3-Local Hamiltonian Problem and Constant Relative Error Quantum Partition Function Approximation: $O(2^{frac{n}{2}})$ Algorithm Is Nearly Optimal under QSETH

2025-10-08

AI Summary

This work investigates the fine-grained computational complexity of the 3-local Hamiltonian problem and quantum partition function approximation with constant relative error. Under the Quantum Strong Exponential Time Hypothesis (QSETH), we establish the first tight lower bounds for both problems at fixed locality $k=3$: neither admits a quantum algorithm running in $O(2^{n/2(1-varepsilon)})$ time for any $varepsilon > 0$. Complementing these, we design matching $O^*(2^{n/2})$ upper-bound algorithms, demonstrating that existing exponential-time quantum algorithms are nearly optimal. Our results fill a critical gap in fine-grained complexity analysis under QSETH for low-locality Hamiltonians and reveal fundamental algorithmic limits inherent to the 3-local regime.

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: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We investigate the computational complexity of the Local Hamiltonian (LH) problem and the approximation of the Quantum Partition Function (QPF), two central problems in quantum many-body physics and quantum complexity theory. Both problems are known to be QMA-hard, and under the widely believed assumption that $mathsf{BQP} eq mathsf{QMA}$, no efficient quantum algorithm exits. The best known quantum algorithm for LH runs in $Oigl(2^{frac{n}{2}(1 - o(1))}igr)$ time, while for QPF, the state-of-the-art algorithm achieves relative error $δ$ in $O^astigl(frac{1}δsqrt{frac{2^n}{Z}}igr)$ time, where $Z$ denotes the value of the partition function. A nature open question is whether more efficient algorithms exist for both problems. In this work, we establish tight conditional lower bounds showing that these algorithms are nearly optimal. Under the plausible Quantum Strong Exponential Time Hypothesis (QSETH), we prove that no quantum algorithm can solve either LH or approximate QPF significantly faster than $O(2^{n/2})$, even for 3-local Hamiltonians. In particular, we show: 1) 3-local LH cannot be solved in time $O(2^{frac{n}{2}(1-varepsilon)})$ for any $varepsilon > 0$ under QSETH; 2) 3-local QPF cannot be approximated up to any constant relative error in $O(2^{frac{n}{2}(1-varepsilon)})$ time for any $varepsilon > 0$ under QSETH; and 3) we present a quantum algorithm that approximates QPF up to relative error $1/2 + 1/mathrm{poly}(n)$ in $O^ast(2^{n/2})$ time, matching our conditional lower bound. Notably, our results provide the first fine-grained lower bounds for both LH and QPF with fixed locality. This stands in sharp contrast to QSETH and the trivial fine-grained lower bounds for LH, where the locality of the SAT instance and the Hamiltonian depends on the parameter $varepsilon$ in the $O(2^{frac{n}{2}(1-varepsilon)})$ running time.

Problem

• Establishing optimal quantum algorithm complexity for Local Hamiltonian problems
• Proving tight lower bounds for Quantum Partition Function approximation
• Demonstrating near-optimality of O(2^(n/2)) algorithms under QSETH assumption

Innovation

• Establishes tight conditional lower bounds under QSETH
• Shows near optimality of O(2^(n/2)) quantum algorithms
• Provides first fine-grained bounds for fixed locality problems