{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T04:55:57Z","timestamp":1776920157048,"version":"3.51.2"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,1,26]],"date-time":"2023-01-26T00:00:00Z","timestamp":1674691200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,1,26]],"date-time":"2023-01-26T00:00:00Z","timestamp":1674691200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator"},{"name":"U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator"},{"name":"U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator"},{"name":"U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Quantum Mach. Intell."],"published-print":{"date-parts":[[2023,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Tensor network quantum machine learning (QML) models are promising applications on near-term quantum hardware. While decoherence of qubits is expected to decrease the performance of QML models, it is unclear to what extent the diminished performance can be compensated for by adding ancillas to the models and accordingly increasing the virtual bond dimension of the models. We investigate here the competition between decoherence and adding ancillas on the classification performance of two models, with an analysis of the decoherence effect from the perspective of regression. We present numerical evidence that the fully decohered unitary tree tensor network (TTN) with two ancillas performs at least as well as the non-decohered unitary TTN, suggesting that it is beneficial to add at least two ancillas to the unitary TTN regardless of the amount of decoherence may be consequently introduced.<\/jats:p>","DOI":"10.1007\/s42484-022-00095-9","type":"journal-article","created":{"date-parts":[[2023,1,26]],"date-time":"2023-01-26T08:02:58Z","timestamp":1674720178000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Decohering tensor network quantum machine learning models"],"prefix":"10.1007","volume":"5","author":[{"given":"Haoran","family":"Liao","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ian","family":"Convy","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhibo","family":"Yang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"K. Birgitta","family":"Whaley","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,1,26]]},"reference":[{"key":"95_CR1","doi-asserted-by":"publisher","first-page":"043001","DOI":"10.1088\/2058-9565\/ab4eb5","volume":"4","author":"M Benedetti","year":"2019","unstructured":"Benedetti M, Lloyd E, Sack S, Fiorentini M (2019) Parameterized quantum circuits as machine learning models. Quantum Sci Technol 4:043001. ISSN 2058-9565, https:\/\/doi.org\/10.1088%2F2058-9565%2Fab4eb5","journal-title":"Quantum Sci Technol"},{"key":"95_CR2","unstructured":"Biamonte J, Bergholm V (2017) Tensor networks in a nutshell. arXiv:1708.00006"},{"key":"95_CR3","doi-asserted-by":"crossref","unstructured":"Bridgeman JC, Chubb CT (2017) Hand-waving and interpretive dance: an introductory course on tensor networks. J Phys A: Math Theor 50. ISSN 17518121, https:\/\/iopscience.iop.org\/article\/10.1088\/1751-8121\/aa6dc3\/meta","DOI":"10.1088\/1751-8121\/aa6dc3"},{"key":"95_CR4","unstructured":"Clanuwat T, Bober-Irizar M, Kitamoto A, Lamb A, Yamamoto K, Ha D (2018) Deep learning for classical Japanese literature. arXiv:1812.01718"},{"key":"95_CR5","unstructured":"Cohen N, Shashua A (2016) Convolutional rectifier networks as generalized tensor decompositions. In: Proceedings of ICML. http:\/\/proceedings.mlr.press\/v48\/cohenb16.pdf, pp 955\u2013963"},{"key":"95_CR6","doi-asserted-by":"publisher","first-page":"1273","DOI":"10.1038\/s41567-019-0648-8","volume":"15","author":"I Cong","year":"2019","unstructured":"Cong I, Choi S, Lukin MD (2019) Quantum convolutional neural networks. Nat Phys 15:1273\u20131278. https:\/\/doi.org\/10.1038\/s41567-019-0648-8","journal-title":"Nat Phys"},{"key":"95_CR7","doi-asserted-by":"publisher","first-page":"015017","DOI":"10.1088\/2632-2153\/ac44a9","volume":"3","author":"I Convy","year":"2022","unstructured":"Convy I, Huggins WJ, Liao H, Whaley KB (2022) Mutual information scaling for tensor network machine learning. Mach Learn Sci Technol 3:015017. https:\/\/doi.org\/10.1088%2F2632-2153%2Fac44a9","journal-title":"Mach Learn Sci Technol"},{"key":"95_CR8","unstructured":"Eisert J (2013) Entanglement and tensor network states. In: Pavarini E, Koch E, Schollw\u00f6ck U (eds) Emergent phenomena in correlated matter modeling and simulation. Chap. 17, ISBN 978-3-89336-884-6 https:\/\/www.cond-mat.de\/events\/correl13\/manuscripts\/, vol 3. Verlag des Forschungszentrum J\u00fclich"},{"key":"95_CR9","doi-asserted-by":"crossref","unstructured":"Evenbly G, Vidal G (2009) Algorithms for entanglement renormalization. Phys Rev B 79. ISSN 10980121, https:\/\/journals.aps.org\/prb\/abstract\/10.1103\/PhysRevB.79.144108","DOI":"10.1103\/PhysRevB.79.144108"},{"key":"95_CR10","doi-asserted-by":"publisher","first-page":"891","DOI":"10.1007\/s10955-011-0237-4","volume":"145","author":"G Evenbly","year":"2011","unstructured":"Evenbly G, Vidal G (2011) Tensor network states and geometry. J Stat Phys 145:891. https:\/\/doi.org\/10.1007%2Fs10955-011-0237-4","journal-title":"J Stat Phys"},{"key":"95_CR11","unstructured":"Glasser I, Sweke R, Pancotti N, Eisert J, Cirac JI (2019) Expressive power of tensor-network factorizations for probabilistic modeling, with applications from hidden Markov models to quantum machine learning. In: Proceedings of NIPS, pp 1498\u20131510. arXiv:1907.03741"},{"key":"95_CR12","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1038\/s41534-018-0116-9","volume":"4","author":"E Grant","year":"2018","unstructured":"Grant E, Benedetti M, Cao S, Hallam A, Lockhart J, Stojevic V, Green AG, Severini S (2018) Hierarchical quantum classifiers. NPJ Quantum Inf 4:65. ISSN 2056-6387. http:\/\/www.nature.com\/articles\/s41534-018-0116-9","journal-title":"NPJ Quantum Inf"},{"key":"95_CR13","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1038\/s41586-019-0980-2","volume":"567","author":"V Havl\u00ed\u010dek","year":"2019","unstructured":"Havl\u00ed\u010dek V, C\u00f3rcoles AD, Temme K, Harrow AW, Kandala A, Chow JM, Gambetta JM (2019) Supervised learning with quantum-enhanced feature spaces. Nature 567:209\u2013212. https:\/\/doi.org\/10.1038\/s41586-019-0980-2","journal-title":"Nature"},{"key":"95_CR14","doi-asserted-by":"crossref","unstructured":"He K, Zhang X, Ren S, Sun J (2016) Deep residual learning for image recognition. In: Proceedings of CVPR, pp 770\u2013778. arXiv:1512.03385","DOI":"10.1109\/CVPR.2016.90"},{"key":"95_CR15","doi-asserted-by":"publisher","first-page":"024001","DOI":"10.1088\/2058-9565\/aaea94","volume":"4","author":"WJ Huggins","year":"2019","unstructured":"Huggins WJ, Patil P, Mitchell B, Whaley KB, Stoudenmire EM (2019) Towards quantum machine learning with tensor networks. Quantum Sci Technol 4:024001. https:\/\/doi.org\/10.1088%2F2058-9565%2Faaea94","journal-title":"Quantum Sci Technol"},{"key":"95_CR16","unstructured":"Kingma DP, Adam JB (2015) A method for stochastic optimization. arXiv:1412.6980"},{"key":"95_CR17","doi-asserted-by":"publisher","first-page":"1708","DOI":"10.1109\/TIT.2008.917696","volume":"54","author":"D Kretschmann","year":"2008","unstructured":"Kretschmann D, Schlingemann D, Werner RF (2008) The information-disturbance tradeoff and the continuity of Stinespring\u2019s representation. IEEE Trans Inf Theory 54:1708. https:\/\/ieeexplore.ieee.org\/document\/4475375","journal-title":"IEEE Trans Inf Theory"},{"key":"95_CR18","doi-asserted-by":"publisher","first-page":"032420","DOI":"10.1103\/PhysRevA.102.032420","volume":"102","author":"R Larose","year":"2020","unstructured":"Larose R, Coyle B (2020) Robust data encodings for quantum classifiers. Phys Rev A 102:032420. arXiv:2003.01695","journal-title":"Phys Rev A"},{"key":"95_CR19","unstructured":"LeCun Y, Cortes C, Burges CJ (2010) MNIST handwritten digit database. http:\/\/yann.lecun.com\/exdb\/mnist"},{"key":"95_CR20","unstructured":"Levine Y, Yakira D, Cohen N, Shashua A (2018) Deep learning and quantum entanglement: fundamental connections with implications to network design. In: Proceedings of ICLR. arXiv:1704.01552"},{"key":"95_CR21","doi-asserted-by":"publisher","first-page":"042427","DOI":"10.1103\/PhysRevA.103.042427","volume":"103","author":"H Liao","year":"2021","unstructured":"Liao H, Convy I, Huggins WJ, Whaley KB (2021) Robust in practice: adversarial attacks on quantum machine learning. Phys Rev A 103:042427. https:\/\/doi.org\/10.1103\/PhysRevA.103.042427","journal-title":"Phys Rev A"},{"key":"95_CR22","unstructured":"Liaw R, Liang E, Nishihara R, Moritz P, Gonzalez JE, Stoica I (2018) Tune: a research platform for distributed model selection and training. arXiv:1807.05118"},{"key":"95_CR23","unstructured":"Lu S, Kan\u00e1sz-Nagy M, Kukuljan I, Cirac JI (2021) Tensor networks and efficient descriptions of classical data. arXiv:2103.06872"},{"key":"95_CR24","unstructured":"Miller J, Roeder G, Bradley T-D (2021) Probabilistic graphical models and tensor networks: a hybrid framework. arXiv:2106.15666"},{"key":"95_CR25","doi-asserted-by":"publisher","first-page":"032309","DOI":"10.1103\/PhysRevA.98.032309","volume":"98","author":"K Mitarai","year":"2018","unstructured":"Mitarai K, Negoro M, Kitagawa M, Fujii K (2018) Quantum circuit learning. Phys Rev A 98:032309. https:\/\/doi.org\/10.1103\/PhysRevA.98.032309","journal-title":"Phys Rev A"},{"key":"95_CR26","doi-asserted-by":"publisher","first-page":"79","DOI":"10.22331\/q-2018-08-06-79","volume":"2","author":"J Preskill","year":"2018","unstructured":"Preskill J (2018) Quantum computing in the NISQ era and beyond. Quantum 2:79. ISSN 2521-327X, arXiv:1801.00862","journal-title":"Quantum"},{"key":"95_CR27","doi-asserted-by":"publisher","first-page":"035036","DOI":"10.1088\/2632-2153\/abffe8","volume":"2","author":"JA Reyes","year":"2021","unstructured":"Reyes JA, Stoudenmire EM (2021) Multi-scale tensor network architecture for machine learning. Mach Learn Sci Technol 2:035036. https:\/\/doi.org\/10.1088\/2632-2153\/abffe8","journal-title":"Mach Learn Sci Technol"},{"key":"95_CR28","doi-asserted-by":"crossref","unstructured":"Robeva E, Seigal A (2019) Duality of graphical models and tensor networks. arXiv:1710.01437","DOI":"10.1093\/imaiai\/iay009"},{"key":"95_CR29","doi-asserted-by":"publisher","first-page":"022320","DOI":"10.1103\/PhysRevA.74.022320","volume":"74","author":"Y Shi","year":"2006","unstructured":"Shi Y, Duan L, Vidal G (2006) Classical simulation of quantum many-body systems with a tree tensor network. Phys Rev A 74:022320. https:\/\/doi.org\/10.1103\/PhysRevA.74.022320","journal-title":"Phys Rev A"},{"key":"95_CR30","doi-asserted-by":"publisher","first-page":"034003","DOI":"10.1088\/2058-9565\/aaba1a","volume":"3","author":"EM Stoudenmire","year":"2018","unstructured":"Stoudenmire EM (2018) Learning relevant features of data with multi-scale tensor networks. Quantum Sci Technol 3:034003. https:\/\/doi.org\/10.1088%2F2058-9565%2Faaba1a","journal-title":"Quantum Sci Technol"},{"key":"95_CR31","unstructured":"Stoudenmire EM, Schwab DJ (2016) Supervised learning with quantum-inspired tensor networks. In: Proceedings of NIPS, pp 4799\u20134807. arXiv:1605.05775"},{"key":"95_CR32","doi-asserted-by":"publisher","first-page":"8485","DOI":"10.1088\/0305-4470\/34\/41\/307","volume":"34","author":"G Tanner","year":"2001","unstructured":"Tanner G (2001) Unitary-stochastic matrix ensembles and spectral statistics. J Phys A: Mat Gen 34:8485. https:\/\/doi.org\/10.1088%2F0305-4470%2F34%2F41%2F307","journal-title":"J Phys A: Mat Gen"},{"key":"95_CR33","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1103\/PhysRevLett.99.220405","volume":"99","author":"G Vidal","year":"2007","unstructured":"Vidal G (2007) Entanglement renormalization. Phys Rev Lett 99:1\u20134. https:\/\/doi.org\/10.1103\/PhysRevLett.99.220405","journal-title":"Phys Rev Lett"},{"key":"95_CR34","doi-asserted-by":"publisher","first-page":"042408","DOI":"10.1103\/PhysRevA.104.042408","volume":"104","author":"ML Wall","year":"2021","unstructured":"Wall ML, D\u2019Aguanno G (2021) Tree-tensor-network classifiers for machine learning: from quantum inspired to quantum assisted. Phys Rev A 104:042408. https:\/\/doi.org\/10.1103\/PhysRevA.104.042408","journal-title":"Phys Rev A"},{"key":"95_CR35","doi-asserted-by":"publisher","DOI":"10.1017\/9781316848142","volume-title":"The theory of quantum information","author":"J Watrous","year":"2018","unstructured":"Watrous J (2018) The theory of quantum information. Cambridge University Press, Cambridge. https:\/\/cs.uwaterloo.ca\/~watrous\/TQI\/"},{"key":"95_CR36","unstructured":"Xiao H, Rasul K, Vollgraf R (2017) Fashion-MNIST: a novel image dataset for benchmarking machine learning algorithms. arXiv:1708.07747"},{"key":"95_CR37","doi-asserted-by":"publisher","first-page":"3425","DOI":"10.1088\/0305-4470\/36\/12\/333","volume":"36","author":"K Zyczkowski","year":"2003","unstructured":"Zyczkowski K, Kus M, Slomczynski W, Sommers H-J (2003) Random unistochastic matrices. J Phys A: Math Gen 36:3425. https:\/\/doi.org\/10.1088%2F0305-4470%2F3%2F12%2F333","journal-title":"J Phys A: Math Gen"}],"container-title":["Quantum Machine Intelligence"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s42484-022-00095-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s42484-022-00095-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s42484-022-00095-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,19]],"date-time":"2023-06-19T07:28:41Z","timestamp":1687159721000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s42484-022-00095-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,26]]},"references-count":37,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,6]]}},"alternative-id":["95"],"URL":"https:\/\/doi.org\/10.1007\/s42484-022-00095-9","relation":{},"ISSN":["2524-4906","2524-4914"],"issn-type":[{"value":"2524-4906","type":"print"},{"value":"2524-4914","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,26]]},"assertion":[{"value":"24 September 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 December 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 January 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Competing interests"}}],"article-number":"7"}}