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bits of the original problem, whereas n−k is the number of
and instability of quantum characteristics.
input bits for the decomposed subfunctions.
• We conduct extensive experiments to demonstrate the
Next,wediscussresourceallocationproblemsindistributed
importance and effectiveness of the proposed optimal
computing, quantum computing, and DQC, which is the mo-
resource allocation scheme, which achieves the lowest
tivation for our work, as follows:
total deployment cost.
1) Resource Allocation in Distributed Computing: To en-
hance the performance of distributed computing, the au-
II. RELATEDWORK
thors [16] reviewed existing resource allocation schemes
Quantumcomputingwasdevelopedtoincreasethecapabil- under dynamic environments in distributed computing
ityofexistingcomputationalresources[12].Oneofchallenges withvarioustypesofclassicalresources,e.g.,computing,
inquantumcomputingisquantumalgorithmsimplementedon power, and storage resources. In particular, the authors
quantum computers, for example, Shor’s [6] and Grover’s [7] in [17] formulated the problem of resource allocation in
algorithms are requiring a massive number of qubits to exe- cloud computing as a stochastic programming model by
cute. For instance, Shor’s algorithm was developed to handle considering the uncertainty of user requirements.
a factorization problem with around 106 qubits, which is too 2) ResourceAllocationinQuantumComputing:Theauthors
complex for classical computers [6]. In addition, Grover’s in[18],[19]proposedandanalyzedthesignificanceofre-
algorithmwaspresentedtosearchunordereddatabyencoding source allocation problems and adaptive resource alloca-
√
inputs with dimension N as superposition with N qubits, tionproblems,respectively,inquantumcloudcomputing,
quantum computers to exchange qubits, as shown in Fig. 2.
Oracle diffuser
operator The quantum teleportation transfers a pair of qubits executing
Quantum Computer 1 Quantum gates
fromthesourcetothedestinationquantumcomputerstolocal
|0⟩
operationsandthenmeasuresthequbitsprevioustothequbits’
|0⟩ decoherence (i.e., loss of information in qubits due to the
instability of quantum characteristics). Consequently, multiple
Quantum teleportation
|0000⟩ Quantum Computer 2 quantum computers can connect and collaborate to complete
|0⟩ computational tasks. The quantum computers interconnected
via links are directional with a fixed capacity, indicated by
|0⟩
C >0, where i and j are two connected quantum comput-
i,j
Measurement ers.However,errorsmayoccurwhilepreservingtheentangled
qubits between quantum computers. In particular, the fidelity
Fig. 2: A procedure of distributed quantum computing of two
of the shared Bell pair, indicated by q , is a performance
quantum computers with quantum teleportation. i,j
that measures the effectiveness of the entanglement between
thedesiredandactualstatesofthequantumteleportation[21].
The range of fidelity is [0,1], where 1 refers to the best
usingquantumandcloudcharacteristicssuchasexecution
performance that the maximally entangled qubits can be
times, cloud query times, and circuit compilation times.
reached [20].
3) Resource Allocation in Distributed Quantum Computing:
In [20], the authors proposed network flow optimization
C. Uncertainty
for DQC. The authors used a weighted round-robin
We classified the uncertainty in DQC into three types, i.e.,
algorithmtopre-computetrafficflowsforallthepossible
i) the demand of the computational task, ii) the availability
paths for each application and then allocate resources to
and computing power of the quantum computers, and iii) the
the applications in the round-robin, where the maximum
fidelityoftheentangledqubits.LetΩ={ω ,...,ω }denote
net rate of the application is a ratio of the round size 1 |Ω|
thesetofscenariosthatdescribesthedemandofcomputational
proportional to its weight.
tasks, the computing power of quantum computers, and the
However, all the existing studies overlook the issue of
fidelity of the Bell pairs, where |Ω| is the number of total
quantum resource allocation, e.g., quantum computers and
scenarios.Letπ(ω)betheprobabilitythatthescenarioω ∈Ω
channels,inadditiontotheuncertaintyofquantumcomputing
is realized, where π(ω) can be calculated based on historical
demands, computational power, and fidelity in DQC, which
data [17].
directly affects the use of quantum resources in DQC.
i) The actual demands of computational tasks are unknown
III. SYSTEMMODEL at the time of deploying the quantum computer. Differ-
ent computational tasks such as minimization, material
A. System Overview
science, and machine learning problems may require
We consider the system model of a quantum computer various qubits [22]. For example, 58 qubits are required
operator that provisions a quantum computing task by using inmaterialscienceproblems[23].Letn˜(ω)beaninteger
quantum computers as shown in Fig.1. Thereare twooptions parameter indicating the demand of computational tasks.
for the quantum computer operator to compute computational ii) The precise availability and computing power of the
tasks on DQC, i.e., using the deployed quantum computers or quantum computer are unknown since it could be set
usingon-demandquantumcomputersfromotherorganizations aside for other applications or because its backend might
such as Amazon Braket [11]. Let J = {1,...,j,...,J} not support all of them [24]. Let k˜ (ω) be an integer
j
be a set of quantum computers. The number of quantum parameterindicatingthecomputingpowerofthequantum
bits, or qubits (a quantum computation unit), is required to computer j in qubits.
complete the computational task, denoted by n. In addition, iii) Specifically, the fidelity of the shared entangled qubits,
n qubits mean that n classical bits can represent up to 2n also known as the Bell pair, in DQC is also not known
differentpossibilities[12].Indetail,weconsiderthatquantum exactly [21]. Let q˜ (ω) denote the fidelity of the Bell
i,j
computerj ownsk j qubits.Alternatively,on-demandquantum pair between quantum computers i and j.
computerscanbepurchasedfromotherorganizationsandthen For example, n˜(ω) = 1,k˜ (ω) = 2, and q˜ (ω) = 0.5
1 1,2
connectedacrossthedeployedquantumcomputerstocompute mean that the demand of the computational task is 1 qubit