Quantum simulations of one dimensional quantum systems
Abstract
We present quantum algorithms for the simulation of quantum systems in one spatial dimension, which result in quantum speedups that range from superpolynomial to polynomial. We first describe a method to simulate the evolution of the quantum harmonic oscillator (QHO) based on a refined analysis of the TrotterSuzuki formula that exploits the Lie algebra structure. For total evolution time $t$ and precision $\epsilon>0$, the complexity of our method is $ O(\exp(\gamma \sqrt{\log(N/\epsilon)}))$, where $\gamma>0$ is a constant and $N$ is the quantum number associated with an "energy cutoff" of the initial state. Remarkably, this complexity is subpolynomial in $N/\epsilon$. We also provide a method to prepare discrete versions of the eigenstates of the QHO of complexity polynomial in $\log(N)/\epsilon$, where $N$ is the dimension or number of points in the discretization. This method may be of independent interest as it provides a way to prepare, e.g., quantum states with Gaussianlike amplitudes. Next, we consider a system with a quartic potential. Our numerical simulations suggest a method for simulating the evolution of sublinear complexity $\tilde O(N^{1/3+o(1)})$, for constant $t$ and $\epsilon$. We also analyze complex onedimensional systems and prove a complexity bound $\tilde O(N)$, under fairly general assumptions. Our quantum algorithms may find applications in other problems. As an example, we discuss the fractional Fourier transform, a generalization of the Fourier transform that is useful for signal analysis and can be formulated in terms of the evolution of the QHO.
 Publication:

arXiv eprints
 Pub Date:
 March 2015
 arXiv:
 arXiv:1503.06319
 Bibcode:
 2015arXiv150306319S
 Keywords:

 Quantum Physics
 EPrint:
 25 pages, 9 figs