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Improved Hardness Results for the Guided Local Hamiltonian Problem

Chris Cade, Marten Folkertsma, Sevag Gharibian, Ryu Hayakawa, François Le Gall, Tomoyuki Morimae, Jordi Weggemans

TL;DR

This work analyzes the Guided k-local Hamiltonian problem (GLH), where a guiding state with fidelity $\delta$ to the ground space is provided to aid energy estimation. It proves that BQP-hardness persists for constant $k\ge 2$ even when $\delta=1-1/\mathrm{poly}(n)$ and extends hardness to excited-state energies via the GLHLE framework, including physically motivated 2-local Hamiltonians on 2D lattices such as XY and Heisenberg models. The authors introduce semi-classical subset and encoded states to realize high-fidelity guiding states under perturbative reductions and to enable tractable classical sampling, enabling hardness results for a broad class of Hamiltonians. They also develop GLHLE to address low-energy (excited-state) estimation, showing hardness in that regime and thereby broadening the landscape of quantum hardness relevant to quantum chemistry applications. Collectively, these results advance our theoretical understanding of when quantum computers can outperform classical methods in estimating energies for practically motivated quantum systems.

Abstract

Estimating the ground state energy of a local Hamiltonian is a central problem in quantum chemistry. In order to further investigate its complexity and the potential of quantum algorithms for quantum chemistry, Gharibian and Le Gall (STOC 2022) recently introduced the guided local Hamiltonian problem (GLH), which is a variant of the local Hamiltonian problem where an approximation of a ground state (which is called a guiding state) is given as an additional input. Gharibian and Le Gall showed quantum advantage (more precisely, BQP-completeness) for GLH with $6$-local Hamiltonians when the guiding state has fidelity (inverse-polynomially) close to $1/2$ with a ground state. In this paper, we optimally improve both the locality and the fidelity parameter: we show that the BQP-completeness persists even with 2-local Hamiltonians, and even when the guiding state has fidelity (inverse-polynomially) close to 1 with a ground state. Moreover, we show that the BQP-completeness also holds for 2-local physically motivated Hamiltonians on a 2D square lattice or a 2D triangular lattice. Beyond the hardness of estimating the ground state energy, we also show BQP-hardness persists when considering estimating energies of excited states of these Hamiltonians instead. Those make further steps towards establishing practical quantum advantage in quantum chemistry.

Improved Hardness Results for the Guided Local Hamiltonian Problem

TL;DR

This work analyzes the Guided k-local Hamiltonian problem (GLH), where a guiding state with fidelity to the ground space is provided to aid energy estimation. It proves that BQP-hardness persists for constant even when and extends hardness to excited-state energies via the GLHLE framework, including physically motivated 2-local Hamiltonians on 2D lattices such as XY and Heisenberg models. The authors introduce semi-classical subset and encoded states to realize high-fidelity guiding states under perturbative reductions and to enable tractable classical sampling, enabling hardness results for a broad class of Hamiltonians. They also develop GLHLE to address low-energy (excited-state) estimation, showing hardness in that regime and thereby broadening the landscape of quantum hardness relevant to quantum chemistry applications. Collectively, these results advance our theoretical understanding of when quantum computers can outperform classical methods in estimating energies for practically motivated quantum systems.

Abstract

Estimating the ground state energy of a local Hamiltonian is a central problem in quantum chemistry. In order to further investigate its complexity and the potential of quantum algorithms for quantum chemistry, Gharibian and Le Gall (STOC 2022) recently introduced the guided local Hamiltonian problem (GLH), which is a variant of the local Hamiltonian problem where an approximation of a ground state (which is called a guiding state) is given as an additional input. Gharibian and Le Gall showed quantum advantage (more precisely, BQP-completeness) for GLH with -local Hamiltonians when the guiding state has fidelity (inverse-polynomially) close to with a ground state. In this paper, we optimally improve both the locality and the fidelity parameter: we show that the BQP-completeness persists even with 2-local Hamiltonians, and even when the guiding state has fidelity (inverse-polynomially) close to 1 with a ground state. Moreover, we show that the BQP-completeness also holds for 2-local physically motivated Hamiltonians on a 2D square lattice or a 2D triangular lattice. Beyond the hardness of estimating the ground state energy, we also show BQP-hardness persists when considering estimating energies of excited states of these Hamiltonians instead. Those make further steps towards establishing practical quantum advantage in quantum chemistry.
Paper Structure (5 sections, 2 theorems, 5 equations)

This paper contains 5 sections, 2 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semi-classical states
  4. Non-2SLD Hamiltonian and geometry of interaction
  5. GLHLE hardness constructions

Key Result

Theorem 1

For any $\delta\in (0,1-\Omega(1/\mathrm{poly}(n)))$, constant $k \geq 2$ and some integer $0\leq c \leq \mathcal{O}(poly(n))$, there exist $a,b\in [-1,1]$ with $b-a\in \Omega(1/\mathrm{poly}(n))$ such that k-$\text{GLHLE}$ is $\textup{BQP}$-hard. Moreover, it is still $\textup{BQP}$-hard if the 2-l

Theorems & Definitions (6)

  • Theorem 1: Main result
  • Definition 2: Semi-classical subset state
  • Definition 3: Semi-classical encoded state
  • Lemma 4
  • Definition 5: 2SLD interaction cubitt2016complexity
  • Definition 6: Guided Local Hamiltonian Low Energy