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

Vilmart, Renaud

doi:10.48550/arxiv.1812.09114

Cited by 8 publications

(12 citation statements)

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“…Two complete axiomatizations for full qubits quantum mechanic have been given in [28] and [29]. We will rely on the compact axiomatization of [30]. It correspond to GA with G = Z 2 with the Hadamard gate .…”

Section: E the Monochromatic Zx-calculusmentioning

confidence: 99%

When Only Topology Matters

Carette

2021

Preprint

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Graphical languages are symmetric monoidal categories presented by generators and equations. The string diagrams notation allows to transform numerous axioms into low dimension topological rules we are comfortable with as three dimensional space citizens. This aspect is often referred to by the Only Topology Matters paradigm (OTM). However OTM remains quite informal and its exact meaning in terms of rewriting rules is ambiguous. In this paper we define three precise aspects of the OTM paradigm, namely flexsymmetry, flexcyclicity and flexibility of Frobenius algebras. We investigate how this new framework can simplify the presentation of known graphical languages based on Frobenius algebras.

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Section: E the Monochromatic Zx-calculusmentioning

confidence: 99%

When Only Topology Matters

Carette

2021

Preprint

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“…The ZX-calculus is a graphical language for manipulating linear maps. It was originally introduced in [4] and proven to be complete for qubit quantum computation [12,11,16,25]. A general introduction to the language alongside the overview of the main applications is available in [24].…”

Section: Introductionmentioning

confidence: 99%

Addition and Differentiation of ZX-diagrams

Jeandel¹,

Perdrix²,

Veshchezerova³

2022

Preprint

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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 linear combinations of arbitrary ZX-diagrams. The universality of the formalism guarantees however that for any two ZX-diagrams, the sum of their interpretations can be represented by a ZX-diagram. We introduce a general, inductive definition of the addition of ZX-diagrams, relying on the construction of controlled diagrams. Based on this addition technique, we provide an inductive differentiation of ZX-diagrams.Indeed, given a ZX-diagram with variables in the description of its angles, one can differentiate the diagram according to one of these variables. Differentiation is ubiquitous in quantum mechanics and quantum computing (e.g. for solving optimization problems). Technically, differentiation of ZX-diagrams is strongly related to summation as witnessed by the product rules.We also introduce an alternative, non inductive, differentiation technique rather based on the isolation of the variables. Finally, we apply our results to deduce a diagram for an Ising Hamiltonian.

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“…A brief introduction to the zx-calculus is found in [19]; for a complete treatment see [14]. For reasons of space we omit the zx-calculus inference rules, however we use the complete set of Vilmart [33].…”

Section: Introductionmentioning

confidence: 99%

“…We regret that at the time of writing this feature is not in the publicly available pytket release; it is planned for a future release.3 This is a consequence of the completeness of the zx-calculus[33].…”

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confidence: 99%

Phase Gadget Synthesis for Shallow Circuits

Cowtan,

Dilkes,

Duncan

et al. 2019

Preprint

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We give an overview of the circuit optimisation methods used by t|ket , a compiler system for quantum software developed by Cambridge Quantum Computing Ltd. We focus on a novel technique based around phase gadgets, a family of multi-qubit quantum operations which occur naturally in a wide range of quantum circuits of practical interest. The phase gadgets have a simple presentation in the zx-calculus, which makes it easy to reason about them. Taking advantage of this, we present an efficient method to translate the phase gadgets back to ∧X gates and single qubit operations suitable for execution on a quantum computer with significant reductions in gate count and circuit depth. We demonstrate the effectiveness of these methods on a quantum chemistry benchmarking set based on variational circuits for ground state estimation of small molecules.

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

Cited by 8 publications

References 0 publications

When Only Topology Matters

When Only Topology Matters

Addition and Differentiation of ZX-diagrams

Phase Gadget Synthesis for Shallow Circuits

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