scholarly journals Comment on: A corrected exponential power series expansion of the position matrix elements of the time evolution operator for a system in the presence of a vector potential

1991 ◽  
Vol 94 (6) ◽  
pp. 4682-4683 ◽  
Author(s):  
R. A. Harris ◽  
C. J. Grayce ◽  
N. Makri ◽  
W. H. Miller
Keyword(s):  
Power Series ◽  
Time Evolution ◽  
Series Expansion ◽  
Vector Potential ◽  
Evolution Operator ◽  
Power Series Expansion ◽  
Matrix Elements ◽  
Time Evolution Operator ◽  
Exponential Power

COLORED EXTENSIONS OF GLq(2) QUANTUM GROUP AND RELATED NONCOMMUTATIVE PLANES

1995 ◽  
Vol 10 (19) ◽  
pp. 2851-2864 ◽  
Author(s):  
B. BASU-MALLICK
Keyword(s):  
Power Series ◽  
Quantum Group ◽  
Series Expansion ◽  
Braid Group ◽  
Group Representation ◽  
Monodromy Matrix ◽  
Power Series Expansion ◽  
Quadratic Algebra ◽  
Matrix Elements ◽  
Infinite Dimensional

An infinite-dimensional quantum group, containing the standard GLq(2) and GLp,q(2) cases as different subalgebras, is constructed by using a colored braid group representation. It turns out that all algebraic relations occurring in this “colored” quantum group can be expressed in the Heisenberg-Weyl form, for a nontrivial choice of corresponding basis elements. Moreover a novel quadratic algebra, defined through Kac-Moody-like generators, is obtained by making some power series expansion of related monodromy matrix elements. The structure of invariant noncommutative planes associated with this “colored” quantum group has also been investigated.

scholarly journals Quantum Algorithm for Simulating Hamiltonian Dynamics with an Off-diagonal Series Expansion

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 426
Author(s):  
Amir Kalev ◽  
Itay Hen
Keyword(s):  
Series Expansion ◽  
Evolution Operator ◽  
State Of The Art ◽  
Hamiltonian Dynamics ◽  
Power Series Expansion ◽  
Quantum Algorithm ◽  
Time Evolution Operator ◽  
Diagonal Component ◽  
Current State ◽  
Diagonal Part

We propose an efficient quantum algorithm for simulating the dynamics of general Hamiltonian systems. Our technique is based on a power series expansion of the time-evolution operator in its off-diagonal terms. The expansion decouples the dynamics due to the diagonal component of the Hamiltonian from the dynamics generated by its off-diagonal part, which we encode using the linear combination of unitaries technique. Our method has an optimal dependence on the desired precision and, as we illustrate, generally requires considerably fewer resources than the current state-of-the-art. We provide an analysis of resource costs for several sample models.

Tunneling Matrix Elements

2021 ◽  
pp. 77-92
Author(s):  
C. Julian Chen
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Spherical Harmonics ◽  
Power Series Expansion ◽  
Matrix Elements ◽  
Modified Bessel Functions ◽  
Surface Integral ◽  
Analytic Expressions ◽  
Derivatives Of ◽  
Tunneling Theory

This chapter presents systematic methods to evaluate the tunneling matrix elements in the Bardeen tunneling theory. A key problem in applying the Bardeen tunneling theory to STM is the evaluation of the tunneling matrix elements, which is a surface integral of the wavefunctions of the tip and the sample, roughly in the middle of the tunneling gap. By expanding the tip wavefunction in terms of spherical harmonics and spherical modified Bessel functions, very simple analytic expressions for the tunneling matrix elements are derived: the tunneling matrix elements are proportional to the amplitudes or the corresponding x-, y-, or z-derivatives of the sample wavefunction at the center of the tip. Two proofs are presented. The first proof is based on the Green’s function of the Schrödinger’s equation in vacuum. The second proof is based on a power-series expansion of the tip wavefunctions.

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.

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