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arXiv:2205.01959

Birkhoff-von Neumann Quantum Logic as an Assertion Language for Quantum Programs

Mingsheng Ying

Published May 4, 2022• 1 citation
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Abstract

A first-order logic with quantum variables is needed as an assertion language for specifying and reasoning about various properties (e.g. correctness) of quantum programs. Surprisingly, such a logic is missing in the literature, and the existing first-order Birkhoff-von Neumann quantum logic deals with only classical variables and quantifications over them. In this paper, we fill in this gap by introducing a first-order extension of Birkhoff-von Neumann quantum logic with universal and existential quantifiers over quantum variables. Examples are presented to show our logic is particularly suitable for specifying some important properties studied in quantum computation and quantum information. We further incorporate this logic into quantum Hoare logic as an assertion logic so that it can play a role similar to that of first-order logic for classical Hoare logic and BI-logic for separation logic. In particular, we show how it can be used to define and derive quantum generalisations of some adaptation rules that have been applied to significantly simplify verification of classical programs. It is expected that the assertion logic defined in this paper - first-order quantum logic with quantum variables - can be combined with various quantum program logics to serve as a solid logical foundation upon which verification tools can be built using proof assistants such as Coq and Isabelle/HOL.

DOI
10.48550/arxiv.2205.01959

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A first-order logic with quantum variables is needed as an assertion language for specifying and reasoning about various properties (e.g. correctness) of quantum programs. Surprisingly, such a logic is missing in the literature, and the existing first-order Birkhoff-von Neumann quantum logic deals with only classical variables and quantifications over them. In this paper, we fill in this gap by introducing a first-order extension of Birkhoff-von Neumann quantum logic with universal and existential quantifiers over quantum variables. Examples are presented to show our logic is particularly suitable for specifying some important properties studied in quantum computation and quantum information. We further incorporate this logic into quantum Hoare logic as an assertion logic so that it can play a role similar to that of first-order logic for classical Hoare logic and BI-logic for separation logic. In particular, we show how it can be used to define and derive quantum generalisations of some adaptation rules that have been applied to significantly simplify verification of classical programs. It is expected that the assertion logic defined in this paper - first-order quantum logic with quantum variables - can be combined with various quantum program logics to serve as a solid logical foundation upon which verification tools can be built using proof assistants such as Coq and Isabelle/HOL.
Hoare logic provides a syntax-oriented method to reason about program correctness and has been proven effective in the verification of classical and probabilistic programs. Existing proposals for quantum Hoare logic either lack completeness or support only quantum variables, thus limiting their capability in practical use. In this article, we propose a quantum Hoare logic for a simple while language that involves both classical and quantum variables. Its soundness and relative completeness are proven for both partial and total correctness of quantum programs written in the language. Remarkably, with novel definitions of classical-quantum states and corresponding assertions, the logic system is quite simple and similar to the traditional Hoare logic for classical programs. Furthermore, to simplify reasoning in real applications, auxiliary proof rules are provided that support standard logical operation in the classical part of assertions and super-operator application in the quantum part. Finally, a series of practical quantum algorithms, in particular the whole algorithm of Shor’s factorisation, are formally verified to show the effectiveness of the logic.
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.
CoqQ is a framework for reasoning about quantum programs in the Coq proof assistant. Its main components are: a deeply embedded quantum programming language, in which classic quantum algorithms are easily expressed, and an expressive program logic for proving properties of programs. CoqQ is foundational: the program logic is formally proved sound with respect to a denotational semantics based on state-of-art mathematical libraries (MathComp and MathComp Analysis). CoqQ is also practical: assertions can use Dirac expressions, which eases concise specifications, and proofs can exploit local and parallel reasoning, which minimizes verification effort. We illustrate the applicability of CoqQ with many examples from the literature.
In quantum information and computation research, symbolic methods have been widely used for human specification and reasoning about quantum states and operations. At the same time, they are essential for ensuring the scalability and efficiency of automated reasoning and verification tools for quantum algorithms and programs. However, a formal theory for symbolic specification and reasoning about quantum data and operations is still lacking, which significantly limits the practical applicability of automated verification techniques in quantum computing. In this paper, we present a general logical framework, called Symbolic Operator Logic SOL\mathbf{SOL}SOL, which enables symbolic specification and reasoning about quantum data and operations. Within this framework, a classical first-order logical language is embedded into a language of formal operators used to specify quantum data and operations, including their recursive definitions. This embedding allows reasoning about their properties modulo a chosen theory of the underlying classical data (e.g., Boolean algebra or group theory), thereby leveraging existing automated verification tools developed for classical computing. It should be emphasised that this embedding of classical first-order logic into SOL\mathbf{SOL}SOL is precisely what makes the symbolic method possible. We envision that this framework can provide a conceptual foundation for the formal verification and automated theorem proving of quantum computation and information in proof assistants such as Lean, Coq, and related systems.
We develop a formal language, a syntax, for specifying quantum predicates, which are yes/no properties of quantum systems. Specifically we study classical-quantum predicates using projector-based Birkhoff-von Neumann quantum logic. We give an extensional definition of a weakest precondition calculus that preserves projector-based predicates fora simple loop-free classical-quantum programming language. We prove that our weakest precondition calculus is relatively complete in the sense of Cook for the classical-quantum programming language we specify. To achieve this, we show that for any post condition expressible in our formal language, the corresponding weakest precondition with respect to any program is also expressible in our formal language. This result establishes a foundation for reasoning about classical-quantum programs using an intensional approach.
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.