Qubit Systems from Colored Toric Geometry and Hypercube Graph Theory ^*

Abstract

We develop a new geometric approach to deal with qubit information systems using colored graph theory. More precisely, we present a one to one correspondence between graph theory, and qubit systems, which may be explored to attack qubit information problems using toric geometry considered as a powerful tool to understand modern physics including string theory. Concretely, we examine in some details the cases of one, two, and three qubits, and we find that they are associated with CP1, CP1 × CP1 and CP1 × CP1 × CP1 toric varieties respectively. Using a geometric procedure referred to as a colored toric geometry, we show that the qubit physics can be converted into a scenario handling toric data of such manifolds by help of hypercube graph theory. Operations on toric information can produce universal quantum gates.

Four-qubit systems and dyonic black Hole–Black branes in superstring theory

“…It has been shown that the first factor SO(4,20) SO( 4)×SO (20) represents the geometric deformations of the K3 surface in the presence of the antisymmetric B-field of the NS-NS sector and it is linked to the symmetry group G 1 = U (1) 24 . However, the second factor SO(1, 1) represents the dilaton scalar field which is associated with G 2 = U (1) [35,36].…”

“…The main obtained relations are between the entropy formulas for specific black hole solutions in supergravity theories and entanglement measures for certain multiqubit systems [17,18]. Alternative studies have been conducted using toric geometry and graph theory [24,25,26,27].…”

“…The black objects, embedded in superstring theory compactifications, can be connected to quantum information theory using the qubit analysis [12][13][14][15][16][17][18][19][20][21][22][23][24]. Precisely, a fascinating correspondence has been discovered between quantum information theory and superstring theory.…”

Compactifications of deformed conifolds, branes and the geometry of qubits

“…The calculations on CP 2 in [9] were aimed at evaluating the Aharonov-Anandan phase for a 3-state spin-1 system using the Kähler connection. More recently there have been interesting applications using ideas from toric geometry [23]. In this paper we shall further elaborate on applications of this formalism, including the study of the quantum entanglement of qubits.…”

Stringy dyonic solutions and clifford structures

“…Concretely, a remarkable correspondence between qubit systems and black holes in superstring theory has been established. The relevant findings concern a nice link between the N = 2 STU black hole with eight charges and threequbit system states [13,17,18,19]. The analysis has been developed to describe structures going beyond such extremal black hole solutions in terms of higher dimensional qubit systems.…”