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A Paraconsistent Approach to Quantum Computing

Juan C. Agudelo, Walter Alexandre Carnielli

Published Feb 1, 2008• 0 citations
OpenAlex

Abstract

We propose a method to define axiomatic theories for deterministic Turing machine computations. This method, when applied to axiomatizing computations in non-deterministic Turing machines, produces (in some cases) contradictory theories, therefore trivial theories (considering classical logic as the underlying logic). Substituting in such theories the underlying logic by the paraconsistent logic LFI1 ∗ permits us to define a new model of computation which we call paraconsistent Turing machine. We show that this initial model of computation allows the simulation of important quantum computing features. In particular, it allows to simulate the quantum solution of the well-known Deutsch’s and Deutsch-Jozsa problems. However, we show that this initial model of computation does not adequately represent the notion of entangled states, a key feature in quantum computing. In this way, the construction is refined by defining a paraconsistent logic with a connective expressing entangled states in a logical fashion, and this logic is used to define a more sharpened model of paraconsistent Turing machines, better approaching the quantum computing features. Finally, we define complexity classes for the models introduced and establish some surprising relationships with classical complexity classes.

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We propose a method to define axiomatic theories for deterministic Turing machine computations. This method, when applied to axiomatizing computations in non-deterministic Turing machines, produces (in some cases) contradictory theories, therefore trivial theories (considering classical logic as the underlying logic). Substituting in such theories the underlying logic by the paraconsistent logic LFI1 ∗ permits us to define a new model of computation which we call paraconsistent Turing machine. We show that this initial model of computation allows the simulation of important quantum computing features. In particular, it allows to simulate the quantum solution of the well-known Deutsch’s and Deutsch-Jozsa problems. However, we show that this initial model of computation does not adequately represent the notion of entangled states, a key feature in quantum computing. In this way, the construction is refined by defining a paraconsistent logic with a connective expressing entangled states in a logical fashion, and this logic is used to define a more sharpened model of paraconsistent Turing machines, better approaching the quantum computing features. Finally, we define complexity classes for the models introduced and establish some surprising relationships with classical complexity classes.
We describe a method to axiomatize computations in deterministic Turing machines (TMs). When applied to computations in non-deterministic TMs, this method may produce contradictory (and therefore trivial) theories, considering classical logic as the underlying logic. By substituting in such theories the underlying logic by a paraconsistent logic we define a new computation model, the paraconsistent Turing machine. This model allows a partial simulation of superposed states of quantum computing. Such a feature allows the definition of paraconsistent algorithms which solve (with some restrictions) the well-known Deutsch's and Deutsch-Jozsa problems. This first model of computation, however, does not adequately represent the notions of entangled states and relative phase, which are key features in quantum computing. In this way, a more sharpened model of paraconsistent TMs is defined, which better approaches quantum computing features. Finally, we define complexity classes for such models, and establish some relationships with classical complexity classes.
We present the new model of paraconsistent Turing machines and revise the models of quantum Turing machines and quantum circuits, stressing the concepts of entangled states and quantum parallelism which are important features for efficient quantum algorithms. We revise a quantum circuit that solves the so-called Deutsch’s problem, and then provide another solution to Deutsch’s problem by means of paraconsistent Turing machines. This raises some stimulating questions, and we close the paper by discussing some relationships between paraconsistent Turing machines and quantum Turing machines.
Lots of efforts in the last decades have been done to prove or disprove whether the set of polynomially bounded problems is equal to the set of polynomially verifiable problems. This paper will present an overview of the current beliefs of quantum complexity theorists and discussion detail the impacts these beliefs may have on the future of the field. We introduce a new form of Turing machine based on the concepts of quantum mechanics and investigate the domain of this novel definition for computation. The main object is to show the domain of these computation machines and the new definition of algorithms using these automata. In section one we give a brief fundamental overview of classical complexity theory, Turing machine and all the fundamental concepts required for the next chapters. In section two we describe various classes of classical complexity automata. In next chapter we introduce quantum complexity featuring quantum Turing machines and discuss its relationship to classical complexity. Section four presents a relationship between quantum complexity classes based on the quantum Turing machines and quantum oracles, and their relationship with corresponding classical complexity classes. And section five presents a discussion on the practical phenomena in problem solving using quantum computers. Finally in section five we speculate briefly on the direction we believe the field to be headed, and what might reasonably be expected in the future
A model of quantum computation based on unitary matrix operations was introduced by Feynman and Deutsch. It has been asked whether the power of this model exceeds that of classical Turing machines. We show here that a significant class of these quantum computations can be simulated classically in polynomial time. In particular we show that two-bit operations characterized by 4 \times 4 matrices in which the sixteen entries obey a set of five polynomial relations can be composed according to certain rules to yield a class of circuits that can be simulated classically in polynomial time. This contrasts with the known universality of two-bit operations, and demonstrates that efficient quantum computation of restricted classes is reconcilable with the Polynomial Time Turing Hypothesis. In other words it is possible that quantum phenomena can be used in a scalable fashion to make computers but that they do not have superpolynomial speedups compared to Turing machines for any problem. The techniques introduced bring the quantum computational model within the realm of algebraic complexity theory. In a manner consistent will one view of quantum physics, the wave function is simulated deterministically, and randomization arises only in the course of making measurements. The results generalize the quantum model in that they do not require the matrices to be unitary. In a different direction these techniques also yield deterministic polynomial time algorithms for the decision and parity problems for certain classes of read-twice Boolean formulae. All our results are based on the use of gates that are defined in terms of their graph matching properties.
Contrary to the classical case, the relation between quantum programming languages and quantum Turing Machines (QTM) has not been fully investigated. In particular, there are features of QTMs that have not been exploited, a notable example being the intrinsic infinite nature of any quantum computation. In this paper, we propose a definition of QTM, which extends and unifies the notions of Deutsch and Bernstein & Vazirani. In particular, we allow both arbitrary quantum input, and meaningful superpositions of computations, where some of them are “terminated” with an “output”, while others are not. For some infinite computations an “output” is obtained as a limit of finite portions of the computation. We propose a natural and robust observation protocol for our QTMs, which does not modify the probability of the possible outcomes of the machines. Finally, we use QTMs to define a class of quantum computable functions—any such function is a mapping from a general quantum state to a probability distribution of natural numbers. We expect that our class of functions, when restricted to classical input-output, will not be different from the set of the recursive functions.
Deutsch, Feynman, and Manin viewed quantum computing as a kind of universal physical simulation procedure. Much of the writing about quantum Turing machines has shown how these machines can simulate an arbitrary unitary transformation on a finite number of qubits. This interesting problem has been addressed most famously in a paper by Deutsch, and later by Bernstein and Vazirani. Quantum Turing machines form a class closely related to deterministic and probabilistic Turing machines and one might hope to find a universal machine in this class. A universal machine is the basis of a notion of programmability. The extent to which universality has in fact been established by the pioneers in the field is examined and a key notion in theoretical computer science (universality) is scrutinised. In a forthcoming paper, the authors will also consider universality in the quantum gate model.