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The Living Review (https://qppqlivingreview.github.io/review) also includes a list of standard (static) reviews within. The references have been grouped into several categories to simplify the search process for the user (see Section 2). Some papers may be listed under multiple categories. The inclusion of a paper in the review does not indicate endorsement or validation of its content, as this is to be determined by the community and through peer review. The classi cation system may have limitations and we welcome feedback from the community on any papers that should be included, papers that have been misclassi ed or errors in citations or journal information.
Quantum Walk Hydrodynamics . In: Scienti c Reports 9.1 (Feb. 2019), p. 2989. issn: 2045-2322. doi: 10.1038/s41598-019-40059-x. Hsin-Yuan Huang, Kishor Bharti, and Patrick Rebentrost. Near-Term Quan- tum Algorithms for Linear Systems of Equations with Regression Loss Func- tions .
In: New Journal of Physics 23.11 (Nov. 2021), p. 113021. issn: 1367- 2630. doi: 10.1088/1367-2630/ac325f. [HBR21] [HHL09] [JL22] [JLL22] [JLY22a] [JLY22b] [JLY22c] [JLY22d] [Jos+22] [Jos20] [Jou22] [Kou+22] [KPE21]
Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum Algorithm for Linear Systems of Equations . In:
Physical Review Letters 103.15 (Oct. 2009), p. 150502. issn: 0031-9007, 1079-7114. doi: 10 . 1103 / PhysRevLett . 103.150502.
Shi Jin and Nana Liu. Quantum Algorithms for Computing Observables of Non- linear Partial Di erential Equations. Feb. 2022. doi: 10.48550/arXiv.2202. 07834. arXiv: 2202.07834 [physics, physics:quant-ph].
Shi Jin, Xiantao Li, and Nana Liu. Quantum Simulation in the Semi-Classical Regime . In: Quantum 6 (June 2022), p. 739. doi: 10.22331/q-2022-06-17- 739.
Shi Jin, Nana Liu, and Yue Yu. Quantum Simulation of Partial Di erential Equations via Schrodingerisation. Dec. 2022. doi: 10 . 48550 / arXiv . 2212 . 13969. arXiv: 2212.13969 [quant-ph].
Shi Jin, Nana Liu, and Yue Yu. Quantum Simulation of Partial Di erential Equations via Schrodingerisation: Technical Details.
Dec. 2022. doi: 10.48550/ arXiv.2212.14703. arXiv: 2212.14703 [quant-ph].
Shi Jin, Nana Liu, and Yue Yu. Time Complexity Analysis of Quantum Algo- rithms via Linear Representations for Nonlinear Ordinary and Partial Di eren- tial Equations .
The Living Review is hosted as a Github project. It can be consulted either (see Fig. 1) as a Markdown/HTML webpage, where the URL links for each reference are available; or as a LATEX PDF le, which is also downloadable directly from the webpage. The GitHub repository naturally also includes the BibTeX (.bib) le, which authors can freely use when writing new articles.
In: (2022). doi: 10.48550/ARXIV.2209.08478. Shi Jin, Nana Liu, and Yue Yu.
Time Complexity Analysis of Quantum Di er- ence Methods for Linear High Dimensional and Multiscale Partial Di erential Equations . In: Journal of Computational Physics 471 (Dec. 2022), p. 111641. issn: 00219991. doi: 10.1016/j.jcp.2022.111641.
I. Joseph et al. Quantum Computing for Fusion Energy Science Applications. Dec. 2022. doi: 10.48550/arXiv.2212.05054. arXiv: 2212.05054 [math-ph, physics:physics, physics:quant-ph].
Ilon Joseph. Koopman von Neumann Approach to Quantum Simulation of Nonlinear Classical Dynamics . In: Physical Review Research 2.4 (Oct. 2020), p. 043102. doi: 10.1103/PhysRevResearch.2.043102.
Lo c Joubert-Doriol. A Variational Approach for Linearly Dependent Mov- ing Bases in Quantum Dynamics: Application to Gaussian Functions . In: arXiv:2205.02358 [physics, physics:quant-ph] (May 2022). arXiv: 2205.02358 [physics, physics:quant-ph]. Efstratios Koukoutsis et al.
Dyson Maps and Unitary Evolution for Maxwell Equations in Tensor Dielectric Media. Sept. 2022. doi: 10 . 48550 / arXiv . 2209.08523. arXiv: 2209.08523 [physics, physics:quant-ph].
Oleksandr Kyriienko, Annie E. Paine, and Vincent E. Elfving. Solving Nonlin- ear Di erential Equations with Di erentiable Quantum Circuits .
In: Physical Review A 103.5 (May 2021), p. 052416. doi: 10.1103/PhysRevA.103.052416.
[Kro22] Hari Krovi. Improved Quantum Algorithms for Linear and Nonlinear Di eren- tial Equations . In: arXiv:2202.01054 [physics, physics:quant-ph] (Feb. 2022). arXiv: 2202.01054 [physics, physics:quant-ph]. [Kub+20] Kenji Kubo et al. Variational Quantum Simulations of Stochastic Di erential Equations . In: arXiv:2012.04429 [quant-ph] (Dec. 2020). arXiv: 2012.04429 [quant-ph]. [Lap22] Leigh Lapworth.
A Hybrid Quantum-Classical CFD Methodology with Bench- mark HHL Solutions. June 2022. doi: 10.48550/arXiv.2206.00419. arXiv: 2206.00419 [quant-ph]. [LEK22] Fong Yew Leong, Wei-Bin Ewe, and Dax Enshan Koh.
This pre-print will serve as a static reference to the review. Please check back before you upload your manuscript to arXiv to ensure that you include the latest work on the subject.
Variational Quantum Evolution Equation Solver . In: Scienti c Reports 12.1 (Dec. 2022), p. 10817. issn: 2045-2322. doi: 10 . 1038 / s41598 - 022 - 14906 - 3. arXiv: 2204 . 02912 [physics, physics:quant-ph].
[Lin+22] Yen Ting Lin et al. Koopman von Neumann Mechanics and the Koopman Representation: A Perspective on Solving Nonlinear Dynamical Systems with Quantum Computers. Feb. 2022. doi: 10.48550/arXiv.2202.02188. arXiv: 2202.02188 [quant-ph]. [Liu+21a] Hai-Ling Liu et al. Variational Quantum Algorithm for the Poisson Equation .
In: Physical Review A 104.2 (Aug. 2021), p. 022418. issn: 2469-9926, 2469-9934. doi: 10.1103/PhysRevA.104.022418. Jin-Peng Liu et al. E cient Quantum Algorithm for Dissipative Nonlinear Di erential Equations .
In: Proceedings of the National Academy of Sciences 118.35 (Aug. 2021), e2026805118. doi: 10.1073/pnas.2026805118.
Y. Y. Liu et al. Application of a Variational Hybrid Quantum-Classical Al- gorithm to Heat Conduction Equation and Analysis of Time Complexity .
In: Physics of Fluids 34.11 (Nov. 2022), p. 117121. issn: 1070-6631. doi: 10.1063/ 5.0121778. [Liu+21b] [Liu+22] [Lju22]
[Llo+20] Budinski Ljubomir. Quantum Algorithm for the Navier Stokes Equations by Using the Streamfunction-Vorticity Formulation and the Lattice Boltzmann Method . In: International Journal of Quantum Information 20.02 (Mar. 2022), p. 2150039. issn: 0219-7499. doi: 10.1142/S0219749921500398. Seth Lloyd et al. Quantum Algorithm for Nonlinear Di erential Equations. Dec. 2020. doi: 10 . 48550 / arXiv . 2011 . 06571. arXiv: 2011 . 06571 [nlin, physics:quant-ph]. [LMS22] [LO08] Noah Linden, Ashley Montanaro, and Changpeng Shao.
Quantum vs. Classical Algorithms for Solving the Heat Equation . In: Communications in Mathemati- cal Physics (Aug. 2022). issn: 1432-0916. doi: 10.1007/s00220-022-04442-6.
Sarah K. Leyton and Tobias J. Osborne. A Quantum Algorithm to Solve Non- linear Di erential Equations. Dec. 2008. doi: 10.48550/arXiv.0812.4423. arXiv: 0812.4423 [quant-ph]. [Lub+20] Michael Lubasch et al. Variational Quantum Algorithms for Nonlinear Prob- lems .
In: Physical Review A 101.1 (Jan. 2020), p. 010301. issn: 2469-9926, 2469-9934. doi: 10.1103/PhysRevA.101.010301.
This paper is organized as follows: we brie y introduce the structure of the Living Review (Section 2); then we describe how one can contribute (Section 3); nally we showcase a rst version of the living review (Section 4). The conclusions are in Section 5.
Koichi Miyamoto and Kenji Kubo. Pricing Multi-Asset Derivatives by Finite- Di erence Method on a Quantum Computer .
In: IEEE Transactions on Quan- tum Engineering 3 (2022), pp. 1 25. issn: 2689-1808. doi: 10.1109/TQE.2021. 3128643. [MK22] [MP16] Ashley Montanaro and Sam Pallister.
Quantum Algorithms and the Finite Element Method . In: Physical Review A 93.3 (Mar. 2016), p. 032324. doi: 10.1103/PhysRevA.93.032324. [MS21] [NDS22]
Philip Mocz and Aaron Szasz. Toward Cosmological Simulations of Dark Mat- ter on Quantum Computers . In: The Astrophysical Journal 910.1 (Mar. 2021), p. 29. issn: 0004-637X. doi: 10.3847/1538-4357/abe6ac. I. Novikau, I. Y. Dodin, and E. A. Startsev. Simulation of Linear Non-Hermitian Boundary-Value Problems with Quantum Singular Value Transformation. Dec. 2022. doi: 10 . 48550 / arXiv . 2212 . 09113. arXiv: 2212 . 09113 [physics, physics:quant-ph]. [NSD22] [Oga+15] I. Novikau, E. A. Startsev, and I. Y. Dodin.
Quantum Signal Processing for Simulating Cold Plasma Waves . In: Physical Review A 105.6 (June 2022), p. 062444. doi: 10.1103/PhysRevA.105.062444. Armen Oganesov et al. Unitary Quantum Lattice Gas Algorithm Generated from the Dirac Collision Operator for 1D Soliton Soliton Collisions . In: Radia- tion E ects and Defects in Solids 170.1 (Jan. 2015), pp. 55 64. issn: 1042-0150. doi: 10.1080/10420150.2014.988625. [Oga+16a] Armen Oganesov et al.
Benchmarking the Dirac-generated Unitary Lattice Qubit Collision-Stream Algorithm for 1D Vector Manakov Soliton Collisions . In: Computers & Mathematics with Applications 72.2 (July 2016), pp. 386 393. issn: 08981221. doi: 10.1016/j.camwa.2015.06.001. [Oga+16b] Armen Oganesov et al. Imaginary Time Integration Method Using a Quantum Lattice Gas Approach .
In: Radiation E ects and Defects in Solids 171.1-2 (Feb. 2016), pp. 96 102. issn: 1042-0150. doi: 10.1080/10420150.2015.1137916.
Armen Oganesov et al. E ect of Fourier Transform on the Streaming in Quan- tum Lattice Gas Algorithms . In: Radiation E ects and Defects in Solids 173.3- 4 (Apr. 2018), pp. 169 174. issn: 1042-0150. doi: 10.1080/10420150.2018. 1462364.
[Oga+18] [OMa+22] Daniel O Malley et al. A Near-Term Quantum Algorithm for Solving Linear Systems of Equations Based on the Woodbury Identity . In: arXiv:2205.00645 [quant-ph] (May 2022). arXiv: 2205.00645 [quant-ph].
[Oz+21] Furkan Oz et al. Solving Burgers Equation with Quantum Computing . In: Quantum Information Processing 21.1 (Dec. 2021), p. 30. issn: 1573-1332. doi: 10.1007/s11128-021-03391-8.
2 Categories One of the main goals of this living review is to provide an easily searchable collection of references. These are presented in an itemized format, according to the type of problem the reference covers. In general, brief descriptions for each category of problem are provided. A single reference can be in more than one category, if applicable.
[Pat+22] Raj Patel et al. Quantum-Inspired Tensor Neural Networks for Partial Dif- ferential Equations. Aug. 2022. doi: 10 . 48550 / arXiv . 2208 . 02235. arXiv: 2208.02235 [cond-mat, physics:physics, physics:quant-ph]. [Ram+21] Abhay K. Ram et al.
Re ection and Transmission of Electromagnetic Pulses at a Planar Dielectric Interface: Theory and Quantum Lattice Simulations . In: AIP Advances 11.10 (Oct. 2021), p. 105116. doi: 10.1063/5.0067204. Alexandre C. Ricardo et al.
Alternatives to a Nonhomogeneous Partial Di er- ential Equation Quantum Algorithm . In: Physical Review A 106.5 (Nov. 2022), p. 052431. doi: 10.1103/PhysRevA.106.052431.
Kamal K. Saha et al. Advancing Algorithm to Scale and Accurately Solve Quan- tum Poisson Equation on Near-term Quantum Hardware. Oct. 2022. doi: 10. 48550/arXiv.2210.16668. arXiv: 2210.16668 [quant-ph]. [Ric+22] [Sah+22] [Sar22] Merey M. Sarsengeldin.
A Hybrid Classical-Quantum Framework for Solving Free Boundary Value Problems and Applications in Modeling Electric Contact Phenomena . In: arXiv:2205.02230 [quant-ph] (May 2022). arXiv: 2205.02230 [quant-ph]. [Sat+21] Yuki Sato et al.
Variational Quantum Algorithm Based on the Minimum Po- tential Energy for Solving the Poisson Equation . In: Physical Review A 104.5 (Nov. 2021), p. 052409. issn: 2469-9926, 2469-9934. doi: 10.1103/PhysRevA. 104.052409.
[SGS22] Amit Surana, Abeynaya Gnanasekaran, and Tuhin Sahai. Carleman Lineariza- tion Based E cient Quantum Algorithm for Higher Order Polynomial Di er- ential Equations. Dec. 2022. arXiv: 2212.10775 [quant-ph]. [Shi+18] Yuan Shi et al.
Simulations of Relativistic Quantum Plasmas Using Real-Time Lattice Scalar QED . In: Physical Review E 97.5 (May 2018), p. 053206. doi: 10.1103/PhysRevE.97.053206. [Shi+21] Yuan Shi et al.
Simulating Non-Native Cubic Interactions on Noisy Quantum Machines . In:
Physical Review A 103.6 (June 2021), p. 062608. doi: 10.1103/ PhysRevA.103.062608.
For the same type of problem, it is not uncommon for each researcher to be mostly interested in speci c types of quantum computing techniques, applicable in speci c contexts. Therefore, to facilitate search, we propose the use of tags to mark references with the context under which the problem was considered, as shown in Fig. 1. The following categories are used to organize the papers:
[SM21] Changpeng Shao and Ashley Montanaro. Faster Quantum-Inspired Algorithms for Solving Linear Systems . In: arXiv:2103.10309 [quant-ph] (Mar. 2021). arXiv: 2103.10309 [quant-ph].
[SS19] Siddhartha Srivastava and Veera Sundararaghavan. Box Algorithm for the Solution of Di erential Equations on a Quantum Annealer .
In: Physical Review A 99.5 (May 2019), p. 052355. issn: 2469-9926, 2469-9934. doi: 10 . 1103 / PhysRevA.99.052355.
[SSC21] Adrien Suau, Gabriel Sta elbach, and Henri Calandra. Practical Quantum Computing: Solving the Wave Equation Using a Quantum Approach .
In: ACM Transactions on Quantum Computing 2.1 (Feb. 2021), 2:1 2:35. issn: 2643- 6809. doi: 10.1145/3430030. [SSO19] Yi git Suba s , Rolando D. Somma, and Davide Orsucci.
Quantum Algorithms for Systems of Linear Equations Inspired by Adiabatic Quantum Comput- ing . In: Physical Review Letters 122.6 (Feb. 2019), p. 060504. doi: 10.1103/ PhysRevLett.122.060504.
[Vah+10] George Vahala et al. Unitary Quantum Lattice Gas Algorithms for Quantum to Classical Turbulence . In: 2010 DoD High Performance Computing Mod- ernization Program Users Group Conference. June 2010, pp. 184 191. doi: 10.1109/HPCMP-UGC.2010.15.
[Vah+11] [Vah+19] George Vahala et al. Unitary Qubit Lattice Simulations of Multiscale Phe- nomena in Quantum Turbulence .
In: SC 11: Proceedings of 2011 Interna- tional Conference for High Performance Computing, Networking, Storage and Analysis. Nov. 2011, pp. 1 11. doi: 10.1145/2063384.2063416. Linda Vahala et al.
Unitary Qubit Lattice Algorithm for Three-Dimensional Vortex Solitons in Hyperbolic Self-Defocusing Media . In: Communications in Nonlinear Science and Numerical Simulation 75 (Aug. 2019), pp. 152 159. issn: 1007-5704. doi: 10.1016/j.cnsns.2019.03.016. [Vah+20a] George Vahala et al. Building a Three-Dimensional Quantum Lattice Al- gorithm for Maxwell Equations .
Figure 1: Snapshot of the PDF (left) and Markdown/HTML (right) form of the review with hyperlinks to the papers. NISQ noisy-intermediate scale quantum computing; FTol fault-tolerant quantum computing; QAnn quantum annealing; QIns quantum-inspired.
In: Radiation E ects and Defects in Solids 175.11-12 (Nov. 2020), pp. 986 990. issn: 1042-0150. doi: 10.1080/10420150. 2020.1845685.
[Vah+20b] George Vahala et al. The E ect of the Pauli Spin Matrices on the Quan- tum Lattice Algorithm for Maxwell Equations in Inhomogeneous Media. Oct. 2020. doi: 10 . 48550 / arXiv . 2010 . 12264. arXiv: 2010 . 12264 [physics, physics:quant-ph]. [Vah+20c] George Vahala et al. Unitary Quantum Lattice Simulations for Maxwell Equa- tions in Vacuum and in Dielectric Media .
In: Journal of Plasma Physics 86.5 (Oct. 2020), p. 905860518. issn: 0022-3778, 1469-7807. doi: 10 . 1017 / S0022377820001166. [Vah+21a] George Vahala et al.
One- and Two-Dimensional Quantum Lattice Algorithms for Maxwell Equations in Inhomogeneous Scalar Dielectric Media I: Theory . In: Radiation E ects and Defects in Solids 176.1-2 (Feb. 2021), pp. 49 63. issn: 1042-0150. doi: 10.1080/10420150.2021.1891058.
[Vah+21b] George Vahala et al. One- and Two-Dimensional Quantum Lattice Algorithms for Maxwell Equations in Inhomogeneous Scalar Dielectric Media. II: Simula- tions . In: Radiation E ects and Defects in Solids 176.1-2 (Feb. 2021), pp. 64 72. issn: 1042-0150. doi: 10.1080/10420150.2021.1891059.
[Vah+21c] George Vahala et al. Two Dimensional Electromagnetic Scattering from Dielec- [Vah+22]
tric Objects Using Quantum Lattice Algorithm. SSRN Scholarly Paper. Rochester, NY, Dec. 2021. doi: 10.2139/ssrn.3996913. George Vahala et al. Quantum Lattice Representation for the Curl Equations of Maxwell Equations . In: Radiation E ects and Defects in Solids 177.1-2 (Feb. 2022), pp. 85 94. issn: 1042-0150. doi: 10.1080/10420150.2022.2049784. [VSV20] [VVS20] [VYV03] [Wan+20] [WX22] [Xu+21] [Xue+22] [XWG21] [Yep02] [Yep05] [Yep16] [YL22] [Zan+21] George Vahala, Min Soe, and Linda Vahala. Qubit Unitary Lattice Algorithm for Spin-2 Bose-Einstein Condensates: II - Vortex Reconnection Simulations and Non-Abelain Vortices . In: Radiation E ects and Defects in Solids 175.1-2 (Jan. 2020), pp. 113 119. issn: 1042-0150. doi: 10.1080/10420150.2020.1718136.
George Vahala, Linda Vahala, and Min Soe. Qubit Unitary Lattice Algorithm for Spin-2 Bose Einstein Condensates. I Theory and Pade Initial Conditions . In: Radiation E ects and Defects in Solids 175.1-2 (Jan. 2020), pp. 102 112. issn: 1042-0150. doi: 10.1080/10420150.2020.1718135.
George Vahala, Je rey Yepez, and Linda Vahala. Quantum Lattice Gas Rep- resentation of Some Classical Solitons . In: Physics Letters A 310.2 (Apr. 2003), pp. 187 196. issn: 0375-9601. doi: 10.1016/S0375-9601(03)00334-7. Shengbin Wang et al. Quantum Fast Poisson Solver: The Algorithm and Com- plete and Modular Circuit Design . In: Quantum Information Processing 19.6 (Apr. 2020), p. 170. issn: 1573-1332. doi: 10.1007/s11128-020-02669-7. Hefeng Wang and Hua Xiang. E cient Quantum Algorithms for Solving Quan- tum Linear System Problems. Aug. 2022. doi: 10.48550/arXiv.2208.06763. arXiv: 2208.06763 [quant-ph]. Xiaosi Xu et al. Variational Algorithms for Linear Algebra .
In: Science Bul- letin 66.21 (Nov. 2021), pp. 2181 2188. issn: 2095-9273. doi: 10.1016/j.scib. 2021.06.023. Cheng Xue et al. Quantum Algorithm for Solving a Quadratic Nonlinear Sys- tem of Equations .
Moreover, we also use secondary tags to indicate the type of analysis performed: Theo is a theoretical tag solely for the papers with analytical results, and no considerable numerical or experimental results; Num marks papers with numerical simulations, but no experimental results run on quan- tum devices; Exp marks papers with displayed experimental results.
In: Physical Review A 106.3 (Sept. 2022), p. 032427. issn: 2469-9926, 2469-9934. doi: 10.1103/PhysRevA.106.032427. Cheng Xue, Yuchun Wu, and Guoping Guo. Quantum Newton s Method for Solving the System of Nonlinear Equations .
In: SPIN 11.03 (Sept. 2021), p. 2140004. issn: 2010-3247. doi: 10.1142/S201032472140004X.
Je rey Yepez. An E cient and Accurate Quantum Algorithm for the Dirac Equation. Oct. 2002. doi: 10.48550/arXiv.quant-ph/0210093. arXiv: quant- ph/0210093.
Je rey Yepez. Relativistic Path Integral as a Lattice-based Quantum Algo- rithm . In: Quantum Information Processing 4.6 (Dec. 2005), pp. 471 509. issn: 1573-1332. doi: 10.1007/s11128-005-0009-7.
Je rey Yepez. Quantum Lattice Gas Algorithmic Representation of Gauge Field Theory . In:
Quantum Information Science and Technology II. Vol. 9996. SPIE, Oct. 2016, pp. 66 87. doi: 10.1117/12.2246702.
Erika Ye and Nuno F. G. Loureiro. A Quantum-Inspired Method for Solving the Vlasov-Poisson Equations. May 2022. arXiv: 2205.11990 [physics]. Benjamin Zanger et al. Quantum Algorithms for Solving Ordinary Di eren- tial Equations via Classical Integration Methods .
In: Quantum 5 (July 2021), p. 502. issn: 2521-327X. doi: 10.22331/q- 2021- 07- 13- 502. arXiv: 2012. 09469 [quant-ph]. [Zyl+22] Julien Zylberman et al. Hybrid Quantum-Classical Algorithm for Hydrodynam- ics. Feb. 2022. doi: 10.48550/arXiv.2202.00918. arXiv: 2202.00918 [physics, physics:quant-ph].
We may omit these tags if the paper is referenced and tagged in a subsequent subsection, or if none of them are clearly applicable. Please note that this choice of structuring is somewhat arbitrary and may change as the
eld evolves.
1 v 1 0 0 0 0 . 2 0 3 2 : v i X r a A Living Review of Quantum Computing for Plasma Physics
3 Contributing The community can also become involved in the process of reviewing and expanding this living document. The main channel for contributions is through a pull request (PR) to the
Figure 2: A printscreen of the README.md (left) and CONTRIBUTING.md (right) document describing the repository s structure and detailing the guidelines for contributions from non- maintainers to the review. review s GitHub project. A frequently asked questions (FAQ) and pull-request guide can be found in the repository, with detailed instructions on the recommended procedures and work ow. To help steer new contributions and ensure a smooth PR process and review with the maintainers, a contributions guide is located in the project s Git repository in the form of a CONTRIBUTING.md document (see Fig. 2) a project staple in the Open Source community. Furthermore, suggestions for new features can be submitted to the Living Review by creating a GitHub issue as documented in the project s CONTRIBUTING.md.
4 Showcase of the living review For reference, in this section we showcase the starting version of the Living Review, as it stood at the time it was rst made public. As previously indicated, a continuously updated version of it may be found in its associated GitHub project.
Modern Reviews Below are links to (static) general and specialized reviews. Applications of Quantum Computing to Plasma Simulations [DS21a]. Quantum Computing for Fusion Energy Science Applications [Jos+22]. System of linear equations NISQ Num [HBR21] Exp [Bra+20; Xu+21], FTol Theo [HHL09; CJS13; CKS17; WX22], QAnn Num [SSO19], Exp [BL22], QIns Theo [SM21]. Many problems in plasma physics may be formulated, either exactly or approximately, as a problem of the form Ax = b, where A and b encode the information about the system (including its initial conditions, if applicable), and the goal is to compute x, which encodes the desired data. System of nonlinear equations FTol Theo [DS21b] Num [XWG21; Xue+22]. Nonlinear equations depend nonlinearly on x, and they are generally much harder to solve. As quantum mechanics is inherently linear, many techniques rely on mapping the original nonlinear problem to a (usually approximate) linear one, that is easier to solve. System of polynomial equations QAnn Exp [Cha+19]. System where each equation is a polynomial. Ordinary di erential equations (ODEs) QAnn Exp [Zan+21]. Many plasma systems can be described by ordinary di erential equations. Some techniques attempt to solve them directly, while others map the ODE to (larger) systems of linear equations, and solve those.
Linear FTol Theo [Ber14; Ber+17; CL20; FLT22; JLY22a], Num [JLY22b], QAnn Exp [Zan+21].
As quantum mechanics is inherently linear, linear ODEs are often more straightforward to solve with quantum computers than nonlinear ones. Second-order QAnn Exp [SS19]. The highest derivative appearing in these ODEs is the second derivative.
Quantum harmonic oscillator FTol Num [Ric+22]. Here, the time-independent Schr odinger equation has a Hamiltonian with a poten- tial proportional to x2.
Laguerre QAnn Num [CS22]. The Laguerre equation is of the form d2y dx2 + (1 x) dy dx
+ n y = 0 Nonlinear NISQ [KPE21; Shi+21] FTol Theo [LO08; DS21b; Llo+20] Num [JLY22b; Liu+21b; SGS22], QAnn Num [Zan+21].
There is no general reliable procedure to solve nonlinear ODEs, but some methods have been proposed. Partial di erential equations (PDEs) NISQ [GRG22], FTol Theo [CLO21; Kro22], QIns [Gar21]. In general, plasma systems are described by partial di erential equations. Although some of the equations presented here do not commonly describe plasma systems, the techniques employed to solve them are often general enough that they could be adapted to tackle plasma PDEs. For ease of search, PDEs arising from stochastic processes are indicated both here and in the following section.

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