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Loop quantum gravity (LQG) attempts to unify general relativity with quantum physics to offer a complete description of the universe by quantising spacetime geometry, but the numerical calculations we encounter are extraordinarily…

General Relativity and Quantum Cosmology · Physics 2025-11-21 Ben Priestley

We present a complete optimization procedure for hybrid quantum-classical circuits with classical parity logic. While common optimization techniques for quantum algorithms focus on rewriting solely the pure quantum segments, there is…

Quantum Physics · Physics 2022-06-22 Agustín Borgna , Simon Perdrix , Benoît Valiron

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

From Feynman diagrams to tensor networks, diagrammatic representations of computations in quantum mechanics have catalysed progress in physics. These diagrams represent the underlying mathematical operations and aid physical interpretation,…

Other Condensed Matter · Physics 2022-01-11 Richard D. P. East , John van de Wetering , Nicholas Chancellor , Adolfo G. Grushin

Compact closed categories provide a foundational formalism for a variety of important domains, including quantum computation. These categories have a natural visualisation as a form of graphs. We present a formalism for equational reasoning…

Symbolic Computation · Computer Science 2009-02-04 Lucas Dixon , Ross Duncan

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

Despite outstanding performance in many tasks, language models are notoriously inclined to make factual errors in tasks requiring arithmetic computation. We address this deficiency by creating Calc-X, a collection of datasets that…

Machine Learning · Computer Science 2023-10-24 Marek Kadlčík , Michal Štefánik , Ondřej Sotolář , Vlastimil Martinek

We present a simple and efficient way to reduce the contraction cost of a tensor network to simulate a quantum circuit. We start by interpreting the circuit as a ZX-diagram. We then use simplification and local complementation rules to…

Quantum Physics · Physics 2023-05-05 Tristan Cam , Simon Martiel

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…

The prop formalism allows representation of processes withstring diagrams and has been successfully applied in various areas such as quantum computing, electric circuits and control flow graphs. However, these graphical approaches suffer…

Logic in Computer Science · Computer Science 2020-07-08 Titouan Carette , 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

We introduce a ZX-like diagrammatic language devoted to manipulating real matrices - and rebits -, with its own set of axioms. We prove the necessity of some non trivial axioms of these. We show that some restriction of the language is…

Quantum Physics · Physics 2018-03-05 Emmanuel Jeandel , Simon Perdrix , Renaud Vilmart

The ZX-Calculus is a powerful graphical language for quantum mechanics and quantum information processing. The completeness of the language -- i.e. the ability to derive any true equation -- is a crucial question. In the quest of a complete…

Quantum Physics · Physics 2017-06-27 Emmanuel Jeandel , Simon Perdrix , Renaud Vilmart , Quanlong Wang

We introduce the Spin-ZX calculus as an elevation of Penrose's diagrams and associated binor calculus to the level of a formal diagrammatic language. The power of doing so is illustrated by the variety of scientific areas we apply it to:…

Quantum Physics · Physics 2025-11-11 Quanlong Wang , Richard D. P. East , Razin A. Shaikh , Lia Yeh , Boldizsár Poór , Bob Coecke

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

ZX-calculus has proved to be a useful tool for quantum technology with a wide range of successful applications. Most of these applications are of an algebraic nature. However, other tasks that involve differentiation and integration remain…

Quantum Physics · Physics 2024-10-09 Quanlong Wang , Richie Yeung , Mark Koch

Finite-dimensional quantum theory serves as the theoretical foundation for quantum information and computation. Mathematically, it is formalized in the category FHilb, comprising all finite-dimensional Hilbert spaces and linear maps between…

Quantum Physics · Physics 2026-04-28 Quanlong Wang , Boldizsár Poór , Razin A. Shaikh

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 ZW-calculus is a graphical language capable of representing 2-dimensional quantum systems (qubit) through its diagrams, and manipulating them through its equational theory. We extend the formalism to accommodate finite dimensional…

Quantum Physics · Physics 2024-12-06 Marc de Visme , Renaud Vilmart

The ZX calculus and ZH calculus use diagrams to denote and compute properties of quantum operations, using `rewrite rules' to transform between diagrams which denote the same operator through a functorial semantic map. Different semantic…

Quantum Physics · Physics 2024-08-26 Niel de Beaudrap , Richard D. P. East