Iterative Matrix Product State Simulation for Scalable Grover's Algorithm

TL;DR

This paper introduces an iterative matrix product state (MPS) simulation framework for Grover's algorithm, significantly improving speed and scalability on classical hardware, which aids in quantum algorithm development and hardware testing.

Contribution

The paper presents a novel iterative MPS-based simulation method for Grover's algorithm, achieving faster runtimes and better scalability compared to traditional approaches.

Findings

Iterative MPS Grover simulation runs 15x faster than non-iterative at 29 qubits.

The method maintains low-shot stability for qubits beyond 13, reducing measurement costs.

Significant speedup over statevector backend, enabling large-scale quantum circuit simulation.

Abstract

Grover's algorithm is a cornerstone of quantum search algorithm, offering quadratic speedup for unstructured problems. However, limited qubit counts and noise in today's noisy intermediate-scale quantum (NISQ) devices hinder large-scale hardware validation, making efficient classical simulation essential for algorithm development and hardware assessment. We present an iterative Grover simulation framework based on matrix product states (MPS) to efficiently simulate large-scale Grover's algorithm. Within the NVIDIA CUDA-Q environment, we compare iterative and common (non-iterative) Grover's circuits across statevector and MPS backends. On the MPS backend at 29 qubits, the iterative Grover's circuit runs about 15x faster than the common (non-iterative) Grover's circuit, and about 3-4x faster than the statevector backend. In sampling experiments, Grover's circuits demonstrate strong low-shot stability: as the qubit number increases beyond 13, a single-shot measurement still closely mirrors the results from 4,096 shots, indicating reliable estimates with minimal sampling and significant potential to cut measurement costs. Overall, an iterative MPS design delivers speed and scalability for Grover's circuit simulation, enabling practical large-scale implementations.