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Efficient Decomposition of Single-Qubit Gates into $V$ Basis Circuits · Around

2013

Efficient Decomposition of Single-Qubit Gates into $V$ Basis Circuits

Bocharov, Alex, Gurevich, Yuri, Svore, Krysta M.

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We develop the first constructive algorithms for compiling single-qubit unitary gates into circuits over the universal $V$ basis.

The $V$ basis is an alternative universal basis to the more commonly studied $\{H,T\}$ basis.
We propose two classical algorithms for quantum circuit compilation: the first algorithm has expected polynomial time (in precision $\log(1/\epsilon)$) and offers a depth/precision guarantee that improves upon state-of-the-art methods for compiling into the $\{H,T\}$ basis by factors ranging from 1.86 to $\log_2(5)$.
The second algorithm is analogous to direct search and yields circuits a factor of 3 to 4 times shorter than our first algorithm, and requires time exponential in $\log(1/\epsilon)$; however, we show that in practice the runtime is reasonable for an important range of target precisions.

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M. Rabin and J. Shallit, Comm. Pure and Appl. Math. 39
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A. Lubotsky, R. Phillips, and P. Sarnak, Comm. Pure and Appl. Math. 40
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Cited in the paper.
V. Kliuchnikov, D. Maslov, and M. Mosca, “Fast and efficient exact synthesis of single qubit unitaries generated by clifford and t gates,” (2012b), 1206.5236
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V. Kliuchnikov, D. Maslov, and M. Mosca, “Asymptotically optimal approximation of single qubit unitaries by clifford and t circuits using a constant number of ancillary qubits,” (2012c), 1212.0822
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N. C. Jones, J. D. Whitfield, P. L. McMahon, M.-H. Yung, R. V. Meter, A. Aspuru-Guzik, and Y. Yamamoto, “Simulating chemistry efficiently on fault-tolerant quantum computers,” (2012), 1204.0567
Original
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Later among the works it cites.

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