Quantum Semantic Interaction

Alеxander Raikov

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.

Henry Pierce Stapp

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.

Wenjun Shi, Qinxiang Cao, Yuxin Deng

In order to verify the functional correctness of quantum circuits or algorithms, a prominent approach is to specify them as quantum programs and semi-automatically deduce them in a theorem prover. It is indispensable to first formalize the semantics of the basic quantum language. We formalize in Coq an imperative language which allows for classical and quantum information interactions. We define small-step operational semantics and state-based denotational semantics. Then we prove a consistency theorem between these two semantics. A distribution-based denotational semantics is also defined and related to the state-based one. Finally, we describe a few typical quantum algorithms and utilize the distribution-based denotational semantics to verify their correctness.

Hutchison, David, Naor, Moni, John C. Mitchell +20 more