Foundations of quantum programming

Progress in the techniques of quantum devices has made people widely believe that large-scale and functional quantum computers will be eventually built. By then, super-powered quantum computer will solve many problems affecting economic and social life that cannot be addressed by classical computing. However, our experiences with classical computing suggest that once quantum computers become available in the future, quantum software will play a key role in exploiting their power, and quantum software market will even be much larger than quantum hardware market. Unfortunately, today's software development techniques are not suited to quantum computers due to the essential differences between the nature of the classical world and that of the quantum world. To lay a solid foundation for tomorrow's quantum software industry, it is critically essential to pursue systematic research into quantum programming methodology and techniques. © 2010 Springer-Verlag

Quantitative Robustness Analysis of Quantum Programs (Extended Version)

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 $\epsilon$-robustness, which characterizes possible "distance" 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.Comment: 34 pages, LaTeX; v2: fixed typo

Quantum Hoare logic with classical variables

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 paper, we propose a quantum Hoare logic for a simple while language which 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 which support the introduction of disjunction and quantifiers in the classical part of assertions, and of super-operator application and superposition 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