ZX-calculus (changes) in nLab

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Contents

Idea

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≃ℂ 2QBit \simeq \mathbb{C}^2, namely

1. the “computational basis”

   {|0⟩,|1⟩} \big\{ \left\vert 0 \right\rangle ,\, \left\vert 1 \right\rangle \big\}

   which consists (by common convention) of the eigenstates of the Pauli-Z gate;

2. the Hadamard basis

   {12(|0⟩±|1⟩)} \Big\{ \tfrac{1}{\sqrt{2}} \big( \left\vert 0 \right\rangle \pm \left\vert 1 \right\rangle \big) \Big\}

   which consists of the eigenstates of the Pauli-X gate;

by observing [[Coecke & Duncan, p. 1](#CoeckeDuncan)] that:

1. either equips QBit=ℂ 2QBit = \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)](quantum+measurement#CoeckePavlović08)]);

2. 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:

1. the (co-)multiplication and (co)-unit string diagrams of these two Frobenius algebra-structures are to be denoted

   1. in green for the Z-basis

   2. in red for the X-basis

2. 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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References

Landing page:

• zxcalculus.com

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](https://doi.org/10.1007/978-3-642-14162-1_24)]

• Ross Duncan, A graphical approach to measurement-based quantum computing [[arXiv:1203.6242](https://arxiv.org/abs/1203.6242) video exposition:YT]

The official introduction of the ZX-calculus:

• Bob Coecke, Ross Duncan, A graphical calculus for quantum observables [[pdf](https://www.cs.ox.ac.uk/people/bob.coecke/GreenRed.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](https://doi.org/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](http://arxiv.org/abs/0906.4725), doi:10.1088/1367-2630/13/4/043016]

Textbook account:

• Bob Coecke, Stefano Gogioso, Quantum in Pictures, Quantinuum Publications (2023) [[ISBN 978-1739214715](https://www.amazon.co.uk/dp/1739214714), Quantinuum blog]

Introduction and review:

• John van de Wetering, ZX-calculus for the working quantum computer scientist [[arXiv:2012.13966](https://arxiv.org/abs/2012.13966)]

• Bob Coecke, Basic ZX-calculus for students and professionals [[arXiv:2303.03163](https://arxiv.org/abs/2303.03163)]

Further developments:

• Hector Bombin, Daniel Litinski, Naomi Nickerson, Fernando Pastawski, Sam Roberts, Unifying flavors of fault tolerance with the ZX calculus [[arXiv:2303.08829](https://arxiv.org/abs/2303.08829)]

Relation to braided fusion categories for anyon braiding:

• Fatimah Rita Ahmadi, Aleks Kissinger, Topological Quantum Computation Through the Lens of Categorical Quantum Mechanics [[arXiv:2211.03855]]

On software verification for the ZX-calculus:

• Adrian Lehmann, Ben Caldwell, Bhakti Shah, Robert Rand: VyZX: Formal Verification of a Graphical Quantum Language [[arXiv:2311.11571](https://arxiv.org/abs/2311.11571)]

• Robert Rand: Verifying the ZX-calculus and its Friends, talk at Running HoTT 2024 [video:kt]

See also:

• Razin A. Shaikh, Quanlong Wang, Richie Yeung, How to sum and exponentiate Hamiltonians in ZXW calculus [[arXiv:2212.04462](https://arxiv.org/abs/2212.04462)]

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