Detailed Analysis of Circuit-to-Hamiltonian Mappings

TL;DR

This paper provides tight bounds on the ground state energies of circuit-to-Hamiltonian mappings, improving understanding of their spectral properties and implications for quantum complexity, especially in clock constructions and quantum walks.

Contribution

It offers exponentially tight bounds on ground state energies for standard and dynamic initialisation clock Hamiltonians, and introduces tools for analyzing low-energy subspaces of quantum walks.

Findings

Exponential bounds on ground state energies for clock Hamiltonians

Improved scaling bounds with constant rejection probability

New tools for analyzing quantum walk low-energy subspaces

Abstract

The circuit-to-Hamiltonian construction has found widespread use within the field of Hamiltonian complexity, particularly for proving QMA-hardness results. In this work we examine the ground state energies of the Hamiltonian for standard clock constructions and those which require dynamic initialisation. We put exponentially tight bounds on these ground state energies and also determine improved scaling bounds in the case where there is a constant probability of the computation being rejected. Furthermore, we prove a collection of results concerning the low-energy subspace of quantum walks on a line with energy penalties appearing at any point along the walk and introduce some general tools that may be useful for such analyses.