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arXiv:2409.10153

An Interpretation of Bunched Logic for Reasoning about Heap-Manipulating Quantum Programs

Su Bonan, Li Jun Zhou, Yuan Feng, Mingsheng Ying

Published Sep 16, 2024• 0 citations
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Abstract

We introduce heap manipulation into quantum programming languages to enable flexible quantum memory management, which in turn poses new challenges for reasoning about program correctness. To address these challenges, we develop a novel quantum interpretation of bunched logic by extending Birkhoff--von Neumann quantum logic to Hilbert spaces whose dimensions vary to accommodate the dynamic allocation and deallocation of variables on the quantum heap. Beyond the separating conjunction ($*$), we present, for the first time, a quantum interpretation of the separating implication ($-\mkern-3mu*$) to support backward reasoning. This interpretation preserves the adjunction between ($*$) and ($-\mkern-3mu*$), ensuring that both operators capture spatial properties and are precisely aligned with the semantics of quantum heap manipulations. Building on this foundation, we establish a quantum separation logic that supports local reasoning and is relatively complete.

DOI
10.48550/arxiv.2409.10153

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We introduce heap manipulation into quantum programming languages to enable flexible quantum memory management, which in turn poses new challenges for reasoning about program correctness. To address these challenges, we develop a novel quantum interpretation of bunched logic by extending Birkhoff--von Neumann quantum logic to Hilbert spaces whose dimensions vary to accommodate the dynamic allocation and deallocation of variables on the quantum heap. Beyond the separating conjunction (∗*∗), we present, for the first time, a quantum interpretation of the separating implication (− ⁣∗-\mkern-3mu*−∗) to support backward reasoning. This interpretation preserves the adjunction between (∗*∗) and (− ⁣∗-\mkern-3mu*−∗), ensuring that both operators capture spatial properties and are precisely aligned with the semantics of quantum heap manipulations. Building on this foundation, we establish a quantum separation logic that supports local reasoning and is relatively complete.
We propose a model of the substructural logic of Bunched Implications (BI) that is suitable for reasoning about quantum states. In our model, the separating conjunction of BI describes separable quantum states. We develop a program logic where pre- and post-conditions are BI formulas describing quantum states -- the program logic can be seen as a counterpart of separation logic for imperative quantum programs. We exercise the logic for proving the security of quantum one-time pad and secret sharing, and we show how the program logic can be used to discover a flaw in Google Cirq's tutorial on the Variational Quantum Algorithm (VQA).
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.
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.
We propose a model of the substructural logic of Bunched Implications (BI) that is suitable for reasoning about quantum states. In our model, the separating conjunction of BI describes separable quantum states. We develop a program logic where pre- and post-conditions are BI formulas describing quantum states—the program logic can be seen as a counterpart of separation logic for imperative quantum programs. We exercise the logic for proving the security of quantum one-time pad and secret sharing, and we show how the program logic can be used to discover a flaw in Google Cirq’s tutorial on the Variational Quantum Algorithm (VQA).
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.
We initiate the study of parallel quantum programming by defining the operational and denotational semantics of parallel quantum programs. The technical contributions of this paper include: (1) find a series of useful proof rules for reasoning about correctness of parallel quantum programs; (2) prove a (relative) completeness of our proof rules for partial correctness of disjoint parallel quantum programs; and (3) prove a strong soundness theorem of the proof rules showing that partial correctness is well maintained at each step of transitions in the operational semantics of a general parallel quantum program (with shared variables). This is achieved by partially overcoming the following conceptual challenges that are never present in classical parallel programming: (i) the intertwining of nondeterminism caused by quantum measurements and introduced by parallelism; (ii) entanglement between component quantum programs; and (iii) combining quantum predicates in the overlap of state Hilbert spaces of component quantum programs with shared variables. Applications of the techniques developed in this paper are illustrated by a formal verification of Bravyi-Gosset-König's parallel quantum algorithm solving a linear algebra problem, which gives for the first time an unconditional proof of a computational quantum advantage.