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arXiv:1807.11610 · Formal Aspects of Computing

Toward automatic verification of quantum programs

Mingsheng Ying

Published Aug 7, 2018• 26 citations
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Abstract

Abstract This paper summarises the results obtained by the author and his collaborators in a program logic approach to the verification of quantum programs, including quantum Hoare logic, invariant generation and termination analysis for quantum programs. It also introduces the notion of proof outline and several auxiliary rules for more conveniently reasoning about quantum programs. Some problems for future research are proposed at the end of the paper.

DOI
10.1007/s00165-018-0465-3

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Abstract This paper summarises the results obtained by the author and his collaborators in a program logic approach to the verification of quantum programs, including quantum Hoare logic, invariant generation and termination analysis for quantum programs. It also introduces the notion of proof outline and several auxiliary rules for more conveniently reasoning about quantum programs. Some problems for future research are proposed at the end of the paper.
Hoare logic is a foundation of axiomatic semantics of classical programs and it provides effective proof techniques for reasoning about correctness of classical programs. To offer similar techniques for quantum program verification and to build a logical foundation of programming methodology for quantum computers, we develop a full-fledged Hoare logic for both partial and total correctness of quantum programs. It is proved that this logic is (relatively) complete by exploiting the power of weakest preconditions and weakest liberal preconditions for quantum programs.
Nondeterministic choice is a useful program construct that provides a way to describe the behaviour of a program without specifying the details of possible implementations. It supports the stepwise refinement of programs, a method that has proven useful in software development. Nondeterminism has also been introduced in quantum programming, and termination of nondeterministic quantum programs has been extensively analysed. In this paper, we go beyond termination analysis to investigate the verification of nondeterministic quantum programs where properties are given by sets of hermitian operators on the associated Hilbert space. Hoare-type logic systems for partial and total correctness are proposed which turn out to be both sound and relatively complete with respect to their corresponding semantic correctness. To show the utility of these proof systems, we analyse some quantum algorithms such as quantum error correction scheme, Deutsch algorithm, and a nondeterministic quantum walk. Finally, a proof assistant prototype is implemented to aid in the automated reasoning of nondeterministic quantum programs.
We formalize the theory of quantum Hoare logic (QHL) [TOPLAS 33(6),19], an extension of Hoare logic for reasoning about quantum programs. In particular, we formalize the syntax and semantics of quantum programs in Isabelle/HOL, write down the rules of quantum Hoare logic, and verify the soundness and completeness of the deduction system for partial correctness of quantum programs. As preliminary work, we formalize some necessary mathematical background in linear algebra, and define tensor products of vectors and matrices on quantum variables. As an application, we verify the correctness of Grover’s search algorithm. To our best knowledge, this is the first time a Hoare logic for quantum programs is formalized in an interactive theorem prover, and used to verify the correctness of a nontrivial quantum algorithm.
We survey the landscape of Hoare logics for quantum programs. We review three papers: "Reasoning about imperative quantum programs" by Chadha, Mateus and Sernadas; "A logic for formal verification of quantum programs" by Yoshihiko Kakutani; and "Floyd-hoare logic for quantum programs" by Mingsheng Ying. We compare the mathematical foundations of the logics, their underlying languages, and the expressivity of their assertions. We also use the languages to verify the Deutsch-Jozsa Algorithm, and discuss their relative usability in practice.
Most modern (classical) programming languages support recursion. Recursion has also been successfully applied to the design of several quantum algorithms and introduced in a couple of quantum programming languages. So, it can be expected that recursion will become one of the fundamental paradigms of quantum programming. Several program logics have been developed for verification of non-recursive quantum programs. However, there are as yet no general methods for reasoning about recursive procedures in quantum computing. We fill the gap in this paper by presenting a logic for recursive quantum programs. This logic is an extension of quantum Hoare logic for quantum While-programs. The (relative) completeness of the logic is proved, and its effectiveness is shown by a running example: fixed-point Grover's search.