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arXiv:2109.02196 · Electronic Proceedings in Theoretical Computer Science

Quantum CPOs

Andre Kornell, Bert Lindenhovius, Michael W. Mislove

Published Sep 6, 2021• 5 citations
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Abstract

We introduce the monoidal closed category qCPO of quantum cpos, whose objects are "quantized" analogs of omega-complete partial orders (cpos). The category qCPO is enriched over the category CPO of cpos, and contains both CPO, and the opposite of the category FdAlg of finite-dimensional von Neumann algebras as monoidal subcategories. We use qCPO to construct a sound model for the quantum programming language Proto-Quipper-M (PQM) extended with term recursion, as well as a sound and computationally adequate model for the Linear/Non-Linear Fixpoint Calculus (LNL-FPC), which is both an extension of the Fixpoint Calculus (FPC) with linear types, and an extension of a circuit-free fragment of PQM that includes recursive types.

DOI
10.4204/eptcs.340.9

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We introduce the monoidal closed category qCPO of quantum cpos, whose objects are "quantized" analogs of omega-complete partial orders (cpos). The category qCPO is enriched over the category CPO of cpos, and contains both CPO, and the opposite of the category FdAlg of finite-dimensional von Neumann algebras as monoidal subcategories. We use qCPO to construct a sound model for the quantum programming language Proto-Quipper-M (PQM) extended with term recursion, as well as a sound and computationally adequate model for the Linear/Non-Linear Fixpoint Calculus (LNL-FPC), which is both an extension of the Fixpoint Calculus (FPC) with linear types, and an extension of a circuit-free fragment of PQM that includes recursive types.
Abstract This paper unites two research lines. The first involves finding categorical models of quantum programming languages with recursion and their type systems. The second line concerns the program of quantization of mathematical structures, which amounts to finding noncommutative generalizations (also called quantum generalizations) of these structures. Using a quantization method called discrete quantization , which essentially amounts to the internalization of structures in a category of von Neumann algebras and quantum relations, we find a noncommutative generalization of ω\omegaω -complete partial orders (cpos), called quantum cpos . Cpos are central in domain theory and are widely used to construct categorical models of programming languages with recursion. We show that quantum cpos have similar categorical properties to cpos and are therefore suitable for the construction of categorical models for quantum programming languages, which is illustrated with some examples. Because of their noncommutative character, quantum cpos may form the backbone of a future quantum domain theory that provides structural methods for the denotational semantics of recursive quantum programming languages.
This thesis contains contributions to the mathematical foundations of quantum programming languages. The likely arrival of scalable quantum computers in the not-so-distant future has resulted in a flurry of activity in the development of quantum programming languages. As in classical computing, the transition from a description of a quantum algorithm found in the literature to a hardware-specific set of instructions run on a quantum device is a complex process, prone to errors. This issue is exacerbated in the quantum setting not only by the complexity of quantum algorithms but also by the fragility of quantum information, which renders ineffective some of the classical techniques used to debug programs. In this thesis, we contribute to the solution of some of these issues. We introduce Proto-Quipper-M, a new quantum programming language designed to serve as a testbed for the research and development of sound mathematical semantics and reasoning techniques for quantum programs. We first present Proto-Quipper-M as a formalization of a fragment of Quipper, a high-level functional programming language for describing families of quantum circuits. In particular, we define Proto-Quipper-M as a simply-typed lambda calculus with a special type for quantum circuits and a strong type system designed to enforce linearity on quantum data, and thus prevent violations of the no-cloning property of quantum information. We endow Proto-Quipper-M with computational meaning via a big-step operational semantics and prove that the language is type-safe by showing that it enjoys the type-preservation and error-freeness properties. We also give Proto-Quipper-M a denotational semantics in a suitable class of monoidal categories and show that these categories give rise to linear-non-linear models in the sense of Benton, and thus models of intuitionistic linear logic. Finally, we crystallize the connection between the syntax and the semantics of the language by proving the soundness theorem for Proto-Quipper-M.
Selinger gave a superoperator model of a first-order quantum programming language and proved that it is fully definable and hence fully abstract. This paper proposes an extension of the superoperator model to higher-order programs based on modules over superoperators or, equivalently, enriched presheaves over the category of superoperators. The enriched presheaf category can be easily proved to be a model of intuitionistic linear logic with cofree exponential, from which one can cave out a model of classical linear logic by a kind of bi-orthogonality construction. Although the structures of an enriched presheaf category are usually rather complex, a morphism in the classical model can be expressed simply as a matrix of completely positive maps. The model inherits many desirable properties from the superoperator model. A conceptually interesting property is that our model has only a state whose “total probability” is bounded by 1, i.e. does not have a state where true and false each occur with probability 2 / 3 . Another convenient property inherited from the superoperator model is a ω CPO-enrichment. Remarkably, our model has a sufficient structure to interpret arbitrary recursive types by the standard domain theoretic technique. We introduce Quantum FPC , a quantum λ -calculus with recursive types, and prove that our model is a fully abstract model of Quantum FPC.
Quipper is a practical programming language for describing families of quantum circuits. In this paper, we formalize a small, but useful fragment of Quipper called Proto-Quipper-M. Unlike its parent Quipper, this language is type-safe and has a formal denotational and operational semantics. Proto-Quipper-M is also more general than Quipper, in that it can describe families of morphisms in any symmetric monoidal category, of which quantum circuits are but one example. We design Proto-Quipper-M from the ground up, by first giving a general categorical model of parameters and state. The distinction between parameters and state is also known from hardware description languages. A parameter is a value that is known at circuit generation time, whereas a state is a value that is known at circuit execution time. After finding some interesting categorical structures in the model, we then define the programming language to fit the model. We cement the connection between the language and the model by proving type safety, soundness, and adequacy properties.
Modern quantum programming languages integrate quantum resources and classical control. They must, on the one hand, be linearly typed to reflect the no-cloning property of quantum resources. On the other hand, high-level and practical languages should also support quantum circuits as first-class citizens, as well as families of circuits that are indexed by some classical parameters. Quantum programming languages thus need linear dependent type theory. This paper defines a general semantic structure for such a type theory via certain fibrations of monoidal categories. The categorical model of the quantum circuit description language Proto-Quipper-M by Rios and Selinger (2017) constitutes an example of such a fibration, which means that the language can readily be integrated with dependent types. We then devise both a general linear dependent type system and a dependently typed extension of Proto-Quipper-M, and provide them with operational semantics as well as a prototype implementation.
Modern quantum programming languages integrate quantum resources and classical control. They must, on the one hand, be linearly typed to reflect the no-cloning property of quantum resources. On the other hand, high-level and practical languages should also support quantum circuits as first-class citizens, as well as families of circuits that are indexed by some classical parameters. Quantum programming languages thus need linear dependent type theory. This paper defines a general semantic structure for such a type theory via certain fibrations of monoidal categories. The categorical model of the quantum circuit description language Proto-Quipper-M in [28] constitutes an example of such a fibration, which means that the language can readily be integrated with dependent types. We then devise both a general linear dependent type system and a dependently typed extension of Proto-Quipper-M, and provide them with operational semantics as well as a prototype implementation.