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arXiv:1212.0506 · Physical Review A

Exact synthesis of multiqubit Clifford+ T circuits

Brett Gordon Giles, Peter Selinger

Published Mar 26, 2013• 162 citations
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Abstract

We prove that a unitary matrix has an exact representation over the Clifford+$T$ gate set with local ancillas if and only if its entries are in the ring $\mathbb{Z}[\frac{1}{\sqrt{2}},i]$. Moreover, we show that one ancilla always suffices. These facts were conjectured by Kliuchnikov, Maslov, and Mosca. We obtain an algorithm for synthesizing a exact Clifford+$T$ circuit from any such $n$-qubit operator. We also characterize the Clifford+$T$ operators that can be represented without ancillas.

DOI
10.1103/physreva.87.032332

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We prove that a unitary matrix has an exact representation over the Clifford+TTT gate set with local ancillas if and only if its entries are in the ring Z[12,i]\mathbb{Z}[\frac{1}{\sqrt{2}},i]Z[2​1​,i]. Moreover, we show that one ancilla always suffices. These facts were conjectured by Kliuchnikov, Maslov, and Mosca. We obtain an algorithm for synthesizing a exact Clifford+TTT circuit from any such nnn-qubit operator. We also characterize the Clifford+TTT operators that can be represented without ancillas.
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.
Kliuchnikov, Maslov, and Mosca proved in 2012 that a 2×2 unitary matrix V can be exactly represented by a single-qubit Clifford+ T circuit if and only if the entries of V belong to the ring Z[1/2,i] . Later that year, Giles and Selinger showed that the same restriction applies to matrices that can be exactly represented by a multi-qubit Clifford+ T circuit. These number-theoretic characterizations shed new light upon the structure of Clifford+ T circuits and led to remarkable developments in the field of quantum compiling. In the present paper, we provide number-theoretic characterizations for certain restricted Clifford+ T circuits by considering unitary matrices over subrings of Z[1/2,i] . We focus on the subrings Z[1/2] , Z[1/2] , Z[1/i2] , and Z[1/2,i] , and we prove that unitary matrices with entries in these rings correspond to circuits over well-known universal gate sets. In each case, the desired gate set is obtained by extending the set of classical reversible gates {X,CX,CCX} with an analogue of the Hadamard gate and an optional phase gate.
It is known that the matrices that can be exactly represented by a multiqubit circuit over the Toffoli+Hadamard, Clifford+TTT, or, more generally, Clifford-cyclotomic gate set are precisely the unitary matrices with entries in the ring Z[1/2,ζk]\mathbb{Z}[1/2,\zeta_k]Z[1/2,ζk​], where kkk is a positive integer that depends on the gate set and ζk\zeta_kζk​ is a primitive 2k2^k2k-th root of unity. In the present paper, we establish an analogous correspondence for qutrits. We define the multiqutrit Clifford-cyclotomic gate set of degree 3k3^k3k by extending the classical qutrit gates XXX, CXCXCX, and CCXCCXCCX with the Hadamard gate HHH and the TkT_kTk​ gate Tk=diag(1,ωk,ωk2)T_k=\mathrm{diag}(1,\omega_k, \omega_k^2)Tk​=diag(1,ωk​,ωk2​), where ωk\omega_kωk​ is a primitive 3k3^k3k-th root of unity. This gate set is equivalent to the qutrit Toffoli+Hadamard gate set when k=1k=1k=1, and to the qutrit Clifford+TkT_kTk​ gate set when k>1k>1k>1. We then prove that a 3n×3n3^n\times 3^n3n×3n unitary matrix UUU can be represented by an nnn-qutrit circuit over the Clifford-cyclotomic gate set of degree 3k3^k3k if and only if the entries of UUU lie in the ring Z[1/3,ωk]\mathbb{Z}[1/3,\omega_k]Z[1/3,ωk​].
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.
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.