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There are various gate sets used for describing quantum computation. A particularly popular one consists of Clifford gates and arbitrary single-qubit phase gates. Computations in this gate set can be elegantly described by the ZX-calculus,…

The ZX-calculus is a graphical calculus for reasoning about quantum systems and processes. It is known to be universal for pure state qubit quantum mechanics, meaning any pure state, unitary operation and post-selected pure projective…

Quantum Physics · Physics 2014-09-22 Miriam Backens

In the noisy intermediate-scale quantum (NISQ) era, two-qubit gates in quantum circuits are more susceptible to noise than single-qubit gates. Therefore, reducing the number of two-qubit gates is crucial for improving circuit efficiency and…

Quantum Physics · Physics 2025-07-22 Kai Chen , Wen Liu , GuoSheng Xu , Yangzhi Li , Maoduo Li , Shouli He

The ZX-calculus is a graphical language for reasoning about ZX-diagrams, a type of tensor networks that can represent arbitrary linear maps between qubits. Using the ZX-calculus, we can intuitively reason about quantum theory, and optimise…

Quantum Physics · Physics 2020-05-04 Aleks Kissinger , John van de Wetering

In this paper, we introduce a technique for contracting (i.e. numerically evaluating) ZX-diagrams whose complexity scales with their rank-width, a graph parameter that behaves nicely under ZX rewrite rules. Given a rank-decomposition of…

Quantum Physics · Physics 2026-03-10 Fedor Kuyanov , Aleks Kissinger

We introduce the Scalable ZX-calculus (SZX-calculus for short), a formal and compact graphical language for the design and verification of quantum computations. The SZX-calculus is an extension of the ZX-calculus, a powerful framework that…

Quantum Physics · Physics 2020-07-31 Titouan Carette , Dominic Horsman , Simon Perdrix

Circuit cutting is a promising technique that leverages both quantum and classical computational resources, enabling the practical execution of large quantum circuits on noisy intermediate-scale quantum (NISQ) hardware. Recent approaches…

Quantum Physics · Physics 2026-04-15 Shuwen Kan , Yanni Li , Hao Wang , Sara Mouradian , Ying Mao

This paper presents a novel approach to quantum architecture search by integrating the techniques of ZX-calculus with Genetic Programming (GP) to optimize the structure of parameterized quantum circuits employed in Quantum Machine Learning…

Quantum Physics · Physics 2025-03-21 Tom Ewen , Ivica Turkalj , Patrick Holzer , Mark-Oliver Wolf

We propose several methods for optimizing the number of qubits in a quantum circuit while preserving the number of non-Clifford gates. One of our approaches consists in reversing, as much as possible, the gadgetization of Hadamard gates,…

Quantum Physics · Physics 2024-07-16 Vivien Vandaele

Graphical languages offer intuitive and rigorous formalisms for quantum physics. They can be used to simplify expressions, derive equalities, and do computations. Yet in order to replace conventional formalisms, rigour alone is not…

Quantum Physics · Physics 2016-03-01 Miriam Backens

We start by studying the subgroup structures underlying stabilizer circuits and we use our results to propose a new normal form for stabilizer circuits. This normal form is computed by induction using simple conjugation rules in the…

Quantum Physics · Physics 2021-07-05 Marc Bataille

The ZX-calculus is a powerful framework for reasoning in quantum computing. It provides in particular a compact representation of matrices of interests. A peculiar property of the ZX-calculus is the absence of a formal sum allowing the…

Quantum Physics · Physics 2024-08-07 Emmanuel Jeandel , Simon Perdrix , Margarita Veshchezerova

We present a smorgasbord of results on the stabiliser ZX-calculus for odd prime-dimensional qudits (i.e. qupits). We derive a simplified rule set that closely resembles the original rules of qubit ZX-calculus. Using these rules, we…

Quantum Physics · Physics 2023-09-01 Boldizsár Poór , Robert I. Booth , Titouan Carette , John van de Wetering , Lia Yeh

The ZX-calculus is a graphical calculus for reasoning about pure state qubit quantum mechanics. It is complete for pure qubit stabilizer quantum mechanics, meaning any equality involving only stabilizer operations that can be derived using…

Quantum Physics · Physics 2014-12-31 Miriam Backens

This work shows that minimizing the depth of a quantum circuit composed of commuting operations reduces to a vertex coloring problem on an appropriately constructed graph, where gates correspond to vertices and edges encode…

Quantum Physics · Physics 2026-02-11 Hochang Lee , Kyung Chul Jeong , Panjin Kim

We present a new graphical calculus that is sound and complete for a universal family of quantum circuits, which can be seen as the natural string-diagrammatic extension of the approximately (real-valued) universal family of Hadamard+CCZ…

Quantum Physics · Physics 2019-01-30 Miriam Backens , Aleks Kissinger

Quantum computing carries significant potential for addressing practical problems. However, currently available quantum devices suffer from noisy quantum gates, which degrade the fidelity of executed quantum circuits. Therefore, quantum…

Quantum Physics · Physics 2025-02-25 Ji Liu , Alvin Gonzales , Benchen Huang , Zain Hamid Saleem , Paul Hovland

Graphical calculi such as the ZH-calculus are powerful tools in the study and analysis of quantum processes, with links to other models of quantum computation such as quantum circuits, measurement-based computing, etc. A somewhat compact…

Quantum Physics · Physics 2021-07-05 Renaud Vilmart

The stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics.The language is sound and complete: a stabilizer ZX-diagram can be transformed into another one if and only if these two diagrams represent…

Quantum Physics · Physics 2017-01-04 Miriam Backens , Simon Perdrix , Quanlong Wang

Counting the solutions to Boolean formulae defines the problem #SAT, which is complete for the complexity class #P. We use the ZH-calculus, a universal and complete graphical language for linear maps which naturally encodes counting…

Computational Complexity · Computer Science 2023-09-01 Tuomas Laakkonen , Konstantinos Meichanetzidis , John van de Wetering