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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 $\epsilon>0$, 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 Gaussian-like 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 one-dimensional 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.

Quantum simulations of one dimensional quantum systems