Quantum Algorithms and Oracles with the Scalable ZX-calculus

TL;DR

This paper demonstrates that the scalable ZX-calculus offers a formal, intuitive, and compact graphical framework to describe and verify quantum algorithms, including oracle-based algorithms like Deutsch-Jozsa, Bernstein-Vazirani, Simon, and Grover.

Contribution

It introduces the scalable ZX-calculus as a high-level graphical language for quantum algorithms, enabling formal proofs and intuitive understanding.

Findings

Successfully described and proved quantum algorithms graphically

Provided a formal framework for high-level quantum process reasoning

Enhanced the practical applicability of ZX-calculus in quantum computing

Abstract

The ZX-calculus was introduced as a graphical language able to represent specific quantum primitives in an intuitive way. The recent completeness results have shown the theoretical possibility of a purely graphical description of quantum processes. However, in practice, such approaches are limited by the intrinsic low level nature of ZX calculus. The scalable notations have been proposed as an attempt to recover an higher level point of view while maintaining the topological rewriting rules of a graphical language. We demonstrate that the scalable ZX-calculus provides a formal, intuitive, and compact framework to describe and prove quantum algorithms. As a proof of concept, we consider the standard oracle-based quantum algorithms: Deutsch-Jozsa, Bernstein-Vazirani, Simon, and Grover algorithms, and we show they can be described and proved graphically.