Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing
Abstract
We present two quantum algorithms based on evolution randomization, a simple variant of adiabatic quantum computing, to prepare a quantum state |x⟩ that is proportional to the solution of the system of linear equations $$A\overrightarrow{x} = \overrightarrow{b}$$. The time complexities of our algorithms are O(κ2log(κ)/ε) and O(κ log(κ)/ε), where κ is the condition number of A and ε is the precision. Both algorithms are constructed using families of Hamiltonians that are linear combinations of products of A, the projector onto the initial state |b⟩, and single-qubit Pauli operators. The algorithms are conceptually simple and easy to implement. They are not obtained from equivalences between the gate model and adiabatic quantum computing. They do not use phase estimation or variable-time amplitude amplification, and do not require large ancillary systems. We discuss a gate-based implementation via Hamiltonian simulation and prove that our second algorithm is almost optimal in terms of κ. Like previous methods, our techniques yield an exponential quantum speed-up under some assumptions. Our results emphasize the role of Hamiltonian-based models of quantum computing for the discovery of important algorithms.
- Authors:
-
[1];
[1];
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Univ. Innsbruck (Austria). Dept. of Theoretical Physics
- Publication Date:
- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Sponsoring Org.:
- USDOE Laboratory Directed Research and Development (LDRD) Program
- OSTI Identifier:
- 1781370
- Report Number(s):
- LA-UR-17-20510
Journal ID: ISSN 0031-9007; TRN: US2210296
- Grant/Contract Number:
- 89233218CNA000001
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Physical Review Letters
- Additional Journal Information:
- Journal Volume: 122; Journal Issue: 6; Journal ID: ISSN 0031-9007
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; Computer Science; Information Science; Quantum computing; linear algebra
Citation Formats
Somma, Rolando Diego, Subasi, Yigit, and Orsucci, Davide. Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing. United States: N. p., 2019.
Web. doi:10.1103/PhysRevLett.122.060504.
Somma, Rolando Diego, Subasi, Yigit, & Orsucci, Davide. Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing. United States. https://doi.org/10.1103/PhysRevLett.122.060504
Somma, Rolando Diego, Subasi, Yigit, and Orsucci, Davide. Thu .
"Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing". United States. https://doi.org/10.1103/PhysRevLett.122.060504. https://www.osti.gov/servlets/purl/1781370.
@article{osti_1781370,
title = {Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing},
author = {Somma, Rolando Diego and Subasi, Yigit and Orsucci, Davide},
abstractNote = {We present two quantum algorithms based on evolution randomization, a simple variant of adiabatic quantum computing, to prepare a quantum state |x⟩ that is proportional to the solution of the system of linear equations $A\overrightarrow{x} = \overrightarrow{b}$. The time complexities of our algorithms are O(κ2log(κ)/ε) and O(κ log(κ)/ε), where κ is the condition number of A and ε is the precision. Both algorithms are constructed using families of Hamiltonians that are linear combinations of products of A, the projector onto the initial state |b⟩, and single-qubit Pauli operators. The algorithms are conceptually simple and easy to implement. They are not obtained from equivalences between the gate model and adiabatic quantum computing. They do not use phase estimation or variable-time amplitude amplification, and do not require large ancillary systems. We discuss a gate-based implementation via Hamiltonian simulation and prove that our second algorithm is almost optimal in terms of κ. Like previous methods, our techniques yield an exponential quantum speed-up under some assumptions. Our results emphasize the role of Hamiltonian-based models of quantum computing for the discovery of important algorithms.},
doi = {10.1103/PhysRevLett.122.060504},
journal = {Physical Review Letters},
number = 6,
volume = 122,
place = {United States},
year = {Thu Feb 14 00:00:00 EST 2019},
month = {Thu Feb 14 00:00:00 EST 2019}
}
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