Efficient Fault-Tolerant Single Qubit Gate Approximation And Universal Quantum Computation Without Using The Solovay-Kitaev Theorem

TL;DR

This paper presents a new method for approximating single-qubit gates fault-tolerantly with fewer gates than previous methods, avoiding the Solovay-Kitaev theorem, and achieving near-optimal gate counts.

Contribution

It introduces a recursive approach to approximate phase gates fault-tolerantly with a gate count of O(log(1/ε) log log(1/ε) ...), surpassing previous bounds.

Findings

Achieves approximation with fewer gates than Solovay-Kitaev-based methods.

Provides a straightforward, implementable recursive approximation technique.

Avoids reliance on the Solovay-Kitaev theorem for single-qubit gate approximation.

Abstract

Arbitrarily accurate fault-tolerant (FT) universal quantum computation can be carried out using the Clifford gates Z, S, CNOT plus the non-Clifford T gate. Moreover, a recent improvement of the Solovay-Kitaev theorem by Kuperberg implies that to approximate any single-qubit gate to an accuracy of $\epsilon > 0$ requires $\text{O}(\log^c[1/\epsilon])$ quantum gates with $c > 1.44042$. Can one do better? That was the question asked by Nielsen and Chuang in their quantum computation textbook. Specifically, they posted a challenge to efficiently approximate single-qubit gate, fault-tolerantly or otherwise, using $\Omega(\log[1/\epsilon])$ gates chosen from a finite set. Here I give a partial answer to this question by showing that this is possible using $\text{O}(\log[1/\epsilon] \log\log[1/\epsilon] \log\log\log[1/\epsilon] \cdots)$ FT gates chosen from a finite set depending on the value of $\epsilon$. The key idea is to construct an approximation of any phase gate in a FT way by recursion to any given accuracy $\epsilon > 0$. This method is straightforward to implement, easy to understand, and interestingly does not involve the Solovay-Kitaev theorem.