Generalised Compositional Theories and Diagrammatic Reasoning

Coecke, Bob; Duncan, Ross; Kissinger, Aleks; Wang, Quanlong

doi:10.1007/978-94-017-7303-4_10

Cited by 30 publications

(32 citation statements)

References 41 publications

(70 reference statements)

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“…are both equality in the path-sum framework. While much progress has been made towards computational methods for diagrammatic reasoning [9,11,13,15], our framework allows us to use standard algebraic tools (e.g., rewriting) without explicitly managing structural laws.…”

Section: Definition 26 (Sequential Composition)mentioning

confidence: 99%

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: Definition 26 (Sequential Composition)mentioning

confidence: 99%

Towards Large-scale Functional Verification of Universal Quantum Circuits

Amy

2019

Electron. Proc. Theor. Comput. Sci.

124

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“…While in the convex set framework one generally starts from convex sets associated with individual systems, and builds composites from them, the OPT framework takes the composition of physical processes as primitive. Mathematically, the 'top-down' approach of the OPT framework is underpinned by the graphical language of circuits [75][76][77][78][79][80]. In this section we give an informal presentation, referring the reader to the original articles for a more in-depth discussion.…”

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confidence: 99%

Microcanonical thermodynamics in general physical theories

Chiribella

Scandolo

2017

New J. Phys.

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Microcanonical thermodynamics studies the operations that can be performed on systems with welldefined energy. So far, this approach has been applied to classical and quantum systems. Here we extend it to arbitrary physical theories, proposing two requirements for the development of a general microcanonical framework. We then formulate three resource theories, corresponding to three different sets of basic operations: (i) random reversible operations, resulting from reversible dynamics with fluctuating parameters, (ii) noisy operations, generated by the interaction with ancillas in the microcanonical state, and (iii) unital operations, defined as the operations that preserve the microcanonical state. We focus our attention on a class of physical theories, called sharp theories with purification, where these three sets of operations exhibit remarkable properties. Firstly, each set is contained into the next. Secondly, the convertibility of states by unital operations is completely characterised by a majorisation criterion. Thirdly, the three sets are equivalent in terms of state convertibility if and only if the dynamics allowed by theory satisfy a suitable condition, which we call unrestricted reversibility. Under this condition, we derive a duality between the resource theories of microcanonical thermodynamics and the resource theory of pure bipartite entanglement.1. random unitary channels [43][44][45], arising from unitary dynamics with randomly fluctuating parameters;2. noisy operations [46][47][48], generated by preparing ancillas in the microcanonical state, turning on a unitary dynamics, and discarding the ancillas;3. unital channels [49,50], defined as the quantum processes that preserve the microcanonical state.These three sets are strictly different: the set of random unitary channels is strictly contained in the set of noisy operations [51], and the latter is strictly contained in the set of unital channels [52]. In spite of this, the three sets are equivalent in terms of state convertibility [48]. This means that all the natural candidates for the sets of free operations induce the same notion of resource. This resource is generally called purity, and plays a fundamental role in many quantum protocols [53].In this paper we extend the paradigm of microcanonical thermodynamics from quantum theory to arbitrary physical theories [54][55][56][57][58][59][60][61]. We propose two minimal requirements a probabilistic theory must satisfy in order to support a microcanonical description, and, when these requirements are satisfied, we provide a general operational definition of random reversible, noisy, and unital operations. We then focus on a special class of theories, called sharp theories with purification, which are appealing for the foundations of thermodynamics [62], and have also been studied for their computational power [63, 64] and interference properties [65]. In sharp theories with purification, we show that the three sets of operations satisfy the same inclusion relations as in quantum theory, with...

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“…Given a diagram D : n → m we define its adjoint D † : m → n to be the diagram obtained by reflecting the diagram around the vertical axis and negating all the angles. Thus the terms of the ZX-calculus naturally form a a †-PROP [8,10], just like the Clifford circuits of the previous section. The three types of single vertex shown in Figure 1 can then be seen as the generators of this PROP, which we call ZX.…”

Section: The Zx-calculusmentioning

confidence: 87%

Optimising Clifford Circuits with Quantomatic

Fagan¹,

Duncan²

2019

Electron. Proc. Theor. Comput. Sci.

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We present a system of equations between Clifford circuits, all derivable in the ZX-calculus, and formalised as rewrite rules in the Quantomatic proof assistant. By combining these rules with some non-trivial simplification procedures defined in the Quantomatic tactic language, we demonstrate the use of Quantomatic as a circuit optimisation tool. We prove that the system always reduces Clifford circuits of one or two qubits to their minimal form, and give numerical results demonstrating its performance on larger Clifford circuits.The Quantomatic project files All the proofs which appear in this paper and its appendix are publicly available as a downloadable Quantomatic project at https://gitlab.cis.strath.ac.uk/kwb13215/ Clifford-Quanto/.

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Generalised Compositional Theories and Diagrammatic Reasoning

Cited by 30 publications

References 41 publications

Towards Large-scale Functional Verification of Universal Quantum Circuits

Towards Large-scale Functional Verification of Universal Quantum Circuits

Microcanonical thermodynamics in general physical theories

Optimising Clifford Circuits with Quantomatic

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