Detailed Analysis of Circuit-to-Hamiltonian Mappings

Understand

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.