Complexity of the Guided Local Hamiltonian Problem: Improved Parameters and Extension to Excited States

TL;DR

This paper demonstrates that the guided local Hamiltonian problem remains computationally hard under more practical conditions, including 2-local Hamiltonians, high overlap guiding states, and excited state energy estimation.

Contribution

It extends the BQP-completeness of the guided local Hamiltonian problem to 2-local Hamiltonians, high-overlap guiding states, and excited states, improving previous results.

Findings

The problem is BQP-complete for 2-local Hamiltonians.

High-overlap guiding states still lead to BQP-completeness.

The method extends to estimating energies of excited states.

Abstract

Recently it was shown that the so-called guided local Hamiltonian problem -- estimating the smallest eigenvalue of a $k$-local Hamiltonian when provided with a description of a quantum state ('guiding state') that is guaranteed to have substantial overlap with the true groundstate -- is BQP-complete for $k \geq 6$ when the required precision is inverse polynomial in the system size $n$, and remains hard even when the overlap of the guiding state with the groundstate is close to a constant $\left(\frac12 - \Omega\left(\frac{1}{\mathop{poly}(n)}\right)\right)$. We improve upon this result in three ways: by showing that it remains BQP-complete when i) the Hamiltonian is 2-local, ii) the overlap between the guiding state and target eigenstate is as large as $1 - \Omega\left(\frac{1}{\mathop{poly}(n)}\right)$, and iii) when one is interested in estimating energies of excited states, rather than just the groundstate. Interestingly, iii) is only made possible by first showing that ii) holds.