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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 21 Pith papers
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Finding diagonal logical gates in CSS codes and circuits
Diagonal logical gates of a CSS code or circuit are exactly the kernel of a pullback map on phase functions, and that kernel can be computed in cubic time.
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The Delayed Stabilizer ZX-Calculus
A complete delayed stabilizer ZX-calculus with delay generator, generating-tableau semantics, and unique normal forms via generalized local complementation captures infinite translation-invariant stabilizer processes.
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Formal Verification of Variational Quantum Circuits
The paper introduces an abstract-interpretation framework with interval domains for formally verifying robustness of variational quantum circuit classifiers, and reports certified perturbation bounds on Iris and MNIST.
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Complete Compositional Syntax for Finite Transducers on Finite and Bi-Infinite Words
A sound and complete equational theory of string diagrams is developed for finite-state transducers, covering both finite and bi-infinite words, with a new canonical normal form for sofic subshifts.
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Towards Lattice Surgery Compilation for the Color Code Using Pipe Diagrams
Distance-independent pipe diagrams for the 6.6.6 triangular color code, with ZX correspondence, correlation surfaces, and syndrome extraction, enable spacetime lattice-surgery compilation beyond the surface code.
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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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Layered Monoidal Theories I: Diagrammatic Algebra and Applications
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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Graphical Calculus for Fermionic Tensors
A parity-aware graphical calculus extends the ZX diagram language to fermionic modes, covering Gaussian states, partial traces, purification, fermionization/bosonization, and fermionic error-correcting codes.
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STABSim: A Parallelized Clifford Simulator with Features Beyond Direct Simulation
STABSim is a GPU-accelerated Clifford tableau simulator with new measurement handling, exact T1/T2 noise sampling in a common regime, and a fast Clifford+T to PBC transpiler.
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Generalised Process Theories
A generalised process theory is an algebra for a wiring operad, subsuming traditional, time-neutral, causal, higher-order, and enriched process theories.
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Multi-objective optimization and quantum hybridization of equivariant deep learning interatomic potentials
Inserting a quantum depth-infused layer into Allegro gives the best force accuracy on a copper-lithium dataset, about 13% better than a classical MLP variant, but not on other datasets.
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Double categories for adaptive quantum computation
The paper unifies circuit, MBQC, magic-state, and Pauli measurement models as double categories, with quantum information horizontal and classical control vertical, and recasts the contextual-fraction bound on computi...
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Quantum Circuit Optimization Based on Dynamic Grouping and ZX-Calculus for Reducing 2-Qubit Gate Count
A dynamic grouping plus ZX-calculus lookahead framework reduces two-qubit gate counts in quantum circuits by 18% on average across 25 benchmarks.
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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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One Polynomial Strategy for Computing Local Projections on Square-Lattice Cluster States
The note conjectures a polynomial-time recursive method for computing arbitrary local projections on 2D square-lattice cluster states, but the core 2D recursion is not proved and the numerical evidence is too small to...
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Two imaginarity monotones induced by unified $(\alpha,\beta)$-relative entropy
The paper defines two imaginarity monotones from the unified (α,β)-relative entropy, but the main property theorems are undermined by incorrect inequality directions.
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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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Quantum computing and artificial intelligence: status and perspectives
A broad expert white paper sets a European research agenda for combining quantum computing and AI, spanning quantum machine learning, AI-driven quantum control, and foundational questions.
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