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Quantum Information Processing

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01.10.2021

A normal form for single-qudit Clifford+T operators

verfasst von: Shiroman Prakash, Amolak Ratan Kalra, Akalank Jain

Erschienen in: Quantum Information Processing | Ausgabe 10/2021

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Abstract

Normal forms play a crucial role in the problem of gate synthesis. We propose a normal form for single-qudit gates composed of Clifford and T-gates for qudits of odd prime dimension \(p\ge 5\). We prove that any single-qudit Clifford+T operator can be re-expressed in this normal form in polynomial time. We obtained strong computational evidence that this normal form is unique. Assuming uniqueness, we are able to use this normal form to provide an efficient algorithm for exact synthesis of any single-qudit Clifford+T operator with minimal T-count.

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If we are working with qudits of dimension \(p \equiv 1 \mod 3\), we may instead be interested in the alternative T-count. It is also easy to see that during the process of conversion to the normal form (4.4) the alternative T-count can only decrease or remain unchanged. So this normal form is also optimal with respect to the alternative T-count.

We thank Peter Sarnak for discussions on this point.

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Titel: A normal form for single-qudit Clifford+T operators
verfasst von: Shiroman Prakash
Amolak Ratan Kalra
Akalank Jain
Publikationsdatum: 01.10.2021
Verlag: Springer US
Erschienen in: Quantum Information Processing / Ausgabe 10/2021
Print ISSN: 1570-0755
Elektronische ISSN: 1573-1332
DOI: https://doi.org/10.1007/s11128-021-03280-0

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