Verifying the Smallest Interesting Colour Code with Quantomatic

Garvie, Liam; Duncan, Ross

doi:10.4204/eptcs.266.10

Cited by 11 publications

(14 citation statements)

References 23 publications

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“…Due to the absence of large, universal quantum computers and the inherent difficulty of simulating quantum circuits, testing is generally not a viable option for verification. By contrast, various methods of formal verification have been developed for quantum circuits and programs, including equivalence checking [7,30,31], diagrammatic methods [13,15], model checkers [6,16], program logics [32] and formal proof [27]. However, two questions remain: how can the intended effect of a quantum program be specified in a clear, human readable and verifiable way, and how can we scale automated verification to large circuits?…”

Section: Introductionmentioning

confidence: 99%

“…Typical functional verification methods -verification of the precise input-output relation -either verify equivalence against a simpler circuit or diagrammatic implementation (e.g., [15,30,31]), or a matrix representation such as a unitary or superoperator (e.g., [27]). With either approach, errors can creep in on the specification side, as both circuit and matrix presentations can be difficult for humans to write and understand.…”

Section: Introductionmentioning

confidence: 99%

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Towards Large-scale Functional Verification of Universal Quantum Circuits

Amy

2019

Electron. Proc. Theor. Comput. Sci.

124

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We introduce a framework for the formal specification and verification of quantum circuits based on the Feynman path integral. Our formalism, built around exponential sums of polynomial functions, provides a structured and natural way of specifying quantum operations, particularly for quantum implementations of classical functions. Verification of circuits over all levels of the Clifford hierarchy with respect to either a specification or reference circuit is enabled by a novel rewrite system for exponential sums with free variables. Our algorithm is further shown to give a polynomial-time decision procedure for checking the equivalence of Clifford group circuits. We evaluate our methods by performing automated verification of optimized Clifford+T circuits with up to 100 qubits and thousands of T gates, as well as the functional verification of quantum algorithms using hundreds of qubits. Our experiments culminate in the automated verification of the Hidden Shift algorithm for a class of Boolean functions in a fraction of the time it has taken recent algorithms to simulate.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Towards Large-scale Functional Verification of Universal Quantum Circuits

Amy

2019

Electron. Proc. Theor. Comput. Sci.

124

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“…quantum picturalism). This has provided an intuitive picture of many aspects of quantum information, computation, and a wide variety of other fields [10, 25, 50-52, 54-58, 60, 75, 81, 91, 97, 98, 106, 110] which lends itself well to computational automation [26,59,62,79].…”

Section: Introductionmentioning

confidence: 99%

Reconstructing quantum theory from diagrammatic postulates

Selby

Scandolo

Coecke

2021

Quantum

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A reconstruction of quantum theory refers to both a mathematical and a conceptual paradigm that allows one to derive the usual formulation of quantum theory from a set of primitive assumptions. The motivation for doing so is a discomfort with the usual formulation of quantum theory, a discomfort that started with its originator John von Neumann. We present a reconstruction of finite-dimensional quantum theory where all of the postulates are stated in diagrammatic terms, making them intuitive. Equivalently, they are stated in category-theoretic terms, making them mathematically appealing. Again equivalently, they are stated in process-theoretic terms, establishing that the conceptual backbone of quantum theory concerns the manner in which systems and processes compose. Aside from the diagrammatic form, the key novel aspect of this reconstruction is the introduction of a new postulate, symmetric purification. Unlike the ordinary purification postulate, symmetric purification applies equally well to classical theory as well as quantum theory. Therefore we first reconstruct the full process theoretic description of quantum theory, consisting of composite classical-quantum systems and their interactions, before restricting ourselves to just the ‘fully quantum’ systems as the final step. We propose two novel alternative manners of doing so, ‘no-leaking’ (roughly that information gain causes disturbance) and ‘purity of cups’ (roughly the existence of entangled states). Interestingly, these turn out to be equivalent in any process theory with cups & caps. Additionally, we show how the standard purification postulate can be seen as an immediate consequence of the symmetric purification postulate and purity of cups. Other tangential results concern the specific frameworks of generalised probabilistic theories (GPTs) and process theories (a.k.a. CQM). Firstly, we provide a diagrammatic presentation of GPTs, which, henceforth, can be subsumed under process theories. Secondly, we argue that the ‘sharp dagger’ is indeed the right choice of a dagger structure as this sharpness is vital to the reconstruction.

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“…The rewrite rules of the ZX-calculus allow us to graphically rewrite ZX-diagrams, and this gives us a universal, sound, complete language for qubit linear algebra [7]. The ZX-calculus has found practical use in quantum information, with results in measurementbased quantum computation [8][9][10], topological quantum computation [11][12][13][14], quantum error correction [15,16] and quantum circuit compilation and optimisation [17][18][19][20][21][22]. For practical reasons, we will work in this paper with a variation of this calculus called the ZXH-calculus [23].…”

Section: Introductionmentioning

confidence: 99%

Spin-networks in the ZX-calculus

East¹,

Martin-Dussaud²,

Wetering³

2021

Preprint

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The ZX-calculus, and the variant we consider in this paper (ZXH-calculus), are formal diagrammatic languages for qubit quantum computing. We show that it can also be used to describe SU(2) representation theory. To achieve this, we first recall the definition of Yutsis diagrams, a standard graphical calculus used in quantum chemistry and quantum gravity, which captures the main features of SU(2) representation theory. Second, we show precisely how it embed within Penrose's binor calculus. Third, we subsume both calculus into ZXH-diagrams. In the process we show how the SU(2) invariance of Wigner symbols is trivially provable in the ZXH-calculus. Additionally, we show how we can explicitly diagrammatically calculate 3jm, 4jm and 6j symbols. It has long been thought that quantum gravity should be closely aligned to quantum information theory. In this paper, we present a way in which this connection can be made exact, by writing the spin-networks of loop quantum gravity (LQG) in the ZX-diagrammatic language of quantum computation.

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Verifying the Smallest Interesting Colour Code with Quantomatic

Cited by 11 publications

References 23 publications

Towards Large-scale Functional Verification of Universal Quantum Circuits

Towards Large-scale Functional Verification of Universal Quantum Circuits

Reconstructing quantum theory from diagrammatic postulates

Spin-networks in the ZX-calculus

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