A Near-Minimal Axiomatisation of ZX-Calculus for Pure Qubit Quantum Mechanics

Vilmart, Renaud

doi:10.1109/lics.2019.8785765

Cited by 59 publications

(56 citation statements)

References 24 publications

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“…Another "family" of categories that share this structure is the family of graphical languages for quantum computation: ZX-Calculus, ZW-Calculus and ZH-Calculus [3,7,8]. All three formalisms represent morphisms of Qubit using diagrams, and come with equational theories, proven to be complete for the whole category [3,11,19], i.e. whenever two diagrams represent the same morphism of Qubit, the first can be turned into the other using only the equational theory.…”

Section: Introductionmentioning

confidence: 99%

The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford

Vilmart

2021

Lecture Notes in Computer Science

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We show that the formalism of “Sum-Over-Path” (SOP), used for symbolically representing linear maps or quantum operators, together with a proper rewrite system, has the structure of a dagger-compact PROP. Several consequences arise from this observation:– Morphisms of SOP are very close to the diagrams of the graphical calculus called ZH-Calculus, so we give a system of interpretation between the two– A construction, called the discard construction, can be applied to enrich the formalism so that, in particular, it can represent the quantum measurement.We also enrich the rewrite system so as to get the completeness of the Clifford fragments of both the initial formalism and its enriched version.

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Section: Introductionmentioning

confidence: 99%

The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford

Vilmart

2021

Lecture Notes in Computer Science

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“…For completeness when considering exact equality, see Remark 2.3. Recent extensions to the calculus have been introduced which are complete for the larger Clifford+T family of ZX-diagrams [35], where phases are multiples of π/4, and for all ZX-diagrams [28,34,51].…”

Section: The Zx-calculusmentioning

confidence: 99%

“…Rewriting of patterns or circuits, as well as translations between the two models, can be performed using the ZX-calculus, a graphical language for quantum computation [11]. This language is more flexible than quantum circuit notation and also has multiple complete sets of graphical rewrite rules [28,34,35,51]. While translating a measurement pattern to a quantum circuit can be difficult, the translation between patterns and ZX-diagrams is straightforward [24,26,37].…”

Section: Introductionmentioning

confidence: 99%

There and back again: A circuit extraction tale

Backens

Miller-Bakewell

Felice

et al. 2021

Quantum

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Translations between the quantum circuit model and the measurement-based one-way model are useful for verification and optimisation of quantum computations. They make crucial use of a property known as gflow. While gflow is defined for one-way computations allowing measurements in three different planes of the Bloch sphere, most research so far has focused on computations containing only measurements in the XY-plane. Here, we give the first circuit-extraction algorithm to work for one-way computations containing measurements in all three planes and having gflow. The algorithm is efficient and the resulting circuits do not contain ancillae. One-way computations are represented using the ZX-calculus, hence the algorithm also represents the most general known procedure for extracting circuits from ZX-diagrams. In developing this algorithm, we generalise several concepts and results previously known for computations containing only XY-plane measurements. We bring together several known rewrite rules for measurement patterns and formalise them in a unified notation using the ZX-calculus. These rules are used to simplify measurement patterns by reducing the number of qubits while preserving both the semantics and the existence of gflow. The results can be applied to circuit optimisation by translating circuits to patterns and back again.

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“…2. Extensions of the calculus exist that are complete for larger families of ZX-diagrams/maps, including Clifford+T ZX-diagrams [20], where phases are multiples of π/4, and arbitrary ZX-diagrams where any phase is allowed [21][22][23].…”

Section: A Background: the Zx-calculusmentioning

confidence: 99%

Reducing the number of non-Clifford gates in quantum circuits

2020

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We present a method for reducing the number of non-Clifford quantum gates, in particularly T-gates, in a circuit, an important task for efficiently implementing fault-tolerant quantum computations. This method matches or beats previous approaches to ancillae-free T-count reduction on the majority of our benchmark circuits, in some cases yielding up to 50% improvement. Our method begins by representing the quantum circuit as a ZX-diagram, a tensor networklike structure that can be transformed and simplified according to the rules of the ZX-calculus. We then extend a recent simplification strategy with a different ingredient, phase gadgetization, which we use to propagate non-Clifford phases through a ZX-diagram to find nonlocal cancellations. Our procedure extends unmodified to arbitrary phase angles and to parameter elimination for variational circuits. Finally, our optimization is self-checking, in the sense that the simplification strategy we propose is powerful enough to independently validate equality of the input circuit and the optimized output circuit. We have implemented the routines of this paper in the open-source library PyZX.

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A Near-Minimal Axiomatisation of ZX-Calculus for Pure Qubit Quantum Mechanics

Cited by 59 publications

References 24 publications

The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford

The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford

There and back again: A circuit extraction tale

Reducing the number of non-Clifford gates in quantum circuits

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