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arXiv:2303.10670

Distributed exact quantum algorithms for Bernstein-Vazirani and search problems

Xu Zhou, Daowen Qiu, Lou, Le

Published Mar 19, 2023• 2 citations
arXivHTMLPDF

Abstract

Distributed quantum computation has gained extensive attention since small-qubit quantum computers seem to be built more practically in the noisy intermediate-scale quantum (NISQ) era. In this paper, we give a distributed Bernstein-Vazirani algorithm (DBVA) with $t$ computing nodes, and a distributed exact Grover's algorithm (DEGA) that solve the search problem with only one target item in the unordered databases. Though the designing techniques are simple in the light of BV algorithm, Grover's algorithm, the improved Grover's algorithm by Long, and the distributed Grover's algorithm by Qiu et al, in comparison to BV algorithm, the circuit depth of DBVA is not greater than $2^{\text{max}(n_0, n_1, \cdots, n_{t-1})}+3$ instead of $2^{n}+3$, and the circuit depth of DEGA is $8(n~\text{mod}~2)+9$, which is less than the circuit depth of Grover's algorithm, $1 + 8\left\lfloor \fracπ{4}\sqrt{2^n} \right\rfloor$. In particular, we provide situations of our DBVA and DEGA on MindQuantum (a quantum software) to validate the correctness and practicability of our methods. By simulating the algorithms running in the depolarized channel, it further illustrates that distributed quantum algorithm has superiority of resisting noise.

DOI
10.48550/arxiv.2303.10670

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Distributed quantum computation has gained extensive attention since small-qubit quantum computers seem to be built more practically in the noisy intermediate-scale quantum (NISQ) era. In this paper, we give a distributed Bernstein-Vazirani algorithm (DBVA) with ttt computing nodes, and a distributed exact Grover's algorithm (DEGA) that solve the search problem with only one target item in the unordered databases. Though the designing techniques are simple in the light of BV algorithm, Grover's algorithm, the improved Grover's algorithm by Long, and the distributed Grover's algorithm by Qiu et al, in comparison to BV algorithm, the circuit depth of DBVA is not greater than 2max(n0,n1,⋯ ,nt−1)+32^{\text{max}(n_0, n_1, \cdots, n_{t-1})}+32max(n0​,n1​,⋯,nt−1​)+3 instead of 2n+32^{n}+32n+3, and the circuit depth of DEGA is 8(n mod 2)+98(n~\text{mod}~2)+98(n mod 2)+9, which is less than the circuit depth of Grover's algorithm, 1+8⌊π42n⌋1 + 8\left\lfloor \fracπ{4}\sqrt{2^n} \right\rfloor1+8⌊4π​2n​⌋. In particular, we provide situations of our DBVA and DEGA on MindQuantum (a quantum software) to validate the correctness and practicability of our methods. By simulating the algorithms running in the depolarized channel, it further illustrates that distributed quantum algorithm has superiority of resisting noise.
Distributed quantum computation has gained extensive attention. In this paper, we consider a search problem that includes only one target item in the unordered database. After that, we propose a distributed exact Grover’s algorithm (DEGA), which decomposes the original search problem into ⌊n/2⌋ parts. Specifically, (i) our algorithm is as exact as the modified version of Grover’s algorithm by Long, which means the theoretical probability of finding the objective state is 100%; (ii) the actual depth of our circuit is 8(n mod 2) + 9, which is less than the circuit depths of the original and modified Grover’s algorithms, 1+8⌊π42n⌋1 + 8\left\lfloor {{\pi \over 4}\sqrt {{2^n}} } \right\rfloor 1+8⌊4π​2n​⌋ and 9+8⌊π42n−12⌋9 + 8\left\lfloor {{\pi \over 4}\sqrt {{2^n}} - {1 \over 2}} \right\rfloor 9+8⌊4π​2n​−21​⌋ , respectively. It only depends on the parity of n, and it is not deepened as n increases; (iii) we provide particular situations of the DEGA on MindQuantum (a quantum software) to demonstrate the practicality and validity of our method. Since our circuit is shallower, it will be more resistant to the depolarization channel noise.
Grover's algorithm accelerates unstructured database search quadratically compared to classical algorithms. In the NISQ era, distributed quantum computing can decrease circuit depth and reduce noise. In this paper, an algorithm for constructing query operators for subfunctions is proposed. By dividing the target string of the search problem into several substrings and integrating the query operator of each subfunction, a low-depth distributed exact quantum search algorithm is designed. The contributions of this paper are as follows: (1) The proposed distributed algorithm has a lower circuit depth and can mitigate error accumulation compared to distributed quantum search algorithms; (2) The target can be accurately located by the proposed distributed algorithm; (3) Experiments conducted with the quantum software MindQuantum confirm the effectiveness and feasibility of the proposed distributed algorithm. Moreover, the introduction of noise to the circuit during these experiments indicates that the algorithm possesses an inherent capacity for noise resistance.
Multi-objective search means searching for any one of several objectives in an unstructured database. Grover's algorithm has quadratic acceleration in multi-objection search than classical ones. Iterated operator in Grover's algorithm is a key element and plays an important role in amplitude amplification. In this paper, we design two distributed iterated operators and therefore two new distributed Grover's algorithms are obtained with the following advantages: (1) Compared to Grover's algorithm and the modified Grover's algorithm by Long, our distributed algorithms require fewer qubits; (2) Compared to the distributed Grover's algorithm proposed by Qiu et al., one of our distributed algorithms is exact. Of course, both our distributed algorithms require quite quantum communication and involve a number of more complicated unitary operators as cost, but there still may have certain advantage of physical realizability in the Noisy Intermediate-Scale Quantum (NISQ) era.
In the noisy intermediate-scale quantum (NISQ) era, distributed quantum computation has garnered considerable interest, as it overcomes the physical limitations of single-device architectures and enables scalable quantum information processing. In this study, we focus on the challenge of achieving exact amplitude amplification for quantum states with arbitrary amplitude distributions and subsequently propose a Distributed Exact Quantum Amplitude Amplification Algorithm (DEQAAA). Specifically, (1) it supports partitioning across any number of nodes ttt within the range 2≤t≤n2 \leq t \leq n2≤t≤n; (2) the maximum qubit count required for any single node is expressed as $\max \left(n_0,n_1,\dots,n_{t-1} \right) $, where njn_jnj​ represents the number of qubits at the jjj-th node, with ∑j=0t−1nj=n\sum_{j=0}^{t-1} n_j =n∑j=0t−1​nj​=n; (3) it can realize exact amplitude amplification for multiple targets of a quantum state with arbitrary amplitude distributions; (4) we verify the effectiveness of DEQAAA by resolving a specific exact amplitude amplification task involving two targets (8 and 14 in decimal) via MindSpore Quantum, a quantum simulation software, with tests conducted on 4-qubit, 6-qubit, 8-qubit and 10-qubit systems. Notably, through the decomposition of Cn−1PSC^{n-1}PSCn−1PS gates, DEQAAA demonstrates remarkable advantages in both quantum gate count and circuit depth as the qubit number scales, thereby boosting its noise resilience. In the 10-qubit scenario, for instance, it achieves a reduction of over 97%97\%97% in both indicators compared to QAAA and EQAAA, underscoring its outstanding resource-saving performance.
Abstract Distributed quantum computation has gained significant interest in the noisy intermediate‐scale quantum (NISQ) era. This paradigm requires each computing node to possess a reduced number of qubits and quantum gates. In this study, a Distributed Simon's Algorithm (DSA) is designed to tackle Simon's problem, which entails the discovery of a hidden string of a promised Boolean function , where if and only if or . Specifically, 1) our algorithm is capable of being partitioned into any nodes, where ; 2) the number of queries required by the DSA is , while that of the original Simon's algorithm (SA) is ; 3) the maximum number of qubits required by our approach at a single node is , which is fewer than the qubits required by both the SA and existing distributed Simon's algorithm. Here, denotes the number of computing qubits needed for the ‐th node and satisfies ; 4) the optimal circuit depth of the DSA is , which is reduced compared to the circuit depth of the SA, ; 5) in contrast to currently distributed schemes, the DSA eliminates the need for classical queries; 6) how the DSA solves a specific Simon's problem (e.g., ) is also simulated using MindSpore Quantum, a quantum simulation software. The simulation results show that the DSA features a shallower quantum circuit, thereby demonstrating enhanced resistance to circuit noise. This characteristic makes it more feasible for implementation in the NISQ era.
Grover's search algorithm has attracted great attention due to its quadratic speedup over classical algorithms in unsorted database search problems. However, Grover's algorithm is inefficient in multi-target search problems, except in the case of 1/4 of the data in the database satisfying the search conditions. Long presented a modified version of Grover's search algorithm by introducing a phase-matching condition that can search for the target state with a zero theoretical failure rate. In this work, we present an optimized exact multi-target search algorithm based on the modified Grover's algorithm by transforming the canonical diffusion operator to a more efficient diffusion operator, which can solve the multi-target search problem with a 100% success rate while requiring fewer gate counts and shallower circuit depth. After that, the optimized multi-target algorithm for four different items, including 2-qubit with 2 targets, 5-qubit with 2 targets, 5-qubit with 4 targets, and 6-qubit with 3 targets, are implemented on MindSpore framework. The experimental results show that, compared with Grover's algorithm and the modified Grover's algorithm, the proposed algorithm can reduce the quantum gate count by at least 21.1% and the depth of the quantum circuit by at least 11.4% and maintain a 100% success probability. Our code are available at https://github.com/mindsporelab/models/tree/master/research/arxiv_papers/GROVER-OP. Thanks for the support provided by MindSpore Community.