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Eigenvector approximation leading to exponential speedup of quantum eigenvalue calculation.

Peter Jaksch1, Anargyros Papageorgiou

  • 1Department of Computer Science, Columbia University, New York, New York 10027-6902, USA. petja@cs.columbia.edu

Physical Review Letters
|February 3, 2004
PubMed
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We developed an efficient quantum method to prepare initial states for eigenvalue approximation algorithms. This approach improves solving quantum problems like the Schrödinger equation on a fine grid.

Area of Science:

  • Quantum computing
  • Computational physics

Background:

  • Quantum algorithms require specific initial states.
  • Solving continuous Hermitian eigenproblems, like the Schrödinger equation, on discrete grids is computationally intensive.

Purpose of the Study:

  • To present an efficient method for preparing initial states for the Abrams and Lloyd eigenvalue approximation quantum algorithm.
  • To enable the application of this quantum algorithm to continuous Hermitian eigenproblems discretized on a fine grid.

Main Methods:

  • Starting with a classically obtained eigenvector from a coarse grid discretization.
  • Quantum mechanically constructing an approximation of the eigenvector on a fine grid.
  • Utilizing this approximation as the initial state for the eigenvalue estimation algorithm.

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Main Results:

  • Demonstrated an efficient quantum method for initial state preparation.
  • Established the relationship between the success probability of the eigenvalue estimation algorithm and the size of the coarse grid.

Conclusions:

  • The proposed method provides an efficient way to prepare initial states for quantum eigenvalue approximation algorithms.
  • This technique facilitates the application of quantum algorithms to complex physical problems such as solving the Schrödinger equation.