The Experts below are selected from a list of 14403 Experts worldwide ranked by ideXlab platform
Michail Zak - One of the best experts on this subject based on the ideXlab platform.
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QCQC - Quantum Resonance for Solving NP-complete Problems by Simulations
Quantum Computing and Quantum Communications, 1999Co-Authors: Michail ZakAbstract:Quantum Analog computing is based upon similarity between mathematical formalism of a Quantum phenomenon and phenomena to be analyzed. In this paper, the mathematical formalism of Quantum resonance combined with tensor product decomposability of unitary evolutions is mapped onto a class of NP-complete combinatorial problems.
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Quantum Analog Computing
Chaos Solitons & Fractals, 1999Co-Authors: Michail ZakAbstract:Abstract Quantum Analog computing is based upon similarity between mathematical formalism of Quantum mechanics and phenomena to be computed. It exploits a dynamical convergence of several competing phenomena to an attractor which can represent an extremum of a function, an image, a solution to a system of ODE, or a stochastic process. In this paper, a Quantum version of recurrent neural nets (QRN) as an Analog computing device is discussed. This concept is introduced by incorporating classical feedback loops into conventional Quantum networks. It is shown that the dynamical evolution of such networks, which interleave Quantum evolution with measurement and reset operations, exhibit novel dynamical properties. Moreover, decoherence in Quantum recurrent networks is less problematic than in conventional Quantum network architectures due to the modest phase coherence times needed for network operation. Application of QRN to simulation of chaos, turbulence, NP-problems, as well as data compression demonstrate computational speedup and exponential increase of information capacity.
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Quantum resonance for solving NP-complete problems by simulations
Lecture Notes in Computer Science, 1999Co-Authors: Michail ZakAbstract:Quantum Analog computing is based upon similarity between mathematical formalism of a Quantum phenomenon and phenomena to be analyzed. In this paper, the mathematical formalism of Quantum resonance combined with tensor product decomposability of unitary evolutions is mapped onto a class of NP-complete combinatorial problems.
Lawrence C. Evans - One of the best experts on this subject based on the ideXlab platform.
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Towards a Quantum Analog of Weak KAM Theory
Communications in Mathematical Physics, 2004Co-Authors: Lawrence C. EvansAbstract:We discuss a Quantum Analogue of Mather’s minimization principle for Lagrangian dynamics, and provide some formal calculations suggesting the corresponding Euler–Lagrange equation. We then rigorously construct from the dual eigenfunctions of a certain non-selfadjoint operator a candidate ψ for a minimizer, and recover aspects of ‘‘weak KAM’’ theory in the limit as h→0. Regarding our state ψ as a quasimode, we furthermore derive some error estimates, although it remains an open problem to improve these bounds.
K Nickoladze - One of the best experts on this subject based on the ideXlab platform.
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Quantum mechanical research on nonlinear resonance and the problem of Quantum chaos
Physical Review E, 2004Co-Authors: A Ugulava, L Chotorlishvili, K NickoladzeAbstract:: The Quantum-mechanical investigation of nonlinear resonance in terms of approximation to moderate nonlinearity is reduced to the investigation of eigenfunctions and eigenvalues of the Mathieu-Schrodinger equation. The eigenstates of the Mathieu-Schrodinger equation are nondegenerate in a certain area of pumping amplitude values in the neighborhood of the classical separatrix. Outside this area, the system finds itself in a degenerate state for both small and large pumping amplitude values. Degenerate energy terms arise as a result of merging and branching of pairs of nondegenerate energy terms. Equations are obtained for finding the merging points of energy terms. These equations are solved by numerical methods. The main objective of this paper is to establish a Quantum Analog of the classical stochastic layer formed in the separatrix area. With this end in view, we consider a nonstationary Quantum-mechanical problem of perturbation of the state of the Mathieu-Schrodinger equation. It is shown that in passing through the branching point the system may pass from the pure state to the mixed one. At multiple passages through branching points there develops the irreversible process of "creeping" of the system to Quantum states. In that case, the observed population of a certain number of levels can be considered, in our opinion, to be a Quantum Analog of the stochastic layer. The number of populated levels is defined by a perturbation amplitude.
Wu-ming Liu - One of the best experts on this subject based on the ideXlab platform.
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Quantum Analog of Kolmogorov-Arnold-Moser theorem and bulk-edge correspondence in Fock space in Type-I and Type-II hybrid Sachdev-Ye-Kitaev models
2018Co-Authors: Fadi Sun, Yu Yi-xiang, Wu-ming LiuAbstract:We investigate possible Quantum Analog of Kolmogorov-Arnold-Moser (KAM) theorem in two types of hybrid SYK models which contain both $ q=4 $ SYK with interaction $ J $ and $ q=2 $ SYK with an interaction $ K $ in type-I or $(q=2)^2$ SYK with an interaction $ \sqrt{K} $ in type-II. These models include hybrid Majorana fermion, complex fermion and bosonic SYK. For the Majorana fermion case, we discuss both $ N $ even and $ N $ odd case. We introduce a new universal ratio which is the ratio of the next nearest neighbour (NNN) energy level spacing to characterize the KAM theorem. We make exact symmetry analysis on the possible symmetry class of both types of hybrid SYK in the 10 fold way by random matrix theory (RMT) and also work out the degeneracy of each energy levels.We perform exact diagonalization to evaluate both the known NN ratio and the new NNN ratio, then use both ratios to study chaotic to non-chaotic transitions (CNCT) in both types of hybrid SYK models. After defining the Quantum Analog of the KAM theorem in the context of RMT, we show that the KAM theorem holds when $ J/K \sim \sqrt{N} e^{-N} $ and shrinks exponentially fast in the thermodynamic limit. In the GOE or GSE case in type-I, we find a lower bound of the dual form of the KAM theorem which states the stability of Quantum chaos. In Analogy to the bulk-edge correspondence in the real space of a gapped topological state, we propose a new concept: bulk-edge correspondence in Fock space (BECFS). It establishes some intrinsic connections between the two complementary approaches to Quantum chaos: the RMT and the Lyapunov exponent by the $ 1/N $ expansion in the large $ N $ limit at a suitable temperature range. Some future perspectives, especially the failure of the Zamoloddchikov's c-theorem in 1d CFT are outlined.
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A new universal ratio in Random Matrix Theory and Quantum Analog of Kolmogorov-Arnold-Moser theorem in Type-I and Type-II hybrid Sachdev-Ye-Kitaev models.
arXiv: Strongly Correlated Electrons, 2018Co-Authors: Yu Yi-xiang, Fadi Sun, Wu-ming LiuAbstract:We investigate possible Quantum Analog of Kolmogorov-Arnold-Moser (KAM) theorem in two types of hybrid SYK models which contain both $ q=4 $ SYK with interaction $ J $ and $ q=2 $ SYK with an interaction $ K $ in type-I or $(q=2)^2$ SYK with an interaction $ \sqrt{K} $ in type-II. These models include hybrid Majorana fermion, complex fermion, and bosonic SYK. For the Majorana fermion case, we discuss both $ N $ even and $ N $ odd case. We introduce a new universal ratio which is the ratio of the next nearest neighbour (NNN) energy level spacing to characterize the KAM theorem. We make exact symmetry analysis on the possible symmetry class of both types of hybrid SYK in the 10-fold way by random matrix theory (RMT) and also work out the degeneracy of each energy level. We perform exact diagonalization to evaluate both the known NN ratio and the new NNN ratio, then use both ratios to study Chaotic to Integrable transitions (CIT) in both types of hybrid SYK models. After defining the Quantum Analog of the KAM theorem in the context of RMT, we show that the KAM theorem holds when $ J/K \sim \sqrt{N} e^{-N} $ and shrinks exponentially fast in the thermodynamic limit. In the GOE or GSE case in type-I, we find a lower bound of the dual form of the KAM theorem which states the stability of Quantum chaos. While the stability of Quantum chaos in all the other cases are much more robust than the KAM theorem in the integrable side. We explore some intrinsic connections between the two complementary approaches to Quantum chaos: the RMT and the Lyapunov exponent by the $ 1/N $ expansion in the large $ N $ limit at a suitable temperature range. We stress the crucial difference between the Quantum phase transition (QPT) characterized by renormalization groups at $ N=\infty $, $ 1/N $ expansions at a finite $ N $, and the CIT characterized by the RMT at a finite $ N $.
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Random matrices and Quantum Analog of Kolmogorov-Arnold-Moser theorem in hybrid Sachdev-Ye-Kitaev models
arXiv: Strongly Correlated Electrons, 2018Co-Authors: Yu Yi-xiang, Fadi Sun, Wu-ming LiuAbstract:Here we investigate possible Quantum Analog of Kolmogorov-Arnold-Moser (KAM) theorem in two types of hybrid SYK models which contain both $ q=4 $ SYK with interaction $ J $ and $ q=2 $ SYK with an interaction $ K $ in Type I or $ (q=2)^2 $ SYK with an interaction $ \sqrt{K} $ in Type II . These models include hybrid Majorana fermion, complex fermion and bosonic SYK. We first introduce a new universal ratio which is the ratio of the next nearest neighbour (NNN) energy level spacing to characterize the random matrix behaviours. We make exact symmetry analysis on the possible symmetry class of both types of hybrid SYK in the 10 fold way and also work out the degeneracy of each energy levels. We perform exact diagonalization to evaluate both the known NN ratio and the new NNN ratio. In Type I, as $ K/J $ changes, there is always a chaotic to non-chaotic transition (CNCT) from the GUE to Poisson in all the hybrid fermionic SYK models, but not the hybrid bosonic SYK model. In Type II, there are always CNCT from the corresponding GOE, GUE or GSE dictated by the symmetry of the $ q=4 $ SYK to the Poisson dictated by $ ( q=2 )^2 $ SYK. When the double degeneracy at the $ q=4 $ ( or $ (q=2)^2 $ ) side is broken by the $ q=2 $ ( or $ q=4 $ ) perturbation in Type I ( or Type II), the new NNN ratio can be most effectively to quantify the stability of Quantum chaos ( or the KAM ). We compare the stability of Quantum chaos and KAM theorem near the integrability in all these hybrid SYK models. We also discuss some possible connections between CNCT characterized by the random matrix theory and the Quantum phase transitions (QPT) characterized by renormalization groups. Quantum chaos in both types of hybrid SYK models are also contrasted with that in the $ U(1)/Z_2 $ Dicke model in Quantum optics.
Janis Nötzel - One of the best experts on this subject based on the ideXlab platform.
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Entanglement Transmission and Generation under Channel Uncertainty: Universal Quantum Channel Coding
Communications in Mathematical Physics, 2009Co-Authors: Igor Bjelaković, Holger Boche, Janis NötzelAbstract:We determine the optimal rates of universal Quantum codes for entanglement transmission and generation under channel uncertainty. In the simplest scenario the sender and receiver are provided merely with the information that the channel they use belongs to a given set of channels, so that they are forced to use Quantum codes that are reliable for the whole set of channels. This is precisely the Quantum Analog of the compound channel coding problem. We determine the entanglement transmission and entanglement-generating capacities of compound Quantum channels and show that they are equal. Moreover, we investigate two variants of that basic scenario, namely the cases of informed decoder or informed encoder, and derive corresponding capacity results.
