Meromorphic Quantum Computing

TL;DR

This paper explores a projective approach to quantum mechanics, linking geometric representations with the ZXW-calculus and identifying meromorphic functions that describe quantum circuit behaviors.

Contribution

It introduces a functorial, lax monoidal projectivization of quantum states, providing a new geometric perspective and an alternative derivation of the GHZ/W-calculus.

Findings

Identifies the Bloch sphere with the Riemann sphere for qubits.

Provides a projective interpretation of the ZXW-calculus.

Discovers meromorphic functions characterizing quantum circuit coherence.

Abstract

We consider the kinematic axioms of quantum mechanics projectively. Instead of normalized (pure) states up to global phase, states become one-dimensional subspaces of vector spaces. This process of projectivization is functorial and lax monoidal. For qubits it identifies the Bloch sphere with the Riemann sphere. We interpret a fragment of the ZXW-calculus projectively and thereby provide an alternate derivation of the arithmetic GHZ/W-calculus of Coecke et al. We find meromorphic functions that characterize the coherent behaviour of circuits for logical state preparation of quantum codes and magic state distillation.