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Related papers: Quantum Algorithms and Oracles with the Scalable Z…

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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

Different graphical calculi have been proposed to represent quantum computation. First the ZX- calculus [4], followed by the ZW-calculus [12] and then the ZH-calculus [1]. We can wonder if new Z*-calculi will continue to be proposed…

Logic in Computer Science · Computer Science 2020-08-11 Titouan Carette , Emmanuel Jeandel

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 Physics · Physics 2020-12-29 John van de Wetering

The Scalable ZX-calculus is a compact graphical language used to reason about linear maps between quantum states. These diagrams have multiple applications, but they frequently have to be constructed in a case-by-case basis. In this work we…

Quantum Physics · Physics 2023-11-17 Augustin Borgna , Rafael Romero

Quantum computations are easily represented in the graphical notation known as the ZX-calculus, a.k.a. the red-green calculus. We demonstrate its use in reasoning about measurement-based quantum computing, where the graphical syntax…

Quantum Physics · Physics 2012-03-29 Ross Duncan

Graphical languages are a convenient shorthand to represent computation, with rewrite rules relating one graph to another. In contrast, proof assistants rely heavily on inductive datatypes, particularly when giving semantics to embedded…

Programming Languages · Computer Science 2026-04-09 Adrian Lehmann , Ben Caldwell , Bhakti Shah , William Spencer , Robert Rand

ZX-calculus is graphical language for quantum computing which usually focuses on qubits. In this paper, we generalise qubit ZX-calculus to qudit ZX-calculus in any finite dimension by introducing suitable generators, especially a carefully…

Quantum Physics · Physics 2022-09-20 Quanlong Wang

The ZX-calculus is a graphical language for suitably represented tensor networks, called ZX-diagrams. Calculations are performed by transforming ZX-diagrams with rewrite rules. The ZX-calculus has found applications in reasoning about…

Computational Complexity · Computer Science 2022-06-22 Alex Townsend-Teague , Konstantinos Meichanetzidis

Quantum computing promises significant speed-ups for certain algorithms but the practical use of current noisy intermediate-scale quantum (NISQ) era computers remains limited by resources constraints (e.g., noise, qubits, gates, and circuit…

Quantum Physics · Physics 2026-03-31 Tobias Fischbach , Pierre Talbot , Pascal Bouvry

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

Continuous-variable (CV) quantum information processing is a promising candidate for large-scale fault-tolerant quantum computation. However, analysis of CV quantum process relies mostly on direct computation of the evolution of operators…

Graphical languages, like quantum circuits or ZX-calculus, have been successfully designed to represent (memoryless) quantum computations acting on a finite number of qubits. Meanwhile, delayed traces have been used as a graphical way to…

Quantum Physics · Physics 2021-04-29 Titouan Carette , Marc de Visme , Simon Perdrix

The ZX-calculus is an algebraic formalism that allows quantum computations to be simplified via a small number of simple graphical rewrite rules. Recently, it was shown that, when combined with a family of "sum-over-Cliffords" techniques,…

Quantum Physics · Physics 2025-08-21 Matthew Sutcliffe , Aleks Kissinger

Quantum Error-Correcting Codes (QECCs) play a crucial role in enhancing the robustness of quantum computing and communication systems against errors. Within the realm of QECCs, stabilizer codes, and specifically graph codes, stand out for…

Quantum Physics · Physics 2024-03-29 Zipeng Wu , Song Cheng , Bei Zeng

The ZX-calculus is a graphical language for quantum processes with built-in rewrite rules. The rewrite rules allow equalities to be derived entirely graphically, leading to the question of completeness: can any equality that is derivable…

Quantum Physics · Physics 2015-11-06 Miriam Backens

ZX-Calculus is a versatile graphical language for quantum computation equipped with an equational theory. Getting inspiration from Geometry of Interaction, in this paper we propose a token-machine-based asynchronous model of both pure…

Logic in Computer Science · Computer Science 2022-08-04 Kostia Chardonnet , Benoît Valiron , Renaud Vilmart

ZX-calculus is a graphical language for quantum computing which is complete in the sense that calculation in matrices can be done in a purely diagrammatic way. However, all previous universally complete axiomatisations of ZX-calculus have…

Quantum Physics · Physics 2021-09-07 Quanlong Wang

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

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

The field of quantum algorithms is vibrant. Still, there is currently a lack of programming languages for describing quantum computation on a practical scale, i.e., not just at the level of toy problems. We address this issue by introducing…

Programming Languages · Computer Science 2013-07-08 Alexander S. Green , Peter LeFanu Lumsdaine , Neil J. Ross , Peter Selinger , Benoît Valiron
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