nLab ZX-calculus

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 $QBit \simeq \mathbb{C}^2$, namely

1. the “computational basis”

   $$\big\{ \left\vert 0 \right\rangle ,\, \left\vert 1 \right\rangle \big\}$$

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

2. the Hadamard basis

   $$\Big\{ \tfrac{1}{\sqrt{2}} \big( \left\vert 0 \right\rangle \pm \left\vert 1 \right\rangle \big) \Big\}$$

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

by observing [Coecke & Duncan, p. 1] that:

1. 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)]);

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]

• 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:

• Bob Coecke, Stefano Gogioso, Quantum in Pictures, Quantinuum Publications (2023) [ISBN 978-1739214715, Quantinuum blog]

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:

• Hector Bombin, Daniel Litinski, Naomi Nickerson, Fernando Pastawski, Sam Roberts, Unifying flavors of fault tolerance with the ZX calculus [arXiv: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]

• 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]

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