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Algorithmic Cluster Expansions for Quantum Problems

Ryan L. Mann, Romy M. Minko

TL;DR

The paper develops a general framework based on the cluster expansion of abstract polymer models to approximate a class of counting problems arising in quantum information and quantum many-body physics. By enforcing a decay bound on polymer weights and exploiting convergence criteria, it yields fully polynomial-time approximation schemes for amplitudes, expectation values, partition functions, and thermal expectations in high-temperature or identity-near regimes, while also proving hardness results that establish complexity transitions. The approach extends and sharpens prior results, providing streamlined proofs and tighter bounds that apply to bounded-degree, bounded-rank multihypergraphs and to complex or time-evolved quantities. The work also connects to Ising-model hardness reductions and discusses zero freeness as a notion of optimality for certain observables. Overall, the framework offers a cohesive, algorithmically powerful lens for identifying tractable quantum counting problems and delineating their computational boundaries.

Abstract

We establish a general framework for developing approximation algorithms for a class of counting problems. Our framework is based on the cluster expansion of abstract polymer models formalism of Kotecký and Preiss. We apply our framework to obtain efficient algorithms for (1) approximating probability amplitudes of a class of quantum circuits close to the identity, (2) approximating expectation values of a class of quantum circuits with operators close to the identity, (3) approximating partition functions of a class of quantum spin systems at high temperature, and (4) approximating thermal expectation values of a class of quantum spin systems at high temperature with positive-semidefinite operators. Further, we obtain hardness of approximation results for approximating probability amplitudes of quantum circuits and partition functions of quantum spin systems. This establishes a computational complexity transition for these problems and shows that our algorithmic conditions are optimal under complexity-theoretic assumptions. Finally, we show that our algorithmic condition is almost optimal for expectation values and optimal for thermal expectation values in the sense of zero freeness.

Algorithmic Cluster Expansions for Quantum Problems

TL;DR

The paper develops a general framework based on the cluster expansion of abstract polymer models to approximate a class of counting problems arising in quantum information and quantum many-body physics. By enforcing a decay bound on polymer weights and exploiting convergence criteria, it yields fully polynomial-time approximation schemes for amplitudes, expectation values, partition functions, and thermal expectations in high-temperature or identity-near regimes, while also proving hardness results that establish complexity transitions. The approach extends and sharpens prior results, providing streamlined proofs and tighter bounds that apply to bounded-degree, bounded-rank multihypergraphs and to complex or time-evolved quantities. The work also connects to Ising-model hardness reductions and discusses zero freeness as a notion of optimality for certain observables. Overall, the framework offers a cohesive, algorithmically powerful lens for identifying tractable quantum counting problems and delineating their computational boundaries.

Abstract

We establish a general framework for developing approximation algorithms for a class of counting problems. Our framework is based on the cluster expansion of abstract polymer models formalism of Kotecký and Preiss. We apply our framework to obtain efficient algorithms for (1) approximating probability amplitudes of a class of quantum circuits close to the identity, (2) approximating expectation values of a class of quantum circuits with operators close to the identity, (3) approximating partition functions of a class of quantum spin systems at high temperature, and (4) approximating thermal expectation values of a class of quantum spin systems at high temperature with positive-semidefinite operators. Further, we obtain hardness of approximation results for approximating probability amplitudes of quantum circuits and partition functions of quantum spin systems. This establishes a computational complexity transition for these problems and shows that our algorithmic conditions are optimal under complexity-theoretic assumptions. Finally, we show that our algorithmic condition is almost optimal for expectation values and optimal for thermal expectation values in the sense of zero freeness.
Paper Structure (16 sections, 30 theorems, 65 equations, 1 table)

This paper contains 16 sections, 30 theorems, 65 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph Theory
  4. Abstract Polymer Models
  5. Abstract Cluster Expansion
  6. Approximation Schemes
  7. Computational Complexity
  8. General Framework
  9. Approximation Algorithms
  10. Hardness of Approximation
  11. Applications
  12. Probability Amplitudes
  13. Expectation Values
  14. Partition Functions
  15. Thermal Expectation Values
  16. ...and 1 more sections

Key Result

Lemma 1

Let $f$ be a function defined on the subsets of finite set $E$, then

Theorems & Definitions (61)

  • Lemma 1: Principle of inclusion-exclusion
  • proof
  • Theorem 2: Kotecký and Preiss kotecky1986cluster
  • Theorem 3
  • Lemma 4
  • proof
  • Remark
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 51 more