While constructing practical quantum computers by big companies remains a challenge, the application of quantum communication and cryptography has made remarkable progress. Therefore, it is crucial to verify quantum protocols before they can be trusted in safety and security-critical applications. We have proposed Basic Dynamic Quantum Logic (BDQL) to formalize and verify sequential models of quantum protocols with a support tool developed in Maude. However, BDQL does not support concurrency in its formalization. This article introduces Concurrent Dynamic Quantum Logic (CDQL), an extension of BDQL, to formalize and verify concurrent models of quantum protocols. CDQL is more expressive than BDQL by considering concurrent behavior and communication among participants in quantum protocols. Since CDQL is a conservative extension of BDQL, we extend the syntax of BDQL to CDQL and make a transformation from CDQL to BDQL without interrupting the semantics of BDQL. We also extend the implementation of BDQL to support CDQL, making a new support tool in Maude. The new support tool is equipped with a lazy rewriting strategy to make the verification process significantly faster. Several quantum communication protocols are successfully formalized and verified in BDQL/CDQL, demonstrating the effectiveness of our automated approach and tool in verifying quantum protocols.

Although various dynamic or temporal logics have been proposed to verify quantum protocols and systems, these two viewpoints have not been studied comprehensively enough. We propose Linear Temporal Quantum Logic (LTQL), a linear temporal extension of quantum logic with a quantum implication, and extend it to Dynamic Linear Temporal Quantum Logic (DLTQL). This logic has temporal operators to express transitions by unitary operators (quantum gates) and dynamic ones to express those by projections (projective measurement). We then prove some logical properties of the relationship between these two transitions expressed by LTQL and DLTQL. A drawback in applying LTQL to the verification of quantum protocols is that these logics cannot express the future operator in linear temporal logic. We propose a way to mitigate this drawback by using a translation from (D)LTQL to Linear Temporal Modal Logic (LTML) and a simulation. This translation reduces the satisfiability problem of (D)LTQL formulas to that of LTML with the classical semantics over quantum states.