Floating Point Representations in Quantum Circuit Synthesis

TL;DR

This paper introduces a quantum protocol for approximating single qubit rotations using floating point-like representations, analyzing its efficiency, bounds, and potential for parallelization in quantum circuit synthesis.

Contribution

It presents a novel floating point approach to quantum circuit synthesis, providing bounds and methods to optimize T-count and T-depth, including ancilla-assisted improvements.

Findings

The method approximates rotations with controlled T-count and T-depth.

Ancilla use can surpass lower bounds with high probability.

Most circuit cost can be parallelized.

Abstract

We provide a non-deterministic quantum protocol that approximates the single qubit rotations R_x(2a^2 b^2)$ using R_x(2a) and R_x(2b) and a constant number of Clifford and T operations. We then use this method to construct a "floating point" implementation of a small rotation wherein we use the aforementioned method to construct the exponent part of the rotation and also to combine it with a mantissa. This causes the cost of the synthesis to depend more strongly on the relative (rather than absolute) precision required. We analyze the mean and variance of the \Tcount required to use our techniques and provide new lower bounds for the T-count for ancilla free synthesis of small single-qubit axial rotations. We further show that our techniques can use ancillas to beat these lower bounds with high probability. We also discuss the T-depth of our method and see that the vast majority of the cost of the resultant circuits can be shifted to parallel computation paths.