We present an exact synthesis algorithm for qutrit unitaries in $\mathcal{U}_{3^n}(\mathbb{Z}[1/3,e^{2πi/3}])$ over the Clifford $+T$ gate set with at most one ancilla. This extends the already known result of qutrit metaplectic gates being a subset of Clifford $+T$ gate set with one ancilla. As an intermediary step, we construct an algorithm to convert 3-level unitaries into multiply-controlled gates, analogous to Gray codes converting 2-level unitaries into multiply-controlled gates. Finally, using catalytic embeddings, we present an algorithm to exactly synthesize unitaries $\mathcal{U}_{3^n}(\mathbb{Z}[1/3,e^{2πi/9}])$ over the Clifford $+T$ gate set with at most 2 ancillas. This, in particular, gives an exact synthesis algorithm of single-qutrit Clifford $+\mathcal{D}$ over the multi-qutrit Clifford $+T$ gate set with at most two ancillas.We present an exact synthesis algorithm for qutrit unitaries in $\mathcal{U}_{3^n}(\mathbb{Z}[1/3,e^{2πi/3}])$ over the Clifford$+T$ gate set with at most one ancilla. This extends the already known result of qutrit metaplectic gates being a subset of Clifford$+T$ gate set with one ancilla. As an intermediary step, we construct an algorithm to convert 3-level unitaries into multiply-controlled gates, analogous to Gray codes converting 2-level unitaries into multiply-controlled gates. Finally, using catalytic embeddings, we present an algorithm to exactly synthesize unitaries $\mathcal{U}_{3^n}(\mathbb{Z}[1/3,e^{2πi/9}])$ over the Clifford$+T$ gate set with at most 2 ancillas. This, in particular, gives an exact synthesis algorithm of single-qutrit Clifford$+\mathcal{D}$ over the multi-qutrit Clifford$+T$ gate set with at most two ancillas.

arXiv:2405.08147May 13, 20241 citation

Clifford algebra and quantum logic gates

Wuming Li

Advances in Applied Clifford AlgebrasOct 1, 20043 citations

Building Qutrit Diagonal Gates from Phase Gadgets

John van de Wetering, Lia Yeh

Phase gadgets have proved to be an indispensable tool for reasoning about ZX-diagrams, being used in optimisation and simulation of quantum circuits and the theory of measurement-based quantum computation.In this paper we study phase gadgets for qutrits.We present the flexsymmetric variant of the original qutrit ZX-calculus, which allows for rewriting that is closer in spirit to the original (qubit) ZX-calculus.In this calculus phase gadgets look as you would expect, but there are non-trivial differences in their properties.We devise new qutrit-specific tricks to extend the graphical Fourier theory of qubits, resulting in a translation between the 'additive' phase gadgets and a 'multiplicative' counterpart we dub phase multipliers.This enables us to generalise the qubit notion of multiple-control to qutrits in two ways.The first type is controlling on a single tritstring, while the second type applies the gate a number of times equal to the tritwise multiplication modulo 3 of the control qutrits.We show how both types of control can be implemented for any qutrit Z or X phase gate, ancilla-free, and using only Clifford and phase gates.The first requires a polynomial number of gates and exponentially small phases, while the second requires an exponential number of gates, but constant sized phases.This is interesting, because such a construction is not possible in the qubit setting.As an application of these results we find a construction for emulating arbitrary qubit diagonal unitaries, and specifically find an ancilla-free emulation for the qubit CCZ gate that only requires three single-qutrit non-Clifford gates -provably lower than the four T gates needed for qubits with ancilla.Phase gadgets have proved to be an indispensable tool for reasoning about ZX-diagrams, being used in optimisation and simulation of quantum circuits and the theory of measurement-based quantum computation.In this paper we study phase gadgets for qutrits.We present the flexsymmetric variant of the original qutrit ZX-calculus, which allows for rewriting that is closer in spirit to the original (qubit) ZX-calculus.In this calculus phase gadgets look as you would expect, but there are non-trivial differences in their properties.We devise new qutrit-specific tricks to extend the graphical Fourier theory of qubits, resulting in a translation between the 'additive' phase gadgets and a 'multiplicative' counterpart we dub phase multipliers.This enables us to generalise the qubit notion of multiple-control to qutrits in two ways.The first type is controlling on a single tritstring, while the second type applies the gate a number of times equal to the tritwise multiplication modulo 3 of the control qutrits.We show how both types of control can be implemented for any qutrit Z or X phase gate, ancilla-free, and using only Clifford and phase gates.The first requires a polynomial number of gates and exponentially small phases, while the second requires an exponential number of gates, but constant sized phases.This is interesting, because such a construction is not possible in the qubit setting.As an application of these results we find a construction for emulating arbitrary qubit diagonal unitaries, and specifically find an ancilla-free emulation for the qubit CCZ gate that only requires three single-qutrit non-Clifford gates -provably lower than the four T gates needed for qubits with ancilla.

arXiv:2204.13681Electronic Proceedings in Theoretical Computer ScienceNov 14, 20237 citations

Canonical forms for single-qutrit Clifford+ T operators

Andrew N. Glaudell, Neil J. Ross, Jacob M. Taylor

arXiv:1803.05047Annals of PhysicsApr 9, 201930 citations

Quantum Gates and Quantum Algorithms with Clifford Algebra Technique

M. Gregorič, N. S. Mankoč Borštnik

arXiv:0801.3201International Journal of Theoretical PhysicsSep 2, 20086 citations

We present an exact synthesis algorithm for qutrit unitaries in

\mathcal{U}_{3^n}(\mathbb{Z}[1/3,e^{2πi/3}])

over the Clifford

+T

gate set with at most one ancilla. This extends the already known result of qutrit metaplectic gates being a subset of Clifford

+T

gate set with one ancilla. As an intermediary step, we construct an algorithm to convert 3-level unitaries into multiply-controlled gates, analogous to Gray codes converting 2-level unitaries into multiply-controlled gates. Finally, using catalytic embeddings, we present an algorithm to exactly synthesize unitaries

\mathcal{U}_{3^n}(\mathbb{Z}[1/3,e^{2πi/9}])

over the Clifford

+T

gate set with at most 2 ancillas. This, in particular, gives an exact synthesis algorithm of single-qutrit Clifford

+\mathcal{D}

over the multi-qutrit Clifford

+T

gate set with at most two ancillas.