Xu Zhou, Yuchen Wang, Wenxuan Tao, Zhuojun Zhou, Le Luo
{"title":"Distributed Quantum Algorithm for the NISQ Era: A Novel Approach to Solving Simon's Problem with Reduced Resources","authors":"Xu Zhou, Yuchen Wang, Wenxuan Tao, Zhuojun Zhou, Le Luo","doi":"10.1002/qute.202500067","DOIUrl":null,"url":null,"abstract":"<p>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 <span></span><math>\n <semantics>\n <mrow>\n <mi>s</mi>\n <mo>∈</mo>\n <msup>\n <mrow>\n <mo>{</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mn>1</mn>\n <mo>}</mo>\n </mrow>\n <mi>n</mi>\n </msup>\n </mrow>\n <annotation>$s \\in \\lbrace 0,1\\rbrace ^n$</annotation>\n </semantics></math> of a promised Boolean function <span></span><math>\n <semantics>\n <mrow>\n <mi>f</mi>\n <mo>:</mo>\n <msup>\n <mrow>\n <mo>{</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mn>1</mn>\n <mo>}</mo>\n </mrow>\n <mi>n</mi>\n </msup>\n <mo>→</mo>\n <msup>\n <mrow>\n <mo>{</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mn>1</mn>\n <mo>}</mo>\n </mrow>\n <mi>m</mi>\n </msup>\n </mrow>\n <annotation>$f: \\lbrace 0,1\\rbrace ^n \\rightarrow \\lbrace 0,1\\rbrace ^m$</annotation>\n </semantics></math>, where <span></span><math>\n <semantics>\n <mrow>\n <mi>f</mi>\n <mo>(</mo>\n <mi>x</mi>\n <mo>)</mo>\n <mo>=</mo>\n <mi>f</mi>\n <mo>(</mo>\n <mi>y</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$f(x)=f(y)$</annotation>\n </semantics></math> if and only if <span></span><math>\n <semantics>\n <mrow>\n <mi>x</mi>\n <mo>=</mo>\n <mi>y</mi>\n </mrow>\n <annotation>$x=y$</annotation>\n </semantics></math> or <span></span><math>\n <semantics>\n <mrow>\n <mi>x</mi>\n <mi>⊕</mi>\n <mi>y</mi>\n <mo>=</mo>\n <mi>s</mi>\n </mrow>\n <annotation>$x \\oplus y = s$</annotation>\n </semantics></math>. Specifically, 1) our algorithm is capable of being partitioned into any <span></span><math>\n <semantics>\n <mi>t</mi>\n <annotation>$t$</annotation>\n </semantics></math> nodes, where <span></span><math>\n <semantics>\n <mrow>\n <mn>2</mn>\n <mo>≤</mo>\n <mi>t</mi>\n <mo>≤</mo>\n <mi>n</mi>\n </mrow>\n <annotation>$2 \\le t \\le n$</annotation>\n </semantics></math>; 2) the number of queries required by the DSA is <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>−</mo>\n <mi>t</mi>\n </mrow>\n <annotation>$n-t$</annotation>\n </semantics></math>, while that of the original Simon's algorithm (SA) is <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$n-1$</annotation>\n </semantics></math>; 3) the maximum number of qubits required by our approach at a single node is <span></span><math>\n <semantics>\n <mrow>\n <mi>max</mi>\n <mfenced>\n <msub>\n <mi>n</mi>\n <mn>0</mn>\n </msub>\n <mo>,</mo>\n <msub>\n <mi>n</mi>\n <mn>1</mn>\n </msub>\n <mo>,</mo>\n <mi>⋯</mi>\n <mo>,</mo>\n <msub>\n <mi>n</mi>\n <mrow>\n <mi>t</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msub>\n </mfenced>\n <mo>+</mo>\n <mi>m</mi>\n </mrow>\n <annotation>$\\max \\left(n_0,n_1,\\dots,n_{t-1} \\right) + m$</annotation>\n </semantics></math>, which is fewer than the qubits required by both the SA and existing distributed Simon's algorithm. Here, <span></span><math>\n <semantics>\n <msub>\n <mi>n</mi>\n <mi>j</mi>\n </msub>\n <annotation>$n_j$</annotation>\n </semantics></math> denotes the number of computing qubits needed for the <span></span><math>\n <semantics>\n <mi>j</mi>\n <annotation>$j$</annotation>\n </semantics></math>-th node and satisfies <span></span><math>\n <semantics>\n <mrow>\n <msubsup>\n <mo>∑</mo>\n <mrow>\n <mi>j</mi>\n <mo>=</mo>\n <mn>0</mn>\n </mrow>\n <mrow>\n <mi>t</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msubsup>\n <msub>\n <mi>n</mi>\n <mi>j</mi>\n </msub>\n <mo>=</mo>\n <mi>n</mi>\n </mrow>\n <annotation>$\\sum _{j=0}^{t-1} n_j =n$</annotation>\n </semantics></math>; 4) the optimal circuit depth of the DSA is <span></span><math>\n <semantics>\n <mrow>\n <mrow>\n <mo>(</mo>\n <mi>m</mi>\n <mo>+</mo>\n <mn>1</mn>\n <mo>)</mo>\n </mrow>\n <mo>·</mo>\n <msup>\n <mn>2</mn>\n <mfenced>\n <mfrac>\n <mi>n</mi>\n <mi>t</mi>\n </mfrac>\n </mfenced>\n </msup>\n <mo>+</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$(m + 1) \\cdot 2^{\\left\\lceil \\frac{n}{t} \\right\\rceil } + 2$</annotation>\n </semantics></math>, which is reduced compared to the circuit depth of the SA, <span></span><math>\n <semantics>\n <mrow>\n <mrow>\n <mo>(</mo>\n <mi>m</mi>\n <mo>+</mo>\n <mn>1</mn>\n <mo>)</mo>\n </mrow>\n <mo>·</mo>\n <msup>\n <mn>2</mn>\n <mi>n</mi>\n </msup>\n <mo>+</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$(m+1) \\cdot 2^n+2$</annotation>\n </semantics></math>; 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., <span></span><math>\n <semantics>\n <mrow>\n <mi>s</mi>\n <mo>=</mo>\n <mn>1000</mn>\n </mrow>\n <annotation>$s =1000$</annotation>\n </semantics></math>) 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.</p>","PeriodicalId":72073,"journal":{"name":"Advanced quantum technologies","volume":"8 5","pages":""},"PeriodicalIF":4.4000,"publicationDate":"2025-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advanced quantum technologies","FirstCategoryId":"1085","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/qute.202500067","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"OPTICS","Score":null,"Total":0}
引用次数: 0
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.