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Related Experiment Video

Updated: Jun 18, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Quantum algorithm for linear systems of equations.

Aram W Harrow1, Avinatan Hassidim, Seth Lloyd

  • 1Department of Mathematics, University of Bristol, Bristol, BS8 1TW, United Kingdom.

Physical Review Letters
|November 13, 2009
PubMed
Summary
This summary is machine-generated.

This study introduces a quantum algorithm for estimating expectation values from linear systems. It significantly outperforms classical algorithms, especially for well-conditioned matrices.

Related Experiment Videos

Last Updated: Jun 18, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Area of Science:

  • Quantum Computing
  • Numerical Analysis
  • Computational Complexity

Background:

  • Solving linear systems (Ax = b) is fundamental in science and engineering.
  • Estimating expectation values (x†Mx) is crucial for many complex problems.
  • Classical algorithms for sparse matrices have limitations in speed, scaling with matrix size (N) and condition number (κ).

Purpose of the Study:

  • To develop a quantum algorithm for efficiently estimating expectation values related to linear systems.
  • To demonstrate a potential speedup over the best-known classical algorithms.

Main Methods:

  • Development of a quantum algorithm tailored for expectation value estimation.
  • Analysis of the algorithm's runtime complexity in terms of matrix size (N) and condition number (κ).

Main Results:

  • The proposed quantum algorithm achieves a runtime polynomial in log(N) and κ.
  • For well-conditioned matrices (κ = poly log N), the quantum approach offers an exponential speedup over classical methods, assuming standard complexity-theoretic assumptions.

Conclusions:

  • Quantum computation offers a significant advantage for estimating expectation values in linear systems.
  • This work highlights the potential of quantum algorithms to solve specific computational problems more efficiently than classical counterparts.