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

• URL: http://arxiv.org/abs/2406.04846v2
• Date: Sun, 30 Jun 2024 07:24:25 GMT
• Title: Efficient Fault-Tolerant Single Qubit Gate Approximation And Universal Quantum Computation Without Using The Solovay-Kitaev Theorem
• Authors: H. F. Chau,
• Abstract summary: A recent improvement of the Solovay-Kitaev theorem implies that to approximate any single-qubit gate to an accuracy of $epsilon > 0$ requires $textO(logc[1/epsilon)$ quantum gates with $c > 1.44042$. Here I give a partial answer to this question by showing that this is possible using $textO(log[/epsilon] loglog[/epsilon] cdots)$ FT gates chosen from a finite set depending on the value of $
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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