Tensor Network Formulation of Dequantized Algorithms for Ground State Energy Estimation

TL;DR

This paper introduces a tensor network-based dequantization method for ground state energy estimation that removes sampling overhead, enabling practical classical simulations and aiding the verification of quantum advantage in many-body systems.

Contribution

It develops a tensor network framework for dequantization that replaces sampling with bond dimension growth, improving practicality and reflecting physical structures.

Findings

Efficiently constructs high-degree polynomials for large Hamiltonians.

Reveals the crossover between classical tractability and quantum advantage.

Demonstrates practical dequantization for systems with up to 100 qubits.

Abstract

Verifying quantum advantage for practical problems, particularly the ground state energy estimation (GSEE) problem, is one of the central challenges in quantum computing theory. For that purpose, dequantization algorithms play a central role in providing a clear theoretical framework to separate the complexity of quantum and classical algorithms. However, existing dequantized algorithms typically rely on sampling procedures, leading to prohibitively large computational overheads and hindering their practical implementation on classical computers. In this work, we propose a tensor network-based dequantization framework for GSEE that eliminates the sampling process while preserving the asymptotic complexity of prior dequantized algorithms. In our formulation, the overhead arising from sampling is replaced by the growth of the bond dimension required to represent Chebyshev vectors as tensor network states. Consequently, physical structure, such as entanglement and locality, is naturally reflected in the computational cost. By combining this approach with tensor network approximations, such as Matrix Product States (MPS), we construct a practical dequantization algorithm that is executable within realistic computational resources. Numerical simulations demonstrate that our method can efficiently construct high-degree polynomials up to $d=10^4$ for Hamiltonians with up to $100$ qubits, explicitly revealing the crossover between classically tractable and quantum advantaged regimes. These results indicate that tensor network-based dequantization provides a crucial tool toward the rigorous, quantitative verification of quantum advantage in realistic many-body systems.