Efficient Fault-Tolerant Single Qubit Gate Approximation And Universal Quantum Computation Without Using The Solovay-Kitaev Theorem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909233813913600 |
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| author | Chau, H. F. |
| author_facet | Chau, H. F. |
| contents | 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 $ε> 0$ requires $\text{O}(\log^c[1/ε])$ 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 $Ω(\log[1/ε])$ gates chosen from a finite set. Here I give a partial answer to this question by showing that this is possible using $\text{O}(\log[1/ε] \log\log[1/ε] \log\log\log[1/ε] \cdots)$ FT gates chosen from a finite set depending on the value of $ε$. The key idea is to construct an approximation of any phase gate in a FT way by recursion to any given accuracy $ε> 0$. This method is straightforward to implement, easy to understand, and interestingly does not involve the Solovay-Kitaev theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_04846 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Efficient Fault-Tolerant Single Qubit Gate Approximation And Universal Quantum Computation Without Using The Solovay-Kitaev Theorem Chau, H. F. Quantum Physics 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 $ε> 0$ requires $\text{O}(\log^c[1/ε])$ 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 $Ω(\log[1/ε])$ gates chosen from a finite set. Here I give a partial answer to this question by showing that this is possible using $\text{O}(\log[1/ε] \log\log[1/ε] \log\log\log[1/ε] \cdots)$ FT gates chosen from a finite set depending on the value of $ε$. The key idea is to construct an approximation of any phase gate in a FT way by recursion to any given accuracy $ε> 0$. This method is straightforward to implement, easy to understand, and interestingly does not involve the Solovay-Kitaev theorem. |
| title | Efficient Fault-Tolerant Single Qubit Gate Approximation And Universal Quantum Computation Without Using The Solovay-Kitaev Theorem |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2406.04846 |