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arXiv:2301.07747

An Automata-based Framework for Verification and Bug Hunting in Quantum Circuits (Technical Report)

Yu‐Fang Chen, Kai-Min Chung, Ondřej Lengál, Jyun-Ao Lin, Tsai, Wei-Lun, Di-De Yen

Published Jan 18, 2023• 1 citation
arXivHTMLPDF

Abstract

We introduce a new paradigm for analysing and finding bugs in quantum circuits. In our approach, the problem is given by a triple $\{P\}\,C\,\{Q\}$ and the question is whether, given a set $P$ of quantum states on the input of a circuit $C$, the set of quantum states on the output is equal to (or included in) a set $Q$. While this is not suitable to specify, e.g., functional correctness of a quantum circuit, it is sufficient to detect many bugs in quantum circuits. We propose a technique based on tree automata to compactly represent sets of quantum states and develop transformers to implement the semantics of quantum gates over this representation. Our technique computes with an algebraic representation of quantum states, avoiding the inaccuracy of working with floating-point numbers. We implemented the proposed approach in a prototype tool and evaluated its performance against various benchmarks from the literature. The evaluation shows that our approach is quite scalable, e.g., we managed to verify a large circuit with 40 qubits and 141,527 gates, or catch bugs injected into a circuit with 320 qubits and 1,758 gates, where all tools we compared with failed. In addition, our work establishes a connection between quantum program verification and automata, opening new possibilities to exploit the richness of automata theory and automata-based verification in the world of quantum computing.

DOI
10.48550/arxiv.2301.07747

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We introduce a new paradigm for analysing and finding bugs in quantum circuits. In our approach, the problem is given by a triple {P} C {Q}\{P\}\,C\,\{Q\}{P}C{Q} and the question is whether, given a set PPP of quantum states on the input of a circuit CCC, the set of quantum states on the output is equal to (or included in) a set QQQ. While this is not suitable to specify, e.g., functional correctness of a quantum circuit, it is sufficient to detect many bugs in quantum circuits. We propose a technique based on tree automata to compactly represent sets of quantum states and develop transformers to implement the semantics of quantum gates over this representation. Our technique computes with an algebraic representation of quantum states, avoiding the inaccuracy of working with floating-point numbers. We implemented the proposed approach in a prototype tool and evaluated its performance against various benchmarks from the literature. The evaluation shows that our approach is quite scalable, e.g., we managed to verify a large circuit with 40 qubits and 141,527 gates, or catch bugs injected into a circuit with 320 qubits and 1,758 gates, where all tools we compared with failed. In addition, our work establishes a connection between quantum program verification and automata, opening new possibilities to exploit the richness of automata theory and automata-based verification in the world of quantum computing.
We introduce a new paradigm for analysing and finding bugs in quantum circuits. In our approach, the problem is given by a ‍triple { P } C { Q } and the question is whether, given a set P of quantum states on the input of a circuit C , the set of quantum states on the output is equal to (or included in) a set Q . While this is not suitable to specify, e.g., functional correctness of a quantum circuit, it is sufficient to detect many bugs in quantum circuits. We propose a technique based on tree automata to compactly represent sets of quantum states and develop transformers to implement the semantics of quantum gates over this representation. Our technique computes with an algebraic representation of quantum states, avoiding the inaccuracy of working with floating-point numbers. We implemented the proposed approach in a prototype tool and evaluated its performance against various benchmarks from the literature. The evaluation shows that our approach is quite scalable, e.g., we managed to verify a large circuit with 40 qubits and 141,527 gates, or catch bugs injected into a circuit with 320 qubits and 1,758 gates, where all tools we compared with failed. In addition, our work establishes a connection between quantum program verification and automata, opening new possibilities to exploit the richness of automata theory and automata-based verification in the world of quantum computing.
As quantum computing hardware advances, the demand for scalable, precise, and fully automated verification techniques for quantum circuits grows. This paper introduces a novel automata-based framework tailored for the verification of quantum circuits. In our approach, the problem is framed as a triple { P } C { S } and the question is whether, given a set P of quantum states on the input of a circuit C , the set of quantum states on the output is equal to (or included in) a set S . Our framework leverages tree automata to compactly represent sets of quantum states and we develop transformers to implement the semantics of quantum gates over this representation. We implemented the proposed approach in a prototype tool and evaluated its performance against various benchmarks from the literature. The evaluation shows that our approach is quite scalable, for example, we managed to verify a large circuit with 40 qubits and 141,527 gates, or catch bugs injected into a circuit with 320 qubits and 1,758 gates, where all tools we compared with failed. Additionally, our work bridges quantum program verification and automata, opening new possibilities to exploit the richness of automata theory in quantum computing.
We present a new method for the verification of quantum circuits based on a novel symbolic representation of sets of quantum states using level-synchronized tree automata (LSTAs). LSTAs extend classical tree automata by labeling each transition with a set of choices, which are then used to synchronize subtrees of an accepted tree. Compared to the traditional tree automata, LSTAs have an incomparable expressive power while maintaining important properties, such as closure under union and intersection, and decidable language emptiness and inclusion. We have developed an efficient and fully automated symbolic verification algorithm for quantum circuits based on LSTAs. The complexity of supported gate operations is at most quadratic, dramatically improving the exponential worst-case complexity of an earlier tree automata-based approach. Furthermore, we show that LSTAs are a promising model for parameterized verification, i.e., verifying the correctness of families of circuits with the same structure for any number of qubits involved, which principally lies beyond the capabilities of previous automated approaches. We implemented this method as a C++ tool and compared it with three symbolic quantum circuit verifiers and two simulators on several benchmark examples. The results show that our approach can solve problems with sizes orders of magnitude larger than the state of the art.
We present the first fully automatic framework for verifying relational properties of parameterized quantum programs, i.e., a program that, given an input size, generates a corresponding quantum circuit. We focus on verifying input-output correctness as well as equivalence. At the core of our approach is a new automata model, synchronized weighted tree automata (SWTAs), which compactly and precisely captures the infinite families of quantum states produced by parameterized programs. We introduce a class of transducers to model quantum gate semantics and develop composition algorithms for constructing transducers of parameterized circuits. Verification is reduced to functional inclusion or equivalence checking between SWTAs, for which we provide decision procedures. Our implementation demonstrates both the expressiveness and practical efficiency of the framework by verifying a diverse set of representative parameterized quantum programs with verification times ranging from milliseconds to seconds.
We present a new method for the verification of quantum circuits based on a novel symbolic representation of sets of quantum states using level-synchronized tree automata (LSTAs). LSTAs extend classical tree automata by labeling each transition with a set of choices , which are then used to synchronize subtrees of an accepted tree. Compared to the traditional tree automata, LSTAs have an incomparable expressive power while maintaining important properties, such as closure under union and intersection, and decidable language emptiness and inclusion. We have developed an efficient and fully automated symbolic verification algorithm for quantum circuits based on LSTAs. The complexity of supported gate operations is at most quadratic, dramatically improving the exponential worst-case complexity of an earlier tree automata-based approach. Furthermore, we show that LSTAs are a promising model for parameterized verification , i.e., verifying the correctness of families of circuits with the same structure for any number of qubits involved, which principally lies beyond the capabilities of previous automated approaches.We implemented this method as a C++ tool and compared it with three symbolic quantum circuit verifiers and two simulators on several benchmark examples. The results show that our approach can solve problems with sizes orders of magnitude larger than the state of the art.
Quantum computing exploits the principles of quantum mechanics to perform computation by manipulating qubit states through sequences of quantum gates, known as quantum circuits. As quantum hardware continues to scale in both complexity and qubit count, conventional verification techniques quickly become computationally impractical, motivating the need for more advanced verification approaches. Formal methods, including theorem proving, model checking, and symbolic reasoning, offer systematic and mathematically rigorous techniques for verifying functional correctness, equivalence, and implementation, while enabling early detection of design errors. This review explores how formal methods are currently applied to the verification of quantum circuits, with a particular focus on barrier certificates, abstract interpretation, model checking, theorem proving, and emerging hybrid approaches. Examining the theoretical foundations and practical applications of these techniques, discussing their strengths, limitations, and comparative effectiveness through representative case studies, review hopes to provide a holistic understanding of the topic. Highlighting the open challenges and outlining promising directions for future research, the review aims to provide a roadmap towards more scalable and robust verification frameworks for quantum computing. For readers seeking deeper engagement, the review offers an extensive set of references to support further study and exploration.