Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1803.03228 · Physical Review A

Normal form for single-qutrit Clifford+ T operators and synthesis of single-qutrit gates

Shiroman Prakash, Akalank Jain, Bhakti Kapur, Shubangi Seth

Published Sep 5, 2018• 40 citations
arXivHTMLPDF

Abstract

We study single-qutrit gates composed of Clifford and $T$ gates, using the qutrit version of the $T$ gate proposed by Howard and Vala [M. Howard and J. Vala, Phys. Rev. A 86, 022316 (2012)]. We propose a normal form for single-qutrit gates analogous to the Matsumoto-Amano normal form for qubits. We prove that the normal form is optimal with respect to the number of $T$ gates used and that any string of qutrit Clifford+$T$ operators can be put into this normal form in polynomial time. We also prove that this form is unique and provide an algorithm for exact synthesis of any single-qutrit Clifford+$T$ operator.

DOI
10.1103/physreva.98.032304

Related papers

A Normal Form for Single-Qutrit Clifford+T Operators

Shiroman Prakash, Akalank Jain, Bhakti Kapur +1 more

We propose a normal form for single-qudit gates composed of Clifford and $T$-gates for qudits of odd prime dimension $p\geq 5$. We prove that any single-qudit Clifford+$T$ operator can be re-expressed in this normal form in polynomial time. We also provide strong numerical evidence that this normal form is unique. Assuming uniqueness, we are able to use this normal form to provide an algorithm for exact synthesis of any single-qudit Clifford+$T$ operator with minimal $T$-count.

•Mar 8, 2018•0 citations

Remarks on Matsumoto and Amano's normal form for single-qubit Clifford+T operators

Brett Gordon Giles, Peter Selinger

Matsumoto and Amano (2008) showed that every single-qubit Clifford+T operator can be uniquely written of a particular form, which we call the Matsumoto-Amano normal form. In this mostly expository paper, we give a detailed and streamlined presentation of Matsumoto and Amano's results, simplifying some proofs along the way. We also point out some corollaries to Matsumoto and Amano's work, including an intrinsic characterization of the Clifford+T subgroup of SO(3), which also yields an efficient T-optimal exact single-qubit synthesis algorithm. Interestingly, this also gives an alternative proof of Kliuchnikov, Maslov, and Mosca's exact synthesis result for the Clifford+T subgroup of U(2).

arXiv:1312.6584•Dec 23, 2013•19 citations

A Normal Form for Single-Qudit Clifford+TTT Operators

Akalank Jain, Amolak Ratan Kalra, Shiroman Prakash

We propose a normal form for single-qudit gates composed of Clifford and $T$-gates for qudits of odd prime dimension $p\geq 5$. We prove that any single-qudit Clifford+$T$ operator can be re-expressed in this normal form in polynomial time. We also provide strong numerical evidence that this normal form is unique. Assuming uniqueness, we are able to use this normal form to provide an algorithm for exact synthesis of any single-qudit Clifford+$T$ operator with minimal $T$-count.

arXiv:2011.07970•Nov 16, 2020•0 citations

Synthesis and Arithmetic of Single Qutrit Circuits

Amolak Ratan Kalra, Michele Mosca, Dinesh Valluri

In this paper we study single qutrit circuits consisting of words over the Clifford$+D$ cyclotomic gate set, where $D=\text{diag}(\pmξ^{a},\pmξ^{b},\pmξ^{c})$, $ξ$ 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 $\mathbb{Z}[ξ, \frac{1}χ]$ 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 $\mathbb{Z}[ξ]$ 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 $U(3,\mathbb{Z}[ξ, \frac{1}χ])$ of $3 \times 3$ unitaries with entries in $\mathbb{Z}[ξ, \frac{1}χ]$. 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.

arXiv:2311.08696•Nov 15, 2023•1 citation

Synthesis and Arithmetic of Single Qutrit Circuits

Amolak Ratan Kalra, Michele Mosca, Dinesh Valluri

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.

Quantum•Feb 26, 2025•4 citations

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

Simon Forest, David Gosset, Vadym Kliuchnikov +1 more

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%.

arXiv:1501.04944•Journal of Mathematical Physics•Aug 1, 2015•44 citations
We study single-qutrit gates composed of Clifford and TTT gates, using the qutrit version of the TTT gate proposed by Howard and Vala [M. Howard and J. Vala, Phys. Rev. A 86, 022316 (2012)]. We propose a normal form for single-qutrit gates analogous to the Matsumoto-Amano normal form for qubits. We prove that the normal form is optimal with respect to the number of TTT gates used and that any string of qutrit Clifford+TTT operators can be put into this normal form in polynomial time. We also prove that this form is unique and provide an algorithm for exact synthesis of any single-qutrit Clifford+TTT operator.
We propose a normal form for single-qudit gates composed of Clifford and TTT-gates for qudits of odd prime dimension p≥5p\geq 5p≥5. We prove that any single-qudit Clifford+TTT operator can be re-expressed in this normal form in polynomial time. We also provide strong numerical evidence that this normal form is unique. Assuming uniqueness, we are able to use this normal form to provide an algorithm for exact synthesis of any single-qudit Clifford+TTT operator with minimal TTT-count.
Matsumoto and Amano (2008) showed that every single-qubit Clifford+T operator can be uniquely written of a particular form, which we call the Matsumoto-Amano normal form. In this mostly expository paper, we give a detailed and streamlined presentation of Matsumoto and Amano's results, simplifying some proofs along the way. We also point out some corollaries to Matsumoto and Amano's work, including an intrinsic characterization of the Clifford+T subgroup of SO(3), which also yields an efficient T-optimal exact single-qubit synthesis algorithm. Interestingly, this also gives an alternative proof of Kliuchnikov, Maslov, and Mosca's exact synthesis result for the Clifford+T subgroup of U(2).
We propose a normal form for single-qudit gates composed of Clifford and TTT-gates for qudits of odd prime dimension p≥5p\geq 5p≥5. We prove that any single-qudit Clifford+TTT operator can be re-expressed in this normal form in polynomial time. We also provide strong numerical evidence that this normal form is unique. Assuming uniqueness, we are able to use this normal form to provide an algorithm for exact synthesis of any single-qudit Clifford+TTT operator with minimal TTT-count.
In this paper we study single qutrit circuits consisting of words over the Clifford+D+D+D cyclotomic gate set, where D=diag(±ξa,±ξb,±ξc)D=\text{diag}(\pmξ^{a},\pmξ^{b},\pmξ^{c})D=diag(±ξa,±ξb,±ξc), ξξξ is a primitive 999-th root of unity and a,b,ca,b,ca,b,c are integers. We characterize classes of qutrit unit vectors zzz with entries in Z[ξ,1χ]\mathbb{Z}[ξ, \frac{1}χ]Z[ξ,χ1​] based on the possibility of reducing their smallest denominator exponent (sde) with respect to χ:=1−ξ,χ:= 1 - ξ,χ:=1−ξ, by acting an appropriate gate in Clifford+D+D+D. We do this by studying the notion of `derivatives mod 333' of an arbitrary element of Z[ξ]\mathbb{Z}[ξ]Z[ξ] and using it to study the smallest denominator exponent of HDzHDzHDz where HHH is the qutrit Hadamard gate and DDD. 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+D+D gates naturally arise as gates with sde 000 and 333 in the group U(3,Z[ξ,1χ])U(3,\mathbb{Z}[ξ, \frac{1}χ])U(3,Z[ξ,χ1​]) of 3×33 \times 33×3 unitaries with entries in Z[ξ,1χ]\mathbb{Z}[ξ, \frac{1}χ]Z[ξ,χ1​]. We illustrate the general applicability of these methods to obtain an exact synthesis algorithm for Clifford+R+R+R and recover the previous exact synthesis algorithm in \cite{kmm}. The framework developed to formulate qutrit gate synthesis for Clifford+D+D+D extends to qudits of arbitrary prime power.
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.
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%.