Statistical Properties of the Time Evolution Operator for Two Coupled Oscillators

1993 ◽  
Vol 40 (7) ◽  
pp. 1369-1385 ◽  
Author(s):  
M. Sebawe Abdalla
Keyword(s):  
Time Evolution ◽  
Coupled Oscillators ◽  
Evolution Operator ◽  
Statistical Properties ◽  
Time Evolution Operator

TIME EVOLUTION OF A CENTRAL OSCILLATOR IN HEAT BATH DERIVED BY VIRTUE OF THE IWOP TECHNIQUE

2006 ◽  
Vol 20 (06) ◽  
pp. 295-304 ◽  
Author(s):  
HONG-YI FAN ◽  
XUE-FEN XU
Keyword(s):  
Partition Function ◽  
Energy Conservation ◽  
Time Evolution ◽  
Coupled Oscillators ◽  
Coherent States ◽  
Evolution Operator ◽  
Heat Bath ◽  
Iwop Technique ◽  
Time Evolution Operator ◽  
Central Oscillator

By virtue of the technique of integration within an ordered product of operators we derive the normally ordered expansion of time evolution operator for the case of a central oscillator immersed in a heat bath composed of a large number of oscillators. The time evolution of the system and heat bath into coherent states is discussed based on energy conservation. As a by-product, the partition function of two coupled oscillators is also calculated in this way.

scholarly journals Complexity growth in integrable and chaotic models

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Vijay Balasubramanian ◽  
Matthew DeCross ◽  
Arjun Kar ◽  
Yue Li ◽  
Onkar Parrikar
Keyword(s):  
Time Evolution ◽  
Evolution Operator ◽  
Linear Growth ◽  
Conjugate Points ◽  
Majorana Fermions ◽  
Time Evolution Operator ◽  
Free Theory ◽  
Group Manifold ◽  
Geodesic Length ◽  
Evolution Trajectory

Abstract We use the SYK family of models with N Majorana fermions to study the complexity of time evolution, formulated as the shortest geodesic length on the unitary group manifold between the identity and the time evolution operator, in free, integrable, and chaotic systems. Initially, the shortest geodesic follows the time evolution trajectory, and hence complexity grows linearly in time. We study how this linear growth is eventually truncated by the appearance and accumulation of conjugate points, which signal the presence of shorter geodesics intersecting the time evolution trajectory. By explicitly locating such “shortcuts” through analytical and numerical methods, we demonstrate that: (a) in the free theory, time evolution encounters conjugate points at a polynomial time; consequently complexity growth truncates at O($$ \sqrt{N} $$ N ), and we find an explicit operator which “fast-forwards” the free N-fermion time evolution with this complexity, (b) in a class of interacting integrable theories, the complexity is upper bounded by O(poly(N)), and (c) in chaotic theories, we argue that conjugate points do not occur until exponential times O(eN), after which it becomes possible to find infinitesimally nearby geodesics which approximate the time evolution operator. Finally, we explore the notion of eigenstate complexity in free, integrable, and chaotic models.

scholarly journals Preparation of Approximate Eigenvector by Unitary Operations on Eigenstate in Abrams-Lloyd Quantum Algorithm

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Latha S. Warrier
Keyword(s):  
Time Evolution ◽  
Programming Language ◽  
Spin Chain ◽  
Evolution Operator ◽  
Unitary Operator ◽  
Language Model ◽  
Quantum Algorithm ◽  
Basis Vector ◽  
Unitary Matrix ◽  
Time Evolution Operator

The Abrams-Lloyd quantum algorithm computes an eigenvalue and the corresponding eigenstate of a unitary matrix from an approximate eigenvector Va. The eigenstate is a basis vector in the orthonormal eigenspace. Finding another eigenvalue, using a random approximate eigenvector, may require many trials as the trial may repeatedly result in the eigenvalue measured earlier. We present a method involving orthogonalization of the eigenstate obtained in a trial. It is used as the Va for the next trial. Because of the orthogonal construction, Abrams-Lloyd algorithm will not repeat the eigenvalue measured earlier. Thus, all the eigenvalues are obtained in sequence without repetitions. An operator that anticommutes with a unitary operator orthogonalizes the eigenvectors of the unitary. We implemented the method on the programming language model of quantum computation and tested it on a unitary matrix representing the time evolution operator of a small spin chain. All the eigenvalues of the operator were obtained sequentially. Another use of the first eigenvector from Abrams-Lloyd algorithm is preparing a state that is the uniform superposition of all the eigenvectors. This is possible by nonorthogonalizing the first eigenvector in all dimensions and then applying the Abrams-Lloyd algorithm steps stopping short of the last measurement.

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