Product-State Approximation Algorithms for the Transverse Field Ising Model

2026-01-19

AI Summary

This work proposes three hierarchically enhanced polynomial-time classical algorithms for approximating the quantum ground-state energy of the transverse-field Ising model (TFIM) with arbitrary ferromagnetic and antiferromagnetic interactions and non-negative transverse fields, using product states. By exploiting the anticommutation between X and ZZ observables, parameter interpolation, and an improved rounding strategy, the algorithms surpass existing approximation ratio bounds, achieving ratios of approximately 0.71, 0.7860, and 0.8156 in the general case. Furthermore, the authors construct a three-qubit instance demonstrating that, for purely ferromagnetic TFIM, no product-state approximation can exceed an approximation ratio of 0.9389, thereby establishing a tight theoretical upper bound on the performance of such methods.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Learning to Search

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• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Privacy-enhancing technologies
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We study classical polynomial-time approximation algorithms for the transverse-field Ising model (TFIM) Hamiltonian, allowing a mixture of ferromagnetic and anti-ferromagnetic interactions between pairs of qbits, alongside transverse field terms with arbitrary non-negative weights. Our main results are a series of approximation algorithms (all approximation ratios with respect to the true quantum optimum): (i) a simple maximum of two product state rounding algorithm achieving an approximation ratio $\gamma\approx 0.71$ , (ii) a strengthened rounding, inspired by the anticommutation property of the two $X_i, Z_iZ_j$ observables achieving ratio $\gamma\approx 0.7860$, and (iii) a further improvement by interpolation achieving ratio $\gamma \approx 0.8156$. We also give an explicit (purely ferromagnetic) TFIM instance on three qbits for which every product state achieves at most $169/180\approx 0.9389$ of the true optimum, yielding an upper bound for all algorithms producing product state approximations, even in the purely ferromagnetic case.

Problem

• Transverse Field Ising Model
• Product-State Approximation
• Quantum Optimum
• Approximation Algorithms
• Ferromagnetic and Antiferromagnetic Interactions

Innovation

• Transverse Field Ising Model
• Product-State Approximation
• Approximation Algorithms
• Anticommutation-inspired Rounding
• Interpolation Method