ZX-calculus publicationshttp://zxcalculus.com/publications.rssAn up to date list of the newest publications related to the ZX-calculusen-USSun, 03 Jan 2021 19:44:19 GMTrfeed v1.0.0https://github.com/svpino/rfeed/blob/master/README.mdZX-calculus for the working quantum computer scientisthttp://arxiv.org/abs/2012.13966The 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 computation, and quantum foundations. The first half of this review gives a gentle introduction to the ZX-calculus suitable for those familiar with the basics of quantum computing. The aim here is to make the reader comfortable enough with the ZX-calculus that they could use it in their daily work for small computations on quantum circuits and states. The latter sections give a condensed overview of the literature on the ZX-calculus. We discuss Clifford computation and graphically prove the Gottesman-Knill theorem, we discuss a recently introduced extension of the ZX-calculus that allows for convenient reasoning about Toffoli gates, and we discuss the recent completeness theorems for the ZX-calculus that show that, in principle, all reasoning about quantum computation can be done using ZX-diagrams. Additionally, we discuss the categorical and algebraic origins of the ZX-calculus and we discuss several extensions of the language which can represent mixed states, measurement, classical control and higher-dimensional qudits.John van de WeteringSun, 27 Dec 2020 00:00:00 GMThttp://arxiv.org/abs/2012.13966A note on diagonal gates in SZX-calculushttp://arxiv.org/abs/2012.09540This note describes how the scalable ZXH calculus can be used to represent in a compact way the quantum gates that are diagonal in the computational basis. This includes controlled and multi-controlled Z gates, their generalizations, respectively graph and hypergraph operators, and also phase gadgets.Titouan CaretteThu, 17 Dec 2020 00:00:00 GMThttp://arxiv.org/abs/2012.09540Constructing quantum circuits with global gateshttp://arxiv.org/abs/2012.09061There are various gate sets that can be used to describe a quantum computation. A particularly popular gate set in the literature on quantum computing consists of arbitrary single-qubit gates and 2-qubit CNOT gates. A CNOT gate is however not always the natural multi-qubit interaction that can be implemented on a given physical quantum computer, necessitating a compilation step that transforms these CNOT gates to the native gate set. A particularly interesting case where compilation is necessary is for ion trap quantum computers, where the natural entangling operation can act on more than 2 qubits and can even act globally on all qubits at once. This calls for an entirely different approach to constructing efficient circuits. In this paper we study the problem of converting a given circuit that uses 2-qubit gates to one that uses global gates. Our three main contributions are as follows. First, we find an efficient algorithm for transforming an arbitrary circuit consisting of Clifford gates and arbitrary phase gates into a circuit consisting of single-qubit gates and a number of global interactions proportional to the number of non-Clifford phases present in the original circuit. Second, we find a general strategy to transform a global gate that targets all qubits into one that targets only a subset of the qubits. This approach scales linearly with the number of qubits that are not targeted, in contrast to the exponential scaling reported in (Maslov & Nam, N. J. Phys. 2018). Third, we improve on the number of global gates required to synthesise an arbitrary n-qubit Clifford circuit from the 12n-18 reported in (Maslov & Nam, N. J. Phys. 2018) to 6n-8.John van de WeteringWed, 16 Dec 2020 00:00:00 GMThttp://arxiv.org/abs/2012.09061Foundations for Near-Term Quantum Natural Language Processinghttp://arxiv.org/abs/2012.03755We provide conceptual and mathematical foundations for near-term quantum natural language processing (QNLP), and do so in quantum computer scientist friendly terms. We opted for an expository presentation style, and provide references for supporting empirical evidence and formal statements concerning mathematical generality. We recall how the quantum model for natural language that we employ canonically combines linguistic meanings with rich linguistic structure, most notably grammar. In particular, the fact that it takes a quantum-like model to combine meaning and structure, establishes QNLP as quantum-native, on par with simulation of quantum systems. Moreover, the now leading Noisy Intermediate-Scale Quantum (NISQ) paradigm for encoding classical data on quantum hardware, variational quantum circuits, makes NISQ exceptionally QNLP-friendly: linguistic structure can be encoded as a free lunch, in contrast to the apparently exponentially expensive classical encoding of grammar. Quantum speed-up for QNLP tasks has already been established in previous work with Will Zeng. Here we provide a broader range of tasks which all enjoy the same advantage. Diagrammatic reasoning is at the heart of QNLP. Firstly, the quantum model interprets language as quantum processes via the diagrammatic formalism of categorical quantum mechanics. Secondly, these diagrams are via ZX-calculus translated into quantum circuits. Parameterisations of meanings then become the circuit variables to be learned. Our encoding of linguistic structure within quantum circuits also embodies a novel approach for establishing word-meanings that goes beyond the current standards in mainstream AI, by placing linguistic structure at the heart of Wittgenstein's meaning-is-context.Bob Coecke, Giovanni de Felice, Konstantinos Meichanetzidis and Alexis ToumiMon, 07 Dec 2020 00:00:00 GMThttp://arxiv.org/abs/2012.03755AKLT-states as ZX-diagrams: diagrammatic reasoning for quantum stateshttp://arxiv.org/abs/2012.01219From 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, but cannot generally be computed with directly. In this paper we introduce the ZXH-calculus, a graphical language based on the ZX-calculus, that we use to represent and reason about many-body states entirely graphically. As a demonstration, we express the 1D AKLT state, a symmetry protected topological state, in the ZXH-calculus by developing a representation of spins higher than 1/2 within the calculus. By exploiting the simplifying power of the ZXH-calculus rules we show how this representation straightforwardly recovers two important properties, the existence of topologically protected edge states, and the non-vanishing of a string order parameter. We furthermore show how the AKLT matrix-product state representation can be recovered from our diagrams. In addition, we provide an alternative proof that the 2D AKLT state on a hexagonal lattice can be reduced to a graph state, demonstrating that it is a universal quantum computing resource. Our results show that the ZXH-calculus is a powerful language for representing and computing with physical states entirely graphically, paving the way to develop more efficient many-body algorithms.Richard D. P. East, John van de Wetering, Nicholas Chancellor and Adolfo G. GrushinWed, 02 Dec 2020 00:00:00 GMThttp://arxiv.org/abs/2012.01219Graphical calculi and their conjecture synthesishttps://arxiv.org/abs/2010.03914Categorical Quantum Mechanics, and graphical calculi in particular, has proven to be an intuitive and powerful way to reason about quantum computing. This work continues the exploration of graphical calculi, inside and outside of the quantum computing setting, by investigating the algebraic structures with which we label diagrams. The initial aim for this was Conjecture Synthesis; the algorithmic process of creating theorems. To this process we introduce a generalisation step, which itself requires the ability to infer and then verify parameterised families of theorems. This thesis introduces such inference and verification frameworks, in doing so forging novel links between graphical calculi and fields such as Algebraic Geometry and Galois Theory. These frameworks inspired further research into the design of graphical calculi, and we introduce two important new calculi here. First is the calculus RING, which is initial among ring-based qubit graphical calculi, and in turn inspired the introduction and classification of phase homomorphism pairs also presented here. The second is the calculus ZQ, an edge-decorated calculus which naturally expresses arbitrary qubit rotations, eliminating the need for non-linear rules such as (EU) of ZX. It is expected that these results will be of use to those creating optimisation schemes and intermediate representations for quantum computing, to those creating new graphical calculi, and for those performing conjecture synthesis.Hector Miller-BakewellThu, 08 Oct 2020 00:00:00 GMThttps://arxiv.org/abs/2010.03914ZX-Calculus and Extended Hypergraph Rewriting Systems I: A Multiway Approach to Categorical Quantum Information Theoryhttp://arxiv.org/abs/2010.02752Categorical quantum mechanics and the Wolfram model offer distinct but complementary approaches to studying the relationship between diagrammatic rewriting systems over combinatorial structures and the foundations of physics; the objective of the present article is to begin elucidating the formal correspondence between the two methodologies in the context of the ZX-calculus formalism of Coecke and Duncan for reasoning diagrammatically about linear maps between qubits. After briefly summarizing the relevant formalisms, and presenting a categorical formulation of the Wolfram model in terms of adhesive categories and double-pushout rewriting systems, we illustrate how the diagrammatic rewritings of the ZX-calculus can be embedded and realized within the broader context of Wolfram model multiway systems, and illustrate some of the capabilities of the software framework (ZXMultiwaySystem) that we have developed specifically for this purpose. Finally, we present a proof (along with an explicitly computed example) based on the methods of Dixon and Kissinger that the multiway evolution graphs and branchial graphs of the Wolfram model are naturally endowed with a monoidal structure based on rulial composition that is, furthermore, compatible with the monoidal product of ZX-diagrams.Jonathan Gorard, Manojna Namuduri and Xerxes D. ArsiwallaMon, 05 Oct 2020 00:00:00 GMThttp://arxiv.org/abs/2010.02752A Recipe for Quantum Graphical Languageshttps://dx.doi.org/10.4230/LIPIcs.ICALP.2020.118Different 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 forever. This article answers negatively. All those language share a common core structure we call Z*-algebras. We classify Z*-algebras up to isomorphism in two dimensional Hilbert spaces and show that they are all variations of the aforementioned calculi. We do the same for linear relations and show that the calculus of [2] is essentially the unique one.Titouan Carette and Emmanuel JeandelMon, 10 Aug 2020 00:00:00 GMThttp://arxiv.org/abs/2008.04193A Compositional Model of Consciousness based on Consciousness-Onlyhttp://arxiv.org/abs/2007.16138Scientific studies of consciousness rely on objects whose existence is independent of any consciousness. This theoretical-assumption leads to the "hard problem" of consciousness. We avoid this problem by assuming consciousness to be fundamental, and the main feature of consciousness is characterized as being other-dependent. We set up a framework which naturally subsumes the other-dependent feature by defining a compact closed category where morphisms represent conscious processes. These morphisms are a composition of a set of generators, each being specified by their relations with other generators, and therefore other-dependent. The framework is general enough, i.e. parameters in the morphisms take values in arbitrary commutative semi-rings, from which any finitely dimensional system can be dealt with. Our proposal fits well into a compositional model of consciousness and is an important step forward that addresses both the hard problem of consciousness and the combination problem of (proto)-panpsychism.Camilo Miguel Signorelli, Quanlong Wang and Ilyas KhanFri, 31 Jul 2020 00:00:00 GMThttp://arxiv.org/abs/2007.16138Algebraic complete axiomatisation of ZX-calculus with a normal form via elementary matrix operationshttp://arxiv.org/abs/2007.13739In this paper we give a complete axiomatisation of qubit ZX-calculus via elementary transformations which are basic operations in linear algebra. This formalism has two main advantages. First, all the operations of the phases are algebraic ones without trigonometry functions involved, thus paved the way for generalising complete axiomatisation of qubit ZX-calculus to qudit ZX-calculus and ZX-calculus over commutative semirings. Second, we characterise elementary transformations in terms of ZX diagrams, so a lot of linear algebra stuff can be done purely diagrammatically.Quanlong WangSun, 26 Jul 2020 00:00:00 GMThttp://arxiv.org/abs/2007.13739