Classical Physics¶
Use quantum computers to solve linear systems of equations or simulate nonlinear physical systems.
Arithmetic operations¶
About the difficulty of performing simple arithmetic operations on a quantum computer. While factoring a number is much easier on a quantum computer, multiplying two real numbers is much more difficult! One algorithm involves the QFT.
Ruiz-Perez, Quantum arithmetic with the Quantum Fourier Transform, 2014, arXiv:1411.5949, doi:10.1007/s11128-017-1603-1.
Draper, Addition on a Quantum Computer, 2000, arXiv:quant-ph/0008033.
Vedral, Quantum Networks for Elementary Arithmetic Operations, 1995, arXiv:quant-ph/9511018, doi:10.1103/PhysRevA.54.147.
Linear Systems of Equations¶
- Harrow, Hassidim & Lloyd, Quantum algorithm for linear systems of equations, 2009, arXiv:0811.3171, doi:10.1103/PhysRevLett.103.150502
The HHL algorithm, see also the section HHL algorithm.
“We consider the case where one doesn’t need to know the solution x itself, but rather an approximation of the expectation value of some operator associated with x.”
See also the qiskit textbook chapter (HHL).
- Bravo-Prieto, Variational Quantum Linear Solver, 2019, arXiv:1909.05820
See also the qiskit textbook chapter (VQLS).
- Perelshtein, Solving Large-Scale Linear Systems of Equations by a Quantum Hybrid Algorithm, 2022, arXiv:2003.12770, doi:10.1002/andp.202200082.
Nonlinear Problems¶
- Lubasch et al., Variational quantum algorithms for nonlinear problems, 2020 [82] arXiv:1907.09032, doi:10.1103/physreva.101.010301
“Nonlinear problems including nonlinear partial differential equations can be efficiently solved by variational quantum computing.”
Liu, Efficient quantum algorithm for dissipative nonlinear differential equations, 2021, arXiv:2011.03185, doi:10.1073/pnas.2026805118. See also dissertation doi:10.13016/jxl7-ldtm.
Lloyd, Quantum algorithm for nonlinear differential equations, 2020, arXiv:2011.06571.
Kacewicz, Almost optimal solution of initial-value problems by randomized and quantum algorithms, 2006, doi:10.1016/j.jco.2006.03.001.
Special applications:
Givois, Kabel & Gauger, QFT-based Homogenization, 2022, arXiv:2207.12949.
Fluid Dynamics¶
- Ray et al., Solving the Navier-Stokes Equation, 2019 [111] - Los Alamos
- Gaitan, Finding flows of a Navier-Stokes fluid, 2020 [40]
- Gaitan, Finding Solutions of the Navier-Stokes Equations through Quantum Computing—Recent Progress, a Generalization, and Next Steps Forward, 2021, pdf:wiley, doi:10.1002/qute.202100055.
- Steijl, Quantum Algorithms for Nonlinear Equations in Fluid Mechanics, 2020, html:intechopen, doi:10.5772/intechopen.95023.
A key contributing factor to the limited progress in algorithms for non-linear problems is the inherent linearity of quantum mechanics. For quantum algorithms encoding information as amplitudes of a quantum state vector, nonlinear (product) terms cannot be obtained by multiplying these amplitudes by themselves, as a result of the no-cloning theorem that prohibits the copying of an arbitrary quantum state.
Quantum circuits for squaring floating-point numbers
Quantum circuits for multiplication of floating-point numbers
Griffin et al., Quantum algorithms for direct numerical simulation, 2020 [51]
Turbulence¶
Three papers supervised by Givi - Pittsburgh:
Miscellaneous¶
Brassard et al., Quantum algorithm to approximate the mean, 2011 [20] (Brassard is one of the authors of the BB84 protocol)
Summary¶
Using a quantum computer to solve problems of classical physics faces numerous and fundamental problems:
Performing some of the basic arithmetic operations as the multiplication is much more complex and inefficient on a quantum hardware.
Nonlinear product terms cannot be obtained easily, as a result of the no-cloning theorem.
Most classical physics problems involves spatial and temporal discretization that results in a huge amount of (qu)bits to represent the problem with the required accuracy, posing a serious obstacle to the use of a quantum computer in a foreseeable future.
Even using a QPU for subproblems that are “hard” to solve with classical algorithms may require a large number of error-corrected qubits. And these subproblems still have to be identified.
Complements: An Introduction » Quantum Computing » Applications