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.
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.
Reasoning about quantum systems has gained prominence due to a big potential in applications such as information processing, security, distributed systems, and randomized algorithms. This fact has attracted research in formal reasoning about quantum states, programs, and processes. On the other hand, temporal logics have proved to be successful in the verification of classical distributed systems and security protocols. In this chapter we extend exogenous quantum propositional logic with temporal modalities, considering both linear and branching time. We provide a weakly complete Hilbert calculi for the proposed quantum temporal logics and study their SAT and model-checking problems. 10.1 Introduction Reasoning about quantum systems has gained prominence due to their potential applications in information processing, security, distributed systems and randomized algorithms. This has attracted research in formal reasoning about quantum states (see for instance van der Meyden and Patra 2003b, a; Mateus and Sernadas 2006; Chadha et al. 2009; Caleiro et al. 2006) and quantum programs (cf. Knill 1996; Sanders and Zuliani 2000; Abramsky and Coecke 2004; D'Hondt and Panangaden 2004; Altenkirch and Grattage 2005; Selinger and Valiron 2005; Baltag and Smets 2006; Baltazar et al. 2007; Chadha et al. 2006, 2007; Baltazar et al. 2008). On the other hand, formal methods have proved to be successful in design and verification of classical distributed systems and security protocols (e.g., Clarke and Wing 1996; Meadows 2003). Herein, we present branching and linear temporal logics for reasoning about evolution of quantum systems composed of a fixed finite set of qubits.
This paper introduces a novel temporal logic formalism, termed Temporal Quantum Logic (TQL), specifically tailored for representing and reasoning about the continuous evolution of quantum states. The core challenge in applying temporal logic to quantum mechanics lies in effectively capturing the inherent complexities of superposition, entanglement, and decoherence, particularly the non-commutativity of quantum operations. Existing temporal logics lack the necessary tools to accurately model these phenomena. TQL addresses this limitation by employing a multi-valued logic built upon tensor networks and process algebra concepts. This approach enables a precise and explicit representation of quantum operations, measurement outcomes, and their temporal dependencies, incorporating probabilistic semantics directly into the logic. The formalization utilizes a state representation based on tensor network contractions, allowing for the dynamic tracking of quantum state evolution over time. The logic incorporates operators that explicitly handle non-commutativity and decoherence, offering a framework for formally verifying the correctness of quantum algorithms and protocols. The key innovation is the combination of tensor network formalism with temporal logic, providing a rigorous method for describing the evolution of quantum systems and enabling a deeper understanding of their behavior. This work lays the foundation for future research in quantum verification, quantum algorithm design, and the development of quantum software.
Abstract Model-checking techniques have been extended to analyze quantum programs and communication protocols represented as quantum Markov chains, an extension of classical Markov chains. To specify qualitative temporal properties, a subspace-based quantum temporal logic is used, which is built on Birkhoff-von Neumann atomic propositions. These propositions determine whether a quantum state is within a subspace of the entire state space. In this paper, we propose the measurement-based linear-time temporal logic MLTL to check quantitative properties. MLTL builds upon classical linear-time temporal logic (LTL) but introduces quantum atomic propositions that reason about the probability distribution after measuring a quantum state. To facilitate verification, we extend the symbolic dynamics-based techniques for stochastic matrices described by Agrawal et al. (JACM 2015) to handle more general quantum linear operators (super-operators) through eigenvalue analysis. This extension enables the development of an efficient algorithm for approximately model checking a quantum Markov chain against an MLTL formula. To demonstrate the utility of our model-checking algorithm, we use it to simultaneously verify linear-time properties of both quantum and classical random walks. Through this verification, we confirm the previously established advantages discovered by Ambainis et al. (STOC 2001) of quantum walks over classical random walks and discover new phenomena unique to quantum walks.
The main contribution of this paper is the introduction of a dynamic logic formalism for reasoning about information flow in composite quantum systems. This builds on our previous work on a complete quantum dynamic logic for single systems. Here we extend that work to a sound (but not necessarily complete) logic for composite systems, which brings together ideas from the quantum logic tradition with concepts from (dynamic) modal logic and from quantum computation. This Logic of Quantum Programs (LQP) is capable of expressing important features of quantum measurements and unitary evolutions of multi-partite states, as well as giving logical characterisations to various forms of entanglement (for example, the Bell states, the GHZ states etc.). We present a finitary syntax, a relational semantics and a sound proof system for this logic. As applications, we use our system to give formal correctness proofs for the Teleportation protocol and for a standard Quantum Secret Sharing protocol; a whole range of other quantum circuits and programs, including other well-known protocols (for example, superdense coding, entanglement swapping, logic-gate teleportation etc.), can be similarly verified using our logic.
The main contribution of this paper is the introduction of a dynamic logic formalism for reasoning about information flow in composite quantum systems. This builds on our previous work on a complete quantum dynamic logic for single systems. Here we extend that work to a sound (but not necessarily complete) logic for composite systems, which brings together ideas from the quantum logic tradition with concepts from (dynamic) modal logic and from quantum computation. This Logic of Quantum Programs (LQP) is capable of expressing important features of quantum measurements and unitary evolutions of multi-partite states, as well as giving logical characterisations to various forms of entanglement (for example, the Bell states, the GHZ states etc.). We present a finitary syntax, a relational semantics and a sound proof system for this logic. As applications, we use our system to give formal correctness proofs for the Teleportation protocol and for a standard Quantum Secret Sharing protocol; a whole range of other quantum circuits and programs, including other well-known protocols (for example, superdense coding, entanglement swapping, logic-gate teleportation etc.), can be similarly verified using our logic.
This paper introduces a novel approach to quantum state evolution by leveraging temporal logic formalisms. The core idea is to directly represent and reason about quantum state transitions, incorporating the inherent uncertainties of non-deterministic branching and measurement outcomes. We propose a new temporal logic variant, Temporal Probabilistic Logic (TPL), where quantum state transitions are defined as logical rules and measurement collapses are modeled as probabilistic branches within the temporal reasoning process. This purely logical framework contrasts with traditional numerical simulation methods used in quantum control theory, offering potential advantages in automated proof of correctness, algorithm design, and optimization. TPL allows for the formal specification of quantum protocols and the verification of their behavior without relying on computationally intensive simulations. The paper outlines the key components of TPL, including its syntax, semantics, and operational rules, and explores its application to a simplified quantum algorithm. The presented framework provides a foundation for developing a more robust and verifiable approach to quantum computation.