Paulo Mateus

Weakly complete axiomatization of exogenous quantum propositional logic

A finitary axiomatization for EQPL (exogenous quantum propositional logic) is presented. The axiomatization is shown to be weakly complete relative to an oracle for analytical reasoning. The proof is carried out using a non-trivial extension of the Fagin-Halpern-Megiddo technique together with three Henkin style completions.

Probabilistic Situation Calculus

In this article we propose a Probabilistic Situation Calculus logical language to represent and reason with knowledge about dynamic worlds in which actions have uncertain effects. Uncertain effects are modeled by dividing an action into two subparts: a deterministic (agent produced) input and a probabilistic reaction (produced by nature). We assume that the probabilities of the reactions have known distributions.Our logical language is an extension to Situation Calculae in the style proposed by Raymond Reiter. There are three aspects to this work. First, we extend the language in order to accommodate the necessary distinctions (e.g., the separation of actions into inputs and reactions). Second, we develop the notion of Randomly Reactive Automata in order to specify the semantics of our Probabilistic Situation Calculus. Finally, we develop a reasoning system in MATHEMATICA capable of performing temporal projection in the Probabilistic Situation Calculus.

Composition of Cryptographic Protocols in a Probabilistic Polynomial-Time Process Calculus

We describe a probabilistic polynomial-time process calculus for analyzing cryptographic protocols and use it to derive compositionality properties of protocols in the presence of computationally bounded adversaries. We illustrate these concepts on oblivious transfer, an example from cryptography. We also compare our approach with a framework based on interactive Turing machines.

Quantum secret sharing with classical Bobs

Boyer et al (2007 Phys. Rev. Lett. 99 140501) proposed a novel idea of semi-quantum key distribution, where a key can be securely distributed between Alice, who can perform any quantum operation, and Bob, who is classical. Extending the ?semi-quantum? idea to other tasks of quantum information processing is of interest and worth considering. In this paper, we consider the issue of semi-quantum secret sharing, where a quantum participant Alice can share a secret key with two classical participants, Bobs. After analyzing the existing protocol, we propose a new protocol of semi-quantum secret sharing. Our protocol is more realistic, since it utilizes product states instead of entangled states. We prove that any attempt of an adversary to obtain information necessarily induces some errors that the legitimate users could notice.

Reasoning about probabilistic sequential programs

A complete and decidable Hoare-style calculus for iteration-free probabilistic sequential programs is presented using a state logic with truth-functional propositional (not arithmetical) connectives.

Reasoning About Imperative Quantum Programs

A logic for reasoning about states of basic quantum imperative programs is presented. The models of the logic are ensembles obtained by attaching probabilities to pairs of quantum states and classical states. The state logic is used to provide a sound Hoare-style calculus for quantum imperative programs. The calculus is illustrated by proving the correctness of the Deutsch algorithm.

Characterizations of one-way general quantum finite automata

Generally, unitary transformations limit the computational power of quantum finite automata (QFA). In this paper, we study a generalized model named one-way general quantum finite automata (1gQFA), in which each symbol in the input alphabet induces a trace-preserving quantum operation, instead of a unitary transformation. Two different kinds of 1gQFA will be studied: measure-once one-way general quantum finite automata (MO-1gQFA) where a measurement deciding to accept or reject is performed at the end of a computation, and measure-many one-way general quantum finite automata (MM-1gQFA) where a similar measurement is performed after each trace-preserving quantum operation on reading each input symbol. We characterize the measure-once model from three aspects: the closure property, the language recognition power, and the equivalence problem. We prove that MO-1gQFA recognize, with bounded error, precisely the set of all regular languages. Our results imply that some models of quantum finite automata proposed in the literature, which were expected to be more powerful, still cannot recognize non-regular languages. We prove that MM-1gQFA also recognize only regular languages with bounded error. Thus, MM-1gQFA and MO-1gQFA have the same language recognition power, in sharp contrast with traditional MO-1QFA and MM-1QFA, the former being strictly less powerful than the latter. Finally, we present a necessary and sufficient condition for two MM-1gQFA to be equivalent.

Reasoning About Quantum Systems

A new logic is proposed for reasoning about quantum systems. The logic embodies the postulates of quantum physics and it was designed from the semantics upwards by identifying quantum models with superpositions of classical models. This novel approach to quantum logic is completely different from the traditional approach of Birkhoff and von Neumann. It has the advantage of making quantum logic an extension of classical logic. The key new ingredient of the language of the proposed logic is a rather general modal operator. The logic incorporates probabilistic reasoning (in the style of Nilsson) in order to deal with uncertainty on the outcome of measurements. The logic also incorporates dynamic reasoning (in the style of Hoare) in order to cope with the evolution of quantum systems. A Hilbert calculus for the logic is sketched. A quantum key distribution protocol is specified and analyzed.

QUANTUM COMPUTATION TREE LOGIC — MODEL CHECKING AND COMPLETE CALCULUS

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.

A Process Algebra for Reasoning About Quantum Security

We present a process algebra for specifying and reasoning about quantum security protocols. Since the computational power of the protocol agents must be restricted to quantum polynomial-time, we introduce the logarithmic cost quantum random access machine (QRAM) similar to [S.A. Cook, R.A. Reckhow, Time bounded random access machines, Journal of Computer and System Sciences 7 (1973) 354-375, E. Knill, Conventions for quantum pseudocode, Technical Report LAUR-96-2724, Los Alamos National Laboratory (1996)], and incorporate it in the syntax of the algebra. Probabilistic transition systems give the semantic for the process algebra. Term reduction is stochastic because quantum computation is probabilistic and, moreover, we consider a uniform scheduler to resolve non-deterministic choices. With the purpose of defining security properties, we introduce observational equivalence and quantum computational indistinguishability, and show that the latter is a congruence relation. A simple corollary of this result asserts that any security property defined via emulation is compositional. Finally, we illustrate our approach by establishing the concept of quantum zero-knowledge protocol.