A fast algorithm for approximating the ground state energy on a quantum computer

Abstract

Estimating the ground state energy of a multiparticle system with relative error ε using deterministic classical algorithms has cost that grows exponentially with the number of particles. The problem depends on a number of state variables d that is proportional to the number of particles and suffers from the curse of dimensionality. Quantum computers can vanquish this curse. In particular, we study a ground state eigenvalue problem and exhibit a quantum algorithm that achieves relative error ε using a number of qubits C ′ d log ε −1 with total cost (number of queries plus other quantum operations) Cdε −(3+δ) , where δ > 0 is arbitrarily small and C and C ′ are independent of d and ε.

Quantum algorithm and circuit design solving the Poisson equation

“…Quantum computers take advantage of quantum mechanics to solve certain computational problems faster than classical computers. Indeed in some cases the quantum algorithm is exponentially faster than the best classical algorithm known [1][2][3][4][5][6][7][8][9][10][11][12].…”

Approximating ground and excited state energies on a quantum computer

“…This increases the matrix size. For example, in the estimation of the ground state energy (smallest eigenvalue) of the time-independent Schrödinger equation, equations (1) and (2), with V uniformly bounded by 1, the finite difference discretization on a grid will yield a matrix of size m d × m d , m = 2 ⌈− log 2 ε⌉ − 1 [29]. This means that the cost of the matrix eigenvalue algorithms mentioned above is bounded from below by a quantity proportional to 1 ε d , i.e.…”

“…To approximate the ground state energy of the problem specified by equations (1) and (2) in the worst case with (relative) error ε, assuming V and its first-order partial derivatives are uniformly bounded by 1, and the function evaluations of V are supplied by an oracle, a much stronger result holds. The complexity (i.e., the minimum cost of any classical algorithm, and not just the eigenvalue algorithms mentioned above) is bounded from below by a quantity proportional to ε −d as dε → 0 [27,29]. So unless d is moderate, the problem is very hard and suffers from the curse of dimensionality.…”

“…We obtain a matrix eigenvalue problem by discretizing (13) on a grid with mesh size h = 1 N +1 , N ∈ N, using finite differences [23,29]. This yields a matrix M h :…”

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