Tensor Network Formulation of Dequantized Algorithms for Ground State Energy Estimation

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.

💡 Research Summary

This paper introduces a novel tensor network-based framework for dequantizing algorithms that solve the Ground State Energy Estimation (GSEE) problem, a central task in quantum computing for which quantum advantage is sought. The primary motivation is to overcome the critical practical limitation of existing dequantization methods: their reliance on Monte Carlo sampling to estimate high-dimensional inner products, which leads to prohibitive computational overhead and prevents concrete implementation.

The core innovation lies in reformulating the dequantization of Quantum Singular Value Transformation (QSVT)-based eigenvalue filtering algorithms using tensor networks. Instead of sampling, the algorithm deterministically computes the required quantities—specifically, the Chebyshev moments μ_k = ⟨ψ|T_k(H)|ψ⟩—by representing the iteratively generated Chebyshev vectors |t_k⟩ = T_k(H)|ψ⟩ as tensor network states (e.g., Matrix Product States) and performing tensor contractions. This eliminates statistical sampling error. The computational bottleneck thus shifts from sampling variance to the entanglement growth in the Chebyshev vectors, which is reflected in the necessary bond dimension of the tensor network representation. Consequently, the physical structure of the problem (locality, dimensionality, interaction strength) directly dictates the algorithmic cost, providing a more nuanced and physically grounded complexity measure.

The authors present two formulations: an exact method using tensor network contraction and a practical variant that employs tensor network approximations. In the practical approach, only a limited set of low-degree Chebyshev vectors are explicitly approximated as tensor network states, and higher-degree moments are extrapolated using linear prediction techniques. This makes the algorithm feasible for realistic classical computation.

Numerical simulations on one- and two-dimensional transverse-field Ising models (TFIM) with up to 100 qubits demonstrate the power of the framework. The method efficiently constructs very high-degree polynomial filters (up to degree d=10^4). The results explicitly visualize the crossover between classically tractable and quantum-advantaged regimes. For the 1D TFIM, high-precision energy estimation is achieved with moderate bond dimensions, indicating its dequantizability within the studied parameters. In contrast, for the 2D TFIM, the same resource constraints fail to achieve high precision, suggesting a regime where quantum resources would be superior. This successfully ties the abstract complexity-theoretic boundary—governed by the scaling of the precision parameter ε—to concrete, observable behavior in many-body systems.

In summary, this work bridges the gap between the theoretical promise of dequantization for verifying quantum advantage and its practical application. By replacing sampling overhead with controlled tensor network approximation, it provides a rigorous, quantitative, and implementable tool for probing the classical-quantum frontier in practical quantum chemistry and condensed matter problems.