{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T20:06:47Z","timestamp":1780085207221,"version":"3.54.0"},"reference-count":23,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,5,15]],"date-time":"2024-05-15T00:00:00Z","timestamp":1715731200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Air Force Research Laboratory, including access to IBM Quantum resources"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Quantum computing has the potential to solve problems that are currently intractable to classical computers with algorithms like Quantum Phase Estimation (QPE); however, noise significantly hinders the performance of today\u2019s quantum computers. Machine learning has the potential to improve the performance of QPE algorithms, especially in the presence of noise. In this work, QPE circuits were simulated with varying levels of depolarizing noise to generate datasets of QPE output. In each case, the phase being estimated was generated with a phase gate, and each circuit modeled was defined by a randomly selected phase. The model accuracy, prediction speed, overfitting level and variation in accuracy with noise level was determined for 5 machine learning algorithms. These attributes were compared to the traditional method of post-processing and a 6x\u201336 improvement in model performance was noted, depending on the dataset. No algorithm was a clear winner when considering these 4 criteria, as the lowest-error model (neural network) was also the slowest predictor; the algorithm with the lowest overfitting and fastest prediction time (linear regression) had the highest error level and a high degree of variation of error with noise. The XGBoost ensemble algorithm was judged to be the best tradeoff between these criteria due to its error level, prediction time and low variation of error with noise. For the first time, a machine learning model was validated using a 2-qubit datapoint obtained from an IBMQ quantum computer. The best 2-qubit model predicted within 2% of the actual phase, while the traditional method possessed a 25% error.<\/jats:p>","DOI":"10.3390\/a17050214","type":"journal-article","created":{"date-parts":[[2024,5,15]],"date-time":"2024-05-15T06:14:42Z","timestamp":1715753682000},"page":"214","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Improving 2\u20135 Qubit Quantum Phase Estimation Circuits Using Machine Learning"],"prefix":"10.3390","volume":"17","author":[{"given":"Charles","family":"Woodrum","sequence":"first","affiliation":[{"name":"Data Analytics Certificate Program, Graduate School of Engineering and Management, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1039-1055","authenticated-orcid":false,"given":"Torrey","family":"Wagner","sequence":"additional","affiliation":[{"name":"Data Analytics Certificate Program, Graduate School of Engineering and Management, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David","family":"Weeks","sequence":"additional","affiliation":[{"name":"Department of Engineering Physics, Graduate School of Engineering and Management, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,15]]},"reference":[{"key":"ref_1","unstructured":"Shankar, R. (2011). Principles of Quantum Mechanics, Plenum Press. [2nd ed.]."},{"key":"ref_2","unstructured":"Nielsen, M., and Chuang, I. (2010). Quantum Computation and Quantum Information, Cambridge University Press. [10th ed.]."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1098\/rspa.1998.0164","article-title":"Quantum Algorithms Revisited","volume":"454","author":"Cleve","year":"1998","journal-title":"Proc. R. Soc. London. Ser. A Math. Phys. Eng. Sci."},{"key":"ref_4","unstructured":"Kitaev, A.Y. (1995). Quantum measurements and the Abelian Stabilizer Problem. arXiv."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2250060","DOI":"10.1142\/S0219477522500602","article-title":"Modeling and Simulation of a Quantum Thermal Noise on the Qubit","volume":"21","year":"2022","journal-title":"Fluct. Noise Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1184","DOI":"10.1038\/s41567-020-0992-8","article-title":"Efficient Learning of Quantum Noise","volume":"16","author":"Harper","year":"2020","journal-title":"Nat. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3912","DOI":"10.1038\/s41598-023-30510-5","article-title":"Efficient noise mitigation technique for quantum computing","volume":"13","author":"Shaib","year":"2023","journal-title":"Sci. Rep."},{"key":"ref_8","unstructured":"IBM Corporation (2011). IBM SPSS Modeler CRISP-DM Guide, IBM Corporation."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"044005","DOI":"10.1088\/2058-9565\/abaa2c","article-title":"Optimizing quantum phase estimation for the simulation of Hamiltonian eigenstates","volume":"5","author":"Cruz","year":"2020","journal-title":"Quantum Sci. Technol."},{"key":"ref_10","unstructured":"Zlokapa, A., and Gheorghiu, A. (2020). A deep learning model for noise prediction on near-term quantum devices. arXiv."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Woodrum, C., and Weeks, D. (2023, January 28\u201331). Machine Learning Approaches to Evaluating Quantum Phase Estimation Algorithm Output. Proceedings of the NAECON 2023-IEEE National Aerospace and Electronics Conference, Dayton, OH, USA.","DOI":"10.1109\/NAECON58068.2023.10365987"},{"key":"ref_12","unstructured":"Qiskit (2023, September 01). Building Noise Models (Qiskit Aer 0.13.1). Available online: https:\/\/qiskit.org\/ecosystem\/aer\/tutorials\/3_building_noise_models.html."},{"key":"ref_13","unstructured":"Qiskit (2024, February 03). Applying Noise to Custom Unitary Gates (Qiskit Aer 0.13.1). Available online: https:\/\/qiskit.org\/ecosystem\/aer\/tutorials\/4_custom_gate_noise.html."},{"key":"ref_14","unstructured":"Woodrum, C. (2023). Methods of Evaluating Quantum Phase Estimation Circuit Output. [Master\u2019s Thesis, Air Force Institute of Technology]."},{"key":"ref_15","unstructured":"G\u00e9ron, A. (2019). Hands-on Machine Learning with Scikit-Learn, Keras, and TensorFlow, O\u2019Riley."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"045129","DOI":"10.1103\/PhysRevB.100.045129","article-title":"Unveiling phase transitions with machine learning","volume":"100","author":"Canabarro","year":"2019","journal-title":"Phys. Rev. B"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Saul, J., Wagner, T., Mbonimpa, E., and Langhals, B. (2023, January 24\u201327). Atmospheric Meteorological Effects on Forecasting Daily Lightning Occurrence at Cape Canaveral Space Force Station. Proceedings of the 2023 Congress in Computer Science, Computer Engineering, & Applied Computing (CSCE), Las Vegas, NV, USA.","DOI":"10.1109\/CSCE60160.2023.00305"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Tucker, T., Wagner, T., Auclair, P., and Langhals, B. (2023, January 24\u201327). Machine Learning Prediction of DoD Personal Property Shipment Costs. Proceedings of the 2023 Congress in Computer Science, Computer Engineering, & Applied Computing (CSCE), Las Vegas, NV, USA.","DOI":"10.1109\/CSCE60160.2023.00303"},{"key":"ref_19","unstructured":"Widrow, B. (1987, January 21). ADALINE and MADALINE. Proceedings of the 1st International Conference on Neural Networks, San Diego, CA, USA."},{"key":"ref_20","unstructured":"IBM Quantum Team (2024, February 02). IBMQ_Perth Quantum Computer, Job ID: 4vm4mnq2dtrqsq566g. Available online: https:\/\/quantum-computing.ibm.com."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"AbuGhanem, M. (2024). Full Quantum Process Tomography of a Universal Entangling Gate on an IBM\u2019s Quantum Computer. arXiv.","DOI":"10.2139\/ssrn.4726035"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Trochatos, T., Xu, C., Deshpande, S., Lu, Y., Ding, Y., and Szefer, J. (2023). Hardware Architecture for a Quantum Computer Trusted Execution Environment. arXiv.","DOI":"10.1109\/HPCA57654.2024.00051"},{"key":"ref_23","unstructured":"IBM Quantum Documentation (2024, May 09). Retired Systems. Available online: https:\/\/docs.quantum.ibm.com\/run\/retired-systems#retrieve."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/5\/214\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:42:43Z","timestamp":1760107363000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/5\/214"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,15]]},"references-count":23,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["a17050214"],"URL":"https:\/\/doi.org\/10.3390\/a17050214","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,15]]}}}