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Building upon recent work on probabilistic programs, we formally define the notion of expected runtime for quantum programs. A representation of the expected runtimes of quantum programs is introduced with an interpretation as an observable in physics. A method for computing the expected runtimes of quantum programs in finite-dimensional state spaces is developed. Several examples are provided as applications of this method; in particular, an open problem of computing the expected runtime of quantum random walks is solved using our method.
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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 noi
Practical error analysis is essential for the design, optimization, and evaluation of Noisy Intermediate-Scale Quantum(NISQ) computing. However, bounding errors in quantum programs is a grand challenge, because the effects of quantum errors depend on exponentially large quantum states. In this work, we present Gleipnir, a novel methodology toward practically computing verified error bounds in quantum programs. Gleipnir introduces the $(hatrho,delta)$-diamond norm, an error metric constrained by a quantum predicate consisting of the approximate state $hatrho$ and its distance $delta$ to the ideal state $rho$. This predicate $(hatrho,delta)$ can be computed adaptively using tensor networks based on the Matrix Product States. Gleipnir features a lightweight logic for reasoning about error bounds in noisy quantum programs, based on the $(hatrho,delta)$-diamond norm metric. Our experimental results show that Gleipnir is able to efficiently generate tight error bounds for real-world quantum programs with 10 to 100 qubits, and can be used to evaluate the error mitigation performance of quantum compiler transformations.
The notion of program sensitivity (aka Lipschitz continuity) specifies that changes in the program input result in proportional changes to the program output. For probabilistic programs the notion is naturally extended to expected sensitivity. A previous approach develops a relational program logic framework for proving expected sensitivity of probabilistic while loops, where the number of iterations is fixed and bounded. In this work, we consider probabilistic while loops where the number of iterations is not fixed, but randomized and depends on the initial input values. We present a sound approach for proving expected sensitivity of such programs. Our sound approach is martingale-based and can be automated through existing martingale-synthesis algorithms. Furthermore, our approach is compositional for sequential composition of while loops under a mild side condition. We demonstrate the effectiveness of our approach on several classical examples from Gamblers Ruin, stochastic hybrid systems and stochastic gradient descent. We also present experimental results showing that our automated approach can handle various probabilistic programs in the literature.
While recent progress in quantum hardware open the door for significant speedup in certain key areas (cryptography, biology, chemistry, optimization, machine learning, etc), quantum algorithms are still hard to implement right, and the validation of such quantum programs is achallenge. Moreover, importing the testing and debugging practices at use in classical programming is extremely difficult in the quantum case, due to the destructive aspect of quantum measurement. As an alternative strategy, formal methods are prone to play a decisive role in the emerging field of quantum software. Recent works initiate solutions for problems occurring at every stage of the development process: high-level program design, implementation, compilation, etc. We review the induced challenges for an efficient use of formal methods in quantum computing and the current most promising research directions.
In this paper, we propose Proq, a runtime assertion scheme for testing and debugging quantum programs on a quantum computer. The predicates in Proq are represented by projections (or equivalently, closed subspaces of the state space), following Birkhoff-von Neumann quantum logic. The satisfaction of a projection by a quantum state can be directly checked upon a small number of projective measurements rather than a large number of repeated executions. On the theory side, we rigorously prove that checking projection-based assertions can help locate bugs or statistically assure that the semantic function of the tested program is close to what we expect, for both exact and approximate quantum programs. On the practice side, we consider hardware constraints and introduce several techniques to transform the assertions, making them directly executable on the measurement-restricted quantum computers. We also propose to achieve simplified assertion implementation using local projection technique with soundness guaranteed. We compare Proq with existing quantum program assertions and demonstrate the effectiveness and efficiency of Proq by its applications to assert two ingenious quantum algorithms, the Harrow-Hassidim-Lloyd algorithm and Shors algorithm.
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