Exact Synthesis of Multiqubit Clifford-Cyclotomic Circuits
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866929312967426048 |
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| author | Amy, Matthew Glaudell, Andrew N. Kelso, Shaun Maxwell, William Mendelson, Samuel S. Ross, Neil J. |
| author_facet | Amy, Matthew Glaudell, Andrew N. Kelso, Shaun Maxwell, William Mendelson, Samuel S. Ross, Neil J. |
| contents | Let $n\geq 8$ be divisible by 4. The Clifford-cyclotomic gate set $\mathcal{G}_n$ is the universal gate set obtained by extending the Clifford gates with the $z$-rotation $T_n = \mathrm{diag}(1,ζ_n)$, where $ζ_n$ is a primitive $n$-th root of unity. In this note, we show that, when $n$ is a power of 2, a multiqubit unitary matrix $U$ can be exactly represented by a circuit over $\mathcal{G}_n$ if and only if the entries of $U$ belong to the ring $\mathbb{Z}[1/2,ζ_n]$. We moreover show that $\log(n)-2$ ancillas are always sufficient to construct a circuit for $U$. Our results generalize prior work to an infinite family of gate sets and show that the limitations that apply to single-qubit unitaries, for which the correspondence between Clifford-cyclotomic operators and matrices over $\mathbb{Z}[1/2,ζ_n]$ fails for all but finitely many values of $n$, can be overcome through the use of ancillas. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_07741 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Exact Synthesis of Multiqubit Clifford-Cyclotomic Circuits Amy, Matthew Glaudell, Andrew N. Kelso, Shaun Maxwell, William Mendelson, Samuel S. Ross, Neil J. Quantum Physics Let $n\geq 8$ be divisible by 4. The Clifford-cyclotomic gate set $\mathcal{G}_n$ is the universal gate set obtained by extending the Clifford gates with the $z$-rotation $T_n = \mathrm{diag}(1,ζ_n)$, where $ζ_n$ is a primitive $n$-th root of unity. In this note, we show that, when $n$ is a power of 2, a multiqubit unitary matrix $U$ can be exactly represented by a circuit over $\mathcal{G}_n$ if and only if the entries of $U$ belong to the ring $\mathbb{Z}[1/2,ζ_n]$. We moreover show that $\log(n)-2$ ancillas are always sufficient to construct a circuit for $U$. Our results generalize prior work to an infinite family of gate sets and show that the limitations that apply to single-qubit unitaries, for which the correspondence between Clifford-cyclotomic operators and matrices over $\mathbb{Z}[1/2,ζ_n]$ fails for all but finitely many values of $n$, can be overcome through the use of ancillas. |
| title | Exact Synthesis of Multiqubit Clifford-Cyclotomic Circuits |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2311.07741 |