constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
quantum algorithms:
The “ZX-calculus” is an elaboration for the purpose of qbit-based quantum computation of the string diagram-calculus used in quantum information theory via dagger-compact categories.
The basic idea is to formalize the co-existence of the ubiquitous pair of measurement bases of qbits $QBit \simeq \mathbb{C}^2$, namely
the “computational basis”
which consists (by common convention) of the eigenstates of the Pauli-Z gate;
the Hadamard basis
which consists of the eigenstates of the Pauli-X gate;
by observing [Coecke & Duncan, p. 1] that:
either equips $QBit = \mathbb{C}^2$ with a Frobenius algebra-structure (associated to the Frobenius monad-structure of the corresponding quantum reader monad as originally observed by [Coecke & Pavlović (2008)]);
jointly these make a certain bialgebra-structure, algebraically reflecting the fact that they are mutually unbiased bases.
It is this bi-algebraic formalizaton of the interaction between the “Pauli-Z basis” and the “Pauli-X basis” for qbits which gives the ZX-calculus its name.
Graphically, the ZX-calculus proceeds to declare that:
the (co-)multiplication and (co)-unit string diagrams of these two Frobenius algebra-structures are to be denoted
in green for the Z-basis
in red for the X-basis
connected sub-string diagrams formed from (co-)multiplications and (co-)units all of the same color are shown as a single box/circle with the given number of in/outputs – then called spiders
(since the Frobenius algebra-property ensures — recalled as Coecke & Duncan (2011), Theorem 1 — that any connected diagrams are equal iff they have the same number of in/outputs).
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Landing page:
The ZX-calculus has its origin as a tool for organizing measurement-based quantum computation-protocols, following Danos, Kahsefi & Panangaden (2007):
Ross Duncan, Simon Perdrix, Rewriting Measurement-Based Quantum Computations with Generalised Flow, in: Automata, Languages and Programming. ICALP 2010, Lecture Notes in Computer Science 6199, Springer (2010) [doi:10.1007/978-3-642-14162-1_24]
Ross Duncan, A graphical approach to measurement-based quantum computing [arXiv:1203.6242 video exposition:YT]
The official introduction of the ZX-calculus:
Bob Coecke, Ross Duncan, A graphical calculus for quantum observables [pdf]
Bob Coecke, Ross Duncan, Interacting Quantum Observables, in Automata, Languages and Programming. ICALP 2008, Lecture Notes in Computer Science 5126, Springer (2008) [doi:10.1007/978-3-540-70583-3_25]
Bob Coecke, Ross Duncan, Interacting Quantum Observables: Categorical Algebra and Diagrammatics, New J. Phys. 13 (2011) 043016 [arXiv:0906.4725, doi:10.1088/1367-2630/13/4/043016]
Textbook account:
Introduction and review:
John van de Wetering, ZX-calculus for the working quantum computer scientist [arXiv:2012.13966]
Bob Coecke, Basic ZX-calculus for students and professionals [arXiv:2303.03163]
Further developments:
Relation to braided fusion categories for anyon braiding:
See also:
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Last revised on August 15, 2023 at 08:36:45. See the history of this page for a list of all contributions to it.