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arXiv:2308.04741

Local Reasoning about Probabilistic Behaviour for Classical-Quantum Programs

Huiling Wu, Yuxin Deng, Ming Xu

Published Aug 9, 2023• 0 citations
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Abstract

Verifying the functional correctness of programs with both classical and quantum constructs is a challenging task. The presence of probabilistic behaviour entailed by quantum measurements, together with the exponential size of quantum state spaces, complicates the verification task greatly. In this paper, we propose a quantum Hoare logic that supports separation-style local reasoning and introduces distribution formulas for specifying probabilistic properties, thereby enabling local reasoning about the probabilistic behaviour of classical-quantum programs. We show that the proof rules in the logic are sound with respect to a denotational semantics. To demonstrate the expressiveness of the logic, we formally verify several non-trivial examples, with a Bell-pair purification protocol serving as the main case study. Moreover, we embed our logic into the Rocq Prover (formerly Coq). The resulting logical framework, called CoqQLR, can facilitate semi-automated reasoning about classical--quantum programs.

DOI
10.48550/arxiv.2308.04741

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Verifying the functional correctness of programs with both classical and quantum constructs is a challenging task. The presence of probabilistic behaviour entailed by quantum measurements, together with the exponential size of quantum state spaces, complicates the verification task greatly. In this paper, we propose a quantum Hoare logic that supports separation-style local reasoning and introduces distribution formulas for specifying probabilistic properties, thereby enabling local reasoning about the probabilistic behaviour of classical-quantum programs. We show that the proof rules in the logic are sound with respect to a denotational semantics. To demonstrate the expressiveness of the logic, we formally verify several non-trivial examples, with a Bell-pair purification protocol serving as the main case study. Moreover, we embed our logic into the Rocq Prover (formerly Coq). The resulting logical framework, called CoqQLR, can facilitate semi-automated reasoning about classical--quantum programs.
It is well-known that quantum programs are not only complicated to design but also challenging to verify because the quantum states can have exponential size and require sophisticated mathematics to encode and manipulate. To tackle the state-space explosion problem for quantum reasoning, we propose a Hoare-style inference framework that supports local reasoning for quantum programs. By providing a quantum interpretation of the separating conjunction, we are able to infuse separation logic into our framework and apply local reasoning using a quantum frame rule that is similar to the classical frame rule. For evaluation, we apply our framework to verify various quantum programs including Deutsch–Jozsa’s algorithm and Grover's algorithm.
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.
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.
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 present a logic for reasoning about pairs of interactive quantum programs – quantum relational Hoare logic (qRHL). This logic follows the spirit of probabilistic relational Hoare logic (Barthe et al. 2009) and allows us to formulate how the outputs of two quantum programs relate given the relationship of their inputs. Probabilistic RHL was used extensively for computer-verified security proofs of classical cryptographic protocols. Since pRHL is not suitable for analyzing quantum cryptography, we present qRHL as a replacement, suitable for the security analysis of post-quantum cryptography and quantum protocols. The design of qRHL poses some challenges unique to the quantum setting, e.g., the definition of equality on quantum registers. Finally, we implemented a tool for verifying proofs in qRHL and developed several example security proofs in it.
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.