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arXiv:2104.14796 · ACM Transactions on Computational Logic

Verification of Distributed Quantum Programs

Yuan Feng, Sanjiang Li, Mingsheng Ying

Published Mar 29, 2022• 16 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1145/3517145

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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.
Nondeterministic choice is a useful program construct that provides a way to describe the behaviour of a program without specifying the details of possible implementations. It supports the stepwise refinement of programs, a method that has proven useful in software development. Nondeterminism has also been introduced in quantum programming, and termination of nondeterministic quantum programs has been extensively analysed. In this paper, we go beyond termination analysis to investigate the verification of nondeterministic quantum programs where properties are given by sets of hermitian operators on the associated Hilbert space. Hoare-type logic systems for partial and total correctness are proposed which turn out to be both sound and relatively complete with respect to their corresponding semantic correctness. To show the utility of these proof systems, we analyse some quantum algorithms such as quantum error correction scheme, Deutsch algorithm, and a nondeterministic quantum walk. Finally, a proof assistant prototype is implemented to aid in the automated reasoning of nondeterministic quantum programs.
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.
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.
Most modern (classical) programming languages support recursion. Recursion has also been successfully applied to the design of several quantum algorithms and introduced in a couple of quantum programming languages. So, it can be expected that recursion will become one of the fundamental paradigms of quantum programming. Several program logics have been developed for verification of non-recursive quantum programs. However, there are as yet no general methods for reasoning about recursive procedures in quantum computing. We fill the gap in this paper by presenting a logic for recursive quantum programs. This logic is an extension of quantum Hoare logic for quantum While-programs. The (relative) completeness of the logic is proved, and its effectiveness is shown by a running example: fixed-point Grover's search.
We provide a formal framework for Continuous-Variable Quantum Computing (CQC). While CQC is supported by photonic quantum hardware, we are not aware of a formal semantics for continuous-variable quantum programs nor of a unary Hoare logic for their verification. There are several technical obstacles to extending to CQC any of the formal frameworks available for Discrete-Variable Quantum Computing (DQC). Most importantly, continuous-variable quantum programs act on {\em infinite-dimensional} Hilbert spaces; their measurement outcomes are often {\em unbounded} and have expected values that are defined by an improper integral (or an infinite series), which may not converge. We overcome these challenges to give a formal semantics to a universal programming language for CQC and to provide the first Hoare logic for CQC. The assertions of our logic are built from polynomials over canonical observables. Besides proving relative completeness, we implement a symbolic weakest-precondition calculator for CQC based on our logic. Our tool has successfully verified CQC algorithms from textbooks and calculated their approximation errors for physically realizable implementations, proved the correctness (i.e., equivalence) of gate decompositions for CQC hardware, and computed the resource requirements (i.e., number of photon-number states) for achieving a desired accuracy in the classical simulation of continuous-variable quantum programs.
Over the past 27 years, quantum computing has seen a huge rise in interest from both academia and industry. At the current rate, quantum computers are growing in size rapidly backed up by the increase of research in the field. Significant efforts are being made to improve the reliability of quantum hardware and to develop suitable software to program quantum computers. In contrast, the verification of quantum programs has received relatively less attention. Verifying programs is especially important in the quantum setting due to how difficult it is to program complex algorithms correctly on resource-constrained and error-prone quantum hardware. Research into creating verification frameworks for quantum programs has seen recent development, with a variety of tools implemented using a collection of theoretical ideas. This survey aims to be a short introduction into the area of formal verification of quantum programs, bringing together theory and tools developed to date. Further, this survey examines some of the challenges that the field may face in the future, namely the development of complex quantum algorithms.