Exponential power series expansion for the quantum time evolution operator

1989 ◽  
Vol 90 (2) ◽  
pp. 904-911 ◽  
Author(s):  
Nancy Makri ◽  
William H. Miller
Keyword(s):  
Power Series ◽  
Time Evolution ◽  
Series Expansion ◽  
Evolution Operator ◽  
Power Series Expansion ◽  
Time Evolution Operator ◽  
Quantum Time ◽  
Exponential Power

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.

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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