Progress in Quantum Algorithms

Peter W. Shor

Xiao‐Hui Ni, Ling-Xiao 凌霄 Li 李, Yanqi Song +3 more

Abstract The quantum alternating operator ansatz algorithm (QAOA+) is widely used for constrained combinatorial optimization problems (CCOPs) due to its ability to construct feasible solution spaces. In this paper, we propose a progressive quantum algorithm (PQA) to reduce qubit requirements for QAOA+ in solving the maximum independent set (MIS) problem. PQA iteratively constructs a subgraph likely to include the MIS solution of the original graph and solves the problem on it to approximate the global solution. Specifically, PQA starts with a small-scale subgraph and progressively expands its graph size utilizing heuristic expansion strategies. After each expansion, PQA solves the MIS problem on the newly generated subgraph using QAOA+. In each run, PQA repeats the expansion and solving process until a predefined stopping condition is reached. Simulation results show that PQA achieves an approximation ratio of 0.95 using only 5.57% (2.17%) of the qubits and 17.59% (6.43%) of the runtime compared with directly solving the original problem with QAOA+ on Erdös–Rényi (3-regular) graphs, highlighting the efficiency and scalability of PQA.

Andris Ambainis

Xiao‐Hui Ni, Lingxiao Li, Yan-Qi Song +3 more

Hadfield et al. proposed a novel Quantum Alternating Operator Ansatz algorithm (QAOA+), and this algorithm has wide applications in solving constrained combinatorial optimization problems (CCOPs) because of the advantages of QAOA+ ansatz in constructing a feasible solution space. In this paper, we propose a Progressive Quantum Algorithm (PQA) with QAOA+ ansatz to solve the Maximum Independent Set (MIS) problem using fewer qubits. The core idea of PQA is to construct a subgraph that is likely to contain the MIS solution of the target graph and then solve the MIS problem on this subgraph to obtain an approximate solution. To construct such a subgraph, PQA starts with a small-scale initial subgraph and progressively expands its graph size utilizing heuristic expansion strategies. After each expansion, PQA solves the MIS problem on the newly generated subgraph. In each run, PQA repeats the expansion and solving process until a predefined stopping condition is reached. Simulation results demonstrate that to achieve an approximation ratio of 0.95, PQA requires only $5.565\%$ ($2.170\%$) of the qubits and $17.59\%$ ($6.430\%$) of the runtime compared with directly solving the original problem using QAOA+ on Erdős-Rényi (3-regular) graphs, highlighting the efficiency of PQA.

Wei Li

This essay mainly introduced Shor algorithm,Grover quantum evolution algorithm and the progress of quantum evolution algorithm based on quantum algorithm theory.It expounded the major problems and research prospect which quantum algorithm had to confront.

Diederik Aerts