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The ZX-calculus is a graphical language for reasoning about quantum computation that has recently seen an increased usage in a variety of areas such as quantum circuit optimisation, surface codes and lattice surgery, measurement-based quantum computation, and quantum foundations.
An algebraic axiomatisation of ZX-calculus
Quanlong Wang · 1911
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On completeness of algebraic ZX-calculus over arbitrary commutative rings and semirings
Quanlong Wang · 1912
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On Frobeniusean algebras II
Tadasi Nakayama · 1941
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Positive functions on C*-algebras
W. Forrest Stinespring · 1955
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Categorical algebra
Saunders MacLane · 1965
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Eulerian lines in finite 4-valent graphs and their transformations
Anton Kotzig · 1968
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Ordinal sums and equational doctrines
F William Lawvere · 1969
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Applications of negative dimensional tensors
Roger Penrose · 1971
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Quantum computation with ions in thermal motion
Anders Sørensen and Klaus Mølmer · 1971
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Coherence for compact closed categories
Gregory M Kelly and Miguel L Laplaza · 1980
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Three qubits can be entangled in two inequivalent ways
Wolfgang Dür, Guifre Vidal, and J Ignacio Cirac · 2000
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A One-Way Quantum Computer
Robert Raussendorf and Hans J. Briegel · 2001
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Both Toffoli and controlled-NOT need little help to do universal quantum computation
Yaoyun Shi · 2002
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A simple proof that Toffoli and Hadamard are quantum universal
Dorit Aharonov · 2003
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Entanglement and Quaternions: The graphical calculus ZQ
Hector Miller-Bakewell · 2003
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Improved simulation of stabilizer circuits
Scott Aaronson and Daniel Gottesman · 2004
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A categorical semantics of quantum protocols
Samson Abramsky and Bob Coecke · 2004
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Tensor Network Rewriting Strategies for Satisfiability and Counting
Niel de Beaudrap, Aleks Kissinger, and Konstantinos Meichanetzidis · 2004
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Frobenius algebras and 2-d topological quantum field theories , volume 59
Joachim Kock · 2004
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Frobenius monads and pseudomonoids
Ross Street · 2004
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Graphical description of the action of local Clifford transformations on graph states
Maarten Van den Nest, Jeroen Dehaene, and Bart De Moor · 2004
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Dagger compact closed categories and completely positive maps
Peter Selinger · 2006
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Generalized flow and determinism in measurement-based quantum computation
Daniel E. Browne, Elham Kashefi, Mehdi Mhalla, and Simon Perdrix · 2007
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A graphical calculus for quantum observables
Bob Coecke and Ross Duncan · 2007
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Interacting quantum observables
Bob Coecke and Ross Duncan · 2008
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Quantum measurements without sums
Bob Coecke and Dusko Pavlovic · 2008
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Graphical description of the action of Clifford operators on stabilizer states
Matthew B Elliott, Bryan Eastin, and Carlton M Caves · 2008
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Optimal synthesis of linear reversible circuits
Ketan Markov, Igor Patel, and John Hayes · 2008
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Classical and Quantum Structuralism , page 29–69
Bob Coecke, Éric Oliver Paquette, and Dusko Pavlovic · 2009
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Graphical reasoning in compact closed categories for quantum computation
Lucas Dixon and Ross Duncan · 2009
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Restrictions on transversal encoded quantum gate sets
Bryan Eastin and Emanuel Knill · 2009
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The compositional structure of multipartite quantum entanglement
Bob Coecke and Aleks Kissinger · 2010
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The GHZ/W-calculus contains rational arithmetic
Bob Coecke, Aleks Kissinger, Alex Merry, and Shibdas Roy · 2010
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Rewriting Measurement-Based Quantum Computations with Generalised Flow
Ross Duncan and Simon Perdrix · 2010
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A survey of graphical languages for monoidal categories
Peter Selinger · 2010
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Interacting quantum observables: categorical algebra and diagrammatics
Bob Coecke and Ross Duncan · 2011
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Quantum picturalism for topological cluster-state computing
Clare Horsman · 2011
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QMA-complete problems
Adam D Bookatz · 2012
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Environment and classical channels in categorical quantum mechanics
Bob Coecke and Simon Perdrix · 2012
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Strong Complementarity and Non-locality in Categorical Quantum Mechanics
Bob Coecke, Ross Duncan, Aleks Kissinger, and Quanlong Wang · 2012
Cited alongside, same era.
Aleks Kissinger · 2012
Cited alongside, same era.
A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits
M. Amy, D. Maslov, M. Mosca, and M. Roetteler · 2013
Cited alongside, same era.
A new description of orthogonal bases
Bob Coecke, Dusko Pavlovic, and Jamie Vicary · 2013
Cited alongside, same era.
A graphical approach to measurement-based quantum computing
Ross Duncan · 2013
Cited alongside, same era.
Low-overhead constructions for the fault-tolerant Toffoli gate
Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics
Kang Feng Ng and Quanlong Wang · 2018
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A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond
Renaud Vilmart · 2018
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Towards large-scale functional verification of universal quantum circuits
Matthew Amy · 2019
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T-count optimization and Reed-Muller codes
Matthew Amy and Michele Mosca · 2019
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ZH: A complete graphical calculus for quantum computations involving classical non-linearity
Miriam Backens and Aleks Kissinger · 2019
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SZX-Calculus: Scalable Graphical Quantum Reasoning
Titouan Carette, Dominic Horsman, and Simon Perdrix · 2019
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Cody Jones · 2013
Cited alongside, same era.
Categories for the working mathematician , volume 5
Saunders Mac Lane · 2013
Cited alongside, same era.
Quantum circuits of T-depth one
Peter Selinger · 2013
Cited alongside, same era.
Polynomial-time T-depth optimization of Clifford+ T circuits via matroid partitioning
Matthew Amy, Dmitri Maslov, and Michele Mosca · 2014
Cited alongside, same era.
The ZX-calculus is complete for the single-qubit Clifford+T group
Miriam Backens · 2014
Cited alongside, same era.
Interacting bialgebras are Frobenius
Filippo Bonchi, Paweł Sobociński, and Fabio Zanasi · 2014
Cited alongside, same era.
Quantum Linear Network Coding as One-way Quantum Computation
Niel de Beaudrap and Martin Roetteler · 2014
Cited alongside, same era.
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Completeness of Graphical Languages for Mixed States Quantum Mechanics
Titouan Carette, Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart · 2019
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Techniques to Reduce π / 4 \pi/4 -Parity-Phase Circuits, Motivated by the ZX Calculus
Niel de Beaudrap, Xiaoning Bian, and Quanlong Wang · 2019
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Pauli Fusion: a Computational Model to Realise Quantum Transformations from ZX Terms
Niel de Beaudrap, Ross Duncan, Dominic Horsman, and Simon Perdrix · 2019
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An Invitation to Applied Category Theory: Seven Sketches in Compositionality
Brendan Fong and David I Spivak · 2019
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Efficient magic state factories with a catalyzed | C C Z ⟩ |CCZ\rangle to 2 | T ⟩ 2|T\rangle transformation
Craig Gidney and Austin G. Fowler · 2019
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A diagrammatic calculus of fermionic quantum circuits
Amar Hadzihasanovic, Giovanni de Felice, and Kang Feng Ng · 2019
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Categories for Quantum Theory: an introduction
Chris Heunen and Jamie Vicary · 2019
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A Generic Normal Form for ZX-Diagrams and Application to the Rational Angle Completeness
Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart · 2019
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Universal MBQC with generalised parity-phase interactions and Pauli measurements
Aleks Kissinger and John van de Wetering · 2019
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PyZX: Large Scale Automated Diagrammatic Reasoning
Aleks Kissinger and John van de Wetering · 2019
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Graphical fourier theory and the cost of quantum addition
Stach Kuijpers, John van de Wetering, and Aleks Kissinger · 2019
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A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery
Daniel Litinski · 2019
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Finite Verification of Infinite Families of Diagram Equations
Hector Miller-Bakewell · 2019
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Completeness of the Phase-free ZH-calculus
John van de Wetering and Sal Wolffs · 2019
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A Near-Minimal Axiomatisation of ZX-Calculus for Pure Qubit Quantum Mechanics
Renaud Vilmart · 2019
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Optimizing T gates in Clifford+T circuit as π / 4 \pi/4 rotations around Paulis
Fang Zhang and Jianxin Chen · 2019
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Quantum linear network coding for entanglement distribution in restricted architectures
Niel de Beaudrap and Steven Herbert · 2020
Closest in time.
String Diagram Rewrite Theory I: Rewriting with Frobenius Structure
Filippo Bonchi, Fabio Gadducci, Aleks Kissinger, Pawel Sobocinski, and Fabio Zanasi · 2020
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A Recipe for Quantum Graphical Languages
Titouan Carette and Emmanuel Jeandel · 2020
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Foundations for Near-Term Quantum Natural Language Processing
Bob Coecke, Giovanni de Felice, Konstantinos Meichanetzidis, and Alexis Toumi · 2020
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Hopf-Frobenius Algebras and a Simpler Drinfeld Double
Joseph Collins and Ross Duncan · 2020
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The ZX& calculus: A complete graphical calculus for classical circuits using spiders
Cole Comfort · 2020
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Phase Gadget Synthesis for Shallow Circuits
Alexander Cowtan, Silas Dilkes, Ross Duncan, Will Simmons, and Seyon Sivarajah · 2020
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AKLT-states as ZX-diagrams: diagrammatic reasoning for quantum states
Richard D. P. East, John van de Wetering, Nicholas Chancellor, and Adolfo G. Grushin · 2020
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The ZX calculus is a language for surface code lattice surgery
Niel de Beaudrap and Dominic Horsman · 2020
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Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus
Ross Duncan, Aleks Kissinger, Simon Perdrix, and John van de Wetering · 2020
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Effective Compression of Quantum Braided Circuits Aided by ZX-Calculus
Michael Hanks, Marta P. Estarellas, William J. Munro, and Kae Nemoto · 2020
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Completeness of the ZX-Calculus
Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart · 2020
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Hypergraph simplification: Linking the path-sum approach to the ZH-calculus
Louis Lemonnier, John van de Wetering, and Aleks Kissinger · 2020
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Algebraic complete axiomatisation of ZX-calculus with a normal form via elementary matrix operations
Quanlong Wang · 2020
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On the controlled-NOT complexity of controlled-NOT–phase circuits
Matthew Amy, Parsiad Azimzadeh, and Michele Mosca · 2058
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Parallelizing quantum circuit synthesis
Olivia Di Matteo and Michele Mosca · 2058
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An efficient quantum compiler that reduces T count
Luke E Heyfron and Earl T Campbell · 2058
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