{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,7]],"date-time":"2026-06-07T05:04:23Z","timestamp":1780808663857,"version":"3.54.1"},"reference-count":69,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,3,28]],"date-time":"2023-03-28T00:00:00Z","timestamp":1679961600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Applied Research Laboratories at the University of Texas at Austin"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Finite-element methods are industry standards for finding numerical solutions to partial differential equations. However, the application scale remains pivotal to the practical use of these methods, even for modern-day supercomputers. Large, multi-scale applications, for example, can be limited by their requirement of prohibitively large linear system solutions. It is therefore worthwhile to investigate whether near-term quantum algorithms have the potential for offering any kind of advantage over classical linear solvers. In this study, we investigate the recently proposed variational quantum linear solver (VQLS) for discrete solutions to partial differential equations. This method was found to scale polylogarithmically with the linear system size, and the method can be implemented using shallow quantum circuits on noisy intermediate-scale quantum (NISQ) computers. Herein, we utilize the hybrid VQLS to solve both the steady Poisson equation and the time-dependent heat and wave equations.<\/jats:p>","DOI":"10.3390\/e25040580","type":"journal-article","created":{"date-parts":[[2023,3,28]],"date-time":"2023-03-28T02:58:21Z","timestamp":1679972301000},"page":"580","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":30,"title":["A Variational Quantum Linear Solver Application to Discrete Finite-Element Methods"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4366-8521","authenticated-orcid":false,"given":"Corey Jason","family":"Trahan","sequence":"first","affiliation":[{"name":"Information and Technology Laboratory, U.S. Army Engineer Research and Development Center, Vicksburg, MS 39180, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mark","family":"Loveland","sequence":"additional","affiliation":[{"name":"Information and Technology Laboratory, U.S. Army Engineer Research and Development Center, Vicksburg, MS 39180, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Noah","family":"Davis","sequence":"additional","affiliation":[{"name":"Applied Research Laboratories, The University of Texas at Austin, Austin, TX 78713, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Elizabeth","family":"Ellison","sequence":"additional","affiliation":[{"name":"Information and Technology Laboratory, U.S. Army Engineer Research and Development Center, Vicksburg, MS 39180, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,28]]},"reference":[{"key":"ref_1","unstructured":"Preskill, J. (2012). Quantum computing and the entanglement frontier. arXiv, Available online: http:\/\/arxiv.org\/abs\/1203.5813."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1038\/nature23458","article-title":"Quantum computational supremacy","volume":"549","author":"Harrow","year":"2017","journal-title":"Nature"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1038\/s41586-019-1666-5","article-title":"Quantum supremacy using a programmable superconducting processor","volume":"574","author":"Arute","year":"2019","journal-title":"Nature"},{"key":"ref_4","unstructured":"Connor, E. (2023, February 20). The New Light-Based Quantum Computer Jiuzhang Has Achieved Quantum Supremacy. Available online: https:\/\/www.sciencenews.org\/article\/new-light-based-quantum-computer-jiuzhang-supremacy."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3061","DOI":"10.1021\/acs.chemrev.0c00620","article-title":"Quantum information and algorithms for correlated quantum matter","volume":"121","author":"Flick","year":"2021","journal-title":"Chem. Rev."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Breuer, H.-P., and Petruccione, F. (2007). The Theory of Open Quantum Systems, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780199213900.001.0001"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1155","DOI":"10.1103\/RevModPhys.82.1155","article-title":"Introduction to quantum noise, measurement, and amplification","volume":"82","author":"Clerk","year":"2010","journal-title":"Rev. Mod. Phys."},{"key":"ref_8","unstructured":"Lidar, D.A. (2019). Lecture notes on the theory of open quantum systems. arXiv."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"021318","DOI":"10.1063\/1.5089550","article-title":"A quantum engineer\u2019s guide to superconducting qubits","volume":"6","author":"Krantz","year":"2019","journal-title":"Appl. Phys. Rev."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1038\/s41586-019-1040-7","article-title":"Error mitigation extends the computational reach of a noisy quantum processor","volume":"567","author":"Kandala","year":"2019","journal-title":"Nature"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"180501","DOI":"10.1103\/PhysRevLett.122.180501","article-title":"Error-mitigated digital quantum simulation","volume":"122","author":"McArdle","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"022517","DOI":"10.1103\/PhysRevA.100.022517","article-title":"Quantum-classical hybrid algorithm using an error-mitigating n -representability condition to compute the mott metal-insulator transition","volume":"100","author":"Smart","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"022405","DOI":"10.1103\/PhysRevA.105.022405","article-title":"Resolving correlated states of benzyne with an error-mitigated contracted quantum eigensolver","volume":"105","author":"Smart","year":"2022","journal-title":"Phys. Rev. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"032001","DOI":"10.7566\/JPSJ.90.032001","article-title":"Hybrid quantum-classical algorithms and quantum error mitigation","volume":"90","author":"Endo","year":"2021","journal-title":"J. Phys. Soc. Jpn."},{"key":"ref_15","first-page":"8","article-title":"Relaxation of stationary states on a quantum computer yields a unique spectroscopic fingerprint of the computer\u2019s noise","volume":"5","author":"Smart","year":"2022","journal-title":"Commun. Phys."},{"key":"ref_16","unstructured":"Aleksandrowicz, G., Alexander, T., Barkoutsosa, P., Bello, L., Ben-Haim, Y., Bucher, D., Cabrera-Hern\u00e1ndez, F., Carballo-Franquis, J., Chen, A., and Chen, C. (2022, May 10). Qiskit: An Open-Source Framework for Quantum Computing. Available online: https:\/\/doi.org\/10.5281\/zenodo.2562111."},{"key":"ref_17","unstructured":"(2022, May 10). Amazon, Amazon Braket. Available online: https:\/\/aws.amazon.com\/braket\/."},{"key":"ref_18","unstructured":"(2021, July 01). IBM, Learning Quantum Computation Using Qiskit. Available online: http:\/\/qiskit.org\/textbook."},{"key":"ref_19","unstructured":"Albornoz, C., Alonso, G., Andrenkov, P.A.M., and Asadi, A. (2022, July 01). Anothers, Xanadu Quantum Codebook. Available online: https:\/\/codebook.xanadu.ai."},{"key":"ref_20","unstructured":"Qbraid (2022, July 10). Qbraid: Cloud-Based ide for Quantum Computing. Available online: https:\/\/qbraid.com."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1038\/nature23474","article-title":"Quantum machine learning","volume":"549","author":"Biamonte","year":"2017","journal-title":"Nature"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2027","DOI":"10.1007\/s11128-012-0506-4","article-title":"Quantum adiabatic machine learning","volume":"12","author":"Pudenz","year":"2013","journal-title":"Quantum Inf. Process."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1038\/nphys3029","article-title":"Quantum principal component analysis","volume":"10","author":"Lloyd","year":"2014","journal-title":"Nat. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"130503","DOI":"10.1103\/PhysRevLett.113.130503","article-title":"Quantum support vector machine for big data classification","volume":"113","author":"Rebentrost","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1080\/00107514.2014.964942","article-title":"An introduction to quantum machine learning","volume":"56","author":"Schuld","year":"2015","journal-title":"Contemp. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"103108","DOI":"10.1063\/1.4943622","article-title":"Towards a feasible implementation of quantum neural networks using quantum dots","volume":"108","author":"Altaisky","year":"2016","journal-title":"Appl. Phys. Lett."},{"key":"ref_27","unstructured":"Dunjko, V., Taylor, J.M., and Briegel, H.J. (2015). Framework for learning agents in quantum environments. arXiv."},{"key":"ref_28","first-page":"1","article-title":"Supervised quantum learning without measurements","volume":"7","author":"Lamata","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-017-01711-6","article-title":"Basic protocols in quantum reinforcement learning with superconducting circuits","volume":"7","author":"Lamata","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"050505","DOI":"10.1103\/PhysRevLett.109.050505","article-title":"Quantum algorithm for data fitting","volume":"109","author":"Wiebe","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"022342","DOI":"10.1103\/PhysRevA.94.022342","article-title":"Prediction by linear regression on a quantum computer","volume":"94","author":"Schuld","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"150502","DOI":"10.1103\/PhysRevLett.103.150502","article-title":"Quantum algorithm for linear systems of equations","volume":"103","author":"Harrow","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Berry, D.W., Childs, A.M., and Kothari, R. (2015, January 17\u201320). Hamiltonian simulation with nearly optimal dependence on all parameters. Proceedings of the 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, Berkeley, CA, USA.","DOI":"10.1109\/FOCS.2015.54"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"052331","DOI":"10.1103\/PhysRevA.99.052331","article-title":"Quantum-assisted Gaussian process regression","volume":"99","author":"Zhao","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"210504","DOI":"10.1103\/PhysRevLett.118.210504","article-title":"Solving systems of linear equations with a superconducting quantum processor","volume":"118","author":"Zheng","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"4778","DOI":"10.1038\/s41598-019-41324-9","article-title":"Hybrid quantum linear equation algorithm and its experimental test on ibm quantum experience","volume":"9","author":"Lee","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"022313","DOI":"10.1103\/PhysRevA.89.022313","article-title":"Experimental realization of quantum algorithm for solving linear systems of equations","volume":"89","author":"Pan","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"30501","DOI":"10.1103\/PhysRevLett.110.230501","article-title":"Experimental quantum computing to solve systems of linear equations","volume":"110","author":"Cai","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"6115","DOI":"10.1038\/srep06115","article-title":"A two-qubit photonic quantum processor and its application to solving systems of linear equations","volume":"4","author":"Barz","year":"2014","journal-title":"Sci. Rep."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"012320","DOI":"10.1103\/PhysRevA.99.012320","article-title":"Experimental realization of quantum algorithms for a linear system inspired by adiabatic quantum computing","volume":"99","author":"Wen","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Feld, S., and Linnhoff-Popien, C. (2019). Quantum Technology and Optimization Problems, Springer International Publishing.","DOI":"10.1007\/978-3-030-14082-3"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"4213","DOI":"10.1038\/ncomms5213","article-title":"A variational eigenvalue solver on a photonic quantum processor","volume":"5","author":"Peruzzo","year":"2014","journal-title":"Nat. Commun."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"10856","DOI":"10.1021\/acs.chemrev.8b00803","article-title":"Quantum chemistry in the age of quantum computing","volume":"119","author":"Cao","year":"2019","journal-title":"Chem. Rev."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"156","DOI":"10.22331\/q-2019-07-01-156","article-title":"Variational quantum computation of excited states","volume":"3","author":"Higgott","year":"2019","journal-title":"Quantum"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"062304","DOI":"10.1103\/PhysRevA.99.062304","article-title":"Variational quantum algorithms for discovering hamiltonian spectra","volume":"99","author":"Jones","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_46","first-page":"021050","article-title":"Efficient variational quantum simulator incorporating active error minimization","volume":"7","author":"Li","year":"2017","journal-title":"Phys. Rev. X"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1038\/s41586-019-1177-4","article-title":"Self-verifying variational quantum simulation of lattice models","volume":"569","author":"Kokail","year":"2019","journal-title":"Nature"},{"key":"ref_48","unstructured":"Heya, K., Nakanishi, K.M., Mitarai, K., and Fujii, K. (2019). Subspace variational quantum simulator. arXiv."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1038\/s41534-020-00302-0","article-title":"Variational fast forwarding for quantum simulation beyond the coherence time","volume":"6","author":"Cirstoiu","year":"2020","journal-title":"npj Quantum Inf."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"191","DOI":"10.22331\/q-2019-10-07-191","article-title":"Theory of variational quantum simulation","volume":"3","author":"Yuan","year":"2019","journal-title":"Quantum"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"045001","DOI":"10.1088\/2058-9565\/aa8072","article-title":"Quantum autoencoders for efficient compression of quantum data","volume":"2","author":"Romero","year":"2017","journal-title":"Quantum Sci. Technol."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1038\/s41534-019-0167-6","article-title":"Variational quantum state diagonalization","volume":"5","author":"LaRose","year":"2019","journal-title":"npj Quantum Inf."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"062310","DOI":"10.1103\/PhysRevA.101.062310","article-title":"Quantum singular value decomposer","volume":"101","author":"Latorre","year":"2020","journal-title":"Phys. Rev. A"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1038\/s41534-022-00611-6","article-title":"Variational quantum state eigensolver","volume":"8","author":"Cerezo","year":"2022","journal-title":"npj Quantum Inf."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"140","DOI":"10.22331\/q-2019-05-13-140","article-title":"Quantum-assisted quantum compiling","volume":"3","author":"Khatri","year":"2019","journal-title":"Quantum"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"628","DOI":"10.22331\/q-2022-01-24-628","article-title":"Robust quantum compilation and circuit optimisation via energy minimisation","volume":"6","author":"Jones","year":"2022","journal-title":"Quantum"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"3438","DOI":"10.1038\/s41467-019-11417-0","article-title":"Variational consistent histories as a hybrid algorithm for quantum foundations","volume":"10","author":"Arrasmith","year":"2019","journal-title":"Nat. Commun."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"248","DOI":"10.22331\/q-2020-03-26-248","article-title":"Variational quantum fidelity estimation","volume":"4","author":"Cerezo","year":"2020","journal-title":"Quantum"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"083038","DOI":"10.1088\/1367-2630\/ab965e","article-title":"Variational-state quantum metrology","volume":"22","author":"Koczor","year":"2020","journal-title":"New J. Phys."},{"key":"ref_60","unstructured":"Bravo-Prieto, C., LaRose, R., Cerezo, M., Subasi, Y., Cincio, L., and Coles, P.J. (2019). Variational Quantum Linear Solver. arXiv."},{"key":"ref_61","unstructured":"Bravo-Prieto, C., LaRose, R., Cerezo, M., Suba\u015f\u0131, Y., Cincio, L., and Coles, P.J. (2020). Variational quantum linear solver: A hybrid algorithm for linear systems. Bull. Am. Phys. Soc."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"13022","DOI":"10.1088\/1367-2630\/aae94a","article-title":"Learning the quantum algorithm for state overlap","volume":"20","author":"Cincio","year":"2018","journal-title":"New J. Phys."},{"key":"ref_63","unstructured":"Pesce, R.M.N., and Stevenson, P.D. (2021). H2zixy: Pauli spin matrix decomposition of real symmetric matrices. arXiv."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1007\/s11128-021-03140-x","article-title":"A comparison of various classical optimizers for a variational quantum linear solver","volume":"20","author":"Sinayskiy","year":"2021","journal-title":"Quantum Inf. Process."},{"key":"ref_65","unstructured":"Hughes, T.J. (2012). The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Courier Corporation."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"012307","DOI":"10.1103\/PhysRevA.73.012307","article-title":"Efficient state preparation for a register of quantum bits","volume":"73","author":"Soklakov","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"160501","DOI":"10.1103\/PhysRevLett.100.160501","article-title":"Quantum Random Access Memory","volume":"100","author":"Giovannetti","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1038\/nphys3272","article-title":"Read the fine print","volume":"11","author":"Aaronson","year":"2015","journal-title":"Nat. Phys."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"012326","DOI":"10.1103\/PhysRevA.99.012326","article-title":"Optimal usage of quantum random access memory in quantum machine learning","volume":"99","author":"Bang","year":"2019","journal-title":"Phys. Rev. A"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/4\/580\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:04:41Z","timestamp":1760123081000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/4\/580"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,28]]},"references-count":69,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["e25040580"],"URL":"https:\/\/doi.org\/10.3390\/e25040580","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,28]]}}}