Hardness of approximation for ground state problems

TL;DR

This paper investigates the computational hardness of approximating properties of quantum ground states, establishing QCMA-hardness for problems like ground state connectivity and entanglement estimation, advancing understanding of quantum complexity.

Contribution

It demonstrates QCMA-hardness of approximating ground state properties, such as GSCON and GSE, and simplifies the proof for NP-completeness of a k-SAT reconfiguration problem.

Findings

QCMA-complete within ratio N^(1-eps) for GSCON

QCMA-hard within ratio N^(1-eps) for GSE

NP-complete approximation for k-SAT reconfiguration

Abstract

After nearly two decades of research, the question of a quantum PCP theorem for quantum Constraint Satisfaction Problems (CSPs) remains wide open. As a result, proving QMA-hardness of approximation for ground state energy estimation has remained elusive. Recently, it was shown [Bittel, Gharibian, Kliesch, CCC 2023] that a natural problem involving variational quantum circuits is QCMA-hard to approximate within ratio N^(1-eps) for any eps > 0 and N the input size. Unfortunately, this problem was not related to quantum CSPs, leaving the question of hardness of approximation for quantum CSPs open. In this work, we show that if instead of focusing on ground state energies, one considers computing properties of the ground space, QCMA-hardness of computing ground space properties can be shown. In particular, we show that it is (1) QCMA-complete within ratio N^(1-eps) to approximate the Ground State Connectivity problem (GSCON), and (2) QCMA-hard within the same ratio to estimate the amount of entanglement of a local Hamiltonian's ground state, denoted Ground State Entanglement (GSE). As a bonus, a simplification of our construction yields NP-completeness of approximation for a natural k-SAT reconfiguration problem, to be contrasted with the recent PCP-based PSPACE hardness of approximation results for a different definition of k-SAT reconfiguration [Karthik C.S. and Manurangsi, 2023, and Hirahara, Ohsaka, STOC 2024].