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IEEE International Symposium on Multiple-Valued Logic (ISMVL)

Representing Fuzzy Structures in Quantum Computation with Mixed States

Héctor Freytes, Roberto Giuntini, Giuseppe Sergioli, Antonio Aricò

Published Jan 1, 2010• 2 citations
Publisher

Abstract

In this work we introduce a particular kind of quantum operations called polynomial quantum operations that allow us to represent the basic operations of the standard Product MV -algebra. Consequently, these operations can be treated as quantum computational gates in the powerful model of quantum computation given by “quantum operations - density operators”.

DOI
10.1109/ismvl.2010.38

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In this work we introduce a particular kind of quantum operations called polynomial quantum operations that allow us to represent the basic operations of the standard Product MV -algebra. Consequently, these operations can be treated as quantum computational gates in the powerful model of quantum computation given by “quantum operations - density operators”.
A model of quantum computation is discussed in (Aharanov et al 1997 Proc. 13th Annual ACM Symp. on Theory of Computation, STOC pp 20–30) and ( Tarasov 2002 J. Phys. A: Math. Gen. 35 5207–35 ) in which quantum gates are represented by quantum operations acting on mixed states. It allows one to use a quantum computational model in which connectives of a four-valued logic can be realized as quantum gates. In this model, we give a representation of certain functions, known as t -norms (Menger 1942 Proc. Natl Acad. Sci. USA 37 57–60), that generalize the triangle inequality for the probability distribution-valued metrics. As a consequence an interpretation of the standard operations associated with the basic fuzzy logic (Hájek 1998 Metamathematics of Fuzzy Logic ( Trends in Logic vol 4) (Dordrecht: Kluwer)) is provided in the frame of quantum computation.
Usually models for quantum computations deal with unitary gates on pure states. In this paper we generalize the usual model. We consider a model of quantum computations in which the state is an operator of density matrix and the gates are quantum operations, not necessarily unitary. A mixed state (operator of density matrix) of n two-level quantum systems is considered as an element of 4n4^{n}4n-dimensional operator Hilbert space. Unitary quantum gates and nonunitary quantum operations for n-qubit system are considered as generalized quantum gates acting on mixed state. In this paper we study universality for quantum computations by quantum operations on mixed states.
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 importantirreversibletransformationsalreadyconsideredintheliterature: theirreversible conjunction IAND and the fuzzy-like Łukasiewicz disjunction⊕.
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 in a pure state) or, more generally, with a mixture of quregisters (called qumix). Following an approach proposed by Domenech and Freytes, we apply residuated structures associated with fuzzy logic to develop certain aspects of information processing in quantum computing from a logical perspective. For this purpose, we introduce an axiomatic system whose natural interpretation is the irreversible quantum Poincaré algebra. Such a system allows to establish a completeness theorem for the treatment of quantum information.
The main motivation for this thesis is given by the open problems regarding the axiomatisation of quantum computational logics. This thesis will be structured as follows: in Chapter 2 we will review some basics of universal algebra and functional analysis. In Chapters 3 through 6 the fundamentals of quantum gate theory will be produced. In Chapter 7 we will introduce quasi-MV algebras, a formal study of a suitable selection of algebraic operations associated with quantum gates. In Chapter 8 quasi-MV algebras will be expanded by a unary operation hereby dubbed square root of the inverse, formalising a quantum gate which allows to induce entanglement states. In Chapter 9 we will investigate some categorial dualities for the classes of algebras introduced in Chapters 7 and 8. In Chapter 10 the discriminator variety of linear Heyting quantum computational structures, an algebraic counterpart of the strong quantum computational logic, will be considered. In Chapter 11, we will list some open problems and, at the same time, draw some tentative conclusions. Lastly, in Chapter 12 we will provide a few examples of the previously investigated structures.