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

TL;DR

This paper strengthens the classification of the guided Local Hamiltonian problem by proving BQP-hardness for 2-local Hamiltonians with guiding-state fidelity as large as $1 - \mathrm{poly}(n)^{-1}$ and extending the hardness to estimating excited-state energies. The authors build on the Feynman-Kitaev circuit-to-Hamiltonian framework, employing pre-idling, gap amplification, and perturbative gadgets to reduce locality while preserving the low-energy structure and fidelity of the guiding state. They also establish containment in BQP for relevant parameter regimes, showing that GLHLE remains tractable under certain locality and fidelity constraints. The work highlights the nuanced source of quantum hardness in Hamiltonian problems, balancing precision requirements with guiding-state properties, and opens avenues for understanding the hardness landscape in relation to physical Hamiltonians and the quantum PCP conjecture.

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 $\left(\frac12 - Ω\left(\frac{1}{\mathop{poly}(n)}\right)\right)$. 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 $1 - Ω\left(\frac{1}{\mathop{poly}(n)}\right)$, 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.

Figures (3)

• Figure 1: Summary of the proof used to lower bound $\lambda_0(H)$. At each step, we 'peel off' a part of the Hilbert space until we are left with a single state spanning the groundspace of $H$. The arrows show the direction of inclusions: e.g. $\mathcal{H} \supset \mathcal{S}_{\text{legal}}$ and $\mathcal{H} \supset \mathcal{S}_{\text{legal}}^\perp$. Underneath the bottom row of subspaces is written the smallest eigenvalue of a particular Hamiltonian within that subspace (i.e. the minimum energy penalty given by that Hamiltonian to states contained in that subspace).
• Figure 2: Visualization of the perturbative gadget for a Hamiltonian consisting of a single $3$-local term. The ancilla registers are indicated with '$a_i$'.
• Figure 3: Visualization of all steps in the reduction used in the proof of theorem \ref{['thm:excited_states']}.

Theorems & Definitions (30)

• Definition 1: $\mathsf{BQP}$
• Definition 2: Semi-classical state - from grilo2015qma
• Definition 3: Guided Local Hamiltonian Low Energy
• Theorem 1: $\mathsf{BQP}$-hardness of $\mathsf{GLHLE}$
• Theorem 2: Containment in $\BQP$ of $\mathsf{GLHLE}$
• Theorem 3: $\mathsf{BQP}$-completeness of $\mathsf{GLHLE}$
• Lemma 1: 'Projection lemma' -- Lemma 1 from kempe2006
• proof
• Lemma 2
• Lemma 3