Charakteristisches Polynom und Minimalpolynom einer Matrix

We describe situations in which chains of N degenerate quantum energy levels, coupled by timedependent external fields, can be replaced by independent sets of chains, of length N , N − 1, . . . , 2, and sets of uncoupled single states. The transformation is a generalization of the two-level MorrisShore transformation [J.R. Morris and B.W. Shore, Phys. Rev. A 27, 906 (1983)]. We illustrate the procedure with examples of three-level chains.

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“…This is known as the Frobenius problem [12]. Because A and X commute, they have the same set of eigenvectors γ n ,…”

Section: The Frobenius Problemmentioning

confidence: 99%

“…Alternatively, we can express the matrix X as a power series in A. The Cayley-Hamilton theorem [12] implies that only N b − 1 of these powers, e.g. A 0 , A 1 , .…”

Section: The Frobenius Problemmentioning

confidence: 99%

Extension of the Morris-Shore transformation to multilevel ladders

2006

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“…In contrast to the gate realization with nonzero detuning, now the adiabatic transition is carried out between the initial rotating computation spin states and the quantum Fourier states. Finally, we point out that instantaneous eigenvectors of Hamiltonian (19) can be found exactly, which allows to combine the gate scheme with the shortcuts to adiabaticity technique (see the Supplement for more details).…”

Section: Controlling the Single Qubit Rabi Frequencymentioning

confidence: 99%

“…This time dependence ensures that ∆ k (0) ≫ J(0), Ω 1 (0), and respectively, ∆ k (t tmax ) ≪ J(t tmax ), Ω 1 (t tmax ). Finally, the adiabatic transition to the Fourier states using Hamiltonian (19) can be carried out by using ∆ 1,2 = 0,…”

Section: Numerical Examplesmentioning

confidence: 99%

Two-qubit quantum gate and entanglement protected by circulant symmetry

Ivanov

Vitanov

2020

Sci Rep

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We propose a method for the realization of the two-qubit quantum Fourier transform (QFT) using a Hamiltonian which possesses the circulant symmetry. Importantly, the eigenvectors of the circulant matrices are the Fourier modes and do not depend on the magnitude of the Hamiltonian elements as long as the circulant symmetry is preserved. The QFT implementation relies on the adiabatic transition from each of the spin product states to the respective quantum Fourier superposition states. We show that in ion traps one can obtain a Hamiltonian with the circulant symmetry by tuning the spin-spin interaction between the trapped ions. We present numerical results which demonstrate that very high fidelity can be obtained with realistic experimental resources. We also describe how the gate can be accelerated by using a "shortcut-to-adiabaticity" field.

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“…Using knowledge from matrix theory, 11 we obtain the characteristic equation from which we find Lyapunov exponents of our system. We consider a model of point-like cars moving on a circular road of length L. The total number of cars is N .…”

Section: -10mentioning

confidence: 99%

Stochastic approach to highway traffic

et al. 2004

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We analyze the characteristic features of jam formation on a circular one-lane road. We have applied an optimal velocity model including stochastic noise, where cars are treated as moving and interacting particles. The motion of N cars is described by the system of 2N stochastic differential equations with multiplicative white noise. Our system of cars behaves in qualitatively different ways depending on the values of control parameters c (dimensionless density), b (sensitivity parameter characterising the fastness of relaxation), and a (dimensionless noise intensity). In analogy to the gas-liquid phase transition in supersaturated vapour at low enough temperatures, we observe three different regimes of traffic flow at small enough values of b < b cr . There is the free flow regime (like gaseous phase) at small densities of cars, the coexistence of a jam and free flow (like liquid and gas) at intermediate densities, and homogeneous dense traffic (like liquid phase) at large densities. The transition from free flow to congested traffic occurs when the homogeneous solution becomes unstable and evolves into the limit cycle. The opposite process takes place at a different density, so that we have a hysteresis effect and phase transition of the first order. A phase transition of second order, characterised by critical exponents, takes place at a certain critical density c = c cr . Inclusion of the stochastic noise allows us to calculate the distribution of headway distances and time headways between the successive cars, as well as the distribution of jam (car cluster) sizes in a congested traffic.

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Charakteristisches Polynom und Minimalpolynom einer Matrix

Cited by 11 publications

References 0 publications

Extension of the Morris-Shore transformation to multilevel ladders

Extension of the Morris-Shore transformation to multilevel ladders

Two-qubit quantum gate and entanglement protected by circulant symmetry

Stochastic approach to highway traffic

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