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Vestnik Moskovskogo Universiteta Seriya 3 Fizika Astronomiya

Parallelisation of Quantum Algorithms Using Quantum Teleportation

S. S. Sysoev

Published Jan 1, 2025• 0 citations
Publisher

Abstract

This study proposes a method for parallelising quantum algorithms in the circuit model based on the technique of quantum gate teleportation. The resulting parallel representation of the algorithm has reduced circuit depth and, consequently, a higher probability of correct execution under conditions of increasing quantum state decoherence over time. The reduction in circuit depth is achieved by increasing circuit width, i.e. the number of qubits simultaneously used in the computation.

DOI
10.55959/msu0579-9392.80.2530406

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This study proposes a method for parallelising quantum algorithms in the circuit model based on the technique of quantum gate teleportation. The resulting parallel representation of the algorithm has reduced circuit depth and, consequently, a higher probability of correct execution under conditions of increasing quantum state decoherence over time. The reduction in circuit depth is achieved by increasing circuit width, i.e. the number of qubits simultaneously used in the computation.
Parallel computation enables multiple processors to execute different parts of a task simultaneously, improving processing speed and efficiency. In quantum computing, parallel gate implementation involves executing gates independently in different registers, directly impacting the circuit depth, the number of sequential quantum gate operations, and thus the algorithm execution time. This work examines a method for reducing circuit depth by introducing auxiliary qubits to enable parallel gate execution, potentially enhancing the performance of quantum simulations on near-term quantum devices. We show that any circuit on nnn qubits with depth O(Mn2)O\left(M n^2\right)O(Mn2), where M=M(n)M = M(n)M=M(n) is some function of nnn, can be transformed into a circuit with depth O(log⁡2(M)n2)O\left(\log_2(M) n^2\right)O(log2​(M)n2) operating on O(Mn)O\left(M n\right)O(Mn) qubits. This technique may be particularly useful in noisy environments, where recent findings indicate that only the final O(log⁡n)O\left(\log n\right)O(logn) layers influence the expectation value of observables. It may also optimize Trotterization by exponentially reducing the number of Trotter steps. Additionally, the method may offer advantages for distributed quantum computing, and the intuition of treating quantum states as gates and operators as vectors used in this work may have broader applications in quantum computation.
The term quantum parallelism is commonly used to refer to a property of quantum computations where an algorithm can act simultaneously on a superposition of states. However, this is not the only aspect of parallelism in quantum computing. Analogously to the classical computing model, every algorithm consists of elementary quantum operations and the application of them could be parallelised itself. This kind of parallelism is explored in this thesis in the one way quantum computing (1WQC) and the quantum circuit model. In the quantum circuit model we explore arithmetic circuits and circuit complexity theory. Two new arithmetic circuits for quantum computers are introduced in this work: an adder and a multiply-adder. The latter is especially interesting because its depth (i.e. the number of parallel steps required to finish the computation) is smaller than for any known classical circuit when applied sequentially. From the complexity theoretical perspective we concentrate on the classes QAC0 and QAC0[2], the quantum counterparts of AC0 and AC0[2]. The class AC0 comprises of constant depth circuits with unbounded fan-in AND and OR gates and AC0[2] is obtained when unbounded fan-in parity gates are added to AC0 circuits. We prove that QAC0 circuits with two layers of multi-qubit gates cannot compute parity exactly. This is a step towards proving QAC0 6= QAC0[2], a relation known to hold for AC0 and AC0[2]. In 1WQC, computation is done through measurements on an entangled state called the resource state. Two well known parallelisation methods exist in this model: signal shifting and finding the maximally delayed general flow. The first one uses the measurement calculus formalism to rewrite the dependencies of an existing computation, whereas the second technique exploits the geometry of the resource state to find the optimal ordering of measurements. We prove that the aforementioned methods result in same depth computations when the input and output sizes are equal. Through showing this equivalence we reveal new properties of 1WQC computations and design a new algorithm for the above mentioned parallelisations.
The main focus of this work is to implement a program that can be used to parallelise quantum circuits. This is done by translating the quantum circuits to the Measurement Based Quantum Computing (MBQC) model, applying a number of optimising techniques to the computation in MBQC, and by translating it back to a quantum circuit. As a side result of implementing this method, a new algorithm for creating and optimising an MBQC graph from a quantum circuit is designed. The implemented program is used to investigate quantum circuits known from the literature: the QFT, and a specific type of circuits. We show that our program will parallelise the circuit to logarithmic depth, and increase the depth of QFT by a logarithmic factor. Finally, three new classes of quantum circuits are presented: the Toffoli staircase circuit, the Toffoli + CNOT staircase circuit, and a circuits consisting of quantum gates ∧Z, CNOT , ω , Z(α), and J(−2 ). In the computational experiments we perform, those classes of circuits will benefit from the parallelisation method implemented in this dissertation.
A distributed Shor’s algorithm is used to factorize a semiprime with two quantum computers using quantum teleportation to transport quantum states of qubits in parallel. We propose another distributed Shor’s algorithm with sequential quantum teleportation to improve the existing one. It explores how to employ distributed quantum computing through a quantum entanglement pair of qubits to transport quantum states of qubits sequentially. We combined two quantum computers to perform the quantum phase estimation (QPE) of Shor’s algorithm. This reduced the required number of qubits for a single quantum computer, decreased quantum circuit depth, mitigated the impact of insufficient decoherence time, and accelerated the process of breaking cryptographic systems such as Rivest-Shamir-Adleman (RSA). The designed quantum circuit was executed on the IBM Quantum Lab platform to verify its correctness. We analyzed and compared the required numbers of qubits to run different versions of Shor’s algorithm for individual quantum computers, as well as the numbers of qubits required to break different RSA cryptographic systems.
Quantum computers are superior to existing classical supercomputers in terms of run time for specific tasks, and various studies on quantum computers are in progress. Despite this, there is a significant gap in technology between available technologies and actual quantum computers on the noisy intermediate-scale quantum-scale. Currently, various quantum algorithms have been implemented and verified using quantum simulations. Therefore, the runtime performance of a quantum circuit can be verified in the same environment as a real quantum machine. A quantum compiler compiles quantum circuits. When a quantum circuit is compiled, logical qubits in a quantum circuit should be mapped to physical qubits, called qubit mapping. As qubit mapping is an NP-Complete problem, the compilation time increases exponentially according to the number of qubits. Therefore, various studies on effective qubit mapping techniques have been developed to accelerate the execution time. This paper proposes a new parallelized Qubit mapping algorithm. The proposed method consists of three-step processes: one circuit is divided into several circuits; the SABRE enhanced algorithm is executed on each circuit in parallel and then merged. The performance was evaluated using 10 benchmarks. The experimental results revealed an acceleration of up to 6.42x and an average speed up of 4.8x, with an average overhead of 1.03x in terms of the number of gates obtained. The proposed method outperformed the well-known modern qubit-mapping algorithm in terms of the execution time with exponential speedup improvement.