Quantum duality

Badis Ydri

C. Y. Li, W. Y. Wang, C. Wang +6 more

Quantum mechanical systems exhibit particle wave duality property. This duality property has been exploited for information processing. A duality quantum computer is a quantum computer on the move and passing through a multi‐slits. It offers quantum wave divider and quantum wave combiner operations in addition to those allowed in an ordinary quantum computer. It has been shown that all linear bounded operators can be realized in a duality quantum computer, and a duality quantum computer with n qubits and d‐slits can be realized in an ordinary quantum computer with n qubits and a qudit in the so‐called duality quantum computing mode. The quantum particle‐wave duality can be used in providing secure communication. In this paper, we will review duality quantum computing and duality quantum key distribution.

Gui Lu Long, Chunyan Li

Quantum mechanics possesses peculiar properties that is hard to comprehend with mundane experience. Exploiting the quantum properties of quantum systems, quantum computer can provide more computing power than classical computer. Duality quantum computer, or duality computer for short, is a new type of quantum computer that exploits the particle wave duality of quantum mechanics. In a duality computer, the computing gates, or generalized quantum gates, need not be unitary. Thus duality computer provides a novel paradigm for quantum computing, and may bring new computing power. In addition, it provides flexibility in computing gates design that is familiar to computer scientists. Here we review the basic concepts of duality quantum computing and give some examples.

Peter J. Riggs

Časlav Brukner

Emilio Cobanera, Gerardo Ortíz, Zohar Nussinov

An algebraic theory of dualities based on the notion of bond algebras is developed. This framework deals with classical and quantum dualities in a unified fashion and its implementation includes discrete lattice, continuum field, and gauge theories. Dualities are revealed to be structure-preserving local mappings between model-specific bond algebras, and can be implemented as unitary (or projective-unitary) transformations. This characterization allows to search systematically for dualities or self-dualities in, finite or infinite, quantum and classical models of arbitrary complexity and number of spatial dimensions. Non-local transformations, like dual or disorder variables and Jordan-Wigner dictionaries, are algorithmically derived from these local duality mappings of bonds. An exact connection between quantum dualities and the low temperature (strong-coupling)/high temperature (weak-coupling) dualities of classical statistical mechanics (or path integrals), as well as the connection between quantum dual and classical disorder variables, is thus established. Our bond-algebraic approach goes beyond the former standard approach to classical dualities, and could help resolve the long standing problem of non-Abelian dualities. In particular, we present new dualities in any spatial dimension for the quantum Heisenberg model. Remarkably, the elimination of gauge constraints can be handled as a bond-algebraic duality (a gauge-reducing duality). In this way, we show that the 2 Higgs model is dual to the extended toric code model in any space dimension. We also show how, in specific models, the phenomenon of dimensional reduction can be accomplished by exact duality mappings. Finally, we discuss various applications including location of phase boundaries, spectral behaviour and, notably, show how bond-algebraic dualities help constrain and realize fermionization in an arbitrary number of spatial dimensions.