Quantum simulations of one dimensional quantum systems
Rolando D. Somma · Quantum Information and Computation · 2016
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 Trotter-Suzuki formula that exploits the Lie algebra structure. For total evolution time t and precision e > 0, the complexity of our method is O(exp(γ√log(N/e))), where γ > 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/e. We also provide a method to prepare discrete versions of the eigenstates of the QHO of complexity polynomial in log(N)/e, where N is the dimension or number of points in the discretization. Next, we consider a system with a quartic potential. Our numerical simulations suggest a method for simulating the evolution of sublinear complexity O(N1/3+o(1)), for constant t and e. We also analyze complex one-dimensional systems and prove a complexity bound O(N), under fairly general assumptions. Our quantum algorithms may find applications in other problems. As an example, we discuss a generalization of the Fourier transform that is useful for signal analysis and can be formulated in terms of the evolution of the QHO.