Skip to main content
SpringerLink
Log in

Parallel hybrid quantum-classical machine learning for kernelized time-series classification

  • Research Article
  • Published:
Quantum Machine Intelligence Aims and scope Submit manuscript

Abstract

Supervised time-series classification garners widespread interest because of its applicability throughout a broad application domain including finance, astronomy, biosensors, and many others. In this work, we tackle this problem with hybrid quantum-classical machine learning, deducing pairwise temporal relationships between time-series instances using a time-series Hamiltonian kernel (TSHK). A TSHK is constructed with a sum of inner products generated by quantum states evolved using a parameterized time evolution operator. This sum is then optimally weighted using techniques derived from multiple kernel learning. Because we treat the kernel weighting step as a differentiable convex optimization problem, our method can be regarded as an end-to-end learnable hybrid quantum-classical-convex neural network, or QCC-net, whose output is a data set-generalized kernel function suitable for use in any kernelized machine learning technique such as the support vector machine (SVM). Using our TSHK as input to a SVM, we classify univariate and multivariate time-series using quantum circuit simulators and demonstrate the efficient parallel deployment of the algorithm to 127-qubit superconducting quantum processors using quantum multi-programming.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Data and code availability

The data used to produce Figs. 3, 4, and 7 can be found at https://doi.org/10.5281/zenodo.7996534 alongside a code example implementing a QCC-net for the Sine vs Cosine classification shown in Fig. 1.

References

  • Agrawal A, Amos B, Barratt S, Boyd S, Diamond S, Kolter JZ (2019a) Differentiable convex optimization layers. In: Wallach H, Larochelle H, Beygelzimer A, d’Alché-Buc F, Fox E, Garnett R (eds) Advances in neural information processing systems, vol 32. Curran Associates, Inc

  • Agrawal A, Barratt S, Boyd S, Busseti E, Moursi WM (2019b) Differentiating through a cone program. https://doi.org/10.48550/ARXIV.1904.09043

  • Aiolli F, Donini M (2015) EasyMKL: a scalable multiple kernel learning algorithm. Neurocomput 169:215. https://doi.org/10.1016/j.neucom.2014.11.078

    Article  Google Scholar 

  • Aiolli F, Martino GDS, Sperduti A (2008) A kernel method for the optimization of the margin distribution. In: Artificial neural networks - ICANN 2008. Springer Berlin Heidelberg, pp 305–314. https://doi.org/10.1007/978-3-540-87536-9_32

  • Badiane M, O’Reilly M, Cunningham P (2018) Kernel methods for time series classification and regression. In: AICS pp 54–65

  • Bailly A (2018) Time series classification algorithms with applications in remote sensing. Ph.D. thesis, Université Rennes 2

  • Baker JS, Horowitz H, Radha SK, Fernandes S, Jones C, Noorani N, Skavysh V, Lamontangne P, Sanders BC (2022) Quantum variational rewinding for time series anomaly detection. https://doi.org/10.48550/ARXIV.2210.16438

  • Bergholm V, Izaac J, Schuld M, Gogolin C, Ahmed S, Ajith V, Alam MS, Alonso-Linaje G, AkashNarayanan B, Asadi A, Arrazola J. M, 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 R. A, Lee C, Loke T, Lowe A, McKiernan K, Meyer JJ, Montañez-Barrera JA, Moyard R, Niu Z, O’Riordan LJ, Oud S, Panigrahi A, Park C-Y, Polatajko D, Quesada N, Roberts C, Sá N, Schoch I, Shi B, Shu S, Sim S, Singh A, Strandberg I, Soni J, Száva A, Thabet S, Vargas-Hernández 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. https://doi.org/10.48550/ARXIV.1811.04968

  • Bharti K, Cervera-Lierta A, Kyaw TH, Haug T, Alperin-Lea S, Anand A, Degroote M, Heimonen H, Kottmann JS, Menke T, Mok W-K, Sim S, Kwek L-C, Aspuru-Guzik A (2022) Noisy intermediate-scale quantum algorithms. Rev Mod Phys 94:015004. https://doi.org/10.1103/RevModPhys.94.015004

    Article  ADS  MathSciNet  CAS  Google Scholar 

  • Bjorken JD, Drell SD (1965) Relativistic quantum fields. McGraw-Hill

    Google Scholar 

  • Blázquez-García A, Conde A, Mori U, Lozano JA (2021) A review on outlier/anomaly detection in time series data. ACM Comput Surv 54:1. https://doi.org/10.1145/3444690

    Article  Google Scholar 

  • Bottou L, Curtis FE, Nocedal J (2018) Optimization methods for large-scale machine learning. SIAM Rev 60:223. https://doi.org/10.1137/16m1080173

    Article  MathSciNet  Google Scholar 

  • Bowles J, Wright VJ, Farkas M, Killoran N, Schuld M (2023) Contextuality and inductive bias in quantum machine learning. arXiv:2302.01365

  • Caro MC, Huang H-Y, Ezzell N, Gibbs J, Sornborger AT, Cincio L, Coles PJ, Holmes Z (2022) Out-of-distribution generalization for learning quantum dynamics. https://doi.org/10.48550/ARXIV.2204.10268

  • Choi K, Yi J, Park C, Yoon S (2021) Deep learning for anomaly detection in time-series data: review, analysis, and guidelines. IEEE Access 9:120043. https://doi.org/10.1109/access.2021.3107975

    Article  Google Scholar 

  • Cîrstoiu C, Holmes Z, Iosue J, Cincio L, Coles P. J, Sornborger A (2020) Variational fast forwarding for quantum simulation beyond the coherence time. npj Quantum Inf 6. https://doi.org/10.1038/s41534-020-00302-0

  • Clark S, Hyndman RJ, Pagendam D, Ryan LM (2020) Modern strategies for time series regression. Int Stat Rev 88. https://doi.org/10.1111/insr.12432

  • Cortes C, Vapnik V (1995) Support-vector networks. Mach Learn 20:273

    Article  Google Scholar 

  • Cross AW, Bishop LS, Sheldon S, Nation PD, Gambetta JM (2019) Validating quantum computers using randomized model circuits. Phys Rev A 100:032328

    Article  ADS  CAS  Google Scholar 

  • Das P, Tannu SS, Nair PJ, Qureshi M (2019) A case for multi-programming quantum computers. In: Proceedings of the 52nd Annual IEEE/ACM international symposium on microarchitecture. pp 291–303

  • Deb C, Zhang F, Yang J, Lee SE, Shah KW (2017) A review on time series forecasting techniques for building energy consumption. Renewa Sustain Energy Rev 74:902. https://doi.org/10.1016/j.rser.2017.02.085

    Article  Google Scholar 

  • Fábregues de los Santos L (2017) Forecasting financial time series using multiple Kernel Learning. Master’s thesis, Universitat Politècnica de Catalunya

  • Farhi E, Goldstone J, Gutmann S (2014) A quantum approximate optimization algorithm. arXiv:1411.4028

  • Fawaz HI, Forestier G, Weber J, Idoumghar L, Muller P-A (2019) Deep learning for time series classification: a review. Data Min Knowl Discov 33:917. https://doi.org/10.1007/s10618-019-00619-1

    Article  MathSciNet  Google Scholar 

  • Ghukasyan A, Baker JS, Goktas O, Carrasquilla J, Radha S. K (2023) Quantum-classical multiple kernel learning. arXiv:2305.17707 [quant-ph]

  • Gibbs J, Holmes Z, Caro MC, Ezzell N, Huang H-Y, Cincio L, Sornborger AT, Coles PJ (2022) Dynamical simulation via quantum machine learning with provable generalization. https://doi.org/10.48550/ARXIV.2204.10269

  • Gönen M, Alpaydın E (2011) Multiple kernel learning algorithms. The J Mach Learn Res 12:2211

    MathSciNet  Google Scholar 

  • Gönen M, Alpaydın E (2011) Multiple kernel learning algorithms. The J Mach Learn Res 12:2211

    MathSciNet  Google Scholar 

  • Hadfield S, Wang Z, O’Gorman B, Rieffel EG, Venturelli D, Biswas R (2019) From the quantum approximate optimization algorithm to a quantum alternating operator ansatz. Algoritm 12

  • Hastie T, Tibshirani R, Friedman JH, Friedman JH (2009) The elements of statistical learning: data mining, inference, and prediction. vol 2. Springer, p 134

  • Havlíček V, Córcoles AD, Temme K, Harrow AW, Kandala A, Chow JM, Gambetta JM (2019) Supervised learning with quantum-enhanced feature spaces. Nat 567:209. https://doi.org/10.1038/s41586-019-0980-2

    Article  ADS  CAS  Google Scholar 

  • Hochreiter S, Schmidhuber J (1997) Long short-term memory. Neural Comput 9:1735. https://doi.org/10.1162/neco.1997.9.8.1735

    Article  PubMed  CAS  Google Scholar 

  • Horowitz H, Rao P, Radha SK (2022) A quantum generative model for multi-dimensional time series using Hamiltonian learning. https://doi.org/10.48550/ARXIV.2204.06150

  • Hubregtsen T, Wierichs D, Gil-Fuster E, Derks P-JHS, Faehrmann PK, Meyer JJ (2022) Training quantum embedding kernels on near-term quantum computers. Phys Rev A 106:042431. https://doi.org/10.1103/PhysRevA.106.042431

    Article  ADS  MathSciNet  CAS  Google Scholar 

  • Keskar NS, Mudigere D, Nocedal J, Smelyanskiy M, Tang PTP (2017) On large-batch training for deep learning: generalization gap and sharp minima. arXiv:1609.04836 [cs.LG]

  • Kingma DP, Ba J (2017) Adam: a method for stochastic optimization. arXiv:1412.6980

  • Kübler JM, Buchholz S, Schölkopf B (2021) The inductive bias of quantum kernels. arXiv:2106.03747

  • Liu L, Dou X (2021) QuCloud: a new qubit mapping mechanism for multi-programming quantum computing in cloud environment. In: 2021 IEEE International symposium on High-Performance Computer Architecture (HPCA). IEEE, pp 167–178

  • Lloyd S, Schuld M, Ijaz A, Izaac J, Killoran N (2020) Quantum embeddings for machine learning. arXiv:2001.03622

  • McClean JR, Boixo S, Smelyanskiy VN, Babbush R, Neven H (2018) Barren plateaus in quantum neural network training landscapes. Nat Commun 9. https://doi.org/10.1038/s41467-018-07090-4

  • Melis G, Kočiský T, Blunsom P (2019) Mogrifier lstm. https://doi.org/10.48550/ARXIV.1909.01792

  • Mitarai K, Negoro M, Kitagawa M, Fujii K (2018) Quantum circuit learning. Phys Rev A 98. https://doi.org/10.1103/physreva.98.032309

  • Murali P, McKay DC, Martonosi M, Javadi-Abhari A (2020) Software mitigation of crosstalk on noisy intermediate-scale quantum computers. In: Proceedings of the twenty-fifth international conference on architectural support for programming languages and operating systems. pp 1001–1016

  • Nguyen T, Baraniuk R, Bertozzi A, Osher S, Wang B (2020) Momentumrnn: integrating momentum into recurrent neural networks. In: Larochelle H, Ranzato M, Hadsell R, Balcan M, Lin H (eds) Advances in neural information processing systems, vol 33. Curran Associates, Inc., pp 1924–1936

  • Niu S, Todri-Sanial A (2021) Enabling multi-programming mechanism for quantum computing in the NISQ era. arXiv:2102.05321

  • Niu S, Todri-Sanial A (2022) How parallel circuit execution can be useful for NISQ computing? In: 2022 Design, Automation & Test in Europe Conference & Exhibition (DATE). IEEE, pp 1065–1070

  • O’Donoghue B (2021) Operator splitting for a homogeneous embedding of the linear complementarity problem. SIAM J Optim 31:1999

    Article  MathSciNet  Google Scholar 

  • O’Donoghue B, Chu E, Parikh N, Boyd S (2016) Conic optimization via operator splitting and homogeneous self-dual embedding. J Optim Theor Appl 169:1042. http://stanford.edu/~boyd/papers/scs.html

  • O’Donoghue B, Chu E, Parikh N, Boyd S (2016a) Conic optimization via operator splitting and homogeneous self-dual embedding. J Optim Theor Appl 169:1042. http://stanford.edu/~boyd/papers/scs.html

  • Ohkura Y (2021) Crosstalk-aware NISQ Multi-programming. Keio Univ, Tokyo, Japan, Fac Policy Manag

    Google Scholar 

  • Ohkura Y, Satoh T, Van Meter R (2022) Simultaneous execution of quantum circuits on current and near-future NISQ systems. IEEE Trans Quantum Eng

  • Park G, Zhang K, Yu K, Korepin V (2023) Quantum multi-programming for Grover’s search. Quantum Inf Process 22:54

    Article  ADS  MathSciNet  Google Scholar 

  • Pesah A, Cerezo M, Wang S, Volkoff T, Sornborger AT, Coles PJ (2021) Absence of barren plateaus in quantum convolutional neural networks. Phys Rev X 11:041011. https://doi.org/10.1103/PhysRevX.11.041011

    Article  CAS  Google Scholar 

  • Radha SK (2021) Quantum constraint learning for quantum approximate optimization algorithm. https://doi.org/10.48550/ARXIV.2105.06770

  • Ratanamahatana CA, Keogh E (2005) Three myths about dynamic time warping data mining. In: Proceedings of the 2005 SIAM international conference on data mining. SIAM, pp 506–510

  • Rüping S (2001) SVM kernels for time series analysis. Tech. Rep. (Technical report 2001)

  • Sakoe H, Chiba S (1978) Dynamic programming algorithm optimization for spoken word recognition. IEEE Trans Acoust, Speech, Signal Process 26:43. https://doi.org/10.1109/TASSP.1978.1163055

    Article  Google Scholar 

  • Schuld M, Bocharov A, Svore KM, Wiebe N (2020) Circuit-centric quantum classifiers. Phys Rev A 101:032308. https://doi.org/10.1103/PhysRevA.101.032308

    Article  ADS  MathSciNet  CAS  Google Scholar 

  • Schuld M, Bergholm V, Gogolin C, Izaac J, Killoran N (2019) Evaluating analytic gradients on quantum hardware. Phys Rev A 99. https://doi.org/10.1103/physreva.99.032331

  • Schuld M, Killoran N (2019) Quantum machine learning in feature Hilbert spaces. Phys Rev Lett 122. https://doi.org/10.1103/physrevlett.122.040504

  • Stone MH (1932) On one-parameter unitary groups in Hilbert space. The Annal Math 33:643. https://doi.org/10.2307/1968538

    Article  MathSciNet  Google Scholar 

  • Tino P (2020) Dynamical systems as temporal feature spaces. The J Mach Learn Res 21:1649

    MathSciNet  Google Scholar 

  • Torres JF, Hadjout D, Sebaa A, Martínez-Álvarez F, Troncoso A (2021) Deep learning for time series forecasting: a survey. Big Data 9:3. https://doi.org/10.1089/big.2020.0159

    Article  PubMed  Google Scholar 

  • Vaswani A, Shazeer N, Parmar N, Uszkoreit J, Jones L, Gomez AN, Kaiser L u, Polosukhin I (2017) Attention is all you need. In: Guyon I, Luxburg UV, Bengio S, Wallach H, Fergus R, Vishwanathan S, Garnett R (eds) Advances in neural information processing systems, vol 30. Curran Associates, Inc

  • Welch J, Greenbaum D, Mostame S, Aspuru-Guzik A (2014) Efficient quantum circuits for diagonal unitaries without ancillas. New J Phys 16:033040. https://doi.org/10.1088/1367-2630/16/3/033040

    Article  Google Scholar 

  • Wiebe N, Berry D, Høyer P, Sanders BC (2010) Higher order decompositions of ordered operator exponentials. J Phys A: Math Theor 43:065203. https://doi.org/10.1088/1751-8113/43/6/065203

    Article  ADS  MathSciNet  Google Scholar 

  • Yang C-HH, Tsai Y-Y, Chen P-Y (2021) Voice2series: reprogramming acoustic models for time series classification. In: International conference on machine learning. PMLR, pp 11808–11819

  • Zerveas G, Jayaraman S, Patel D, Bhamidipaty A, Eickhoff C (2021) A transformer-based framework for multivariate time series representation learning. In: Proceedings of the 27th ACM SIGKDD conference on knowledge discovery; data mining. ACM, https://doi.org/10.1145/3447548.3467401

  • Zhang C, Kuppannagari SR, Kannan R, Prasanna VK (2008) Generative adversarial network for synthetic time series data generation in smart grids. In: 2018 IEEE international conference on communications, control, and computing technologies for smart grids (SmartGridComm). IEEE, https://doi.org/10.1109/smartgridcomm.2018.8587464

Download references

Acknowledgements

We acknowledge the use of IBM Quantum services for this work. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Quantum team. This research used quantum computing resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231 using NERSC award DDR-ERCAP0024165.

Funding

This work was partially supported by the U.S. Department of Energy, Office of Advanced Scientific Computing Research, Grant No. DE-SC-0012704 (GP, KY) and the SBU-BNL Seed Grant 2019, 2023 (KY).

Author information

Authors and Affiliations

  1. Agnostiq Inc., 325 Front St W, Toronto, ON, M5V 2Y1, Canada

    Jack S. Baker, Ara Ghukasyan, Oktay Goktas & Santosh Kumar Radha

  2. Computational Science Initiative, Brookhaven National Laboratory, Upton, New York, NY, 11973, USA

    Gilchan Park & Kwangmin Yu

Authors
  1. Jack S. Baker
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gilchan Park
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kwangmin Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ara Ghukasyan
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Oktay Goktas
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Santosh Kumar Radha
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Jack S. Baker made the figures, conducted the numerical experiments on quantum circuit simulators, wrote the manuscript, and helped develop the theoretical method. Gilchan Park conducted the QMP experiments and contributed to the manuscript. Kwangmin Yu oversaw the QML experiments and contributed to the manuscript and the supplemental information. Ara Ghukasayan helped to develop the theoretical method and reviewed the manuscript. Oktay Goktas oversaw the numerical experiments and reviewed the manuscript. Santosh Kumar Radha seeded the idea of using time evolution operators to create a time dependent quantum kernel function and reviewed the manuscript.

Corresponding author

Correspondence to Kwangmin Yu.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file 1 (pdf 907 KB)

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baker, J.S., Park, G., Yu, K. et al. Parallel hybrid quantum-classical machine learning for kernelized time-series classification. Quantum Mach. Intell. 6, 18 (2024). https://doi.org/10.1007/s42484-024-00149-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s42484-024-00149-0

Keywords

Search

Navigation