Quantum Lower Bound for Estimating Partition Functions

Core Concepts

Any quantum algorithm that estimates the partition function Z(+∞) with relative error ǫ requires Ω(1/ǫ) reflections through the coherent encoding of Gibbs states.

Abstract

The paper studies the complexity of estimating the partition function Z(β) = Σx∈χ e^(-βH(x)) for a Gibbs distribution characterized by the Hamiltonian H(x). The key highlights are: The authors provide a simple and natural lower bound of Ω(1/ǫ) on the number of reflections needed by a quantum algorithm to estimate Z(+∞) with relative error ǫ. This matches the scaling of existing quantum algorithms. The authors also prove a classical lower bound of Ω(1/ǫ^2) on the number of queries to the Hamiltonian H(x) needed by a classical algorithm to estimate Z(+∞) with relative error ǫ. This also matches the scaling of the fastest classical algorithms. The proofs rely on a reduction from the problem of estimating the Hamming weight of an unknown binary string, using quantum query complexity results and fixed-point quantum search. The authors discuss the implications of their results and mention that a stronger quantum lower bound incorporating the size of the ground set χ remains an open question.

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Key Insights Distilled From

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

by Zherui Chen,... at arxiv.org 04-04-2024

https://arxiv.org/pdf/2404.02414.pdf

Deeper Inquiries

How can the quantum lower bound be extended to incorporate the size of the ground set χ in addition to the precision parameter ǫ

To extend the quantum lower bound to incorporate the size of the ground set χ, we can introduce a more intricate construction that involves the interplay between the precision parameter ǫ and the cardinality of χ. One approach could be to design a Hamiltonian that scales in complexity with the size of the ground set χ. By carefully selecting the Hamiltonian function H(x) to have dependencies on the cardinality of χ, we can create a scenario where the number of reflections required by a quantum algorithm increases with the size of χ. This would result in a lower bound that incorporates both the precision parameter ǫ and the size of the ground set χ, providing a more comprehensive understanding of the complexity of estimating partition functions on a quantum computer.

Can the classical lower bound be improved beyond Ω(1/ǫ^2) by using a different approach than the reduction to the Hamming weight problem

To potentially improve the classical lower bound beyond Ω(1/ǫ^2), an alternative approach could involve exploring different reduction strategies or problem formulations. Instead of relying solely on the Hamming weight problem, which has been extensively studied in the context of lower bounds, researchers could investigate other related combinatorial or statistical problems that exhibit similar characteristics to partition functions or Gibbs distributions. By identifying a problem with specific properties that allow for a more intricate reduction, it may be possible to derive a stronger lower bound that surpasses the current Ω(1/ǫ^2) limitation. This approach would involve delving into the nuances of different problem domains and finding connections that lend themselves to more robust lower bound proofs.

Are there other problems related to partition functions or Gibbs distributions where similar lower bound techniques could be applied

The techniques used to establish lower bounds for estimating partition functions or Gibbs distributions can be applied to a variety of related problems in statistical physics, combinatorial optimization, and quantum information theory. For instance, problems involving counting proper k-colorings of graphs, estimating volumes of convex bodies, or sampling log-concave distributions share similarities with partition function estimation and could benefit from similar lower bound techniques. By adapting the constructions and reductions used in the context of partition functions, researchers can explore the complexity of these related problems and provide insights into the fundamental limits of classical and quantum algorithms in various domains. This approach broadens the applicability of lower bound techniques and contributes to a deeper understanding of computational challenges in diverse areas of research.