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

Anargyros Papageorgiou, Iasonas Petras

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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 logε-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, logε-1 and logδ-1.

Original languageEnglish (US)
Pages (from-to)469-494
Number of pages26
JournalJournal of Complexity
Volume30
Issue number4
DOIs
StatePublished - Aug 2014

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Statistics and Probability
  • Numerical Analysis
  • Mathematics(all)
  • Control and Optimization
  • Applied Mathematics

Keywords

  • Eigenvalue problem
  • Numerical approximation
  • Quantum algorithms

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