{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T11:59:12Z","timestamp":1770465552471,"version":"3.49.0"},"reference-count":54,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T00:00:00Z","timestamp":1696982400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Imaging"],"abstract":"<jats:p>Our study explores the feasibility of quantum computing in emission tomography reconstruction, addressing a noisy ill-conditioned inverse problem. In current clinical practice, this is typically solved by iterative methods minimizing a L2 norm. After reviewing quantum computing principles, we propose the use of a commercially available quantum annealer and employ corresponding hybrid solvers, which combine quantum and classical computing to handle more significant problems. We demonstrate how to frame image reconstruction as a combinatorial optimization problem suited for these quantum annealers and hybrid systems. Using a toy problem, we analyze reconstructions of binary and integer-valued images with respect to their image size and compare them to conventional methods. Additionally, we test our method\u2019s performance under noise and data underdetermination. In summary, our method demonstrates competitive performance with traditional algorithms for binary images up to an image size of 32\u00d732 on the toy problem, even under noisy and underdetermined conditions. However, scalability challenges emerge as image size and pixel bit range increase, restricting hybrid quantum computing as a practical tool for emission tomography reconstruction until significant advancements are made to address this issue.<\/jats:p>","DOI":"10.3390\/jimaging9100221","type":"journal-article","created":{"date-parts":[[2023,10,12]],"date-time":"2023-10-12T03:07:47Z","timestamp":1697080067000},"page":"221","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Exploring the Limitations of Hybrid Adiabatic Quantum Computing for Emission Tomography Reconstruction"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0009-0006-6468-0270","authenticated-orcid":false,"given":"Merlin A.","family":"Nau","sequence":"first","affiliation":[{"name":"Pattern Recognition Lab, Department of Computer Science, Friedrich-Alexander Universit\u00e4t Erlangen-N\u00fcrnberg (FAU), Martensstrasse 3, 91058 Erlangen, Germany"},{"name":"Siemens Healthineers GmbH, Siemensstrasse 1, 91301 Forchheim, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5562-7922","authenticated-orcid":false,"given":"A. Hans","family":"Vija","sequence":"additional","affiliation":[{"name":"Siemens Medical Solutions USA, Inc., 2501 Barrington Rd, Hoffman Estates, IL 60192, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3890-3593","authenticated-orcid":false,"given":"Wesley","family":"Gohn","sequence":"additional","affiliation":[{"name":"Siemens Medical Solutions USA, Inc., 2501 Barrington Rd, Hoffman Estates, IL 60192, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9632-1113","authenticated-orcid":false,"given":"Maximilian P.","family":"Reymann","sequence":"additional","affiliation":[{"name":"Pattern Recognition Lab, Department of Computer Science, Friedrich-Alexander Universit\u00e4t Erlangen-N\u00fcrnberg (FAU), Martensstrasse 3, 91058 Erlangen, Germany"},{"name":"Siemens Healthineers GmbH, Siemensstrasse 1, 91301 Forchheim, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9550-5284","authenticated-orcid":false,"given":"Andreas K.","family":"Maier","sequence":"additional","affiliation":[{"name":"Pattern Recognition Lab, Department of Computer Science, Friedrich-Alexander Universit\u00e4t Erlangen-N\u00fcrnberg (FAU), Martensstrasse 3, 91058 Erlangen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Nielsen, M.A., and Chuang, I. (2002). Quantum Computation and Quantum Information, Cambridge University Press.","DOI":"10.1119\/1.1463744"},{"key":"ref_2","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_3","doi-asserted-by":"crossref","first-page":"180501","DOI":"10.1103\/PhysRevLett.127.180501","article-title":"Strong quantum computational advantage using a superconducting quantum processor","volume":"127","author":"Wu","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1137\/080734479","article-title":"Adiabatic quantum computation is equivalent to standard quantum computation","volume":"50","author":"Aharonov","year":"2008","journal-title":"SIAM Rev."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Raymond, J., Stevanovic, R., Bernoudy, W., Boothby, K., McGeoch, C., Berkley, A.J., Farr\u00e9, P., and King, A.D. (2022). Hybrid quantum annealing for larger-than-QPU lattice-structured problems. arXiv.","DOI":"10.1145\/3579368"},{"key":"ref_6","unstructured":"Wernick, M.N., and Aarsvold, J.N. (2004). Emission Tomography: The Fundamentals of PET and SPECT, Elsevier."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1145\/367701.367709","article-title":"An introduction to quantum computing for non-physicists","volume":"32","author":"Rieffel","year":"2000","journal-title":"ACM Comput. Surv. (CSUR)"},{"key":"ref_8","unstructured":"Chow, J., Dial, O., and Gambetta, J. (2023, October 07). IBM Quantum Breaks the 100-Qubit Processor Barrier. Available online: https:\/\/research.ibm.com\/blog\/127-qubit-quantum-processor-eagle."},{"key":"ref_9","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_10","doi-asserted-by":"crossref","unstructured":"Van Dam, W., Mosca, M., and Vazirani, U. (2001, January 14\u201317). How powerful is adiabatic quantum computation?. Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, Washington, DC, USA.","DOI":"10.1109\/SFCS.2001.959902"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1126\/science.1057726","article-title":"A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem","volume":"292","author":"Farhi","year":"2001","journal-title":"Science"},{"key":"ref_12","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. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"102111","DOI":"10.1063\/1.2798382","article-title":"Bounds for the adiabatic approximation with applications to quantum computation","volume":"48","author":"Jansen","year":"2007","journal-title":"J. Math. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/0009-2614(94)00117-0","article-title":"Quantum annealing: A new method for minimizing multidimensional functions","volume":"219","author":"Finnila","year":"1994","journal-title":"Chem. Phys. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"McGeoch, C.C. (2014). Adiabatic Quantum Computation and Quantum Annealing: Theory and Practice, Springer Nature. Synthesis Lectures on Quantum Computing.","DOI":"10.1007\/978-3-031-02518-1"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1109\/JSAIT.2020.3014192","article-title":"Comparison of D-wave quantum annealing and classical simulated annealing for local minima determination","volume":"1","author":"Koshka","year":"2020","journal-title":"IEEE J. Sel. Areas Inf. Theory"},{"key":"ref_17","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_18","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_19","doi-asserted-by":"crossref","first-page":"456","DOI":"10.1038\/s41586-018-0410-x","article-title":"Observation of topological phenomena in a programmable lattice of 1800 qubits","volume":"560","author":"King","year":"2018","journal-title":"Nature"},{"key":"ref_20","unstructured":"D-Wave (2022, September 28). D-Wave Ocean Software. Available online: https:\/\/docs.ocean.dwavesys.com."},{"key":"ref_21","unstructured":"D-Wave (2022, September 28). D-Wave Leap. Available online: https:\/\/cloud.dwavesys.com\/leap\/."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2722","DOI":"10.1063\/1.1729798","article-title":"Representation of a function by its line integrals, with some radiological applications","volume":"34","author":"Cormack","year":"1963","journal-title":"J. Appl. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1109\/TASSP.1987.1165108","article-title":"Discrete radon transform","volume":"35","author":"Beylkin","year":"1987","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1088\/0266-5611\/17\/1\/309","article-title":"Inversion of the attenuated Radon transform","volume":"17","author":"Natterer","year":"2001","journal-title":"Inverse Probl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1315","DOI":"10.1088\/0266-5611\/15\/5\/314","article-title":"Numerical solution of the identification problem for the attenuated Radon transform","volume":"15","author":"Bronnikov","year":"1999","journal-title":"Inverse Probl."},{"key":"ref_26","unstructured":"Barrett, H.H., and Myers, K.J. (2013). Foundations of Image Science, John Wiley & Sons."},{"key":"ref_27","unstructured":"Herman, G.T., and Kuba, A. (2012). Discrete Tomography: Foundations, Algorithms, and Applications, Springer Science & Business Media."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1612","DOI":"10.1109\/JPROC.2003.817871","article-title":"Discrete tomography in medical imaging","volume":"91","author":"Herman","year":"2003","journal-title":"Proc. IEEE"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"123009","DOI":"10.1088\/0266-5611\/25\/12\/123009","article-title":"Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?","volume":"25","author":"Pan","year":"2009","journal-title":"Inverse Probl."},{"key":"ref_30","first-page":"1343","article-title":"Analytic and iterative reconstruction algorithms in SPECT","volume":"43","author":"Bruyant","year":"2002","journal-title":"J. Nucl. Med."},{"key":"ref_31","first-page":"306","article-title":"EM reconstruction algorithms for emission and transmission tomography","volume":"8","author":"Lange","year":"1984","journal-title":"J. Comput. Assist. Tomogr."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/j.2517-6161.1977.tb01600.x","article-title":"Maximum likelihood from incomplete data via the EM algorithm","volume":"39","author":"Dempster","year":"1977","journal-title":"J. R. Stat. Soc. Ser. B (Methodol.)"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"409","DOI":"10.6028\/jres.049.044","article-title":"Methods of conjugate gradients for solving","volume":"49","author":"Hestenes","year":"1952","journal-title":"J. Res. Natl. Bur. Stand."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1177\/016173468400600107","article-title":"Simultaneous algebraic reconstruction technique (SART): A superior implementation of the ART algorithm","volume":"6","author":"Andersen","year":"1984","journal-title":"Ultrason. Imaging"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"W\u00fcrfl, T., Ghesu, F.C., Christlein, V., and Maier, A. (2016, January 17\u201321). Deep learning computed tomography. Proceedings of the Medical Image Computing and Computer-Assisted Intervention-MICCAI 2016: 19th International Conference, Athens, Greece. Proceedings, Part III 19.","DOI":"10.1007\/978-3-319-46726-9_50"},{"key":"ref_36","unstructured":"Penrose, R. (1955). Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"600","DOI":"10.1109\/TIP.2003.819861","article-title":"Image quality assessment: From error visibility to structural similarity","volume":"13","author":"Wang","year":"2004","journal-title":"IEEE Trans. Image Process."},{"key":"ref_38","unstructured":"Caraiman, S., and Manta, V. (2012, January 12\u201314). Image processing using quantum computing. Proceedings of the 2012 16th International Conference on System Theory, Control and Computing (ICSTCC), Sinaia, Romania."},{"key":"ref_39","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_40","unstructured":"Kiani, B.T., Villanyi, A., and Lloyd, S. (2020). Quantum medical imaging algorithms. arXiv."},{"key":"ref_41","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_42","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_43","unstructured":"Farhi, E., Goldstone, J., and Gutmann, S. (2014). A quantum approximate optimization algorithm. arXiv."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"O\u2019Malley, D., Vesselinov, V.V., Alexandrov, B.S., and Alexandrov, L.B. (2018). Nonnegative\/binary matrix factorization with a d-wave quantum annealer. PLoS ONE, 13.","DOI":"10.1371\/journal.pone.0206653"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"10258","DOI":"10.1038\/s41598-019-46729-0","article-title":"Quantum annealing for systems of polynomial equations","volume":"9","author":"Chang","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"265","DOI":"10.3389\/fphy.2020.00265","article-title":"Floating-point calculations on a quantum annealer: Division and matrix inversion","volume":"8","author":"Rogers","year":"2020","journal-title":"Front. Phys."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"748285","DOI":"10.3389\/fphy.2021.748285","article-title":"An application of quantum annealing computing to seismic inversion","volume":"9","author":"Souza","year":"2022","journal-title":"Front. Phys."},{"key":"ref_48","unstructured":"Schielein, R., Basting, M., Dremel, K., Firsching, M., Fuchs, T., Graetz, J., Kasperl, S., Prjamkov, D., Semmler, S., and Suth, D. (December, January 8\u201311). Quantum Computing and Computed Tomography: A Roadmap towards QuantumCT. Proceedings of the 11th Conference on Industrial Computed Tomography, Wels, Austria."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Jun, K. (2022). Highly accurate quantum optimization algorithm for CT image reconstructions based on sinogram patterns. arXiv.","DOI":"10.21203\/rs.3.rs-1855285\/v1"},{"key":"ref_50","unstructured":"D-Wave (2023, September 22). QPU Solvers: Minor-Embedding. Available online: https:\/\/docs.dwavesys.com\/docs\/latest\/handbook_embedding.html."},{"key":"ref_51","unstructured":"D-Wave (2022, October 27). Hybrid Solvers for Quadratic Optimization. Available online: https:\/\/www.dwavesys.com\/media\/soxph512\/hybrid-solvers-for-quadratic-optimization.pdf."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"e453","DOI":"10.7717\/peerj.453","article-title":"scikit-image: Image processing in Python","volume":"2","author":"Boulogne","year":"2014","journal-title":"PeerJ"},{"key":"ref_53","unstructured":"Alpaydin, E., and Kaynak, C. (1998). Optical Recognition of Handwritten Digits, UCI Machine Learning Repository."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"2212","DOI":"10.1038\/s41467-022-29887-0","article-title":"Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian","volume":"13","author":"Yan","year":"2022","journal-title":"Nat. Commun."}],"container-title":["Journal of Imaging"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2313-433X\/9\/10\/221\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:05:10Z","timestamp":1760130310000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2313-433X\/9\/10\/221"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,11]]},"references-count":54,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2023,10]]}},"alternative-id":["jimaging9100221"],"URL":"https:\/\/doi.org\/10.3390\/jimaging9100221","relation":{},"ISSN":["2313-433X"],"issn-type":[{"value":"2313-433X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,11]]}}}