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A Normal Form for Single-Qutrit Clifford+T Operators

Shiroman Prakash, Akalank Jain, Bhakti Kapur, Shubhangi Seth

Published Mar 8, 2018• 0 citations
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Abstract

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.

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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.
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.
We present a complete set of 161616 rewrite rules for multi-qudit Clifford circuits in all odd prime dimensions. Completeness means that any two Clifford circuits representing the same linear map can be rewritten into each other using these rules. Each rule involves at most three qudits and admits an intuitive interpretation. To establish completeness, we first work at the symplectic level, using the isomorphism between the symplectic group Sp(2n,Zp)\mathrm{Sp}(2n,\mathbb{Z}_p)Sp(2n,Zp​) and the quotient of the Clifford group by the Pauli group. We construct a circuit normal form that captures the stabiliser tableau of a Clifford operator and is unique up to Pauli correction. Using this normal form, we derive a complete set of symplectic relations, which we then lift to Clifford relations by incorporating Pauli corrections. We also formally verify completeness in the Agda proof assistant.
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.
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).
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.
We present a complete set of rewrite rules for n-qutrit Clifford circuits where n is any non-negative integer.This is the first completeness result for any fragment of quantum circuits in odd prime dimensions.We first generalize Selinger's normal form for n-qubit Clifford circuits to the qutrit setting.Then, we present a rewrite system by which any Clifford circuit can be reduced to this normal form.We then simplify the rewrite rules in this procedure to a small natural set of rules, giving a clean presentation of the group of qutrit Clifford unitaries in terms of generators and relations.