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 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%.
In this paper we study single qutrit circuits consisting of words over the Clifford +D cyclotomic gate set, where D=diag(±ξa,±ξb,±ξc) , ξ is a primitive 9 -th root of unity and a,b,c are integers. We characterize classes of qutrit unit vectors z with entries in Z[ξ,1χ] based on the possibility of reducing their smallest denominator exponent (sde) with respect to χ:=1–ξ, by acting an appropriate gate in Clifford +D . We do this by studying the notion of `derivatives mod 3 ' of an arbitrary element of Z[ξ] and using it to study the smallest denominator exponent of HDz where H is the qutrit Hadamard gate and D . In addition, we reduce the problem of finding all unit vectors of a given sde to that of finding integral solutions of a positive definite quadratic form along with some additional constraints. As a consequence we prove that the Clifford +D gates naturally arise as gates with sde 0 and 3 in the group U(3,Z[ξ,1χ]) of 3×3 unitaries with entries in Z[ξ,1χ] . We illustrate the general applicability of these methods to obtain an exact synthesis algorithm for Clifford +R and recover the previously known exact synthesis algorithm of Kliuchnikov, Maslov, Mosca (2012). The framework developed to formulate qutrit gate synthesis for Clifford +D extends to qudits of arbitrary prime power.
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[21,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[21,i] holds in the n-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[21,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[21,i] holds in the n-qubit case.
In this paper we study single qutrit circuits consisting of words over the Clifford+D cyclotomic gate set, where D=diag(±ξa,±ξb,±ξc), ξ is a primitive 9-th root of unity and a,b,c are integers. We characterize classes of qutrit unit vectors z with entries in Z[ξ,χ1] based on the possibility of reducing their smallest denominator exponent (sde) with respect to χ:=1−ξ, by acting an appropriate gate in Clifford+D. We do this by studying the notion of `derivatives mod 3' of an arbitrary element of Z[ξ] and using it to study the smallest denominator exponent of HDz where H is the qutrit Hadamard gate and D. In addition, we reduce the problem of finding all unit vectors of a given sde to that of finding integral solutions of a positive definite quadratic form along with some additional constraints. As a consequence we prove that the Clifford+D gates naturally arise as gates with sde 0 and 3 in the group U(3,Z[ξ,χ1]) of 3×3 unitaries with entries in Z[ξ,χ1]. We illustrate the general applicability of these methods to obtain an exact synthesis algorithm for Clifford+R and recover the previous exact synthesis algorithm in \cite{kmm}. The framework developed to formulate qutrit gate synthesis for Clifford+D extends to qudits of arbitrary prime power.
Approximate synthesis of general single-qubit unitaries over the Clifford+T gate set
For the standard Clifford+T gate set, deterministic, ancilla-free synthesis now attains the minimal T-count for general single-qubit unitaries (Morisaki et al., arXiv:2510.05816). The T gate rotates by half the angle of T, 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 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 to 10−8, the cost of Clifford+T circuits scales as 2.4log2(1/ε) compared to 3.0log2(1/ε) for T-count-optimal Clifford+T circuits. Once a one-time catalyst state is amortized, the Clifford+T circuits are never costlier than their Clifford+T counterparts.
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 Z[21,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.