uncertainty. Other methods, such as dynamic resource allocation, and other
factors, such as the deployment of quantum computers across a long distance
and the uncertainty of the time required to accomplish computing tasks, can be
considered for future research.

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