Distributed Quantum Algorithm for the NISQ Era: A Novel Approach to Solving Simon's Problem with Reduced Resources

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 $s\in {\{0,1\}}^n$ of a promised Boolean function $f:{\{0,1\}}^n\to {\{0,1\}}^m$, where $f\left(x\right)=f\left(y\right)$ if and only if $x=y$ or $x\oplus y=s$. Specifically, 1) our algorithm is capable of being partitioned into any $t$ nodes, where $2\le t\le n$; 2) the number of queries required by the DSA is $n-t$, while that of the original Simon's algorithm (SA) is $n-1$; 3) the maximum number of qubits required by our approach at a single node is $\max \left(n_0,,,n_1,,,\cdots ,,,n_{t-1}\right)+m$, which is fewer than the qubits required by both the SA and existing distributed Simon's algorithm. Here, $n_j$ denotes the number of computing qubits needed for the $j$-th node and satisfies ${\sum }_{j=0}^{t-1}{n}_j=n$; 4) the optimal circuit depth of the DSA is $(m+1)·2^{\left(\frac{n}{t}\right)}+2$, which is reduced compared to the circuit depth of the SA, $(m+1)·2^n+2$; 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., $s=1000$) 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.