Tensor Network Formulation of Dequantized Algorithms for Ground State Energy Estimation

TL;DR

The paper introduces a tensor-network-based dequantization framework for ground state energy estimation (GSEE) that eliminates sampling overhead by replacing it with bond-dimension growth in Chebyshev-vector representations. By recasting QSVT-based eigenvalue filtering into tensor-network contractions, the authors maintain the asymptotic complexity of prior dequantized algorithms while enabling practical, scalable classical computation using MPS approximations and linear prediction. Key contributions include a rigorous formulation of exact and approximate tensor-network dequantization, complexity analysis showing poly-time scaling under efficient contraction assumptions, and numerical demonstrations on one- and two-dimensional transverse-field Ising models up to 100 qubits and degree $d=10^4$, illustrating a crossover between classically tractable and quantum-advantaged regimes. The framework provides a quantitative tool for verifying quantum advantage in realistic many-body systems and offers a pathway to apply similar TN-based dequantization to other QSVT-driven quantum algorithms.

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.

Figures (12)

• Figure 1: An overview of the dequantized algorithm with tensor networks for ground state energy estimation. $P$ denotes a polynomial of degree $d$, and $T_k$ denotes the $k$th Chebyshev polynomial. $\ket{\widetilde{t}_k}$ and $\widetilde{\mu}_k$ are approximations of the Chebyshev vector $\ket{t_k}$ and the moment $\mu_k$ respectively, and we assume $N_{\mathrm{max}}<d$. The formal definitions are provided later in the main text.
• Figure 2: The diagram notation of MPS, a tensor network with a simple 1D structure. $\{s_i\}$ is a set of indices that corresponds to the dangling edges, and $\{\alpha_i\}$ is a set of internal indices.
• Figure 3: The process of calculating the inner product of two MPS. The double-layered MPS is contracted from the left-hand side. Each contraction costs $O(D^3)$, and this process is repeated $n$ times.
• Figure 4: An example of polynomial approximation of the shifted sign function $P_{\eta,\Delta,c}$ and the rectangle function $P_{\eta,\Delta,c}^{\mathrm{QSVT}}$. Parameters are set as $\eta=0.1, \Delta=0.2, c=0.5$ and the degree is $20$. The polynomial approximation is obtained by CVXPY diamond_CVXPY_2016.
• Figure 5: The comparison between two polynomial approximations. (a) The polynomials that approximate the shifted sign function are plotted. $P_{\mathrm{CVXPY}}$ is the one obtained by CVXPY and $P_{\mathrm{Cheb}}$ is obtained by Chebyshev truncation. The discontinuity point is set to $c=-0.5$. The inset is the zoom-in version around $x=c$. (b) The Chebyshev coefficients $|a_k|$ are plotted as a function of $k$.

Theorems & Definitions (21)

• Definition 1: Tensor network
• Definition 2: Tensor network states
• Definition 3: Inner product of tensor network states
• Definition 4: Efficiently contractible tensor network states
• Lemma 1: Polynomial approximation of the shifted sign function low_Hamiltonian_2017gilyen_Quantum_2019
• Lemma 2: Polynomial approximation of the rectangle function low_Hamiltonian_2017gilyen_Quantum_2019
• Lemma 3: GSEE via QSVT martyn_Grand_2021
• proof
• Lemma 4: GSEE via dequantization gharibian_Dequantizing_2023gall_Classical_2024
• proof : Proof sketch