Formalization of Quantum Protocols using Coq

Abstract

Quantum Information Processing, which is an exciting area of research at the intersection of physics and computer science, has great potential for influencing the future development of information processing systems. The building of practical, general purpose Quantum Computers may be some years into the future. However, Quantum Communication and Quantum Cryptography are well developed. Commercial Quantum Key Distribution systems are easily available and several QKD networks have been built in various parts of the world. The security of the protocols used in these implementations rely on information-theoretic proofs, which may or may not reflect actual system behaviour. Moreover, testing of implementations cannot guarantee the absence of bugs and errors. This paper presents a novel framework for modelling and verifying quantum protocols and their implementations using the proof assistant Coq. We provide a Coq library for quantum bits (qubits), quantum gates, and quantum measurement. As a step towards verifying practical quantum communication and security protocols such as Quantum Key Distribution, we support multiple qubits, communication and entanglement. We illustrate these concepts by modelling the Quantum Teleportation Protocol, which communicates the state of an unknown quantum bit using only a classical channel.

QWIRE Practice: Formal Verification of Quantum Circuits in Coq

“…QWIRE programs are interpreted as superoperators over density matrices, following the denotational semantics described by Paykin et al [14]. We use a density matrix representation over the standard unit vector representation (used for example by Boender et al [3]) because density matrices represent probability distributions (introduced via measurement) over quantum states directly, as opposed to the unit vector representation which must be embedded inside a probability monad.…”

“…First, every pattern is interpreted as a list of indices indicating the wire number (counting from 0) that each variable refers to. So the pattern (3,0) : Pat [Some Qubit, None, None, Some Bit] (Bit ⊗ Qubit) corresponds to the list [1;0], because variable 3 corresponds the wire numbered 1 and variable 0 corresponds to wire 0 in the circuit. This list ls is then turned into an association list [(0,1);(1,0)], mapping variable i to ls[i].…”

“…In the near future, we will be transitioning the linear algebra back-end of our development to one of several existing projects, in the hopes of making it more efficient. Boender et al [3] present one candidate, a library developed reasoning about the correctness of quantum protocols using pure states. Another candidate is the Coq Effective Algebra Library (CoqEAL) [5], which is designed specifically to allow matrix computation inside Coq.…”

“…Doing so requires a formal semantics for programs, such as unit vectors or density matrices. The specification can then be verified using model-checking [9], Hoare logic [23], proof assistants [3], or other techniques.…”

A Formal Proof of the Expressiveness of Deep Learning

“…For example, the Mathematical Components include comprehensive matrix theories [6], which are naturally expressed using dependent types. The tensor formalization by Boender [10] restricts itself to the Kronecker product on matrices. Bernard et al [5] formalized multivariate polynomials and used them to show the transcendence of e and π. Kam formalized the Lebesgue integral, which is closely related to the Lebesgue measure, to state and prove Markov's inequality [27].…”