Semantic Techniques in Quantum Computation: Classical and Quantum Structuralism

Ichiro Hasuo A, Naohiko Hoshino

While much of the current study on quantum computation employs low-level formalisms such as quantum circuits, several high-level languages/calculi have been recently proposed aiming at structured quantum programming. The current work contributes to the semantical study of such languages, by providing interaction-based semantics of a functional quantum programming language, the latter is based on linear lambda calculus and is equipped with features like the! modality and recursion. The proposed denotational model is the first one that supports the full features of a quantum functional programming language, we also prove adequacy of our semantics. The construction of our model is by a series of existing techniques taken from the semantics of classical computation as well as from process theory. The most notable among them is Girard's Geometry of Interaction (GoI), categorically formulated by Abramsky, Haghverdi and Scott. The mathematical genericity of these techniques - largely dueto their categorical formulation - is exploited for our move from classical to quantum.

María Luisa Dalla Chiara, Roberto Giuntini, Roberto Leporini

María Luisa Dalla Chiara, Roberto Giuntini, Roberto Leporini

Ichiro Hasuo A, Naohiko Hoshino

While much of the current study on quantum computation employs low-level formalisms such as quantum circuits, several high-level languages/calculi have been recently proposed aiming at structured quantum programming. The current work contributes to the semantical study of such languages by providing interaction-based semantics of a functional quantum programming language; the latter is, much like Selinger and Valiron's, based on linear lambda calculus and equipped with features like the ! modality and recursion. The proposed denotational model is the first one that supports the full features of a quantum functional programming language; we prove adequacy of our semantics. The construction of our model is by a series of existing techniques taken from the semantics of classical computation as well as from process theory. The most notable among them is Girard's Geometry of Interaction (GoI), categorically formulated by Abramsky, Haghverdi and Scott. The mathematical genericity of these techniques---largely due to their categorical formulation---is exploited for our move from classical to quantum.

Prakash Panangaden

Martin Avanzini, Georg Moser, Romain Péchoux +2 more

We introduce a new kind of expectation transformer for a mixed classical-quantum programming language. Our semantic approach relies on a new notion of a cost structure, which we introduce and which can be seen as a specialisation of the Kegelspitzen of Keimel and Plotkin. We show that our weakest precondition analysis is both sound and adequate with respect to the operational semantics of the language. Using the induced expectation transformer, we provide formal analysis methods for the expected cost analysis and expected value analysis of classical-quantum programs. We illustrate the usefulness of our techniques by computing the expected cost of several well-known quantum algorithms and protocols, such as coin tossing, repeat until success, entangled state preparation, and quantum walks.