Quantum computational structures

2018

In this work we advance a generalization of quantum computational logics capable of dealing with some important examples of quantum algorithms. We outline an algebraic axiomatization of these structures. 1Universidad Nacional de La Plata, Instituto de F́ısica (IFLP-CCT-CONICET), C.C. 727, 1900 La Plata, Argentina 2Universitá di Cagliari, Via Is Mirrionis 1, 09123, Cagliari-Italia 3Center Leo Apostel for Interdisciplinary Studies and, Department of Mathematics, Brussels Free University Krijgskundestraat 33, 1160 Brussels, Belgium

Handbook of Quantum Logic and Quantum Structures, 2007

Entropy

In this work we advance a generalization of quantum computational logics capable of dealing with some important examples of quantum algorithms. We outline an algebraic axiomatization of these structures.

Mathematical Logic Quarterly, 2013

Using an algebraic framework we solve a problem posed in [5] and [7] about the axiomatizability of a type quantum computational logic related to fuzzy logic. A Hilbert-style calculus is developed obtaining an algebraic strong completeness theorem. 1 a qbit. A qbit state (the quantum counterpart of the classical bit) is represented by a unit vector in C 2 and, generalizing for a positive integer n, n-qbits are pure states represented by unit vectors in C 2 n . They conform the information units in quantum computation. These state spaces only concerned with the "static" part of quantum computing and possible logical systems can be founded in the Birkhoff and von Neumann quantum logic based on the Hilbert lattices L(C 2 n ) [11]. Similarly to the classical computing case, we can introduce and study the behavior of a number of quantum logical gates (hereafter quantum gates for short) operating on qbits, giving rise to "new forms" of quantum logic. These gates are mathematically represented by unitary operators on the appropriate Hilbert spaces of qbits. In other words, standard quantum computation is mathematically founded on "qbits-unitary operators" and only takes into account reversible processes. This framework can be generalized to a powerful mathematical representation of quantum computation in which the qbit states are replaced by density operators over Hilbert spaces and unitary operators by linear operators acting over endomorphisms of Hilbert spaces called quantum operations. The new model "density operators-quantum operations" also called "quantum computation with mixed states" ([1, 32]) is equivalent in computational power to the standard one but gives a place to irreversible processes as measurements in the middle of the computation.

Trends in Logic, 2003

Quantum computation has suggested new forms of quantum logic, called quantum computational logics ([CDCGL02]). The basic semantic idea is the following: the meaning of a sentence is identified with a quregister, a system of qubits, representing a possible pure state of a compound quantum system. The generalization to mixed states, which might be useful to analyse entanglement-phenomena, is due to Gudder ([Gu02]). Quantum computational logics represent non standard examples of unsharp quantum logic, where the non-contradiction principle is violated, while conjunctions and disjunctions are strongly non-idempotent. In this framework, any sentence α of the language gives rise to a quantum tree: a kind of quantum circuit that transforms the quregister associated to the atomic subformulas of α into the quregister associated to α.

International Journal of Theoretical Physics, 2015

We introduce the notion of proper unitary connective-gate and we prove that entanglement cannot be characterized by such gates. We consider then a larger class of gates (called pseudo-unitary gates), which contains both the unitary and the anti-unitary quantum operations. By using a mixed language (a proper extension of the standard quantum computational language), we show how a logical characterization of entanglement is possible in the framework of a mixed semantics, which generalizes both the unitary and the pseudo-unitary quantum computational semantics.

International Journal of Theoretical Physics - INT J THEOR PHYS, 2000

It is shown that computations can be founded on the laws of the genuine(Birkhoff—nvon Neumann) quantum logic treated as a particular version ofLukasiewicz infinite-valued logic. A new way of encoding nonexact data whichencodes both the value of a number and its “fuzziness” is introduced. A simpleexample of a full adder that works in the proposed way is given and it is comparedwith other designs of quantum adders existing in the literature. A controversybetween the meaning of the very term “quantum logic” as used recently withinthe theory of quantum computations and the traditional meaning of this term isbriefly discussed.

International Journal of Quantum Information, 2005

The theory of logical gates in quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence is identified with a quregister (a system of qubits) or, more generally, with a mixture of quregisters (called qumix ). In this framework, any sentence α of the language gives rise to a quantum tree: a kind of quantum circuit that transforms the quregister (qumix) associated to the atomic subformulas of α into the quregister (qumix) associated to α. A variant of the quantum computational semantics is represented by the quantum holistic semantics, which permits us to represent entangled meanings. Physical models of quantum computational logics can be built by means of Mach-Zehnder interferometers.

APPLIED MATHEMATICS & INFORMATION SCIENCES, 2011

In this paper we discuss an approach to quantum computation where the basic information units (qubits and quregisters) are replaced by density operators and the restriction to unitary operators as logical gates is lifted through the introduction of the more general concept of quantum operation ([17],[1]). This perspective is especially suited to provide a physical description of open systems. In particular, we illustrate the advantages of this approach over the standard one and show that it can account for two important irreversible ...

2005

The (meta)logic underlying classical theory of computation is Boolean (twovalued) logic. Quantum logic was proposed by Birkhoff and von Neumann as a logic of quantum mechanics more than sixty years ago. It is currently understood as a logic whose truth values are taken from an orthomodular lattice. 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. We introduce the notion of orthomodular lattice-valued (quantum) automaton. Various properties of automata are carefully reexamined in the framework of quantum logic by employing an approach of semantic analysis. We define the class of regular languages accepted by orthomodular lattice-valued automata. The acceptance abilities of orthomodular lattice-valued nondeterministic automata and their various modifications (such as deterministic automata and automata with ε−moves) are compared. The closure properties of orthomodular lattice-valued regular languages are derived. The Kleene theorem about equivalence of regular expressions and finite automata is generalized into quantum logic. We also present a pumping lemma for orthomodular lattice-valued regular languages. It is found that the universal validity of many properties (for example, the Kleene theorem, the equivalence of deterministic and nondeterministic automata) 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.