On connectivity-dependent resource requirements for digital quantum simulation of $d$-level particles

TL;DR

This paper analyzes how hardware connectivity impacts the resource requirements, specifically SWAP gates, for simulating d-level particles in quantum computers, comparing various encodings and connectivities.

Contribution

It extends previous encoding efficiency studies by incorporating hardware connectivity effects and optimizing SWAP gate schedules for different qudit encodings and connectivities.

Findings

Ladder connectivity reduces SWAP gate counts compared to linear arrays.

Square grid connectivity offers modest advantages over ladder arrays.

Optimization tools effectively minimize resource overheads in quantum simulations.

Abstract

A primary objective of quantum computation is to efficiently simulate quantum physics. Scientifically and technologically important quantum Hamiltonians include those with spin-$s$, vibrational, photonic, and other bosonic degrees of freedom, i.e. problems composed of or approximated by $d$-level particles (qudits). Recently, several methods for encoding these systems into a set of qubits have been introduced, where each encoding's efficiency was studied in terms of qubit and gate counts. Here, we build on previous results by including effects of hardware connectivity. To study the number of SWAP gates required to Trotterize commonly used quantum operators, we use both analytical arguments and automatic tools that optimize the schedule in multiple stages. We study the unary (or one-hot), Gray, standard binary, and block unary encodings, with three connectivities: linear array, ladder array, and square grid. Among other trends, we find that while the ladder array leads to substantial efficiencies over the linear array, the advantage of the square over the ladder array is less pronounced. These results are applicable in hardware co-design and in choosing efficient qudit encodings for a given set of near-term quantum hardware. Additionally, this work may be relevant to the scheduling of other quantum algorithms for which matrix exponentiation is a subroutine.