quantum algorithm formalization

Quantum program correctness assurance lags behind algorithm development, with existing approaches relying predominantly on post-hoc verification. Method: This paper proposes the “Correctness-by-Construction” paradigm for quantum programs: starting from formal pre-/postcondition specifications, it employs sound and complete refinement rules to synthesize quantum programs top-down, ensuring conformance to specifications. Grounded in the quantum while language and an extended quantum Hoare logic, the approach supports explicit modeling and systematic categorization of key design decisions. Contributions/Results: It is the first framework to achieve constructive correctness guarantees for quantum programs—moving beyond traditional verification. We implement a specification-driven refinement framework and successfully construct canonical quantum algorithms—including Deutsch-Jozsa and Grover’s search subroutine—demonstrating feasibility, design guidance, and scalability. This work establishes the first systematic, formal methodology for quantum software engineering.

Existing Hoare-style verification methods for quantum programs struggle to achieve full automation, primarily due to the exponential blowup in translating high-level set-based assertions into automata. This work proposes an extended set-based specification language together with a novel translation algorithm whose complexity scales linearly with the number of qubits. By integrating controlled automaton construction and qubit reordering techniques, the approach achieves—for the first time—linear scalability in translating specifications to automata with respect to qubit count. The method substantially enhances expressiveness while maintaining computational efficiency, thereby overcoming the scalability barrier that has long hindered automated verification of quantum programs. As a result, it enables fully automatic Hoare-style verification of large-scale quantum programs previously deemed infeasible.

This work addresses the low efficiency and insufficient trustworthiness of formal verification for quantum error correction (QEC) procedures. Methodologically: (1) We design a dedicated assertion logic and program logic, and fully formalize and mechanically verify their soundness in Coq; (2) We establish a differentiated verification condition (VC) solving framework—employing SMT solvers for automatic validity checking of Pauli errors, and introducing heuristic algorithms to enhance feasibility for non-Pauli errors; (3) We integrate the Coq proof assistant with an automated toolchain. Contributions include fully automated verification of 14 mainstream stabilizer codes across representative fault-tolerant scenarios—including gate-level fault tolerance, measurement noise, and circuit compilation—thereby significantly improving both verification efficiency and mathematical rigor. Veri-QEC provides a solid formal foundation for fault-tolerant quantum computation.

This work addresses the low efficiency of detecting critical security vulnerabilities—such as use-after-free, null-pointer dereference, and division-by-zero—in classical formal program verification. We propose the first systematic approach that models defect detection as a structured optimization problem amenable to quantum computation. Methodologically, we encode program semantics into SAT instances and map them onto a quantum optimization framework, integrating the Quantum Approximate Optimization Algorithm (QAOA), Grover’s search, and Quantum Singular Value Transformation (QSVT) to establish an end-to-end solution pathway from logical constraints to quantum state evolution. Empirical evaluation on synthetic benchmarks and real-world vulnerability cases demonstrates that our method efficiently recovers satisfying assignments on both quantum simulators and actual quantum hardware, exhibiting asymptotic polynomial speedup potential. This work establishes a novel paradigm for leveraging quantum computing to enhance software trustworthiness and reliability assurance.

Verifying quantum programs with classical variables remains challenging due to high conceptual barriers, lack of integration with classical verification toolchains, and insufficient logical foundations. Method: This paper proposes a lightweight Hoare-style logic framework. It introduces parameterized quantum gates and quantum arrays into quantum Hoare logic with classical variables—the first such extension. It designs novel measurement inference rules ensuring minimal compatibility with classical first-order Hoare logic. Specifications adopt a two-layer structure: classical first-order formulas augmented with parameterized quantum predicates, enabling semantically precise and intuitive assertion expression. Contribution/Results: The framework enables seamless reuse of classical verification tools, drastically reducing learning and adoption costs. It supports expressive, compositional, and mathematically rigorous correctness proofs for quantum programs, thereby advancing the practical engineering of quantum software.

This work addresses the gap between idealized noise-free models and real-world noisy quantum hardware in quantum program verification. It introduces, for the first time, a noise-aware quantum Hoare logic that integrates hardware-specific error models—such as those provided by IBM Qiskit—to define a realistic noisy semantics. The study further demonstrates the critical role of classical probabilistic branching in achieving optimality in quantum programs. Building on this foundation, the authors develop a bounded verification algorithm and an automated synthesis method capable of generating optimal quantum subroutines tailored to specific noise environments, including tasks like parity computation, state preparation, and state discrimination. The efficacy of the proposed approach is validated against actual hardware specifications.

Current quantum programs predominantly rely on single-qubit gate operations and lack high-level abstractions, leading to complex and error-prone designs. This work proposes a structured programming paradigm that treats indivisible quantum registers as fundamental units, advancing computation through semantically precise register-level transformations and entanglement operations. To bridge high-level expressions with low-level semantics, the approach introduces an algebraic formal syntax. By integrating phase-conditioned operations, parallel evaluation mechanisms, and quantum SMT solving techniques, the framework enables a reliable mapping from high-level structured descriptions to low-level quantum semantics. This methodology substantially reduces programming complexity and establishes a foundation for scalable and robust quantum software systems.