Exact synthesis of single-qubit unitaries over Clifford-cyclotomic gate sets

Abstract

We generalize an efficient exact synthesis algorithm for single-qubit unitaries over the Clifford+T gate set which was presented by Kliuchnikov, Maslov and Mosca. Their algorithm takes as input an exactly synthesizable single-qubit unitary--one which can be expressed without error as a product of Clifford and T gates--and outputs a sequence of gates which implements it. The algorithm is optimal in the sense that the length of the sequence, measured by the number of T gates, is smallest possible. In this paper, for each positive even integer $n$ we consider the "Clifford-cyclotomic" gate set consisting of the Clifford group plus a z-rotation by $\frac{\pi }{n}$. We present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of $\frac{\pi }{n}$ z-rotations. For the Clifford+T case $n=4$ the group of exactly synthesizable unitaries was shown to be equal to the group of unitaries with entries over the ring $\mathbb{Z}[e^{i\frac{\pi }{n}},1/2]$. We prove that this characterization holds for a handful of other small values of $n$ but the fraction of positive even integers for which it fails to hold is 100%.

v2: published version

References in corpus (3)

• Asymptotically Optimal Topological Quantum Compiling
• Efficient synthesis of probabilistic quantum circuits with fallback
• Reducing the quantum computing overhead with complex gate distillation

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• Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits
• Enabling Dataflow Optimization for Quantum Programs
• Topological qubits from valence bond solids
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• Generators and Relations for the Group On(Z[1/2])
• CSS-T Codes from Reed Muller Codes
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