# Reducing T-count with the ZX-calculus

## TL;DR

This paper introduces a ZX-calculus-based method for reducing T-gates in quantum circuits, achieving significant improvements over previous methods, especially in ancilla-free scenarios, by employing phase teleportation and circuit simplification techniques.

## Contribution

It presents a novel T-count reduction technique using ZX-calculus and phase teleportation, applicable to arbitrary non-Clifford phases and validated by circuit equality checks.

## Key findings

- Up to 50% T-count reduction on benchmark circuits
- Method matches or exceeds previous T-count reduction approaches
- Implemented in open-source library PyZX

## Abstract

Reducing the number of non-Clifford quantum gates present in a circuit is an important task for efficiently implementing quantum computations, especially in the fault-tolerant regime. We present a new method for reducing the number of T-gates in a quantum circuit based on the ZX-calculus, which matches or beats previous approaches to T-count reduction on the majority of our benchmark circuits in the ancilla-free case, in some cases yielding up to 50% improvement. Our method begins by representing the quantum circuit as a ZX-diagram, a tensor network-like structure that can be transformed and simplified according to the rules of the ZX-calculus. We then show that a recently-proposed simplification strategy can be extended to reduce T-count using a new technique called phase teleportation. This technique allows non-Clifford phases to combine and cancel by propagating non-locally through a generic quantum circuit. Phase teleportation does not change the number or location of non-phase gates and the method also applies to arbitrary non-Clifford phase gates as well as gates with unknown phase parameters in parametrised circuits. Furthermore, the simplification strategy we use is powerful enough to validate equality of many circuits. In particular, we use it to show that our optimised circuits are indeed equal to the original ones. We have implemented the routines of this paper in the open-source library PyZX.

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Source: https://tomesphere.com/paper/1903.10477