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Temporal Logics for Reasoning about Quantum Systems

Paulo Mateus, Jaime Ramos, Amı́lcar Sernadas, Cristina Sernadas

Published Nov 30, 2009• 12 citations

Abstract

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.

DOI: 10.1017/cbo9781139193313.011

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This article is intended as an introduction to the subject of quantum logic, and as a brief survey of the relevant literature. Also discussed here are logics for specification and analysis of quantum information systems, in particular, recent work by P. Mateus and A. Sernadas, and also by R. van der Meyden and M. Patra. Overall, our objective is to provide a high-level presentation of the logical aspects of quantum theory. Mateus’ and Sernadas’ EQPL logic is illustrated with a small example, namely the state of an entangled pair of qubits. The "KT" logic of van der Meyden and Patra is demonstrated briey in the context of the B92 protocol for quantum key distribution.

Logics for reasoning about quantum states and their evolution have been given in the literature. In this paper, we consider quantum computation tree logic (QCTL), which adds temporal modalities to exogenous quantum propositional logic. We give a sound and complete axiomatization of QCTL and combine the standard CTL model-checking algorithm with the dEQPL model-checking algorithm to obtain a model-checking algorithm for QCTL. Finally, we illustrate the use of the logic by reasoning about the BB84 key distribution protocol.

This article is intended as an introduction to the subject of quantum logic, and as a brief survey of the relevant literature. Also discussed here are logics for specification and analysis of quantum information systems, in particular, recent work by P. Mateus and A. Sernadas, and also by R. van der Meyden and M. Patra. Overall, our objective is to provide a high-level presentation of the logical aspects of quantum theory. Mateus' and Sernadas' EQPL logic is illustrated with a small example, namely the state of an entangled pair of qubits. The "KT" logic of van der Meyden and Patra is demonstrated briefly in the context of the B92 protocol for quantum key distribution.

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.

This paper presents a novel approach to verifying the correctness of quantum circuits using temporal logic. Traditional verification methods often rely on gate-level simulation, which can be computationally expensive and insufficient for complex quantum algorithms. This work addresses this limitation by formally modeling quantum circuits as temporal logic systems. We leverage the expressive power of temporal logic to capture the desired behavior of quantum algorithms, including their evolution over time and the effects of quantum phenomena like superposition and entanglement. The core of our method involves mapping the circuit's operations onto temporal logic specifications and utilizing automated theorem proving and model checking techniques, tailored to the probabilistic nature of quantum systems. We demonstrate that this approach provides a more rigorous and scalable verification framework compared to existing methods, offering a pathway to ensuring the reliability of quantum computations. The key contributions of this work are the formalization of quantum circuits with temporal logic, the adaptation of theorem proving and model checking techniques for quantum systems, and the potential for significantly improving the trustworthiness of quantum algorithms.