Underlying theory for quantum computing

Mingsheng Ying

The (meta)logic underlying classical theory of computation is Boolean (two-valued) logic. Quantum logic was proposed by Birkhoff and von Neumann as a logic of quantum mechanics more than sixty years ago. The major difference between Boolean logic and quantum logic is that the latter does not enjoy distributivity in general. The rapid development of quantum computation in recent years stimulates us to establish a theory of computation based on quantum logic. The present paper is the first step toward such a new theory and it focuses on the simplest models of computation, namely finite automata. It is found that the universal validity of many properties of automata depend heavily upon the distributivity of the underlying logic. This indicates that these properties does not universally hold in the realm of quantum logic. On the other hand, we show that a local validity of them can be recovered by imposing a certain commutativity to the (atomic) statements about the automata under consideration. This reveals an essential difference between the classical theory of computation and the computation theory based on quantum logic.

Masahito Hayashi, Satoshi Ishizaka, Akinori Kawachi +2 more

Juan C. Agudelo, Walter Alexandre Carnielli

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.

Juan C. Agudelo, Walter Alexandre Carnielli

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.

Myo Myo Aung