A recipe for quantum graphical languages

Titouan Carette (1) , Emmanuel Jeandel (1)

1 MOCQUA - Designing the Future of Computational Models

Different graphical calculi have been proposed to represent quantum computation. First the ZX-calculus [4], followed by the ZW-calculus [12] and then the ZH-calculus [1]. We can wonder if new Z *-calculi will continue to be proposed forever. This article answers negatively. All those language share a common core structure we call Z *-algebras. We classify Z *-algebras up to isomorphism in two dimensional Hilbert spaces and show that they are all variations of the aforementioned calculi. We do the same for linear relations and show that the calculus of [2] is essentially the unique one. The most common formalization of quantum computing is the circuit model, a diagram-matical language representing unitary matrices in a two dimensional Hilbert space, see [20] for an introduction. Verification of quantum processes requires a sound and complete equational theory for quantum circuits, i.e. a complete presentation of unitaries by generators and relations. This is known to be a difficult open problem. By relaxing the unitarity condition and allowing all linear maps, at least three different complete equational theories were found. The ZX-calculus was introduced in [4] and was designed as a part of the categorical quantum mechanics program. It relies on the interaction between two complementary observables. The ZX-calculus has proven to be a good language to reason about quantum processes [7, 11]. However, finding a set of rules to make it complete has been open for a long time, and part of the solution [15] involved a secondary graphical language: the ZW-calculus [12, 5]. This calculus is built on two tripartite entanglement classes (GHZ and W-states) unraveling new structures. Yet another complete graphical language was later introduced, the ZH-calculus [1], inspired by hyper-graph states. Compared to quantum circuits, these three languages share an important advantage. Processes and matrices are not represented merely by diagrams, but by graphs (hence the term graphical language). Isomorphic graphs represent the same quantum evolution. This peculiarity is embedded in the only topology matters paradigm. This is a subtle feature: a usual diagrammatic language (like quantum circuits) starts with a given set of primitives (usually quantum gates) for which the notion of inputs and outputs is significant. When only topology matters, one can readily switch an input into an output, and conversely. This property follows from some specificities of the building blocks of those languages. One goal of this article is to give a formal definition of these specificities. Then, we will be able to prove that the three existing graphical calculi for quantum computing, ZX, ZH and ZW , are essentially the only possible graphical calculi for quantum computing.

Mots clés: 2012 ACM Subject Classification Theory of computation → Quantum computation theory; Theory of computation → Semantics and reasoning; Mathematics of computing; Theory of computation → Equational logic and rewriting Keywords and phrases Categorical Quantum Mechanics; Quantum Computing; Category Theory Funding ANR-17-CE25-0009; PIA-GDN/Quantex

Domaines: Logique en informatique [cs.LO]; Physique Quantique [quant-ph]

Titouan Carette, Emmanuel Jeandel. A recipe for quantum graphical languages. ICALP 2020, 2020, Saarbrücken, Germany. ⟨hal-02914177⟩