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Product-State Approximation Algorithms for the Transverse Field Ising Model

Vincenzo Lipardi, David Mestel, Georgios Stamoulis

TL;DR

The paper advances classical polynomial-time approximation schemes for the Transverse Field Ising Model (TFIM) with arbitrary sign and weight patterns by focusing on product-state encodings. It introduces a robust PSD-based reformulation H_{TFIM'} and a SOC-SDP relaxation to upper-bound the true optimum, then derives three constant-factor product-state algorithms: a 0.7154-approximation, a 0.7860-approximation via SOC-SDP-driven rounding, and an 0.8156-approximation via interpolation. A concrete 3-qubit ferromagnetic triangle demonstrates an unavoidable product-state optimality gap of 169/180, showing limits of product-state approaches. Together, these results extend multiplicative approximation guarantees beyond ferromagnetic or stoquastic cases and provide explicit, efficiently computable bounds on TFIM ground-state energy.

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 $γ\approx 0.71$ , (ii) a strengthened rounding, inspired by the anticommutation property of the two $X_i, Z_iZ_j$ observables achieving ratio $γ\approx 0.7860$, and (iii) a further improvement by interpolation achieving ratio $γ\approx 0.8156$. 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.

Product-State Approximation Algorithms for the Transverse Field Ising Model

TL;DR

The paper advances classical polynomial-time approximation schemes for the Transverse Field Ising Model (TFIM) with arbitrary sign and weight patterns by focusing on product-state encodings. It introduces a robust PSD-based reformulation H_{TFIM'} and a SOC-SDP relaxation to upper-bound the true optimum, then derives three constant-factor product-state algorithms: a 0.7154-approximation, a 0.7860-approximation via SOC-SDP-driven rounding, and an 0.8156-approximation via interpolation. A concrete 3-qubit ferromagnetic triangle demonstrates an unavoidable product-state optimality gap of 169/180, showing limits of product-state approaches. Together, these results extend multiplicative approximation guarantees beyond ferromagnetic or stoquastic cases and provide explicit, efficiently computable bounds on TFIM ground-state energy.

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 , (ii) a strengthened rounding, inspired by the anticommutation property of the two observables achieving ratio , and (iii) a further improvement by interpolation achieving ratio . We also give an explicit (purely ferromagnetic) TFIM instance on three qbits for which every product state achieves at most of the true optimum, yielding an upper bound for all algorithms producing product state approximations, even in the purely ferromagnetic case.
Paper Structure (17 sections, 18 theorems, 84 equations, 1 figure, 3 algorithms)

This paper contains 17 sections, 18 theorems, 84 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions.
  3. Scope relative to prior work.
  4. The Model and its relaxation
  5. Formulating the approximation problem
  6. Warm-up: a $0.7154$ two-candidate product state approximation algorithm
  7. The Field Solution:
  8. The Ising Solution:
  9. A SOC-SDP relaxation upper-bounding the true quantum optimum
  10. The anticommutation tradeoff inequality
  11. The relaxation
  12. A 0.786 approximation algorithm based on SOC-SDP
  13. A pure $Z$ solution
  14. A mixed $X$, $ZZ$ algorithm
  15. Combining Algorithms \ref{['alg:A']} & \ref{['alg:B']}
  16. ...and 2 more sections

Key Result

Theorem 1

Taking the maximum of the two product state solutions above gives an approximation algorithm for TFIM with approximation ratio $1-1/4\alpha_{\mathrm{GW}} \approx 0.7154$.

Figures (1)

  • Figure 1: Optimal $q$ and resulting approximation factor for varying $p$.

Theorems & Definitions (37)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Definition 1: SOC-SDP relaxation
  • Lemma 2
  • proof
  • ...and 27 more