The ZX-calculus is complete for the single-qubit Clifford+T group

TL;DR

This paper proves that the ZX-calculus is complete for the single-qubit Clifford+T group, enabling pictorial derivations of all single-qubit unitaries including non-stabilizer operations.

Contribution

The authors extend the ZX-calculus completeness to include the T gate, covering the universal single-qubit Clifford+T group.

Findings

ZX-calculus is complete for Clifford+T group

All single-qubit unitaries can be derived pictorially with the extended calculus

The extended calculus supports approximate universality for single-qubit operations

Abstract

The ZX-calculus is a graphical calculus for reasoning about pure state qubit quantum mechanics. It is complete for pure qubit stabilizer quantum mechanics, meaning any equality involving only stabilizer operations that can be derived using matrices can also be derived pictorially. Stabilizer operations include the unitary Clifford group, as well as preparation of qubits in the state |0>, and measurements in the computational basis. For general pure state qubit quantum mechanics, the ZX-calculus is incomplete: there exist equalities involving non-stabilizer unitary operations on single qubits which cannot be derived from the current rule set for the ZX-calculus. Here, we show that the ZX-calculus for single qubits remains complete upon adding the operator T to the single-qubit stabilizer operations. This is particularly interesting as the resulting single-qubit Clifford+T group is approximately universal, i.e. any unitary single-qubit operator can be approximated to arbitrary accuracy using only Clifford operators and T.