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Quantum-Inspired Algorithm for Classical Spin Hamiltonians Based on Matrix Product Operators

Ryo Watanabe, Joseph Tindall, Shohei Miyakoshi, Hiroshi Ueda

TL;DR

The paper presents a quantum-inspired tensor-network approach to classical optimization by spectral amplification of a shifted cost Hamiltonian $\hat{G}=a\hat{H}+b\mathbb{I}$, represented as an MPO and powered to $\hat{G}^K$ before sampling from an embedded MPS in the computational basis. This framework yields amplitudes biased toward low-energy configurations while enabling systematic improvement via bond-dimension growth, and it avoids common local-minima traps that hinder DMRG. Benchmarks on Edwards–Anderson spin glasses and HUBO problems on a heavy-hex lattice show robust performance, with advantages over SA and competitive results against DMRG depending on initialization and schedule. The method provides a practical, scalable baseline for quantum-inspired optimization and a baseline for assessing future quantum algorithms, with clear directions for enhancements such as MPI parallelism and tensor-network architectures tailored to problem structure.

Abstract

We propose a tensor-network (TN) approach for solving classical optimization problems that is inspired by spectral filtering and sampling on quantum states. We first shift and scale an Ising Hamiltonian of the cost function so that all eigenvalues become non-negative and the ground states correspond to the the largest eigenvalues, which are then amplified by power iteration. We represent the transformed Hamiltonian as a matrix product operator (MPO) and form an immense power of this object via truncated MPO-MPO contractions, embedding the resulting operator into a matrix product state for sampling in the computational basis. In contrast to the density-matrix renormalization group, our approach provides a straightforward route to systematic improvement by increasing the bond dimension and is better at avoiding local minima. We also study the performance of this power method in the context of a higher-order Ising Hamiltonian on a heavy-hexagonal lattice, making a comparison with simulated annealing. These results highlight the potential of quantum-inspired algorithms for solving optimization problems and provide a baseline for assessing and developing quantum algorithms.

Quantum-Inspired Algorithm for Classical Spin Hamiltonians Based on Matrix Product Operators

TL;DR

The paper presents a quantum-inspired tensor-network approach to classical optimization by spectral amplification of a shifted cost Hamiltonian , represented as an MPO and powered to before sampling from an embedded MPS in the computational basis. This framework yields amplitudes biased toward low-energy configurations while enabling systematic improvement via bond-dimension growth, and it avoids common local-minima traps that hinder DMRG. Benchmarks on Edwards–Anderson spin glasses and HUBO problems on a heavy-hex lattice show robust performance, with advantages over SA and competitive results against DMRG depending on initialization and schedule. The method provides a practical, scalable baseline for quantum-inspired optimization and a baseline for assessing future quantum algorithms, with clear directions for enhancements such as MPI parallelism and tensor-network architectures tailored to problem structure.

Abstract

We propose a tensor-network (TN) approach for solving classical optimization problems that is inspired by spectral filtering and sampling on quantum states. We first shift and scale an Ising Hamiltonian of the cost function so that all eigenvalues become non-negative and the ground states correspond to the the largest eigenvalues, which are then amplified by power iteration. We represent the transformed Hamiltonian as a matrix product operator (MPO) and form an immense power of this object via truncated MPO-MPO contractions, embedding the resulting operator into a matrix product state for sampling in the computational basis. In contrast to the density-matrix renormalization group, our approach provides a straightforward route to systematic improvement by increasing the bond dimension and is better at avoiding local minima. We also study the performance of this power method in the context of a higher-order Ising Hamiltonian on a heavy-hexagonal lattice, making a comparison with simulated annealing. These results highlight the potential of quantum-inspired algorithms for solving optimization problems and provide a baseline for assessing and developing quantum algorithms.
Paper Structure (12 sections, 48 equations, 3 figures, 1 table)

This paper contains 12 sections, 48 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Operator Basis for classical Spin Hamiltonians
  4. Theoretical Analysis of the Powered-Operators
  5. Method
  6. Numerical Benchmarks and Performance Evaluation
  7. Energy Distributions and Entanglement Entropy
  8. Performance Benchmarking with DMRG
  9. Benchmark for higher-order Ising spin glass on heavy-hexagonal architecture
  10. Conclusion and Outlook
  11. Shifted and scaled cost Hamiltonian
  12. Procedure for constructing the MPO representation

Figures (3)

  • Figure 1: Panel (a) shows the mean energy and its standard deviation obtained from 1,000 samples of $\ket{\Psi}$ for a specific instance with $(D,L)=(2,10)$. The Label “Linear” denotes a linear schedule in which the power $K$ is increased step by step by repeatedly multiplying by $\hat{G}$, whereas label “Double” means using a doubling schedule $K=2^m$ by multiplying the result of the MPO-MPO contraction with itself $m$ times. On the horizontal axis, $C(\bm{z}^\ast)$ denotes the optimal cost value of the instance. Panel (b) shows the $K~(=2^m)$ dependence of the maximum bipartite entanglement entropy $\mathcal{S}$. Panel (c) plots the number of unique bitstrings among $1000$ samples. We used $\chi=64$ to generate data for these plots.
  • Figure 2: Error, $1 - {\rm AR}$, where ${\rm AR}$ is the mean approximation ratio [Eq. \ref{['eq:ar']}] averaged over five problem instances for powered-MPO method and DMRG; $D$ denotes the spatial dimension and $L$ signifies the linear size of $D$-dimensional hypercubic lattice. Stars and circles represent the powered-MPO results using linear and doubling schedules, respectively. Colors indicate the bond dimension used in the calculations. The black line represents the DMRG results starting from the initial state $\ket{+}^{\otimes N}$, which remain invariant across the investigated bond dimensions $\chi \in \{16, 64, 128\}$. Colored dash-dotted lines indicate the DMRG results starting from random MPS. In certain cases the powered-MPO approaches successfully reache the exact solution to the COP and this is indicated by the broken axis and the data lying exactly on the $x$-axis.
  • Figure 3: (a) Schematic of a heavy-hexagonal lattice with size $L=4$, consisting of $L \times L$ unit cells. The vertex indices denote the one-dimensional site ordering adopted for the MPO representation of the Hamiltonian. (b) Average minimum energy obtained with the powered-MPO method as a function of $m$, compared with SA results. Markers represent mean values across five problem instances; error bars (standard deviation) are omitted for clarity. The insets show a magnified view of the results for $m > 10$. All MPO calculations were performed with a maximum bond dimension of $\chi=64$.