Building Qutrit Diagonal Gates from Phase Gadgets

• URL: http://arxiv.org/abs/2204.13681v2
• Date: Wed, 15 Nov 2023 11:02:06 GMT
• Title: Building Qutrit Diagonal Gates from Phase Gadgets
• Authors: John van de Wetering (University of Oxford), Lia Yeh (University of Oxford)
• Abstract summary: 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. 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 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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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