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A quantum circuit may be strongly classically simulated with the aid of ZX-calculus by decomposing its $t$ T-gates into a sum of $2^{\alpha t}$ classically computable stabiliser terms. In this paper, we introduce a general procedure to find…

Quantum Physics · Physics 2024-08-13 Matthew Sutcliffe , Aleks Kissinger

Diagrammatic representations of quantum algorithms and circuits offer novel approaches to their design and analysis. In this work, we describe extensions of the ZX-calculus especially suitable for parameterized quantum circuits, in…

Quantum Physics · Physics 2023-11-16 Tobias Stollenwerk , Stuart Hadfield

Quantum computers allow a near-exponential speed-up for specific applications when compared to classical computers. Despite recent advances in the hardware of quantum computers, their practical usage is still severely limited due to a…

Quantum Physics · Physics 2025-05-09 Tobias Fischbach , Pierre Talbot , Pascal Bouvry

The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum mechanics and quantum information theory. It comes equipped with an equational presentation. We focus here on a very important property of the language:…

Quantum Physics · Physics 2023-06-22 Emmanuel Jeandel , Simon Perdrix , Renaud Vilmart

Systematic discovery of optimization paths in quantum circuit simplification remains a challenge. Today, ZX-calculus, a computing model for quantum circuit transformation, is attracting attention for its highly abstract graph-based…

Programming Languages · Computer Science 2025-11-20 Kayo Tei , Haruto Mishina , Naoki Yamamoto , Kazunori Ueda

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

Recent advances in classical simulation of Clifford+T circuits make use of the ZX calculus to iteratively decompose and simplify magic states into stabiliser terms. We improve on this method by studying stabiliser decompositions of ZX…

Quantum Physics · Physics 2025-09-23 Mark Koch , Richie Yeung , Quanlong Wang

Parametrised quantum circuits contain phase gates whose phase is determined by a classical algorithm prior to running the circuit on a quantum device. Such circuits are used in variational algorithms like QAOA and VQE. In order for these…

Quantum Physics · Physics 2025-08-27 John van de Wetering , Richie Yeung , Tuomas Laakkonen , Aleks Kissinger

Quantum computing is an emerging computational paradigm with the potential to outperform classical computers in solving a variety of problems. To achieve this, quantum programs are typically represented as quantum circuits, which must be…

Quantum Physics · Physics 2025-11-18 Valter Uotila , Cong Yu , Bo Zhao

Classical simulation of quantum circuits is a pivotal part of the quantum computing landscape, specially within the NISQ era, where the constraints imposed by available hardware are unavoidable. The Gottesman-Knill theorem further motivates…

Quantum Physics · Physics 2025-04-23 Fernando Lima , Arcesio Castañeda Medina

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

Quantum circuits are considered more powerful than classical circuits and require exponential resources to simulate classically. Clifford circuits are a special class of quantum circuits that can be simulated in polynomial time but still…

Quantum Physics · Physics 2025-12-09 Yuchen Pang , Edgar Solomonik

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

Flow criteria are used to efficiently extract computations, either in the form of measurement patterns or quantum circuits, from ZX-diagrams. Existing criteria such as causal flow, generalised flow, and Pauli flow, were all originally…

Quantum Physics · Physics 2026-03-11 Aleks Kissinger , John van de Wetering

We introduce a novel method for strong classical simulation of quantum circuits based on optimally k-partitioning ZX-diagrams, reducing each part individually, and then efficiently cross-referencing their results to conclude the overall…

Quantum Physics · Physics 2024-09-04 Matthew Sutcliffe

Recent completeness results on the ZX-Calculus used a third-party language, namely the ZW-Calculus. As a consequence, these proofs are elegant, but sadly non-constructive. We address this issue in the following. To do so, we first describe…

Quantum Physics · Physics 2018-05-15 Emmanuel Jeandel , Simon Perdrix , Renaud Vilmart

Recently, we gave a complete axiomatisation of the ZX-calculus for the overall pure qubit quantum mechanics. Based on this result, here we also obtain a complete axiomatisation of the ZX-calculus for the Clifford+T quantum mechanics by…

Quantum Physics · Physics 2018-01-30 Kang Feng Ng , Quanlong Wang

Quantum circuit optimization - the process of transforming a quantum circuit into an equivalent one with reduced time and space requirements - is crucial for maximizing the utility of current and near-future quantum devices. While most…

Quantum Physics · Physics 2026-01-23 Marcin Szyniszewski , Aleks Kissinger , Noah Linden , Paul Skrzypczyk

The ZX-calculus is a convenient formalism for expressing and reasoning about quantum circuits at a low level, whereas the recently-proposed ZH-calculus yields convenient expressions of mid-level quantum gates such as Toffoli and CCZ. In…

Quantum Physics · Physics 2019-04-17 Stach Kuijpers , John van de Wetering , Aleks Kissinger

We present a quantum compilation algorithm that maps Clifford encoders, encoding maps for stabilizer quantum codes, to a unique graphical representation in the ZX calculus. Specifically, we develop a canonical form in the ZX calculus and…

Quantum Physics · Physics 2025-02-11 Andrey Boris Khesin , Jonathan Z. Lu , Peter W. Shor