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tation, a pair of qubits from the source quantum computer quality of matter qubits is a trade-off. Therefore, the quantum
to the destination quantum computer is transferred for local architectures, including quantum algorithms, need to be well-
operations and measurements before decoherence, the loss of designedandwell-resolvedtoachievethecommunicationand
the shared entangled qubits. However, errors and failures in- local operations for generating matter qubits.
curred in quantum teleportation preserve the entangled qubits
C. QuantumAlgorithmsandDistributedQuantumAlgorithms
of quantum computers. To guarantee the Bell state of the
shared pair of qubits, an entanglement distillation procedure The quantum algorithms to be executed in quantum com-
is developed to iteratively increase fidelity or the probability puterscanbeofseveraltypesdependingonthecomputational
that the qubit will be transferred without changing its state. paradigm. The key issues on quantum algorithms and dis-
However, quantum teleportation requires a lot of iterations, tributed quantum algorithms are as follows.
time,andresourcestoachievetheBellstateofapairofqubits. 1) QuantumAlgorithm: Numerousstudieshavefocusedon
Additionally, the qubits of the shared Bell pairs are called quantumalgorithmsoperatingatexponentiallyfasterspeedson
entangled-qubits (or eqbits), which are stored in quantum quantumcomputersimplementedonquantumcircuits[12].We
memories. discuss the most prevalent quantum algorithms as follows:
2) Distributed Quantum Circuits: Required quantum cir- • Shor’s algorithm was devised to factor prime numbers in
cuitscanbesplitanddistributedamongmultiplequantumpro- polynomial time, which cannot be done polynomially by
cessors of quantum computers, with each quantum computer classical algorithms. The factoring problem is suggested
executing a fragment of quantum circuits. The architecture to be reformulated as the finding-period problem and
of the quantum circuits executes non-local quantum gates solved through quantum phase estimation, which esti-
for the shared entangled qubits. A logically identical set of mates phases or eigenvalues of eigenvectors of unitary
instructions is used to coordinate the additional operations operators.
required for the non-local operation to replace the multiple • Grover’salgorithmsolvesasearchingproblemforanun-
qubit operations. In particular, a partitioning of quantum structured database, which provides quadratically speed-
circuits to achieve the least number of qubits among quantum up computation. An equal superposition of all possible
processors must be determined remotely by exchanging non- solutions is employed as inputs that result in the same
local operations with other quantum gates. amplitudes. The inputs are passed and processed through
3) Distributed Quantum Architectures: Multiple quantum theoraclediffusertoboosttheamplitudesandthenreflect
processing units (QPUs) can be executed for computations thesolutionintermsofthegreatestamountofamplitudes.
in parallel with each other universally to achieve distributed The Grover’s algorithm can be used as a generic algorithm
quantum computing. Qubits are prepared as input data by the to address a variety of problems due to the independence of
the algorithm and the internal structure of lists. As a result,
Oracle diffuser
executing Grover’s algorithm gives many classical problems a operator Quantum gates
quadratic speed-up computation. |0⟩
2) Distributed Quantum Algorithm: Despite the fact that
|0⟩
quantumcomputingissignificantlymoreadvancedthanclassi-
cal computing, only small quantum computers (with a limited |0000⟩ |0⟩
number of qubits) have so far been constructed due to the
noiseanddepthofquantumcircuits.Asanalternativetolarge- |0⟩
scale monolithic designs, a distributed grid of small quan-
Measurement
tum computers has been proposed to help advance quantum
computing. In particular, as distributed quantum computing (a)
necessitates sharing the superposition state among quantum Oracle diffuser
computers, quantum networks through entanglement need to Quantum Computer 1 operator Quantum gates
|0⟩
be considered in the implementation. In addition, swap gates
are implemented in order to teleport qubits between a pair |0⟩
of two quantum computers. We discuss the most prevalent
Quantum teleportation
distributed quantum algorithms as follows: |0000⟩ Quantum Computer 2
|0⟩
• Distributed Shor’s algorithm was implemented to solve
factorization problems by small-capacity quantum com- |0⟩
puters. The architecture of quantum circuits to per- Measurement
form collaboratively is designed to handle quantum
(b)
teleportation and simulate a large capacity quantum
computer [3]. The complexity of distributed Shor’s Fig. 2: A procedure of computing the required four-qubits
algorithm is O((logN)2), while Shor’s algorithm re- quantumtask;(a)onequantumcomputerwithfourqubitsand
quires O(cid:0) (logN)2(loglogN)(logloglogN)(cid:1) , where N (b) two quantum computers with two qubits per each.
denotes the integer to be factored.
• Distributed Grover’s algorithm was also developed to
linksbetweentwointerconnectedquantumcomputersenabling
address the unstructured search problem. According to
quantum teleportation are oriented and have a static capacity.
the expensive query time in Grover’s algorithm, the
However, the number of quantum computers, the number of
distributedGrover’salgorithmcanreducequerytimes[4].
qubitsofeachquantumcomputer,andthecapacityofquantum
Functions that needs to be computed can be divided into
2k subfunctions by the distributed Grover’s algorithm, networks are limited and static. Therefore, the allocation of
these resources needs to provide sufficient quantum deploy-
which then computes one of the usable subfunctions to
ments to support quantum tasks in the distributed quantum
findthesolutiontotheoriginalfunction.Incomparisonto
computing framework.
the original Grover’s algorithm, distributed Grover’s al-
gorithm can significant speed up queries. A procedure of
B. System Model, Decisions, and Costs
Grover’s algorithm solving a four-qubit quantum task for
AsshowninFig.1,thequantumcomputeroperatorchooses
quantum computing and distributed quantum computing
between two alternative stages for allocating resources for
using two quantum computers is shown in Fig. 2.
distributed quantum computing, i.e., utilizing the reserved
Among distributed quantum algorithms, the grid of dis-
quantum computers and deploying on-demand quantum com-
tributed quantum circuits has to be well-designed to make the
puters. To use the reserved quantum computer, the quantum
best utilization of the constrained resources in quantum com-
computeroperatorfollowsinstructionsofthedistributedquan-
puters.Therefore,itbecomesincreasinglyimportanttoaddress
tum computers to complete the quantum task. However, the
resource allocation in distributed quantum computing to solve
reserved quantum computers may not be sufficient to accom-
resource constraints in quantum computing environments.
plishalarge-scalequantumcomputationaltask.Therefore,the