Energy Extrapolation in Quantum Optimization Algorithms

Quantum annealing and the variational quantum eigensolver are two promising quantum algorithms to find the ground state of a target Hamiltonian on near-term quantum devices. However, it is necessary to limit the evolution time or the circuit depth as much as possible since otherwise decoherence will degrade the computation. Even when this is done, there always exists a non-negligible estimation error in the ground state energy. Here we propose a scalable extrapolation approach to mitigate this error. With an appropriate regression, we can significantly improve the estimation accuracy for quantum annealing and variational quantum eigensolver for fixed quantum resources. The inference is achieved by extrapolating the annealing time to infinity or extrapolating the variance to zero. The only additional overhead is an increase in the number of measurements by a constant factor. We verified the validity of our method with the transverse-field Ising model. The method is robust to noise, and the techniques are applicable to other physics problems. Analytic derivations for the quadratic convergence feature of the residual energy in quantum annealing and the linear convergence feature of energy variance are given.