Estimating the ground state energy of the Schrödinger equation for convex potentials

Abstract

In 2011, the fundamental gap conjecture for Schrödinger operators was proven. This can be used to estimate the ground state energy of the time-independent Schrödinger equation with a convex potential and relative error ε. Classical deterministic algorithms solving this problem have cost exponential in the number of its degrees of freedom d. We show a quantum algorithm, that is based on a perturbation method, for estimating the ground state energy with relative error ε. The cost of the algorithm is polynomial in d and ε −1 , while the number of qubits is polynomial in d and log ε −1 . In addition, we present an algorithm for preparing a quantum state that overlaps within 1 − δ, δ ∈ (0, 1), with the ground state eigenvector of the discretized Hamiltonian. This algorithm also approximates the ground state with relative error ε. The cost of the algorithm is polynomial in d, ε −1 and δ −1 , while the number of qubits is polynomial in d, log ε −1 and log δ −1 .

Approximating ground and excited state energies on a quantum computer

“…In [30] we explain why this problem is different from the QMA-complete problems of discrete complexity theory. In [28] we relaxed an important assumption of [29] and extended our results to the ground state energy approximation for the time-independent Schrödinger equation with a convex potential.…”

“…It is worth noting that in certain cases quantum algorithms may be able to break the curse of dimensionality by computing ε-accurate eigenvalue estimates with cost polynomial in ε −1 and d. This was shown in [29,28] where we saw that for smooth nonnegative potentials that are uniformly bounded by a relatively small constant, or are convex, there exists a quantum algorithm approximating the ground state energy with relative error O(ε) and cost polynomial in d and ε −1 .…”

“…We illustrate our results by considering the time-independent Schrödinger equation under weaker assumptions than those of [29,28], as we explain below. For this problem, the cardinality of the set S of trial eigenvectors turns out to be polynomial in the number of degrees of freedom d. We derive cost and probability estimates for approximating a constant number of low order eigenvalues.…”

“…It is important to investigate conditions for the potential V beyond those of [27,28,29,30]. In this paper we pursue this direction.…”

“…In this paper we present an entirely new approach for approximating a constant number of low order eigenvalues. At the same time we relax some of the assumptions of our previous work for approximating the ground state energy [29,28]. We will discuss these papers in Section 2.1.…”