ZX-calculus publicationshttp://zxcalculus.com/publications.rssAn up to date list of the newest publications related to the ZX-calculusen-USThu, 04 Jun 2020 16:32:44 GMTrfeed v1.0.0https://github.com/svpino/rfeed/blob/master/README.mdThe ZX& calculus: A complete graphical calculus for classical circuits using spidershttps://arxiv.org/abs/2004.05287We give a complete presentation for the fragment, ZX&, of the ZX-calculus generated by the Z and X spiders (corresponding to copying and addition) along with the not gate and the and gate. To prove completeness, we freely add units and counits to the category TOF generated by the Toffoli gate and ancillary bits, showing that this yields the strictification of spans of powers of the two element set; and then perform a two way translation between this category and ZX&. A translation to some extension of TOF, as opposed to some fragment of the ZX-calculus, is a natural choice because of the multiplicative nature of the Toffoli gate. To this end, we show that freely adding counits to the semi-Frobenius algebra of a discrete inverse category is the same as computing the "environment structure" of the classical structures of the base discrete inverse category. We show that in this setting, the classical channels and the discrete Cartesian completion are the same constructions. Therefore, in the case of TOF, freely adding a counit, constructing the category of quantum channels, and computing the discrete Cartesian completion are all equivalent to partial functions between powers of the two element set. By glueing together the free counit completion and the free unit completion, this yields the strictification of spans between powers of the two element set. Cole ComfortWed, 29 Apr 2020 00:00:00 GMThttps://arxiv.org/abs/2004.05287Fast and effective techniques for T-count reduction via spider nest identitieshttps://arxiv.org/abs/2004.05164In fault-tolerant quantum computing systems, realising (approximately) universal quantum computation is usually described in terms of realising Clifford+T operations, which is to say a circuit of CNOT, Hadamard, and π/2-phase rotations, together with T operations (π/4-phase rotations). For many error correcting codes, fault-tolerant realisations of Clifford operations are significantly less resource-intensive than those of T gates, which motivates finding ways to realise the same transformation involving T-count (the number of T gates involved) which is as low as possible. Investigations into this problem [arXiv:1206.0758, 1303.2042, 1308.4134, 1601.07363, 1606.01904, 1701.00140] has led to observations that this problem is closely related to NP-hard tensor decomposition problems [arXiv:1712.01557] and is tantamount to the difficult problem of decoding exponentially long Reed-Muller codes [arXiv:1601.07363]. This problem then presents itself as one for which must be content in practise with approximate optimisation, in which one develops an array of tactics to be deployed through some pragmatic strategy. In this vein, we describe techniques to reduce the T-count, based on the effective application of "spider nest identities": easily recognised products of parity-phase operations which are equivalent to the identity operation. We demonstrate the effectiveness of such techniques by obtaining improvements in the T-counts of a number of circuits, in run-times which are typically less than the time required to make a fresh cup of coffee. Niel de Beaudrap, Xiaoning Bian and Quanlong WangFri, 10 Apr 2020 00:00:00 GMThttps://arxiv.org/abs/2004.05164Hypergraph simplification: Linking the path-sum approach to the ZH-calculushttps://arxiv.org/abs/2003.13564The ZH-calculus is a complete graphical calculus for linear maps between qubits that admits, for example, a straightforward encoding of hypergraph states and circuits arising from the Toffoli+Hadamard gate set. In this paper, we establish a correspondence between the ZH-calculus and the path-sum formalism, a technique recently introduced by Amy to verify quantum circuits. In particular, we find a bijection between certain canonical forms of ZH-diagrams and path-sum expressions. We then introduce and prove several new simplification rules for the ZH-calculus, which are in direct correspondence to the simplification rules of the path-sum formalism. The relatively opaque path-sum rules are shown to arise naturally as the convergence of two consequences of the ZH-calculus. The first is the extension of the familiar graph-theoretic simplifications based on local complementation and pivoting to their hypergraph-theoretic analogues: hyper-local complementation and hyper-pivoting. The second is the graphical Fourier transform introduced by Kuijpers et al., which enables effective simplification of ZH-diagrams encoding multi-linear phase polynomials with arbitrary real coefficients.Louis Lemonnier, John van de Wetering and Aleks KissingerMon, 30 Mar 2020 00:00:00 GMThttps://arxiv.org/abs/2003.13564Entanglement and Quaternions: The graphical calculus ZQhttps://arxiv.org/abs/2003.09999Graphical calculi are vital tools for representing and reasoning about quantum circuits and processes. Some are not only graphically intuitive but also logically complete. The best known of these is the ZX-calculus, which is an industry candidate for an Intermediate Representation; a language that sits between the algorithm designer's intent and the quantum hardware's gate instructions. The ZX calculus, built from generalised Z and X rotations, has difficulty reasoning about arbitrary rotations. This contrasts with the cross-hardware compiler TriQ which uses these arbitrary rotations to exploit hardware efficiencies. In this paper we introduce the graphical calculus ZQ, which uses quaternions to represent these arbitrary rotations, similar to TriQ, and the phase-free Z spider to represent entanglement, similar to ZX. We show that this calculus is sound and complete for qubit quantum computing, while also showing that a fully spider-based representation would have been impossible. This new calculus extends the zoo of qubit graphical calculi, each with different strengths, and we hope it will provide a common language for the optimisation procedures of both ZX and TriQ.Hector Miller-BakewellSun, 22 Mar 2020 00:00:00 GMThttps://arxiv.org/abs/2003.09999The Structure of Sum-Over-Paths, its Consequences, and Completeness for Cliffordhttps://arxiv.org/abs/2003.05678We 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 a structure of dagger-compact PROP. Several consequences arise from this observation: 1) 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; 2) 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.Renaud VilmartThu, 12 Mar 2020 00:00:00 GMThttps://arxiv.org/abs/2003.05678There and back again: A circuit extraction talehttps://arxiv.org/abs/2003.01664We 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.Miriam Backens, Hector Miller-Bakewell, Giovanni de Felice, Leo Lobski and John van de WeteringTue, 03 Mar 2020 00:00:00 GMThttps://arxiv.org/abs/2003.01664Efficient compression of quantum braided circuitshttps://arxiv.org/abs/1912.11503Mapping a quantum algorithm to any practical large-scale quantum computer will require a sequence of compilations and optimizations. At the level of fault-tolerant encoding, one likely requirement of this process is the translation into a topological circuit, for which braided circuits represent one candidate model. Given the large overhead associated with encoded circuits, it is paramount to reduce their size in terms of computation time and qubit number through circuit compression. While these optimizations have typically been performed in the language of 3-dimensional diagrams, such a representation does not allow an efficient, general and scalable approach to reduction or verification. We propose the use of the ZX-calculus as an intermediate language for braided circuit compression, demonstrating advantage by comparing results using this approach with those previously obtained for the compression of |A⟩ and |Y⟩ state distillation circuits. We then provide a rough benchmark of our method against a small set of Clifford+T circuits, yielding compression percentages of $∼77%$. Our results suggest that the overheads of braided, defect-based circuits are comparable to those of their lattice-surgery counterparts, restoring the potential of this model for surface code quantum computation. Michael Hanks, Marta P. Estarellas, William J. Munro and Kae NemotoTue, 24 Dec 2019 00:00:00 GMThttps://arxiv.org/abs/1912.11503ZX-calculus over arbitrary commutative rings and semiringshttps://arxiv.org/abs/1912.01003In this extended abstract we give an axiomatisation of ZX-calculus over arbi-trary commutative rings and semirings respectively. By a normal form inspiredfrom matrix elementary operations such as row addition and row multiplication,we can obtain that these versions of ZX-calculus are still universal and complete. Quanlong WangMon, 02 Dec 2019 00:00:00 GMThttps://arxiv.org/abs/1912.01003Techniques to Reduce $π/4$-Parity-Phase Circuits, Motivated by the ZX Calculushttps://dx.doi.org/10.4204/EPTCS.318.9To approximate arbitrary unitary transformations on one or more qubits, one must perform transformations which are outside of the Clifford group. The gate most commonly considered for this purpose is the T=diag(1,exp(iπ/4)) gate. As T gates are computationally expensive to perform fault-tolerantly in the most promising error-correction technologies, minimising the "T-count" (the number of T gates) required to realise a given unitary in a Clifford+T circuit is of great interest. We describe techniques to find circuits with reduced T-count in unitary circuits, which develop on the ideas of Heyfron and Campbell [arXiv:1712.01557] with the help of the ZX calculus. Following [arXiv:1712.01557], we reduce the problem to that of minimising the T count of a CNOT+T circuit. The ZX calculus motivates a further reduction to simplifying a product of commuting "π/4-parity-phase" operations: diagonal unitary transformations which induce a relative phase of exp(iπ/4) depending on the outcome of a parity computation on the standard basis (which motivated Kissinger and van de Wetering [1903.10477] to introduce "phase gadgets"). For a number of standard benchmark circuits, we show that these techniques --- in some cases supplemented by the TODD subroutine of Heyfron and Campbell [arXiv:1712.01557] --- yield T-counts comparable to or better than the best previously known results.Niel de Beaudrap, Xiaoning Bian and Quanlong WangFri, 20 Nov 2020 00:00:00 GMThttps://arxiv.org/abs/1911.09039An algebraic axiomatisation of ZX-calculushttps://arxiv.org/abs/1911.06752In this paper we give an algebraic complete axiomatisation of ZX-calculus in the sense that there are only ring operations involved for phases, without any need of trigonometry functions such as sin and cos, in contrast to previous universally complete axiomatisations of ZX-calculus. With this algebraic axiomatisation of ZX-calculus, we are able to establish an invertible translation from ZH-calculus to ZX-calculus and to derive all the ZX-translated rules of ZH-calculus. As a consequence, we have a great benefit that all techniques obtained in ZH-calculus can be transplanted to ZX-calculus. Quanlong WangFri, 15 Nov 2019 00:00:00 GMThttps://arxiv.org/abs/1911.06752