Quantum HHL algorithm applied to electric circuit and line transmission wave
Keywords:Quantum algorithms; quantum linear solver; electric circuits
The HHL quantum algorithm  is a procedure that addresses the resolution of linear systems of equations (QLSP). Under certain conditions, the algorithm has a logarithmic order in the number of equations, better than the faster classical method. The algorithm manages to find a quantum state proportional to the solution vector, up to a normalization factor. The disadvantage is that to determine each of the coefficients of the solution vector, the algorithm’s output quantum state must be determined with additional statistical methods, thus losing its exponential advantage. There are certain types of problems in which this disadvantage can be circumvented, the statistical treatment is unavoidable, but for certain cases such as electrical circuits, in which the main interest is to find only one of the currents, (for example the load current), we only need to measure one of the qubits of the solution state. In this article we solve the linear system associated with the currents of an electrical circuit with sinusoidal voltage, using the HHL algorithm, simulated in a Scilab numerical environment. An optimized example of an electrical line transmission wave in a real computer on the IBMQ platform, is also solved
Harrow AW, Hassidim A, Lloyd S. Quantum Algorithm for Linear Systems of Equations. Phys Rev Lett. 2009 Oct;103:150502. Available from: https://link.aps.org/doi/10.1103/PhysRevLett.103.150502.
Zheng Y, Song C, Chen MC, Xia B, Liu W, Guo Q, et al. Solving Systems of Linear Equations with a Superconducting Quantum Processor. Phys Rev Lett. 2017 May;118:210504. Available from: https://link.aps.org/doi/10.1103/PhysRevLett.118.210504.
Chen CC, Shiau SY, Wu MF, Wu YR. Hybrid classical-quantum linear solver using Noisy Intermediate-Scale Quantum machines. Scientific Reports. 2019 Nov;9(1):16251. Available from: https://doi.org/10.1038/s41598-019-52275-6.
Nielsen MA, Chuang IL. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press; 2011.
Kitaev AY. Quantum measurements and the Abelian Stabilizer Problem. Electron Colloquium Comput Complex. 1996;3.
Langton C, Levin V. The Intuitive Guide to Fourier Analysis and Spectral Estimation. Mountcastle Company, 1st edition; 2017.
Bharti K, Cervera-Lierta A, Kyaw TH, Haug T, Alperin-Lea S, Anand A, et al. Noisy intermediate-scale quantum algorithms. Rev Mod Phys. 2022 Feb;94:015004. Available from: https://link.aps.org/doi/10.1103/RevModPhys.94.015004.
Chappell JM, Lohe MA, von Smekal L, Iqbal A, Abbott D. A Precise Error Bound for Quantum Phase Estimation. PLOS ONE. 2011 05;6(5):1-4. Available from: https://doi.org/10.1371/journal.pone.0019663.
Berry DW, Ahokas G, Cleve R, Sanders BC. Efficient Quantum Algorithms for Simulating Sparse Hamiltonians. Communications in Mathematical Physics. 2007 03;270(2):359-71.
Saad Y. Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics; 2003.
Alexander C, Sadiku M. Fundamentals of electric circuits. McGraw-Hill Higher Education; 2007.
IBMQ. Qiskit Tutorial; 2022. Available from: https://qiskit.org/textbook/ch-applications/hhl_tutorial.html#4.-Qiskit-Implementation
How to Cite
Copyright (c) 2023 Ariel Mordetzki, Efrain Buksman, André Fonseca de Oliveira
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Authors retain copyright and grant the Revista Mexicana de Física E right of first publication with the work simultaneously licensed under a CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.