{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T10:03:42Z","timestamp":1775469822206,"version":"3.50.1"},"reference-count":61,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,21]],"date-time":"2023-03-21T00:00:00Z","timestamp":1679356800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"US Department of Energy, Advanced Scientific Computing Research program office Quantum Algorithms Team project","award":["ERKJ335"],"award-info":[{"award-number":["ERKJ335"]}]},{"name":"US Department of Energy, Advanced Scientific Computing Research program office Quantum Algorithms Team project","award":["4000175762"],"award-info":[{"award-number":["4000175762"]}]},{"name":"US Department of Energy, Advanced Scientific Computing Research program office Quantum Algorithms Team project","award":["DE-AC05-00OR22725"],"award-info":[{"award-number":["DE-AC05-00OR22725"]}]},{"name":"US Department of Energy, Advanced Scientific Computing Research program office Quantum Algorithms Team project","award":["DE-AC05-00OR22725"],"award-info":[{"award-number":["DE-AC05-00OR22725"]}]},{"name":"Purdue University","award":["ERKJ335"],"award-info":[{"award-number":["ERKJ335"]}]},{"name":"Purdue University","award":["4000175762"],"award-info":[{"award-number":["4000175762"]}]},{"name":"Purdue University","award":["DE-AC05-00OR22725"],"award-info":[{"award-number":["DE-AC05-00OR22725"]}]},{"name":"Purdue University","award":["DE-AC05-00OR22725"],"award-info":[{"award-number":["DE-AC05-00OR22725"]}]},{"name":"UT-Battelle, LLC","award":["ERKJ335"],"award-info":[{"award-number":["ERKJ335"]}]},{"name":"UT-Battelle, LLC","award":["4000175762"],"award-info":[{"award-number":["4000175762"]}]},{"name":"UT-Battelle, LLC","award":["DE-AC05-00OR22725"],"award-info":[{"award-number":["DE-AC05-00OR22725"]}]},{"name":"UT-Battelle, LLC","award":["DE-AC05-00OR22725"],"award-info":[{"award-number":["DE-AC05-00OR22725"]}]},{"name":"Purdue University, College of Science","award":["ERKJ335"],"award-info":[{"award-number":["ERKJ335"]}]},{"name":"Purdue University, College of Science","award":["4000175762"],"award-info":[{"award-number":["4000175762"]}]},{"name":"Purdue University, College of Science","award":["DE-AC05-00OR22725"],"award-info":[{"award-number":["DE-AC05-00OR22725"]}]},{"name":"Purdue University, College of Science","award":["DE-AC05-00OR22725"],"award-info":[{"award-number":["DE-AC05-00OR22725"]}]},{"name":"College of Science, Quantum Seed Grant","award":["ERKJ335"],"award-info":[{"award-number":["ERKJ335"]}]},{"name":"College of Science, Quantum Seed Grant","award":["4000175762"],"award-info":[{"award-number":["4000175762"]}]},{"name":"College of Science, Quantum Seed Grant","award":["DE-AC05-00OR22725"],"award-info":[{"award-number":["DE-AC05-00OR22725"]}]},{"name":"College of Science, Quantum Seed Grant","award":["DE-AC05-00OR22725"],"award-info":[{"award-number":["DE-AC05-00OR22725"]}]},{"name":"U.S. Department of Energy","award":["ERKJ335"],"award-info":[{"award-number":["ERKJ335"]}]},{"name":"U.S. Department of Energy","award":["4000175762"],"award-info":[{"award-number":["4000175762"]}]},{"name":"U.S. Department of Energy","award":["DE-AC05-00OR22725"],"award-info":[{"award-number":["DE-AC05-00OR22725"]}]},{"name":"U.S. Department of Energy","award":["DE-AC05-00OR22725"],"award-info":[{"award-number":["DE-AC05-00OR22725"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Recent advances in quantum hardware offer new approaches to solve various optimization problems that can be computationally expensive when classical algorithms are employed. We propose a hybrid quantum-classical algorithm to solve a dynamic asset allocation problem where a target return and a target risk metric (expected shortfall) are specified. We propose an iterative algorithm that treats the target return as a constraint in a Markowitz portfolio optimization model, and dynamically adjusts the target return to satisfy the targeted expected shortfall. The Markowitz optimization is formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem. The use of the expected shortfall risk metric enables the modeling of extreme market events. We compare the results from D-Wave\u2019s 2000Q and Advantage quantum annealers using real-world financial data. Both quantum annealers are able to generate portfolios with more than 80% of the return of the classical optimal solutions, while satisfying the expected shortfall. We observe that experiments on assets with higher correlations tend to perform better, which may help to design practical quantum applications in the near term.<\/jats:p>","DOI":"10.3390\/e25030541","type":"journal-article","created":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T07:09:31Z","timestamp":1679468971000},"page":"541","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Dynamic Asset Allocation with Expected Shortfall via Quantum Annealing"],"prefix":"10.3390","volume":"25","author":[{"given":"Hanjing","family":"Xu","sequence":"first","affiliation":[{"name":"Department of Computer Science, Purdue University, West Lafayette, IN 47906, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Samudra","family":"Dasgupta","sequence":"additional","affiliation":[{"name":"Department of Physics, Purdue University, West Lafayette, IN 47906, USA"},{"name":"Oak Ridge National Laboratory, Quantum Computing Institute, Oak Ridge, TN 37831, USA"},{"name":"Bredesen Center, University of Tennessee, Knoxville, TN 37996, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alex","family":"Pothen","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Purdue University, West Lafayette, IN 47906, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3088-6071","authenticated-orcid":false,"given":"Arnab","family":"Banerjee","sequence":"additional","affiliation":[{"name":"Department of Physics, Purdue University, West Lafayette, IN 47906, USA"},{"name":"Oak Ridge National Laboratory, Quantum Computing Institute, Oak Ridge, TN 37831, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Glover, F., Kochenberger, G., and Du, Y. (2019). A Tutorial on Formulating and Using QUBO Models. arXiv.","DOI":"10.1007\/s10288-019-00424-y"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/s11128-019-2418-z","article-title":"Quantum Annealing Learning Search for Solving QUBO Problems","volume":"18","author":"Pastorello","year":"2019","journal-title":"Quantum Inf. Process."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1007\/s10878-014-9734-0","article-title":"The Unconstrained Binary Quadratic Programming Problem: A Survey","volume":"28","author":"Kochenberger","year":"2014","journal-title":"J. Comb. Optim."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"5","DOI":"10.3389\/fphy.2014.00005","article-title":"Ising Formulations of Many NP Problems","volume":"2","author":"Lucas","year":"2014","journal-title":"Front. Phys."},{"key":"ref_5","unstructured":"Djidjev, H.N., Chapuis, G., Hahn, G., and Rizk, G. (2018). Efficient Combinatorial Optimization Using Quantum Annealing. arXiv."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"12837","DOI":"10.1038\/s41598-019-49172-3","article-title":"Application of Quantum Annealing to Nurse Scheduling Problem","volume":"9","author":"Ikeda","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1016\/j.disopt.2010.12.001","article-title":"Quantum Annealing of the Graph Coloring Problem","volume":"8","author":"Titiloye","year":"2011","journal-title":"Discret. Optim."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"104001","DOI":"10.1088\/1361-6633\/ac8c54","article-title":"Quantum Annealing for Industry Applications: Introduction and Review","volume":"85","author":"Yarkoni","year":"2022","journal-title":"Rep. Prog. Phys."},{"key":"ref_9","first-page":"77","article-title":"Portfolio Selection","volume":"7","author":"Markowitz","year":"1952","journal-title":"J. Financ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"7","DOI":"10.3905\/jfi.1991.408013","article-title":"Asset Allocation: Combining Investor Views with Market Equilibrium","volume":"1","author":"Black","year":"1991","journal-title":"J. Fixed Income"},{"key":"ref_11","unstructured":"McNeil, A.J., Frey, R., and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools\u2014Revised Edition, Princeton University Press."},{"key":"ref_12","unstructured":"Dasgupta, S., and Banerjee, A. (2020). Quantum Annealing Algorithm for Expected Shortfall Based Dynamic Asset Allocation. arXiv."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/BF00120662","article-title":"Quadratic Programming with One Negative Eigenvalue Is NP-hard","volume":"1","author":"Pardalos","year":"1991","journal-title":"J. Glob. Optim."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Floudas, C.A., and Pardalos, P.M. (2001). Encyclopedia of Optimization, Springer.","DOI":"10.1007\/0-306-48332-7"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"014012","DOI":"10.1103\/PhysRevApplied.15.014012","article-title":"Benchmarking Quantum Annealing Controls with Portfolio Optimization","volume":"15","author":"Grant","year":"2021","journal-title":"Phys. Rev. Appl."},{"key":"ref_16","unstructured":"O\u2019Donnell, R. (2014). Analysis of Boolean Functions, Cambridge University Press."},{"key":"ref_17","unstructured":"Dattani, N. (2019). Quadratization in Discrete Optimization and Quantum Mechanics. arXiv."},{"key":"ref_18","unstructured":"Verma, A., Lewis, M., and Kochenberger, G. (2021). Efficient QUBO Transformation for Higher Degree Pseudo Boolean Functions. arXiv."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Mandal, A., Roy, A., Upadhyay, S., and Ushijima-Mwesigwa, H. (2020, January 24\u201327). Compressed Quadratization of Higher Order Binary Optimization Problems. Proceedings of the 2020 Data Compression Conference (DCC), Snowbird, UT, USA.","DOI":"10.1109\/DCC47342.2020.00090"},{"key":"ref_20","unstructured":"(2020, May 18). Yahoo Finance iShares MSCI Emerging Markets ETF (EEM). Available online: https:\/\/finance.yahoo.com\/quote\/EEM\/history?p=EEM."},{"key":"ref_21","unstructured":"(2020, May 18). Yahoo Finance Invesco QQQ Trust (QQQ). Available online: https:\/\/finance.yahoo.com\/quote\/QQQ\/history?p=QQQ."},{"key":"ref_22","unstructured":"(2020, May 18). Yahoo Finance iShares Silver Trust (SLV). Available online: https:\/\/finance.yahoo.com\/quote\/SLV\/history?p=SLV."},{"key":"ref_23","unstructured":"(2020, May 18). Yahoo Finance SPDR S&P 500 ETF Trust (SPY). Available online: https:\/\/finance.yahoo.com\/quote\/SPY\/history?p=SPY."},{"key":"ref_24","unstructured":"(2020, May 18). Yahoo Finance ProShares UltraPro Short QQQ (SQQQ). Available online: https:\/\/finance.yahoo.com\/quote\/SQQQ\/history?p=SQQQ."},{"key":"ref_25","unstructured":"(2020, May 18). Yahoo Finance Financial Select Sector SPDR Fund (XLF). Available online: https:\/\/finance.yahoo.com\/quote\/XLF\/history?p=XLF."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Uryasev, S., and Pardalos, P.M. (2001). Stochastic Optimization: Algorithms and Applications, Springer. Applied Optimization.","DOI":"10.1007\/978-1-4757-6594-6"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1281","DOI":"10.1007\/s10479-019-03373-1","article-title":"Calculating CVaR and bPOE for Common Probability Distributions with Application to Portfolio Optimization and Density Estimation","volume":"299","author":"Norton","year":"2021","journal-title":"Ann. Oper. Res."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1353","DOI":"10.1016\/S0165-1889(03)00109-X","article-title":"Shortfall as a Risk Measure: Properties, Optimization and Applications","volume":"28","author":"Bertsimas","year":"2004","journal-title":"J. Econ. Dyn. Control"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1126\/science.284.5415.779","article-title":"Quantum Annealing of a Disordered Magnet","volume":"284","author":"Brooke","year":"1999","journal-title":"Science"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2427","DOI":"10.1126\/science.1068774","article-title":"Theory of Quantum Annealing of an Ising Spin Glass","volume":"295","author":"Santoro","year":"2002","journal-title":"Science"},{"key":"ref_31","unstructured":"King, A.D., Raymond, J., Lanting, T., Harris, R., Zucca, A., Altomare, F., Berkley, A.J., Boothby, K., Ejtemaee, S., and Enderud, C. (2022). Quantum Critical Dynamics in a 5000-Qubit Programmable Spin Glass. arXiv."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1080\/00107514.2018.1450720","article-title":"A Cross-Disciplinary Introduction to Quantum Annealing-Based Algorithms","volume":"59","author":"McGeoch","year":"2018","journal-title":"Contemp. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3241","DOI":"10.1088\/0305-4470\/15\/10\/028","article-title":"On the Computational Complexity of Ising Spin Glass Models","volume":"15","author":"Barahona","year":"1982","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1016\/j.jmst.2019.12.009","article-title":"Computational Complexity of Spin-Glass Three-Dimensional (3D) Ising Model","volume":"44","author":"Zhang","year":"2020","journal-title":"J. Mater. Sci. Technol."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41534-017-0037-z","article-title":"Non-Stoquastic Hamiltonians in Quantum Annealing via Geometric Phases","volume":"3","author":"Vinci","year":"2017","journal-title":"Npj Quantum Inf."},{"key":"ref_36","unstructured":"Farhi, E., Goldstone, J., Gutmann, S., and Sipser, M. (2000). Quantum Computation by Adiabatic Evolution. arXiv."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"015002","DOI":"10.1103\/RevModPhys.90.015002","article-title":"Adiabatic Quantum Computation","volume":"90","author":"Albash","year":"2018","journal-title":"Rev. Mod. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/BF01343193","article-title":"Beweis des Adiabatensatzes","volume":"51","author":"Born","year":"1928","journal-title":"Z. f\u00fcr Phys."},{"key":"ref_39","unstructured":"D-Wave Systems Inc. (ICE: Dynamic Ranges in h and J Values, 2021). The Practical Quantum Computing Company, ICE: Dynamic Ranges in h and J Values."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.1109\/JSTSP.2016.2574703","article-title":"Solving the Optimal Trading Trajectory Problem Using a Quantum Annealer","volume":"10","author":"Rosenberg","year":"2016","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s42484-019-00001-w","article-title":"Reverse Quantum Annealing Approach to Portfolio Optimization Problems","volume":"1","author":"Venturelli","year":"2019","journal-title":"Quantum Mach. Intell."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Paszynski, M., Kranzlm\u00fcller, D., Krzhizhanovskaya, V.V., Dongarra, J.J., and Sloot, P.M.A. (2021, January 16\u201318). Portfolio Optimisation Using the D-Wave Quantum Annealer. Proceedings of the Computational Science\u2014ICCS 2021, Krakow, Poland. Lecture Notes in Computer Science.","DOI":"10.1007\/978-3-030-77964-1"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1063\/1.1699114","article-title":"Equation of State Calculations by Fast Computing Machines","volume":"21","author":"Metropolis","year":"1953","journal-title":"J. Chem. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1126\/science.220.4598.671","article-title":"Optimization by Simulated Annealing","volume":"220","author":"Kirkpatrick","year":"1983","journal-title":"Science"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"933","DOI":"10.1016\/j.ejor.2014.10.061","article-title":"A Multi-Period Fuzzy Portfolio Optimization Model with Minimum Transaction Lots","volume":"242","author":"Liu","year":"2015","journal-title":"Eur. J. Oper. Res."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"7121","DOI":"10.1016\/j.eswa.2015.05.020","article-title":"Portfolio Optimization Using a Credibility Mean-Absolute Semi-Deviation Model","volume":"42","author":"Vercher","year":"2015","journal-title":"Expert Syst. Appl."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"518","DOI":"10.1016\/j.ejor.2013.08.035","article-title":"Twenty Years of Linear Programming Based Portfolio Optimization","volume":"234","author":"Mansini","year":"2014","journal-title":"Eur. J. Oper. Res."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1023\/A:1020920706534","article-title":"Local Search Techniques for Constrained Portfolio Selection Problems","volume":"20","author":"Schaerf","year":"2002","journal-title":"Comput. Econ."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"043204","DOI":"10.1103\/PhysRevResearch.4.043204","article-title":"Portfolio Optimization with Digitized-Counterdiabatic Quantum Algorithms","volume":"4","author":"Hegade","year":"2022","journal-title":"Phys. Rev. Res."},{"key":"ref_50","unstructured":"D-Wave Systems Inc. (2020, May 18). The Practical Quantum Computing Company. The D-Wave Advantage System: An Overview. Available online: https:\/\/www.dwavesys.com\/media\/s3qbjp3s\/14-1049a-a_the_d-wave_advantage_system_an_overview.pdf."},{"key":"ref_51","first-page":"031040","article-title":"Quantum Optimization of Fully Connected Spin Glasses","volume":"5","author":"Venturelli","year":"2015","journal-title":"Phys. Rev. X"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1007\/s11128-015-1150-6","article-title":"Fast Clique Minor Generation in Chimera Qubit Connectivity Graphs","volume":"15","author":"Boothby","year":"2016","journal-title":"Quantum Inf. Process."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Pelofske, E., Hahn, G., and Djidjev, H. (2019, January 6\u20138). Optimizing the Spin Reversal Transform on the D-Wave 2000Q. Proceedings of the 2019 IEEE International Conference on Rebooting Computing (ICRC), San Mateo, CA, USA.","DOI":"10.1109\/ICRC.2019.8914719"},{"key":"ref_54","unstructured":"Lanting, T., Amin, M.H., Baron, C., Babcock, M., Boschee, J., Boixo, S., Smelyanskiy, V.N., Foygel, M., and Petukhov, A.G. (2020). Probing Environmental Spin Polarization with Superconducting Flux Qubits. arXiv."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Pudenz, K.L. (2016, January 13\u201315). Parameter Setting for Quantum Annealers. Proceedings of the 2016 IEEE High Performance Extreme Computing Conference (HPEC), Waltham, MA, USA.","DOI":"10.1109\/HPEC.2016.7761619"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1287\/mnsc.3.3.255","article-title":"The Elimination Form of the Inverse and Its Application to Linear Programming","volume":"3","author":"Markowitz","year":"1957","journal-title":"Manag. Sci."},{"key":"ref_57","first-page":"269","article-title":"Bayesian Updating in Causal Probabilistic Networks by Local Computations","volume":"4","author":"Jensen","year":"1990","journal-title":"Comput. Stat. Q."},{"key":"ref_58","first-page":"2909","article-title":"CVXPY: A Python-Embedded Modeling Language for Convex Optimization","volume":"17","author":"Diamond","year":"2016","journal-title":"J. Mach. Learn. Res."},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Barbosa, A., Pelofske, E., Hahn, G., and Djidjev, H.N. (2021). Using Machine Learning for Quantum Annealing Accuracy Prediction. Algorithms, 14.","DOI":"10.3390\/a14060187"},{"key":"ref_60","unstructured":"Farhi, E., Goldstone, J., and Gutmann, S. (2014). A Quantum Approximate Optimization Algorithm. arXiv."},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Hadfield, S., Wang, Z., O\u2019Gorman, B., Rieffel, E.G., Venturelli, D., and Biswas, R. (2019). From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz. Algorithms, 12.","DOI":"10.3390\/a12020034"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/3\/541\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:59:51Z","timestamp":1760122791000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/3\/541"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,21]]},"references-count":61,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["e25030541"],"URL":"https:\/\/doi.org\/10.3390\/e25030541","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,21]]}}}