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Quantum Information and Computation

Fast and efficient exact synthesis of single-qubit unitaries generated by clifford and T gates

KliuchnikovVadym, MaslovDmitri, MoscaMichele

Published Jul 1, 2013• 27 citations
Publisher

Abstract

In this paper, we show the equivalence of the set of unitaries computable by the circuits over the Clifford and T library and the set of unitaries over the ring Z[1/√2, i], in the single-qubit case...
DOI
10.5555/2535649.2535653

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In this paper, we show the equivalence of the set of unitaries computable by the circuits over the Clifford and T library and the set of unitaries over the ring Z[12,i]\mathbb{Z}[\frac{1}{\sqrt{2}},i]Z[2​1​,i], in the single-qubit case. We report an efficient synthesis algorithm, with an exact optimality guarantee on the number of Hadamard and T gates used. We conjecture that the equivalence of the sets of unitaries implementable by circuits over the Clifford and T library and unitaries over the ring Z[12,i]\mathbb{Z}[\frac{1}{\sqrt{2}},i]Z[2​1​,i] holds in the nnn-qubit case.
In this paper, we show the equivalence of the set of unitaries computable by the circuits over the Clifford and T library and the set of unitaries over the ring Z[12,i]\mathbb{Z}[\frac{1}{\sqrt{2}},i]Z[2​1​,i], in the single-qubit case. We report an efficient synthesis algorithm, with an exact optimality guarantee on the number of Hadamard gates used. We conjecture that the equivalence of the sets of unitaries implementable by circuits over the Clifford and T library and unitaries over the ring Z[12,i]\mathbb{Z}[\frac{1}{\sqrt{2}},i]Z[2​1​,i] holds in the nnn-qubit case.
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 [Quantum Inf. Comput. 13(7,8), 607–630 (2013)]. 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 πn. We present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of πn 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 Z[eiπn,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%.
We describe a new method for approximating an arbitrary nnn qubit unitary with precision ε\varepsilonε using a Clifford and T circuit with O(4nn(log⁡(1/ε)+n))O(4^{n}n(\log(1/\varepsilon)+n))O(4nn(log(1/ε)+n)) gates. The method is based on rounding off a unitary to a unitary over the ring Z[i,1/2]\mathbb{Z}[i,1/\sqrt{2}]Z[i,1/2​] and employing exact synthesis. We also show that any nnn qubit unitary over the ring Z[i,1/2]\mathbb{Z}[i,1/\sqrt{2}]Z[i,1/2​] with entries of the form (a+b2+ic+id2)/2k(a+b\sqrt{2}+ic+id\sqrt{2})/2^{k}(a+b2​+ic+id2​)/2k can be exactly synthesized using O(4nnk)O(4^{n}nk)O(4nnk) Clifford and T gates using two ancillary qubits. This new exact synthesis algorithm is an improvement over the best known exact synthesis method by B. Giles and P. Selinger requiring O(32nnk)O(3^{2^{n}}nk)O(32nnk) elementary gates.
Approximate synthesis of general single-qubit unitaries over the Clifford+T\sqrt{T}T​ gate set
For the standard Clifford+TTT gate set, deterministic, ancilla-free synthesis now attains the minimal TTT-count for general single-qubit unitaries (Morisaki et al., arXiv:2510.05816). The T\sqrt{T}T​ gate rotates by half the angle of TTT, generating a finer lattice of implementable operations. Access to this magic state is expected to lower the cost of deterministic and ancilla-free synthesis of general single-qubit unitaries (Kliuchnikov et al., arXiv:2203.10064), but no direct Clifford+T\sqrt{T}T​ algorithm existed for this case. We provide one by extending the integer lattice-point enumeration method of Morisaki et al. We adopt a resource state cost model based on the magic-state catalysis approach of Gidney and Fowler (arXiv:1812.01238). On Haar-random targets synthesized to precisions ranging from ε=10−3\varepsilon=10^{-3}ε=10−3 to 10−810^{-8}10−8, the cost of Clifford+T\sqrt{T}T​ circuits scales as 2.4log⁡2(1/ε)2.4\log_2(1/\varepsilon)2.4log2​(1/ε) compared to 3.0log⁡2(1/ε)3.0\log_2(1/\varepsilon)3.0log2​(1/ε) for TTT-count-optimal Clifford+TTT circuits. Once a one-time catalyst state is amortized, the Clifford+T\sqrt{T}T​ circuits are never costlier than their Clifford+TTT counterparts.
Exact synthesis of arbitrary unitary quantum gates into a Fault Tolerant (FT) gate library is generally impossible due to the limitation of available libraries. However, exact synthesis methods for special quantum gates can be found in the literature. In this paper, two approaches for exact synthesis of n-qubit ring multiplexed- RzR_{\text{z}}Rz​ and n-qubit ring diagonal gates Z(1/2, i)\mathbb{Z}(\pmb{1}/\sqrt{2},\ i)Z(1/2​, i) are presented. The proposed approaches use QSD to synthesize diagonal and multiplexed- RzR_{z}Rz​ gates into intermediate gates and then synthesis those to generate single-qubit ring gates. By using QSD, an n-qubit ring diagonal gate is decomposed into CNOT and RzR_{z}Rz​ gates that have an exact synthesis into CTL. The results show that the proposed approach for the diagonal ring gates generates O(2n+1)O(2^{n+1})O(2n+1) gates instead of O(n9n)O(n9^{n})O(n9n) gates produced by other methods.