# 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.103.150502The

**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:1909.05820See 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