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

TL;DR

Fine-grained complexity lower bounds for approximating the quantum partition function (QPF) with an arbitrary constant relative error are demonstrated and a quantum algorithm is provided that runs in $O(\sqrt{2^n})$ time for an arbitrary $1/\mathrm{poly}(n)$ relative error, matching the lower bounds and improving the state-of-the-art algorithm.

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.

Theorems & Definitions (42)

• Theorem 1.1: Informal version of \ref{['theorem:main']}
• Theorem 1.2: Informal version of \ref{['thm:SETH hardness for 3LH']}
• Remark 1
• Theorem 1.3: Informal version of \ref{['thm:main_qpf']}
• Theorem 1.4: Informal version of \ref{['thm:main_qpf_alg']}
• Definition 1: Multi-control-NOT gate and Toffoli gate
• Remark 2: Decompose $C^k\textsc{not}$ into Toffolis NC10
• Remark 3: Decompose Toffoli into elementary gates NC10
• Definition 2: Local Hamiltonian
• Definition 3: $k$-local Hamiltonian ($k\textrm{-}\mathrm{LH}$) problem