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ZX-calculus for the working quantum computer scientist
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ZX-calculus for the working quantum computer scientist
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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. The first half of this review gives a gentle introduction to the ZX-calculus suitable for those familiar with the basics of quantum computing. The aim here is to make the reader comfortable enough with the ZX-calculus that they could use it in their daily work for small computations on quantum circuits and states. The latter sections give a condensed overview of the literature on the ZX-calculus. We discuss Clifford computation and graphically prove the Gottesman-Knill theorem, we discuss a recently introduced extension of the ZX-calculus that allows for convenient reasoning about Toffoli gates, and we discuss the recent completeness theorems for the ZX-calculus that show that, in principle, all reasoning about quantum computation can be done using ZX-diagrams. Additionally, we discuss the categorical and algebraic origins of the ZX-calculus and we discuss several extensions of the language which can represent mixed states, measurement, classical control and higher-dimensional qudits.
Forward citations
Cited by 26 Pith papers
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The Delayed Stabilizer ZX-Calculus
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Minimality of the Stabilizer ZX Calculus
The stabilizer ZX calculus rule set is minimal because the red/green compact-structure coincidence rule and the bialgebra law are each individually necessary relative to the connectivity meta-rule.
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Current-state opacity is formalized in safe partially observed quantum Petri nets with true-concurrency semantics and verified exactly via stabilizer formalism and targeted unfolding.
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Hybrid Fourier Neural Operator for Surrogate Modeling of Laser Processing with a Quantum-Circuit Mixer
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Generating one-way computations with flow: flow-preserving rewriting that ignores the interpretation
Three flow-preserving ZX rewrite rules (IO, LC, ZL) are necessary and sufficient to generate any labelled open graph with Pauli flow or gflow from a trivial diagram of matching inputs and outputs.
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Floquet Abelian Multicycle Codes
Floquet Abelian multicycle codes encode logical qubits in measurement-only schedules derived from higher-dimensional chain complexes, with compact examples at [[108,6,5]], [[144,6,8]], and [[324,6,10]].
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Fractal quantum many-body scars and Hamiltonian inverse design from ZX-calculus
Sierpiński-triangle ZX-diagrams yield exact quantum many-body scars in local chaotic Hamiltonians, with ZX identities certifying the annihilation.
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Generating one-way computations with flow: flow-preserving rewriting that ignores the interpretation
Three flow-preserving rewrite rules — (IO), (LC), (ZL) — are complete and minimal for generating any labelled open graph with Pauli flow or gflow from a trivial diagram.
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Gauging the Spacetime Code
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Quokka#: Quantum Computing with #SAT
Quokka# is a Python library that converts quantum circuit analysis tasks into #SAT problems, offering multiple encodings, approximate equivalence checking, and depth-optimal synthesis.
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Synthesis and Optimization of Encoding Circuits for Fault-Tolerant Quantum Computation
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Linear-Time T-Gate Optimization via Random Abstraction
A linear-time randomized static analysis that propagates constant-width bitstrings enables phase folding and T-count optimization matching SOTA tools on large circuits.
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String Diagrams for Quantum Foundations, Computing and Natural Language Processing
String diagrams formalize constructor theory with locality-composition conflicts, enable wave-based Boolean logic design and optimization, and map Urdu text circuits equivalently to English ones up to gate translation...
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Meromorphic Quantum Computing
Projectivizing quantum kinematics produces meromorphic functions characterizing coherent circuits for quantum error correction and magic state distillation.
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FTPrimitiveBench: A Benchmark Suite For Logical Computation Under Hardware-Motivated and Biased Noise Models
FTPrimitiveBench is a new benchmark suite for testing surface-code logical primitives under Pauli-biased, measurement-biased, and spatially non-uniform noise models, revealing that noise structure interacts distinctly...
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Layered monoidal theories let different abstraction levels of a system live in one string diagram with formal translations between layers.
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FeynmanDD: Quantum Circuit Analysis with Classical Decision Diagrams
FeynmanDD maps Feynman path integral sums onto classical decision diagrams, so quantum circuit amplitudes, probabilities, and equivalence checks become BDD counting tasks that run very fast on structured circuits.
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FTPrimitiveBench: A Benchmark Suite For Logical Computation Under Hardware-Motivated and Biased Noise Models
FTPrimitiveBench is an open-source pipeline that connects parameterized hardware-motivated noise models to surface-code logical primitive circuits, enabling reproducible cross-primitive QEC benchmarking under Pauli bi...
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Multi-objective optimization and quantum hybridization of equivariant deep learning interatomic potentials
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Double categories for adaptive quantum computation
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Categorical Tensor-Graph Semantics for Quantum Algorithms
Quantum algorithms (Bernstein–Vazirani, Simon, qutrit Deutsch–Jozsa, single-shot Grover) and entanglement states are recast as tensor-graph diagrams in FHilb, with CNOT via complementary Frobenius algebras.
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Categorical Tensor-Graph Semantics for Quantum Algorithms
Standard and qutrit quantum algorithms are recast as categorical tensor diagrams, with a claimed distribution criterion for single-shot Grover that does not withstand scrutiny.
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Quantum Purification for Amplitude Damping Noise
Postselecting on the no-jump measurement in a one- or two-ancilla circuit improves state and channel fidelity under amplitude-damping noise, leaving a residual amplitude attenuation.
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