While recent progress in quantum hardware open the door for significant speedup in certain key areas, quantum algorithms are still hard to implement right, and the validation of such quantum programs is a challenge. Early attempts either suffer from the lack of automation or parametrized reasoning, or target high-level abstract algorithm description languages far from the current de facto consensus of circuit-building quantum programming languages. As a consequence, no significant quantum algorithm implementation has been currently verified in a scale-invariant manner. We propose Qbricks, the first formal verification environment for circuit-building quantum programs, featuring clear separation between code and proof, parametric specifications and proofs, high degree of proof automation and allowing to encode quantum programs in a natural way, i.e. close to textbook style. Qbricks builds on best practice of formal verification for the classical case and tailor them to the quantum case: we bring a new domain-specific circuit-building language for quantum programs, namely Qbricks-DSL, together with a new logical specification language Qbricks-Spec and a dedicated Hoare-style deductive verification rule named Hybrid Quantum Hoare Logic. Especially, we introduce and intensively build upon HOPS, a higher-order extension of the recent path-sum symbolic representation, used for both specification and automation. To illustrate the opportunity of Qbricks, we implement the first verified parametric implementations of several famous and non-trivial quantum algorithms, including the quantum part of Shor integer factoring (Order Finding - Shor-OF), quantum phase estimation (QPE) - a basic building block of many quantum algorithms, and Grover search. These breakthroughs were amply facilitated by the specification and automated deduction principles introduced within Qbricks.
While recent progress in quantum hardware open the door for significant speedup in certain key areas, quantum algorithms are still hard to implement right, and the validation of such quantum programs is a challenge. Early attempts either suffer from the lack of automation or parametrized reasoning, or target high-level abstract algorithm description languages far from the current de facto consensus of circuit-building quantum programming languages. As a consequence, no significant quantum algorithm implementation has been currently verified in a scale-invariant manner. We propose Qbricks, the first formal verification environment for circuit-building quantum programs, featuring clear separation between code and proof, parametric specifications and proofs, high degree of proof automation and allowing to encode quantum programs in a natural way, i.e. close to textbook style. Qbricks builds on best practice of formal verification for the classical case and tailor them to the quantum case: we bring a new domain-specific circuit-building language for quantum programs, namely Qbricks-DSL, together with a new logical specification language Qbricks-Spec and a dedicated Hoare-style deductive verification rule named Hybrid Quantum Hoare Logic. Especially, we introduce and intensively build upon HOPS, a higher-order extension of the recent path-sum symbolic representation, used for both specification and automation. To illustrate the opportunity of Qbricks, we implement the first verified parametric implementations of several famous and non-trivial quantum algorithms, including the quantum part of Shor integer factoring (Order Finding - Shor-OF), quantum phase estimation (QPE) - a basic building block of many quantum algorithms, and Grover search. These breakthroughs were amply facilitated by the specification and automated deduction principles introduced within Qbricks.
While recent progress in quantum hardware open the door for significant speedup in certain key areas, quantum algorithms are still hard to implement right, and the validation of such quantum programs is a challenge. Early attempts either suffer from the lack of automation or parametrized reasoning, or target high-level abstract algorithm description languages far from the current de facto consensus of circuit-building quantum programming languages. As a consequence, no significant quantum algorithm implementation has been currently verified in a scale-invariant manner. We propose Qbricks, the first formal verification environment for circuit-building quantum programs, featuring clear separation between code and proof, parametric specifications and proofs, high degree of proof automation and allowing to encode quantum programs in a natural way, i.e. close to textbook style. Qbricks builds on best practice of formal verification for the classical case and tailor them to the quantum case: we bring a new domain-specific circuit-building language for quantum programs, namely Qbricks-DSL, together with a new logical specification language Qbricks-Spec and a dedicated Hoare-style deductive verification rule named Hybrid Quantum Hoare Logic. Especially, we introduce and intensively build upon HOPS, a higher-order extension of the recent path-sum symbolic representation, used for both specification and automation. To illustrate the opportunity of Qbricks, we implement the first verified parametric implementations of several famous and non-trivial quantum algorithms, including the quantum part of Shor integer factoring (Order Finding - Shor-OF), quantum phase estimation (QPE) - a basic building block of many quantum algorithms, and Grover search. These breakthroughs were amply facilitated by the specification and automated deduction principles introduced within Qbricks.
Abstract While recent progress in quantum hardware open the door for significant speedup in certain key areas, quantum algorithms are still hard to implement right, and the validation of such quantum programs is a challenge. In this paper we propose Qbricks, a formal verification environment for circuit-building quantum programs, featuring both parametric specifications and a high degree of proof automation. We propose a logical framework based on first-order logic, and develop the main tool we rely upon for achieving the automation of proofs of quantum specification: PPS, a parametric extension of the recently developed path sum semantics. To back-up our claims, we implement and verify parametric versions of several famous and non-trivial quantum algorithms, including the quantum parts of Shor’s integer factoring, quantum phase estimation (QPE) and Grover’s search.
We formalize the theory of quantum Hoare logic (QHL) [TOPLAS 33(6),19], an extension of Hoare logic for reasoning about quantum programs. In particular, we formalize the syntax and semantics of quantum programs in Isabelle/HOL, write down the rules of quantum Hoare logic, and verify the soundness and completeness of the deduction system for partial correctness of quantum programs. As preliminary work, we formalize some necessary mathematical background in linear algebra, and define tensor products of vectors and matrices on quantum variables. As an application, we verify the correctness of Grover’s search algorithm. To our best knowledge, this is the first time a Hoare logic for quantum programs is formalized in an interactive theorem prover, and used to verify the correctness of a nontrivial quantum algorithm.
The field of quantum mechanics predates computer science by at least ten years, the time between the publication of the Schrodinger equation and the Church-Turing thesis. It took another fifty years for Feynman to recognize that harnessing quantum mechanics is necessary to efficiently simulate physics and for David Deutsch to propose the quantum Turing machine. After thirty more years, we are finally getting close to the first general-purpose quantum computers based upon prototypes by IBM, Intel, Google and others.\nWhile physicists and engineers have worked on building scalable quantum computers, theoretical computer scientists have made their own advances. Complexity theorists introduced quantum complexity classes like BQP and QMA; Shor and Grover developed their famous algorithms for factoring and unstructured search. Programming languages researchers pursued two main research directions: Small-scale languages like QPL and the quantum lambda-calculi for reasoning about quantum computation and large-scale languages like Quipper and Q# for industrial-scale quantum software development. This thesis aims to unify these two threads while adding a third one: formal verification.\nWe argue that quantum programs demand machine-checkable proofs of correctness. We justify this on the basis of the complexity of programs manipulating quantum states, the expense of running quantum programs, and the inapplicability of traditional debugging techniques to programs whose states cannot be examined. We further argue that the existing mathematical models of quantum computation make this an easier task than one could reasonably expect. In light of these observations we introduce QWIRE, a tool for writing verifiable, large scale quantum programs.\nQWIRE is not merely a language for writing and verifying quantum circuits: it is a verified circuit description language. This means that the semantics of QWIRE circuits are verified in the Coq proof assistant. We also implement verified abstractions, like ancilla management and reversible circuit compilation. Finally, we turn QWIRE and Coq's abilities outwards, towards verifying popular quantum algorithms like quantum teleportation. We argue that this tool provides a solid foundation for research into quantum programming languages and formal verification going forward.
Quantum mechanics based computing systems are expected to have high capabilities and are considered good candidates to replace classical cryptography and supercomputing systems. Among many implementations, quantum optics systems provide a promising platform to implement universal quantum computers, since they link quantum computation and quantum communication in the same framework. Recently, several quantum gates, circuits, and protocols have been experimentally realized using optics. Despite the fact that big advances in building the physical quantum computers were achieved, there are no currently available industrial computer aided tools that can perform the modeling, analysis, and verification of optical quantum computing systems. In this thesis, we tackle the idea of design automation for quantum circuits, where we use a sound language, higher order logic, to model and reason about quantum circuits formally. In particular, we propose a framework for the hierarchical modeling and automated verification of quantum computing circuits. The modeling approach captures quantum models built hierarchically from quantum gates, which models are readily available in a library. The analysis and verification of composed circuits is done seamlessly based on dedicated mathematical foundations formalized in the theorem prover. \nSpecifically, the tensor product and linear projection are used to extract the quantum circuit outputs. Subsequently, a rich library of quantum gates which includes 1-qubit, 2-qubit, and 3-qubit gates is formalized. In order to automate the analysis process, we developed a decision procedure to eliminate the need of user guidance throughout the formal proofs. To demonstrate the effectiveness of the proposed framework, we conduct the formal analysis of a benchmark of quantum circuits including the Shor's integer factorization algorithm, the Grover's oracle, and the quantum full adder.
Quantum Hoare Logic (QHL) was introduced in Ying's work to specify and reason about quantum programs. In this paper, we implement a theorem prover for QHL based on Isabelle/HOL. By applying the theorem prover, verifying a quantum program against a specification is transformed equivalently into an order relation between matrices. Due to the limitation of Isabelle/HOL, the calculation of the order relation is solved by calling an outside oracle written in Python. To the best of our knowledge, this is the first theorem prover for quantum programs. To demonstrate its power, the correctness of two well-known quantum algorithms, i.e., Grover Quantum Search and Quantum Phase Estimation (the key step in Shor's quantum algorithm of factoring in polynomial time) are proved using the theorem prover. These are the first mechanized proofs for both of them.