Fine-Grained Complexity for Quantum Problems from Size-Preserving Circuit-to-Hamiltonian Constructions

2026-02-16

AI Summary

This work investigates fine-grained time complexity lower bounds for the local Hamiltonian problem and the approximation of quantum partition functions under both classical and quantum computational models. Assuming the Strong Exponential Time Hypothesis (SETH) and its quantum analogue (QSETH), we establish matching upper and lower bounds for any constant relative error: specifically, we prove that the 3-local Hamiltonian problem requires time at least $O(2^{(1-\varepsilon)n})$ classically and $O(2^{(1-\varepsilon)n/2})$ quantumly. Furthermore, we design a quantum algorithm that approximates the partition function in $O(\sqrt{2^n})$ time, matching the lower bound and outperforming the best known algorithms in the low-temperature regime. A key technical contribution is the first size-preserving circuit-to-Hamiltonian construction, which encodes a $T$-step quantum circuit into a $(d+1)$-local Hamiltonian acting on $N + O(T^{1/d})$ qubits, substantially improving upon the traditional unary clock method.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Local Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

The local Hamiltonian (LH) problem is the canonical $\mathsf{QMA}$-complete problem introduced by Kitaev. In this paper, we show its hardness in a very strong sense: we show that the 3-local Hamiltonian problem on $n$ qubits cannot be solved classically in time $O(2^{(1-\varepsilon)n})$ for any $\varepsilon>0$ under the Strong Exponential-Time Hypothesis (SETH), and cannot be solved quantumly in time $O(2^{(1-\varepsilon)n/2})$ for any $\varepsilon>0$ under the Quantum Strong Exponential-Time Hypothesis (QSETH). These lower bounds give evidence that the currently known classical and quantum algorithms for LH cannot be significantly improved. Furthermore, we are able to demonstrate fine-grained complexity lower bounds for approximating the quantum partition function (QPF) with an arbitrary constant relative error. Approximating QPF with relative error is known to be equivalent to approximately counting the dimension of the solution subspace of $\mathsf{QMA}$ problems. We show the SETH and QSETH hardness to estimate QPF with constant relative error. We then provide a quantum algorithm that runs in $O(\sqrt{2^n})$ time for an arbitrary $1/\mathrm{poly}(n)$ relative error, matching our lower bounds and improving the state-of-the-art algorithm by Bravyi, Chowdhury, Gosset, and Wocjan (Nature Physics 2022) in the low-temperature regime. To prove our fine-grained lower bounds, we introduce the first size-preserving circuit-to-Hamiltonian construction that encodes the computation of a $T$-time quantum circuit acting on $N$ qubits into a $(d+1)$-local Hamiltonian acting on $N+O(T^{1/d})$ qubits. This improves the standard construction based on the unary clock, which uses $N+O(T)$ qubits.

Problem

• Local Hamiltonian
• Quantum Partition Function
• Fine-Grained Complexity
• SETH
• QSETH

Innovation

• size-preserving circuit-to-Hamiltonian
• fine-grained complexity
• quantum partition function
• Strong Exponential-Time Hypothesis
• local Hamiltonian problem