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ACM/IEEE Symposium on Logic in Computer Science (LICS)

A Quantum Interpretation of Bunched Logic & Quantum Separation Logic

Li Zhou, Gilles Barthe, Justin Hsu, Mingsheng Ying, Nengkun Yu

Published Jun 29, 2021• 18 citations
Publisher

Abstract

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).

DOI
10.1109/lics52264.2021.9470673

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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 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 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.
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.
© 2020 Copyright held by the owner/author(s). Relational verification of quantum programs has many potential applications in quantum and post-quantum security and other domains. We propose a relational program logic for quantum programs. The interpretation of our logic is based on a quantum analogue of probabilistic couplings. We use our logic to verify non-trivial relational properties of quantum programs, including uniformity for samples generated by the quantum Bernoulli factory, reliability of quantum teleportation against noise (bit and phase flip), security of quantum one-time pad and equivalence of quantum walks.
Relational verification of quantum programs has many potential applications in quantum and post-quantum security and other domains. We propose a relational program logic for quantum programs. The interpretation of our logic is based on a quantum analogue of probabilistic couplings. We use our logic to verify non-trivial relational properties of quantum programs, including uniformity for samples generated by the quantum Bernoulli factory, reliability of quantum teleportation against noise (bit and phase flip), security of quantum one-time pad and equivalence of quantum walks.