{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T15:14:11Z","timestamp":1773155651275,"version":"3.50.1"},"reference-count":49,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,9,18]],"date-time":"2025-09-18T00:00:00Z","timestamp":1758153600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,9,18]],"date-time":"2025-09-18T00:00:00Z","timestamp":1758153600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Quantum Mach. Intell."],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Recent proposals for quantum generative adversarial networks (GANs) suffer from the issue of mode collapse, analogous to classical GANs, wherein the distribution learnt by the GAN fails to capture the high mode complexities of the target distribution. Mode collapse can arise due to the use of uninformed prior distributions in the generative learning task. To alleviate the issue of mode collapse for quantum GANs, this work presents a novel hybrid quantum-classical generative model, the VAE-QWGAN, which combines the strengths of a classical variational autoencoder (VAE) with a hybrid quantum Wasserstein GAN (QWGAN). The VAE-QWGAN fuses the VAE decoder and QWGAN generator into a single quantum model and utilizes the VAE encoder for data-dependent latent vector sampling during training. This, in turn, enhances the diversity and quality of generated images. To generate new data from the trained model at inference, we sample from a Gaussian mixture model (GMM) prior that is learnt on the latent vectors generated during training. We conduct extensive experiments for image generation QGANs on MNIST\/Fashion-MNIST datasets and compute a range of metrics that measure the diversity and quality of generated samples. We show that VAE-QWGAN demonstrates significant improvement over existing QGAN approaches.<\/jats:p>","DOI":"10.1007\/s42484-025-00314-z","type":"journal-article","created":{"date-parts":[[2025,9,18]],"date-time":"2025-09-18T12:52:35Z","timestamp":1758199955000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["VAE-QWGAN: addressing mode collapse in quantum GANs via autoencoding priors"],"prefix":"10.1007","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-4313-9650","authenticated-orcid":false,"given":"Aaron Mark","family":"Thomas","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0009-0007-3004-6367","authenticated-orcid":false,"given":"Harry","family":"Youel","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8872-3462","authenticated-orcid":false,"given":"Sharu Theresa","family":"Jose","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,9,18]]},"reference":[{"issue":"6","key":"314_CR1","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1038\/s43588-021-00084-1","volume":"1","author":"A Abbas","year":"2021","unstructured":"Abbas A, Sutter D, Zoufal C, Lucchi A, Figalli A, Woerner S (2021) The power of quantum neural networks. Nat Comput Sci 1(6):403\u2013409","journal-title":"Nat Comput Sci"},{"key":"314_CR2","unstructured":"Arjovsky M, Chintala S, Bottou L (2017) Wasserstein generative adversarial networks. In: Precup D, Teh YW (eds) Proceedings of the 34th international conference on machine learning. proceedings of machine learning Research, vol 70. PMLR, pp 214\u2013223. https:\/\/proceedings.mlr.press\/v70\/arjovsky17a.html"},{"key":"314_CR3","unstructured":"Arjovsky M, Chintala S, Bottou L (2017a) Wasserstein generative adversarial networks. In: Precup D, Teh YW (eds) Proceedings of the 34th international conference on machine learning. Proceedings of Machine Learning Research, vol 70. PMLR, pp 214\u2013223. https:\/\/proceedings.mlr.press\/v70\/arjovsky17a.html"},{"key":"314_CR4","doi-asserted-by":"crossref","unstructured":"Barthe A, Grossi M, Vallecorsa S, Tura J, Dunjko V (2024) Expressivity of parameterized quantum circuits for generative modeling of continuous multivariate distributions. arXiv:2402.09848","DOI":"10.1038\/s41534-025-01064-3"},{"key":"314_CR5","unstructured":"Bergholm V, Izaac J, Schuld M, Gogolin C, Ahmed S, Ajith V, Sohaib Alam M, Alonso-Linaje G, AkashNarayanan B, Asadi A, Arrazola JM, Azad U, Banning S, Blank C, Bromley TR, Cordier BA, Ceroni J, Delgado A, Di Matteo O, Dusko A, Garg T, Guala D, Hayes A, Hill R, Ijaz A, Isacsson T, Ittah D, Jahangiri S, Jain P, Jiang E, Khandelwal A, Kottmann K, Lang RA, Lee C, Loke T, Lowe A, McKiernan K, Meyer JJ, Monta\u00f1ez-Barrera JA, Moyard R, Niu Z, O\u2019Riordan LJ, Oud S, Panigrahi A, Park C-Y, Polatajko D, Quesada N, Roberts C, S\u00e1 N, Schoch I, Shi B, Shu S, Sim S, Singh A, Strandberg I, Soni J, Sz\u00e1va A, Thabet S, Vargas-Hern\u00e1ndez RA, Vincent T, Vitucci N, Weber M, Wierichs D, Wiersema R, Willmann M, Wong V, Zhang S, Killoran N (2018) PennyLane: automatic differentiation of hybrid quantum-classical computations. arXiv:1811.04968 [quant-ph]"},{"key":"314_CR6","unstructured":"Bhattacharyya A, Hanselmann M, Fritz M, Schiele B, Straehle C-N (2019) Conditional flow variational autoencoders for structured sequence prediction. arXiv:1908.09008"},{"issue":"7671","key":"314_CR7","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1038\/nature23474","volume":"549","author":"J Biamonte","year":"2017","unstructured":"Biamonte J, Wittek P, Pancotti N, Rebentrost P, Wiebe N, Lloyd S (2017) Quantum machine learning. Nature 549(7671):195\u2013202. https:\/\/doi.org\/10.1038\/nature23474","journal-title":"Nature"},{"key":"314_CR8","doi-asserted-by":"publisher","first-page":"777","DOI":"10.22331\/q-2022-08-17-777","volume":"6","author":"C Bravo-Prieto","year":"2022","unstructured":"Bravo-Prieto C, Baglio J, C\u00e8 M, Francis A, Grabowska DM, Carrazza S (2022) Style-based quantum generative adversarial networks for Monte Carlo events. Quantum 6:777. https:\/\/doi.org\/10.22331\/q-2022-08-17-777","journal-title":"Quantum"},{"key":"314_CR9","unstructured":"Chang SY, Thanasilp S, Le Saux B, Vallecorsa S, Grossi M (2024) Latent style-based quantum GAN for high-quality image generation. In: arXiv e-prints. pp 2406\u201302668. arXiv:2406.02668 [quant-ph]"},{"issue":"1","key":"314_CR10","doi-asserted-by":"publisher","first-page":"012324","DOI":"10.1103\/PhysRevA.98.012324","volume":"98","author":"P-L Dallaire-Demers","year":"2018","unstructured":"Dallaire-Demers P-L, Killoran N (2018) Quantum generative adversarial networks. Phys Rev A 98(1):012324","journal-title":"Phys Rev A"},{"key":"314_CR11","doi-asserted-by":"crossref","unstructured":"De\u00a0Falco F, Ceschini A, Sebastianelli A, Saux BL, Panella M (2024) Towards efficient quantum hybrid diffusion models. arXiv:2402.16147","DOI":"10.1007\/s42484-024-00224-6"},{"key":"314_CR12","doi-asserted-by":"publisher","unstructured":"Falcon W, The PyTorch Lightning team. PyTorch Lightning. https:\/\/doi.org\/10.5281\/zenodo.3828935. https:\/\/github.com\/Lightning-AI\/lightning","DOI":"10.5281\/zenodo.3828935"},{"issue":"11","key":"314_CR13","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1145\/3422622","volume":"63","author":"I Goodfellow","year":"2020","unstructured":"Goodfellow I, Pouget-Abadie J, Mirza M, Xu B, Warde-Farley D, Ozair S, Courville A, Bengio Y (2020) Generative adversarial networks. Commun ACM 63(11):139\u2013144. https:\/\/doi.org\/10.1145\/3422622","journal-title":"Commun ACM"},{"key":"314_CR14","unstructured":"Gulrajani I, Ahmed F, Arjovsky M, Dumoulin V, Courville A (2017) Improved training of Wasserstein GANs. In: Proceedings of the 31st International Conference on Neural Information Processing Systems. NIPS\u201917. Curran Associates Inc., Red Hook, NY, USA. pp 5769\u20135779"},{"key":"314_CR15","doi-asserted-by":"crossref","unstructured":"Han D (2013) Comparison of commonly used image interpolation methods. In: Conference of the 2nd International Conference on Computer Science and Electronics Engineering (ICCSEE 2013). Atlantis Press, pp 1556\u20131559","DOI":"10.2991\/iccsee.2013.391"},{"key":"314_CR16","unstructured":"Heusel M, Ramsauer H, Unterthiner T, Nessler B, Hochreiter S (2017) GANs trained by a two time-scale update rule converge to a local Nash equilibrium. Adv Neural Inf Process Syst 30"},{"key":"314_CR17","doi-asserted-by":"publisher","unstructured":"He K, Zhang X, Ren S, Sun J (2015) Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In: 2015 IEEE International Conference on Computer Vision (ICCV). pp 1026\u20131034. https:\/\/doi.org\/10.1109\/ICCV.2015.123","DOI":"10.1109\/ICCV.2015.123"},{"issue":"1","key":"314_CR18","doi-asserted-by":"publisher","first-page":"2761","DOI":"10.1126\/sciadv.aav2761","volume":"5","author":"L Hu","year":"2019","unstructured":"Hu L, Wu S-H, Cai W, Ma Y, Mu X, Xu Y, Wang H, Song Y, Deng D-L, Zou C-L et al (2019) Quantum generative adversarial learning in a superconducting quantum circuit. Sci Adv 5(1):2761","journal-title":"Sci Adv"},{"issue":"2","key":"314_CR19","doi-asserted-by":"publisher","first-page":"024051","DOI":"10.1103\/PhysRevApplied.16.024051","volume":"16","author":"H-L Huang","year":"2021","unstructured":"Huang H-L, Du Y, Gong M, Zhao Y, Wu Y, Wang C, Li S, Liang F, Lin J, Xu Y et al (2021) Experimental quantum generative adversarial networks for image generation. Phys Rev Appl 16(2):024051","journal-title":"Phys Rev Appl"},{"key":"314_CR20","unstructured":"Kingma DP, Welling M (2014) Auto-encoding variational Bayes. In: 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, 2014, Conference Track Proceedings"},{"key":"314_CR21","unstructured":"Kingma D, Ba J ((2015)) Adam: a method for stochastic optimization. In: International Conference on Learning Representations (ICLR), San Diega, CA, USA"},{"key":"314_CR22","doi-asserted-by":"crossref","unstructured":"K\u00f6lle M, Stenzel G, Stein J, Zielinski S, Ommer B, Linnhoff-Popien C (2024) Quantum denoising diffusion models. In: 2024 IEEE International Conference on Quantum Software (QSW). IEEE, pp 88\u201398","DOI":"10.1109\/QSW62656.2024.00023"},{"key":"314_CR23","unstructured":"Larsen ABL, S\u00f8nderby SK, Larochelle H, Winther O (2016) Autoencoding beyond pixels using a learned similarity metric. In: International conference on machine learning. PMLR, pp 1558\u20131566"},{"issue":"11","key":"314_CR24","doi-asserted-by":"publisher","first-page":"2278","DOI":"10.1109\/5.726791","volume":"86","author":"Y Lecun","year":"1998","unstructured":"Lecun Y, Bottou L, Bengio Y, Haffner P (1998) Gradient-based learning applied to document recognition. Proc IEEE 86(11):2278\u20132324. https:\/\/doi.org\/10.1109\/5.726791","journal-title":"Proc IEEE"},{"key":"314_CR25","unstructured":"Liu Y, Li Y (2021) Metrics of GANs. https:\/\/github.com\/yhlleo\/GAN-Metrics. Accessed on 09 Feb 2024, Online"},{"issue":"4","key":"314_CR26","doi-asserted-by":"publisher","first-page":"040502","DOI":"10.1103\/PhysRevLett.121.040502","volume":"121","author":"S Lloyd","year":"2018","unstructured":"Lloyd S, Weedbrook C (2018) Quantum generative adversarial learning. Phys Rev Lett 121(4):040502","journal-title":"Phys Rev Lett"},{"issue":"8","key":"314_CR27","doi-asserted-by":"publisher","first-page":"5493","DOI":"10.1109\/TPAMI.2024.3367532","volume":"46","author":"Y Luo","year":"2024","unstructured":"Luo Y, Yang Z (2024) DynGAN: solving mode collapse in GANs with dynamic clustering. IEEE Trans Pattern Anal Mach Intell 46(8):5493\u20135503. https:\/\/doi.org\/10.1109\/TPAMI.2024.3367532","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"key":"314_CR28","unstructured":"Maaten L, Hinton G (2008) Visualizing data using t-SNE. J Mach Learn Res 9(11)"},{"key":"314_CR29","doi-asserted-by":"crossref","unstructured":"Mustafa WA, Abdul\u00a0Kader MMM (2018) A review of histogram equalization techniques in image enhancement application. In: Journal of Physics: Conference Series, vol 1019. IOP Publishing, p 012026","DOI":"10.1088\/1742-6596\/1019\/1\/012026"},{"issue":"2","key":"314_CR30","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1002\/wics.199","volume":"4","author":"AA Neath","year":"2012","unstructured":"Neath AA, Cavanaugh JE (2012) The Bayesian information criterion: background, derivation, and applications. Wiley Interdisc Rev Comput Stat 4(2):199\u2013203","journal-title":"Wiley Interdisc Rev Comput Stat"},{"key":"314_CR31","unstructured":"Paszke A, Gross S, Massa F, Lerer A, Bradbury J, Chanan G, Killeen T, Lin Z, Gimelshein N, Antiga L et al (2019) Pytorch: an imperative style, high-performance deep learning library. Adv Neural Inf Process Syst 32"},{"key":"314_CR32","doi-asserted-by":"publisher","unstructured":"Preskill J (2018) Quantum computing in the NISQ era and beyond. Quantum 2:79. https:\/\/doi.org\/10.22331\/q-2018-08-06-79","DOI":"10.22331\/q-2018-08-06-79"},{"key":"314_CR33","doi-asserted-by":"crossref","unstructured":"Reynolds DA et al (2009) Gaussian mixture models Encyclopedia of biometrics. 741(659\u2013663):3","DOI":"10.1007\/978-0-387-73003-5_196"},{"key":"314_CR34","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1023\/B:VLSI.0000028532.53893.82","volume":"38","author":"AM Reza","year":"2004","unstructured":"Reza AM (2004) Realization of the contrast limited adaptive histogram equalization (CLAHE) for real-time image enhancement. J VLSI Signal Process Syst Signal Image Video Technol 38:35\u201344","journal-title":"J VLSI Signal Process Syst Signal Image Video Technol"},{"key":"314_CR35","unstructured":"Richardson, E., Weiss, Y.: On GANs and GMMs. In: Proceedings of the 32nd International Conference on Neural Information Processing Systems. NIPS\u201918, pp. 5852\u20135863. Curran Associates Inc., Red Hook, NY, USA (2018)"},{"key":"314_CR36","doi-asserted-by":"crossref","unstructured":"Silver D, Patel T, Cutler W, Ranjan A, Gandhi H, Tiwari D (2023) MosaiQ: quantum generative adversarial networks for image generation on NISQ computers. In: Proceedings of the IEEE\/CVF international conference on computer vision. pp 7030\u20137039","DOI":"10.1109\/ICCV51070.2023.00647"},{"key":"314_CR37","unstructured":"Srivastava A, Valkov L, Russell C, Gutmann MU, Sutton C (2017) VEEGAN: reducing mode collapse in GANs using implicit variational learning. Adv Neural Inf Process Syst 30"},{"key":"314_CR38","doi-asserted-by":"publisher","unstructured":"Szegedy C, Vanhoucke V, Ioffe S, Shlens J, Wojna Z (2016) Rethinking the inception architecture for computer vision. In: 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). pp 2818\u20132826. https:\/\/doi.org\/10.1109\/CVPR.2016.308","DOI":"10.1109\/CVPR.2016.308"},{"key":"314_CR39","doi-asserted-by":"crossref","unstructured":"Thanh-Tung H, Tran T (2020) Catastrophic forgetting and mode collapse in GANs. In: 2020 International Joint Conference on Neural Networks (ijcnn). IEEE, pp 1\u201310","DOI":"10.1109\/IJCNN48605.2020.9207181"},{"issue":"10","key":"314_CR40","doi-asserted-by":"publisher","first-page":"12321","DOI":"10.1109\/TPAMI.2023.3272029","volume":"45","author":"J Tian","year":"2023","unstructured":"Tian J, Sun X, Du Y, Zhao S, Liu Q, Zhang K, Yi W, Huang W, Wang C, Wu X, Hsieh M-H, Liu T, Yang W, Tao D (2023) Recent advances for quantum neural networks in generative learning. IEEE Trans Pattern Anal Mach Intell 45(10):12321\u201312340. https:\/\/doi.org\/10.1109\/TPAMI.2023.3272029","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"key":"314_CR41","doi-asserted-by":"crossref","unstructured":"Tomar S, Gupta A (2023) A review on mode collapse reducing GANs with GAN\u2019s algorithm and theory. GANs for Data Augmentation in Healthcare, pp 21\u201340","DOI":"10.1007\/978-3-031-43205-7_2"},{"key":"314_CR42","first-page":"1","volume":"4","author":"SL Tsang","year":"2022","unstructured":"Tsang SL, West MT, Erfani SM, Usman M (2022) Hybrid quantum\u2013classical generative adversarial network for high-resolution image generation. IEEE Trans Quant Eng 4:1\u201319","journal-title":"IEEE Trans Quant Eng"},{"issue":"1","key":"314_CR43","doi-asserted-by":"publisher","first-page":"6961","DOI":"10.1038\/s41467-021-27045-6","volume":"12","author":"S Wang","year":"2021","unstructured":"Wang S, Fontana E, Cerezo M, Sharma K, Sone A, Cincio L, Coles PJ (2021) Noise-induced barren plateaus in variational quantum algorithms. Nat Commun 12(1):6961","journal-title":"Nat Commun"},{"key":"314_CR44","doi-asserted-by":"crossref","unstructured":"Wang Z, Simoncelli E, Bovik A (2003) Multi-scale structural similarity for image quality assessment. Conference record of the asilomar conference on signals systems and computers 2:1398\u20131402. Conference Record of the Thirty-Seventh Asilomar Conference on Signals, Systems and Computers; Conference date: 09-11-2003 Through 12-11-2003","DOI":"10.1109\/ACSSC.2003.1292216"},{"key":"314_CR45","unstructured":"Wehenkel A, Louppe, G (2021) Diffusion priors in variational autoencoders. arXiv:2106.15671"},{"key":"314_CR46","unstructured":"Xiao H, Rasul K, Vollgraf R (2017) Fashion-MNIST: a novel image dataset for benchmarking machine learning algorithms. arXiv:1708.07747 [cs.LG]"},{"key":"314_CR47","unstructured":"Xu B, Wang N, Chen T, Li M (2015) Empirical evaluation of rectified activations in convolutional network"},{"key":"314_CR48","doi-asserted-by":"publisher","unstructured":"Zhang K (2021) On mode collapse in generative adversarial networks. In: Artificial Neural Networks and Machine Learning \u2013 ICANN 2021: 30th International Conference on Artificial Neural Networks, Bratislava, Slovakia, September 14\u201317, 2021, Proceedings, Part II. Springer, Berlin, Heidelberg, pp. 563\u2013574. https:\/\/doi.org\/10.1007\/978-3-030-86340-1_45","DOI":"10.1007\/978-3-030-86340-1_45"},{"issue":"1","key":"314_CR49","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1038\/s41534-019-0223-2","volume":"5","author":"C Zoufal","year":"2019","unstructured":"Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. NPJ Quant Inf 5(1):103","journal-title":"NPJ Quant Inf"}],"container-title":["Quantum Machine Intelligence"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s42484-025-00314-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s42484-025-00314-z","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s42484-025-00314-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T09:38:25Z","timestamp":1769679505000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s42484-025-00314-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,18]]},"references-count":49,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["314"],"URL":"https:\/\/doi.org\/10.1007\/s42484-025-00314-z","relation":{},"ISSN":["2524-4906","2524-4914"],"issn-type":[{"value":"2524-4906","type":"print"},{"value":"2524-4914","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,18]]},"assertion":[{"value":"11 April 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 September 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 September 2025","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":"Conflict of interest"}}],"article-number":"91"}}