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.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:2011.07970Nov 16, 20200 citations

A Complete and Natural Rule Set for Multi-Qudit Clifford Circuits in All Odd Prime Dimensions

Xiaoning Bian, Sarah Meng Li, Neil J. Ross +2 more

We present a complete set of $16$ 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 $\mathrm{Sp}(2n,\mathbb{Z}_p)$ 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 present a complete set of $16$ 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 $\mathrm{Sp}(2n,\mathbb{Z}_p)$ 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.

quant-pharXiv:2609.40106Oct 1, 20260 citations

Normal forms of quadratic bose operators

V. V. Kucherenko, Виктор Павлович Маслов

Mathematical NotesJan 1, 19972 citations

A Normal Form for Single-Qutrit Clifford+T Operators

Shiroman Prakash, Akalank Jain, Bhakti Kapur +1 more

Mar 8, 20180 citations

Quantization from Clifford Algebra

Justin Dressel

Bulletin of the American Physical SocietyMar 7, 20180 citations

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.

We present a complete set of

16

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

\mathrm{Sp}(2n,\mathbb{Z}_p)

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 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

T

operator with minimal

T

-count.