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Relational proofs for quantum programs

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

Published Jan 1, 2020• 27 citations
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Abstract

© 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.

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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.

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© 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.
Relational verification of quantum programs has many potential applications in 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), and equivalence of quantum random walks.
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.
Abstract Quantum computation is inevitably subject to imperfections in its implementation. These imperfections arise from various sources, including environmental noise at the hardware level and the introduction of approximate implementations by quantum algorithm designers, such as lower-depth computations. Given the significant advantage of relational logic in program reasoning and the importance of assessing the robustness of quantum programs between their ideal specifications and imperfect implementations, we design a proof system to verify the approximate relational properties of quantum programs. We demonstrate the effectiveness of our approach by providing the first formal verification of the renowned low-depth approximation of the quantum Fourier transform. Furthermore, we validate the approximate correctness of the repeat-until-success algorithm. From the technical point of view, we develop approximate quantum coupling as a fundamental tool to study approximate relational reasoning for quantum programs, a novel generalization of the widely used approximate probabilistic coupling in probabilistic programs, answering a previously posed open question for projective predicates.
Quantum computation is a topic of significant recent interest, with practical advances coming from both research and industry.A major challenge in quantum programming is dealing with errors (quantum noise) during execution.Because quantum resources (e.g., qubits) are scarce, classical error correction techniques applied at the level of the architecture are currently cost-prohibitive.But while this reality means that quantum programs are almost certain to have errors, there as yet exists no principled means to reason about erroneous behavior.This paper attempts to fill this gap by developing a semantics for erroneous quantum while programs, as well as a logic for reasoning about them.This logic permits proving a property we have identified, called ϵ-robustness, which characterizes possible łdistancež between an ideal program and an erroneous one.We have proved the logic sound, and showed its utility on several case studies, notably: (1) analyzing the robustness of noisy versions of the quantum Bernoulli factory (QBF) and quantum walk (QW); (2) demonstrating the (in)effectiveness of different error correction schemes on single-qubit errors; and (3) analyzing the robustness of a fault-tolerant version of QBF.
Distributed quantum systems and especially the Quantum Internet have the ever-increasing potential to fully demonstrate the power of quantum computation. This is particularly true given that developing a general-purpose quantum computer is much more difficult than connecting many small quantum devices. One major challenge of implementing distributed quantum systems is programming them and verifying their correctness. In this paper, we propose a CSP-like distributed programming language to facilitate the specification and verification of such systems. After presenting its operational and denotational semantics, we develop a Hoare-style logic for distributed quantum programs and establish its soundness and (relative) completeness with respect to both partial and total correctness. The effectiveness of the logic is demonstrated by its applications in the verification of quantum teleportation and local implementation of non-local CNOT gates, two important algorithms widely used in distributed quantum systems.