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A simple lower bound for the complexity of estimating partition functions on a quantum computer

Zherui Chen, Giacomo Nannicini

TL;DR

This work studies the complexity of estimating the partition function $Z(\beta)$ for Gibbs distributions defined by a Hamiltonian $H(x)$, focusing on the low-temperature regime. It proves a simple quantum lower bound of $\Omega(1/\epsilon)$ on the number of reflections through coherent Gibbs states required to estimate $Z(+\infty)$ to relative error $\epsilon$, by reducing the problem to the quantum Hamming-weight estimation and applying fixed-point quantum search. It also establishes a classical lower bound of $\Omega(1/\epsilon^2)$ queries in the same model, using a biased-coin reduction and Chernoff bounds. Together, these results imply near-optimality of current quantum algorithms in the reflection-access setting and clarify baseline limits for classical approaches, while leaving open questions about incorporating the configuration-space size $|\chi|$ into tighter bounds.

Abstract

We study the complexity of estimating the partition function $\mathsf{Z}(β)=\sum_{x\inχ} e^{-βH(x)}$ for a Gibbs distribution characterized by the Hamiltonian $H(x)$. We provide a simple and natural lower bound for quantum algorithms that solve this task by relying on reflections through the coherent encoding of Gibbs states. Our primary contribution is a $\varOmega(1/ε)$ lower bound for the number of reflections needed to estimate the partition function with a quantum algorithm. The proof is based on a reduction from the problem of estimating the Hamming weight of an unknown binary string.

A simple lower bound for the complexity of estimating partition functions on a quantum computer

TL;DR

This work studies the complexity of estimating the partition function for Gibbs distributions defined by a Hamiltonian , focusing on the low-temperature regime. It proves a simple quantum lower bound of on the number of reflections through coherent Gibbs states required to estimate to relative error , by reducing the problem to the quantum Hamming-weight estimation and applying fixed-point quantum search. It also establishes a classical lower bound of queries in the same model, using a biased-coin reduction and Chernoff bounds. Together, these results imply near-optimality of current quantum algorithms in the reflection-access setting and clarify baseline limits for classical approaches, while leaving open questions about incorporating the configuration-space size into tighter bounds.

Abstract

We study the complexity of estimating the partition function for a Gibbs distribution characterized by the Hamiltonian . We provide a simple and natural lower bound for quantum algorithms that solve this task by relying on reflections through the coherent encoding of Gibbs states. Our primary contribution is a lower bound for the number of reflections needed to estimate the partition function with a quantum algorithm. The proof is based on a reduction from the problem of estimating the Hamming weight of an unknown binary string.
Paper Structure (12 sections, 9 theorems, 29 equations)

This paper contains 12 sections, 9 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Basic definitions
  4. Existing lower bounds
  5. Contributions
  6. Quantum lower bound
  7. Preliminaries
  8. Construction of the Hamiltonian
  9. Classical lower bound
  10. Preliminaries
  11. Proof
  12. Conclusions

Key Result

Theorem 1

Any quantum algorithm that computes $\hat{Z}$ satisfying $|\hat{Z}-\mathsf{Z}(+\infty)|\leq \epsilon \cdot\mathsf{Z}(+\infty)$ for a given Hamiltonian $H$, where $H$ is accessed via the reflection through Gibbs states eq:mu_beta_state at different temperatures, takes $\varOmega({1/\epsilon})$ such r

Theorems & Definitions (12)

  • Theorem 1: Informal
  • Theorem 2: Informal
  • Proposition 1: Quantum query tight bound for the Hamming weight problem nayak1999quantum childs2022quantum
  • Corollary 1
  • proof
  • Proposition 2: Fixed-point quantum search yoder2014fixed
  • Theorem 3: Quantum lower bound for estimating partition function
  • proof
  • Proposition 3: Biased coin model, Claim D.1 of ge2020estimating
  • Proposition 4: Chernoff bound, Theorem 2 of Lecture3
  • ...and 2 more