Interacting quantum observables: categorical algebra and diagrammatics

By Bob Coecke, Ross Duncan. arXiv.org, 2009

578 citations recorded by the source corpus.

This paper has two tightly intertwined aims: (i) to introduce an intuitive and universal graphical calculus for multi-qubit systems, the ZX-calculus, which greatly simplifies derivations in the area of quantum computation and information. (ii) To axiomatize complementarity of quantum observables within a general framework for physical theories in terms of dagger symmetric monoidal categories. We also axiomatize phase shifts within this framework. Using the well-studied canonical correspondence between graphical calculi and dagger symmetric monoidal categories, our results provide a purely graphical formalisation of complementarity for quantum observables. Each individual observable, represented by a commutative special dagger Frobenius algebra, gives rise to an Abelian group of phase shifts, which we call the phase group. We also identify a strong form of complementarity, satisfied by the Z- and X-spin observables, which yields a scaled variant of a bialgebra.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5aee63873bc5cde7022fc38f6036e10ce465bcc5

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Miriam Backens
2. Ross Duncan
3. Bob Coecke

In Atomic and Molecular Physics, and Optics

1. Miriam Backens
2. Ross Duncan
3. Bob Coecke

In Mathematical Physics

1. Miriam Backens
2. Ross Duncan

Citations in the corpus

• The ZX-calculus is complete for stabilizer quantum mechanics (2013, methodology): “Unlike quantum circuit notation, the ZX-calculus developed in [5, 6] is not just a graphical notation: It has built-in rewrite rules, which transform one diagram into a different diagram representing the same overall process.”
• Categories in Control (2014, unknown): “However, it is surprising that most of these structures are present in a seemingly very different context: the so-called ‘ZX calculus’, a diagrammatic formalism for working with complementary observables in quantum theory [8].”
• t|ket⟩: a retargetable compiler for NISQ devices (2020, methodology): “The zx-calculus [71] is already incorporated into t|ket〉 and has proven an effective tool for MBQC calculations in the past [73, 93, 94].”