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Title: Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems

Abstract

We present a quantum eigenstate filtering algorithm based on quantum signal processing (QSP) and minimax polynomials. The algorithm allows us to efficiently prepare a target eigenstate of a given Hamiltonian, if we have access to an initial state with non-trivial overlap with the target eigenstate and have a reasonable lower bound for the spectral gap. We apply this algorithm to the quantum linear system problem (QLSP), and present two algorithms based on quantum adiabatic computing (AQC) and quantum Zeno effect respectively. Both algorithms prepare the final solution as a pure state, and achieves the near optimal $$\mathcal{\tilde{O}}$$(dκlog(1/ϵ)) query complexity for a d -sparse matrix, where κ is the condition number, and ϵ is the desired precision. Neither algorithm uses phase estimation or amplitude amplification.

Authors:
ORCiD logo [1]; ORCiD logo [2]
  1. Univ. of California, Berkeley, CA (United States); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
  2. Univ. of California, Berkeley, CA (United States)
Publication Date:
Research Org.:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC); US Air Force Office of Scientific Research (AFOSR)
OSTI Identifier:
1763676
Grant/Contract Number:  
AC02-05CH11231; SC0017867; FA9550-18-1-0095
Resource Type:
Accepted Manuscript
Journal Name:
Quantum
Additional Journal Information:
Journal Volume: 4; Journal ID: ISSN 2521-327X
Publisher:
Quantum Science Open Community
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

Citation Formats

Lin, Lin, and Tong, Yu. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. United States: N. p., 2020. Web. https://doi.org/10.22331/q-2020-11-11-361.
Lin, Lin, & Tong, Yu. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. United States. https://doi.org/10.22331/q-2020-11-11-361
Lin, Lin, and Tong, Yu. Wed . "Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems". United States. https://doi.org/10.22331/q-2020-11-11-361. https://www.osti.gov/servlets/purl/1763676.
@article{osti_1763676,
title = {Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems},
author = {Lin, Lin and Tong, Yu},
abstractNote = {We present a quantum eigenstate filtering algorithm based on quantum signal processing (QSP) and minimax polynomials. The algorithm allows us to efficiently prepare a target eigenstate of a given Hamiltonian, if we have access to an initial state with non-trivial overlap with the target eigenstate and have a reasonable lower bound for the spectral gap. We apply this algorithm to the quantum linear system problem (QLSP), and present two algorithms based on quantum adiabatic computing (AQC) and quantum Zeno effect respectively. Both algorithms prepare the final solution as a pure state, and achieves the near optimal $\mathcal{\tilde{O}}$(dκlog(1/ϵ)) query complexity for a d-sparse matrix, where κ is the condition number, and ϵ is the desired precision. Neither algorithm uses phase estimation or amplitude amplification.},
doi = {10.22331/q-2020-11-11-361},
journal = {Quantum},
number = ,
volume = 4,
place = {United States},
year = {2020},
month = {11}
}

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