The Feynman-Kitaev computer's clock: bias, gaps, idling and pulse tuning

TL;DR

This paper analyzes the spectral properties and optimization of the clock in Feynman's quantum computer, improving gap bounds, success probabilities, and implementation costs for quantum computational models.

Contribution

It introduces new bounds on the Hamiltonian gap, a fast mixing idling clock construction, and optimized pulse clock implementation, advancing quantum computational clock design.

Findings

Better lower bound on the Feynman Hamiltonian gap

High success probability with logarithmic clock qubits

Improved costs for pulse clock implementation

Abstract

We present a collection of results about the clock in Feynman's computer construction and Kitaev's Local Hamiltonian problem. First, by analyzing the spectra of quantum walks on a line with varying endpoint terms, we find a better lower bound on the gap of the Feynman Hamiltonian, which translates into a less strict promise gap requirement for the QMA-complete Local Hamiltonian problem. We also translate this result into the language of adiabatic quantum computation. Second, introducing an idling clock construction with a large state space but fast Cesaro mixing, we provide a way for achieving an arbitrarily high success probability of computation with Feynman's computer with only a logarithmic increase in the number of clock qubits. Finally, we tune and thus improve the costs (locality, gap scaling) of implementing a (pulse) clock with a single excitation.