An Algebra of Quantum Processes

TL;DR

This paper introduces qCCS, an algebraic framework for modeling pure quantum processes with physical state movement and super-operator computations, providing tools for analyzing process equivalences and robustness.

Contribution

It presents a novel algebraic formalism for quantum processes, including semantics, bisimulation, and approximate equivalences, enabling rigorous analysis of quantum system interactions.

Findings

Defined strong bisimulation and proved its algebraic properties.

Introduced strong reduction-bisimulation for sequential quantum computation.

Established preservation of approximate bisimulation under process constructors.

Abstract

We introduce an algebra qCCS of pure quantum processes in which no classical data is involved, communications by moving quantum states physically are allowed, and computations is modeled by super-operators. An operational semantics of qCCS is presented in terms of (non-probabilistic) labeled transition systems. Strong bisimulation between processes modeled in qCCS is defined, and its fundamental algebraic properties are established, including uniqueness of the solutions of recursive equations. To model sequential computation in qCCS, a reduction relation between processes is defined. By combining reduction relation and strong bisimulation we introduce the notion of strong reduction-bisimulation, which is a device for observing interaction of computation and communication in quantum systems. Finally, a notion of strong approximate bisimulation (equivalently, strong bisimulation distance) and its reduction counterpart are introduced. It is proved that both approximate bisimilarity and approximate reduction-bisimilarity are preserved by various constructors of quantum processes. This provides us with a formal tool for observing robustness of quantum processes against inaccuracy in the implementation of its elementary gates.