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computational tasks. Let R = {1,...,r,...,R} represent a (or 21 in bit), the computing power of the quantum computer
set of on-demand quantum computers. 1 is 2 qubits, and the fidelity of the Bell pair for quantum
teleportation between the quantum computers 1 and 2 is 0.5.
B. Quantum Teleportation Across Quantum Networks
D. Cost
Due to the no-cloning theorem in quantum mechanics,
qubits cannot be duplicated or cloned across quantum com- One of the main obstacles to the practical deployment of
puters[14].Theconceptofquantumteleportationenablestwo DQC is the increased costs of using quantum computers,
requiring computing power, and using the Bell pair for the are defined in the second stage. The first-stage and second-
entangled qubits [25]. Four associated costs of quantum com- stagedecisionsaremadebeforeandafterobservingtheactual
puting are described. c(dep) represents the deployment cost of demand,computingpowerofthequantumcomputers,andthe
j
thequantumcomputerj.c(com) representstheunitcostofthe fidelity of the Bell pairs, respectively. Decision variables are
j
computingpowerofthequantumcomputerj inqubits.c(Bell) listed as follows:
i,j
represents the cost of the Bell pair for the shared entangled • x˜ j(ω) indicates whether the deployed quantum computer
qubits of the quantum computers i and j. c(ond) represents j is used to compute the computational task.
r
the additional cost associated with the deployment of an on- • y r(ω) indicates whether the on-demand quantum com-
puter r is deployed to compute the computational task.
demand quantum computer r.
The stochastic formulation is defined as follows:
IV. PROBLEMFORMULATION
J
Thissectionpresentsthedeterministicintegerprogramming min :(cid:88) c( jdep)x j +E[Q(x˜ j,y r,ω)], (5)
xj,x˜j(ω),yr(ω)
and stochastic integer programming formulations to minimize j=1
the total deployment cost for the quantum computer operator.
(cid:18) J
A. Deterministic Integer Programming E[Q(y ,ω)]= (cid:88) π(ω) (cid:88) c(com)x˜ (ω)
r j j
If the demand of the computational task, computing power ω∈Ω j=1
of quantum computers, and fidelity of the Bell pairs are com- J J\{j} R (cid:19)
pletely known, quantum computers can certainly be deployed. +(cid:88) (cid:88) c( i,B jell)x˜ i(ω)x˜ j(ω)+(cid:88) c( rond)y r(ω) , (6)
Therefore, an on-demand quantum computer is not necessary. j=1 i=1 r=1
This problem can be formulated as deterministic integer non- subject to
linear programming [26]. A decision variable is listed below.
x˜ (ω)≤x , ∀j ∈J,∀ω ∈Ω, (7)
• x j indicates whether thequantum computer j is reserved j j
to compute the computational task. (cid:88)J (ω)+(cid:88)R
k˜ (ω)x˜ g y (ω)≥2n˜(ω), ∀ω ∈Ω, (8)
j j r r
The deterministic formulation is defined as follows:
j=1 r=1
min:(cid:88)J +(cid:88)J +(cid:88)J C i,jq˜ i,j(ω)≥min{k˜ i(ω)x˜ i(ω),k˜ j(ω)x˜ j(ω)},
c(dep)x c(com)x c(Bell)x x , (1)
j j j j i,j i j
xj ∀i,j ∈J,i(cid:54)=j,∀ω ∈Ω, (9)
j=1 j=1 j=1
x˜ (ω),y (ω)∈{0,1}, ∀j ∈J,∀ω ∈Ω. (10)
subject to j j
The objective in (6) is to minimize the total deployment
J
(cid:88)
k x ≥2n, (2) costforthequantumcomputeroperatorunderuncertainty.The
j j
constraintin(7)ensuresthattheutilizedquantumcomputerfor
j=1
computingthecomputationaltaskdoesnotexceedtheselected
C q ≥min{x k ,x k }, ∀i,j ∈J,i(cid:54)=j, (3)
i,j i,j i i j j
quantum computer. The constraint in (8) enforces that the
x ∈{0,1}, ∀j ∈J. (4)
j computingpowerofboththeutilizedandon-demandquantum
The objective in (1) is to minimize the total deployment computers can complete the computational task requiring n
cost for the quantum computer operator that depends on qubits, where g r indicates the computing power of the on-
the deployment of the quantum computers. The constraint in demand quantum computer. The constraint in (9) ensures that
(2) enforces that the computing power of the used quantum the computing power of the quantum computers does not
computersmustmeetthedemandofthecomputationaltaskin exceed the capability of the link capacity. The constraint in
n qubits indicating 2n bits. The constraint in (3) ensures that (10) indicates that x˜ j(ω) and y j(ω) are a binary variable.
the entanglement of qubits can be successful if the computing In addition, the stochastic non-linear programming can
power of the used quantum computers does not exceed the be reformulated as a deterministic non-linear equivalent for-
link capacity [20]. The constraint in (4) indicates that x is a mulation [26], i.e., mixed-integer non-linear programming
j
binary variable. (MINLP), which can be solved by conventional optimization
solver software using synergies with mixed-integer program-
B. Stochastic Integer Programming ming and non-linear programming methods [27].
Inreality,thedemandsofthecomputationaltask,computing
V. PERFORMANCEEVALUATION
power,andfidelityoftheBellpairscannotbepredictedatthe
A. Parameter Setting
time of quantum computer deployment. Thus, deterministic
integer programming is unsuitable. Therefore, we formulate WeconsiderthesystemmodelofDQC,wherethequantum
a two-stage stochastic non-linear integer programming [26]. computer operator consists of 10 quantum computers. We set
The deployment of quantum computers is defined in the first the cost values, measured in normalized monetary, for the
stage,andtheirutilizationandon-demandquantumcomputers deploymentofquantumcomputers,computingpower,andBell
1e5 1e5 1e5
1.2
Total Cost Total Cost Total Cost
2.0 D Coe mp. p Q . C C oC so tst First-Stage Cost 2.0 D Coe mp. p Q . C C oC so tst
Comm. Cost 1.0 Second-Stage Cost Comm. Cost
tsoC 1.5 On-demand QC Cost tsoC tsoC 1.5 On-demand QC Cost
0.8
latoTtsoC latoTtsoC latoTtsoC
1.0 1.0
0.6
0.5 0.5
0.4
0.0 0.0