Optimally generating $\mathfrak{su}(2^N)$ using Pauli strings

TL;DR

This paper identifies the minimal set of Pauli-based Hamiltonians needed to generate the entire su(2^N) algebra, provides an optimal algorithm for their implementation, and explores how to optimize their generation rate.

Contribution

It introduces the minimal set of $2N+1$ Pauli Hamiltonians for su(2^N), offers an optimal algorithm for their sequence generation, and discusses rate optimization through anticommuting pairs.

Findings

Minimal generating set contains 2N+1 elements.

An optimal algorithm for generating any Pauli rotation.

Generation rate can be improved by tuning anticommuting pairs.

Abstract

Any quantum computation consists of a sequence of unitary evolutions described by a finite set of Hamiltonians. When this set is taken to consist of only products of Pauli operators, we show that the minimal such set generating $\mathfrak{su}(2^{N})$ contains $2N+1$ elements. We provide a number of examples of such generating sets and furthermore provide an algorithm for producing a sequence of rotations corresponding to any given Pauli rotation, which is shown to have optimal complexity. We also observe that certain sets generate $\mathfrak{su}(2^{N})$ at a faster rate than others, and we show how this rate can be optimized by tuning the fraction of anticommuting pairs of generators. Finally, we briefly comment on implications for measurement-based and trapped ion quantum computation as well as the construction of fault-tolerant gate sets.