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. Thus, the number of qubits and the total cost are linear in the number of particles.
As an example, in REF , a quantum algorithm for approximating the ground state energy of a continuous-variable quantum system is provided.
A fast algorithm for approximating the ground state energy on a quantum computer
Mathematics of Computation, 82 (2013), 2293-2304
Physics Mathematics Computer Science