Iterative Matrix Product State Simulation for Scalable Grover's Algorithm

TL;DR

This work tackles the challenge of classically simulating Grover's algorithm at scale on hardware-limited platforms. It introduces an Iterative Grover circuit implemented with a Matrix Product State (MPS) backend, updating the state with a single Grover gate per iteration to avoid deep circuit construction and achieve memory efficiency of O(nχ^2) with χ_max = 64. Empirically, the Iterative MPS method delivers up to about 3–4× speedups over state-vector backends and up to 15× over common MPS at n = 29, while maintaining amplitude fidelity with FP64 precision; low-shot measurements remain reliable for n ≥ 13, reducing measurement costs. These results demonstrate a scalable, accurate framework for large-scale Grover simulations on classical hardware, enabling practical benchmarking and hardware assessment for quantum search implementations.

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.

Figures (4)

• Figure 1: Structure and comparison of Grover’s quantum circuits. (A) Internal composition of the Grover gate $G$, consisting of the oracle $O$ and the diffusion operator $D$. In the circuit, filled circles connected by vertical lines represent multi-controlled operations while the letters denote standard quantum gates: $H$ for Hadamard, $X$ for Pauli-X, and $Z$ for Pauli-Z gates. (B) Common Grover’s Quantum Circuit: the standard implementation repeats $G$ for $O(\sqrt{2^n})$ iterations, leading to a very deep quantum circuit as the number of qubits increases. (C) Iteration Grover’s Quantum Circuit: the proposed iterative approach updates the state vector $|s\rangle \leftarrow G|s\rangle$ at each step, avoiding the multi-layer circuit construction.
• Figure 2: Runtime of Grover’s algorithm versus the number of qubits for different methods, simulation backends and precisions. At small system sizes, the proposed Iterative method and the Common implementation show similar performance, and the state-vector backend runs faster than MPS. As the number of qubits increases, the runtime of state-vector simulation grows most rapidly, followed by Common (MPS), whereas Iterative (MPS) scales the slowest. Beyond 27 qubits, the MPS Iterative method outperforms state-vector simulation, and the observed trend suggests a crossing point between Common (MPS) and state-vector performance at even larger scales.
• Figure 3: Target amplitude of Grover’s algorithm obtained from different methods, backends and precisions. Float64 state-vector simulations (lighter bars) serve as the reference and yield identical amplitudes of 1.0 across all setups. When using float32 (solid bars), the target amplitude remains consistent between the Common and Iterative implementations for small systems, but deviations appear as the qubit number increases. MPS amplitudes start to decrease at $n=18$ and drop below 0.9 at $n=28$, whereas state-vector amplitudes begin to decrease only at $n=23$ and remain above 0.98.
• Figure 4: Effect of measurement shot number on the target amplitude in Grover’s algorithm, obtained using the MPS Iterative simulation with FP64 precision. For small qubit systems ($n \le 7$), Low-shot measurements (1 or 8 shots) lead to visible amplitude fluctuations, whereas for larger qubit numbers $(n \geq 13)$ the results from all shot numbers converge to $1.0$.