Phase gadget compilation for diagonal qutrit gates

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 build different types of qutrit multiplecontrolled phase gates.As an application of these results we find a construction for emulating arbitrary qubit diagonal unitaries, and specifically find an emulation for the qubit CCZ gate that only requires three singlequtrit non-Clifford gates to implement -provably lower than the four T gates needed using just qubit gates.

Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus

“…Recently, these qudit graphical calculi have been applied to achieve new results in qudit circuit synthesis. By leveraging the qutrit ZX-calculus, novel qutrit constructions have been elucidated from their qubit ZX-calculus counterparts, for instance, discovering a qutrit unitary that implements the qubit CCZ gate using only three single-qudit non-Clifford gates [57]. Another recent example is the application of qutrit ZX-calculus to synthesizing two-qutrit controlled phase gates on superconducting qutrits [13].…”

Light-Matter Interaction in the ZXW Calculus

“…Based on this form, it becomes evident that the diagram corresponds to the operator iτ n ˆ1n ˆ2, which needs to be exponentiated. We employ the technique described by van de Wetering and Yeh [51] for constructing diagonal qudit gates using phase gadgets [20,38]. Firstly, note that the polynomial equation (x + y) 2 = x 2 + 2xy + y 2 implies that e iτ xy = e i τ 2 ((x+y) 2 −x 2 −y 2 ) = e i τ 2 (x+y) 2 e −i τ 2 x 2 e −i τ 2 y 2 .…”

The Qupit Stabiliser ZX-travaganza: Simplified Axioms, Normal Forms and Graph-Theoretic Simplification

“…There exist several variations of the ZX-calculus that extend it to higher-dimensional qudits. Many have focused on the specific case of qutrit systems [65,39,65,62], with applications in quantum computation [70,63], and complexity theory [62]. Recent papers have focused on the stabiliser fragment of odd prime dimensional qudits, including Ref.…”