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.

Quantum technologies are progressing at an extraordinary pace and are poised to transform numerous sectors both nationally and globally. Among them, quantum computing stands out for its potential to revolutionize areas such as cryptography, optimization, and the simulation of quantum systems, offering dramatic speed-ups for specific classes of problems. As quantum devices evolve and become increasingly pervasive, guaranteeing their correctness is of paramount importance. This necessitates the development of rigorous methods and tools to analyze and verify their behavior. However, the construction of such verification frameworks presents fundamental challenges. Quantum phenomena such as superposition and entanglement give rise to computational behaviors that differ profoundly from those of classical systems, leading to inherently probabilistic models and exponentially large state spaces, even for relatively small programs. Addressing these challenges requires building on the extensive expertise of the formal methods community in classical program verification, while incorporating recent advances and collaborative efforts in quantum systems. An interesting challenge for the verification community is to design and implement novel verification frameworks that transfer the key strengths of classical verification, such as expressive specification, precise error detection, automation, and scalability, to the quantum domain. We expect that the results of this research will play a crucial role in enabling the dependable deployment of quantum technologies across a wide range of future applications.

Abstract We present a verifier of quantum programs called AutoQ 2.0. Quantum programs extend quantum circuits (the domain of AutoQ 1.0) by classical control flow constructs, which enable users to describe advanced quantum algorithms in a formal and precise manner. The extension is highly non-trivial, as we needed to tackle both theoretical challenges (such as the treatment of measurement, the normalization problem, and lifting techniques for verification of classical programs with loops to the quantum world), and engineering issues (such as extending the input format with a support for specifying loop invariants). We have successfully used AutoQ 2.0 to verify two types of advanced quantum programs that cannot be expressed using only quantum circuits: the repeat-until-success (RUS) algorithm and the weak-measurement-based version of Grover’s search algorithm. AutoQ 2.0 can efficiently verify all our benchmarks: all RUS algorithms were verified instantly and, for the weak-measurement-based version of Grover’s search, we were able to handle the case of 100 qubits in $$\sim $$ ∼ 20 minutes.

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.

Abstract We present a specification language and a fully automated tool named AutoQ for verifying quantum circuits symbolically. The tool implements the automata-based algorithm from [14] and extends it with the capabilities for symbolic reasoning. The extension allows to specify relational properties, i.e., relationships between states before and after executing a circuit. We present a number of use cases where we used AutoQ to fully automatically verify crucial properties of several quantum circuits, which have, to the best of our knowledge, so far been proved only with human help.

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.