Representation of binary classification trees with binary features by quantum circuits

Raoul Heese1, Patricia Bickert1, and Astrid Elisa Niederle2

1Fraunhofer ITWM, 67663 Kaiserslautern, Germany
2BASF SE, 67063 Ludwigshafen, Germany

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We propose a quantum representation of binary classification trees with binary features based on a probabilistic approach. By using the quantum computer as a processor for probability distributions, a probabilistic traversal of the decision tree can be realized via measurements of a quantum circuit. We describe how tree inductions and the prediction of class labels of query data can be integrated into this framework. An on-demand sampling method enables predictions with a constant number of classical memory slots, independent of the tree depth. We experimentally study our approach using both a quantum computing simulator and actual IBM quantum hardware. To our knowledge, this is the first realization of a decision tree classifier on a quantum device.

Decision trees are well-known predictive models commonly used in data mining and machine learning for a wide area of applications. We propose a quantum representation of binary decision tree classifiers by loading a classical decision tree into a quantum circuit. This approach allows us to train quantum decision trees and use them to make memory-efficient predictions. We experimentally study our approach using both a quantum computing simulator and actual IBM quantum hardware.

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► References

[1] S.B. Kotsiantis ``Decision trees: a recent overview'' Artif Intell Rev 39, 261-283 (2013).

[2] O.Z. Maimonand L. Rokach ``Data Mining With Decision Trees: Theory And Applications'' World Scientific Publishing Company (2014).

[3] L. Breiman ``Classification and Regression Trees'' CRC Press (2017).

[4] Laurent Hyafiland Ronald L. Rivest ``Constructing optimal binary decision trees is NP-complete'' Information Processing Letters 5, 15–17 (1976).

[5] Dimitris Bertsimasand Jack Dunn ``Optimal classification trees'' Machine Learning 106, 1039–1082 (2017).

[6] Arman Zharmagambetov, Suryabhan Singh Hada, Miguel Á. Carreira-Perpiñán, and Magzhan Gabidolla, ``An Experimental Comparison of Old and New Decision Tree Algorithms'' (2020).

[7] J.R. Quinlan ``Induction of decision trees'' Machine Learning 1, 81–706 (1986).

[8] Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd, ``Quantum machine learning'' Nature 549, 195–202 (2017).

[9] Edward Farhiand Sam Gutmann ``Quantum computation and decision trees'' Phys. Rev. A 58, 915–928 (1998).

[10] Songfeng Luand Samuel L. Braunstein ``Quantum decision tree classifier'' Quantum Information Processing 13, 757–770 (2014).

[11] Maria Schuld, Ilya Sinayskiy, and Francesco Petruccione, ``An introduction to quantum machine learning'' Contemporary Physics 56, 172–185 (2015).

[12] Kamil Khadiev, Ilnaz Mannapov, and Liliya Safina, ``The Quantum Version Of Classification Decision Tree Constructing Algorithm C5.0'' (2019).

[13] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone, ``Quantum Random Access Memory'' Physical Review Letters 100 (2008).

[14] Iordanis Kerenidisand Anupam Prakash ``Quantum Recommendation Systems'' (2016).

[15] Howard Barnum, M Saks, and M Szegedy, ``Quantum Decision Trees and Semidefinite Programming'' (2001).

[16] Harry Buhrmanand Ronald de Wolf ``Complexity measures and decision tree complexity: a survey'' Theoretical Computer Science 288, 21–43 (2002) Complexity and Logic.

[17] Yaoyun Shi ``Entropy lower bounds for quantum decision tree complexity'' Information Processing Letters 81, 23–27 (2002).

[18] Salman Beigiand Leila Taghavi ``Quantum Speedup Based on Classical Decision Trees'' Quantum 4, 241 (2020).

[19] Cristian S. Calude, Michael J. Dinneen, Monica Dumitrescu, and Karl Svozil, ``Experimental evidence of quantum randomness incomputability'' Physical Review A 82 (2010).

[20] Johannes Koflerand Anton Zeilinger ``Quantum Information and Randomness'' European Review 18, 469–480 (2010).

[21] Sourabh Katoch, Sumit Singh Chauhan, and Vijay Kumar, ``A review on genetic algorithm: past, present, and future'' Multimedia Tools and Applications 80, 8091–8126 (2021).

[22] Guang Hao Low, Theodore J. Yoder, and Isaac L. Chuang, ``Quantum inference on Bayesian networks'' Phys. Rev. A 89, 062315 (2014).

[23] Sima E. Borujeni, Nam H. Nguyen, Saideep Nannapaneni, Elizabeth C. Behrman, and James E. Steck, ``Experimental evaluation of quantum Bayesian networks on IBM QX hardware'' 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) 372–378 (2020).

[24] Michael de Oliveiraand Luis Soares Barbosa ``Quantum Bayesian Decision-Making'' Foundations of Science (2021).

[25] S. Akers ``Binary Decision Diagrams'' IEEE Transactions on Computers 27, 509–516 (1978).

[26] Foster Provostand Pedro Domingos ``Tree Induction for Probability-Based Ranking'' Machine Learning 52, 199–215 (2003).

[27] Melanie Mitchell ``An Introduction to Genetic Algorithms'' MIT Press, Cambridge (1999).

[28] Alvaro H. C. Correia, Robert Peharz, and Cassio de Campos, ``Joints in Random Forests'' (2020).

[29] M. A. Nielsenand I. L. Chuang ``Quantum Computation and Quantum Information: 10th Anniversary Edition'' Cambridge University Press (2010).

[30] Lov Groverand Terry Rudolph ``Creating superpositions that correspond to efficiently integrable probability distributions'' (2002).

[31] Maris Ozols, Martin Roetteler, and Jérémie Roland, ``Quantum rejection sampling'' Proceedings of the 3rd Innovations in Theoretical Computer Science Conference on - ITCS'12 (2012).

[32] Maria Schuldand Francesco Petruccione ``Supervised Learning with Quantum Computers'' Springer International Publishing (2018).

[33] Mikko Möttönen, Juha J. Vartiainen, Ville Bergholm, and Martti M. Salomaa, ``Transformation of Quantum States Using Uniformly Controlled Rotations'' Quantum Info. Comput. 5, 467–473 (2005).

[34] V.V. Shende, S.S. Bullock, and I.L. Markov, ``Synthesis of quantum-logic circuits'' IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006).

[35] Martin Pleschand Časlav Brukner ``Quantum-state preparation with universal gate decompositions'' Physical Review A 83 (2011).

[36] Yuval R. Sanders, Guang Hao Low, Artur Scherer, and Dominic W. Berry, ``Black-Box Quantum State Preparation without Arithmetic'' Physical Review Letters 122 (2019).

[37] Johannes Bausch ``Fast Black-Box Quantum State Preparation'' (2021).

[38] Pierre-Luc Dallaire-Demersand Nathan Killoran ``Quantum generative adversarial networks'' Physical Review A 98 (2018).

[39] Marcello Benedetti, Erika Lloyd, Stefan Sack, and Mattia Fiorentini, ``Parameterized quantum circuits as machine learning models'' Quantum Science and Technology 4, 043001 (2019).

[40] Ling Hu, Shu-Hao Wu, Weizhou Cai, Yuwei Ma, Xianghao Mu, Yuan Xu, Haiyan Wang, Yipu Song, Dong-Ling Deng, Chang-Ling Zou, and Luyan Sun, ``Quantum generative adversarial learning in a superconducting quantum circuit'' Science Advances 5 (2019).

[41] Christa Zoufal, Aurélien Lucchi, and Stefan Woerner, ``Quantum Generative Adversarial Networks for learning and loading random distributions'' npj Quantum Information 5, 103 (2019).

[42] Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter, ``Elementary gates for quantum computation'' Physical Review A 52, 3457–3467 (1995).

[43] Israel F. Araujo, Daniel K. Park, Francesco Petruccione, and Adenilton J. da Silva, ``A divide-and-conquer algorithm for quantum state preparation'' Scientific Reports 11, 6329 (2021).

[44] Xiaoming Sun, Guojing Tian, Shuai Yang, Pei Yuan, and Shengyu Zhang, ``Asymptotically Optimal Circuit Depth for Quantum State Preparation and General Unitary Synthesis'' (2021).

[45] Xiao-Ming Zhang, Man-Hong Yung, and Xiao Yuan, ``Low-depth Quantum State Preparation'' (2021).

[46] Ji Liu, Luciano Bello, and Huiyang Zhou, ``Relaxed Peephole Optimization: A Novel Compiler Optimization for Quantum Circuits'' (2020).

[47] Laxmidhar Biswal, Debjyoti Bhattacharjee, Anupam Chattopadhyay, and Hafizur Rahaman, ``Techniques for fault-tolerant decomposition of a multicontrolled Toffoli gate'' Physical Review A 100 (2019).

[48] Mehdi Saeediand Massoud Pedram ``Linear-depth quantum circuits forn-qubit Toffoli gates with no ancilla'' Physical Review A 87 (2013).

[49] Manabendra Nath Bera, Antonio Acín, Marek Kuś, Morgan W Mitchell, and Maciej Lewenstein, ``Randomness in quantum mechanics: philosophy, physics and technology'' Reports on Progress in Physics 80, 124001 (2017).

[50] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles, ``Variational Quantum Algorithms'' (2020).

[51] W. Zhai, P. Kelly, and W.-B. Gong, ``Genetic algorithms with noisy fitness'' Mathematical and Computer Modelling 23, 131–142 (1996).

[52] Donald A. Sofge ``Toward a Framework for Quantum Evolutionary Computation'' 2006 IEEE Conference on Cybernetics and Intelligent Systems 1–6 (2006).

[53] Gexiang Zhang ``Quantum-inspired evolutionary algorithms: a survey and empirical study'' Journal of Heuristics 17, 303–351 (2011).

[54] Rafael Lahoz-Beltra ``Quantum Genetic Algorithms for Computer Scientists'' Computers 5 (2016).

[55] Harper R. Grimsley, Sophia E. Economou, Edwin Barnes, and Nicholas J. Mayhall, ``An adaptive variational algorithm for exact molecular simulations on a quantum computer'' Nature Communications 10, 3007 (2019).

[56] Arthur G. Rattew, Shaohan Hu, Marco Pistoia, Richard Chen, and Steve Wood, ``A Domain-agnostic, Noise-resistant, Hardware-efficient Evolutionary Variational Quantum Eigensolver'' (2020).

[57] L. Franken, B. Georgiev, S. Muecke, M. Wolter, N. Piatkowski, and C. Bauckhage, ``Gradient-free quantum optimization on NISQ devices'' (2021).

[58] Andrew Arrasmith, M. Cerezo, Piotr Czarnik, Lukasz Cincio, and Patrick J. Coles, ``Effect of barren plateaus on gradient-free optimization'' (2020).

[59] David W. Aha ``Tic-Tac-Toe Endgame Data Set'' (1991) accessed June 2021.

[60] Dheeru Duaand Casey Graff ``UCI Machine Learning Repository'' (2017).

[61] ``Qiskit: An Open-source Framework for Quantum Computing'' (2019).

[62] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay, ``Scikit-learn: Machine Learning in Python'' Journal of Machine Learning Research 12, 2825–2830 (2011).

[63] John Preskill ``Quantum Computing in the NISQ era and beyond'' Quantum 2, 79 (2018).

[64] IBM ``IBM Quantum'' (2021).

[65] Ryan LaRose ``Overview and Comparison of Gate Level Quantum Software Platforms'' Quantum 3, 130 (2019).

[66] Andrew W. Cross, Lev S. Bishop, Sarah Sheldon, Paul D. Nation, and Jay M. Gambetta, ``Validating quantum computers using randomized model circuits'' Physical Review A 100 (2019).

[67] Leo Breiman ``Random Forests'' Machine Learning 45, 5–32 (2001).

[68] Chin-Yao Chang, Eric Jones, and Peter Graf, ``On Quantum Computing for Mixed-Integer Programming'' (2020).

[69] Jin-Kao Kochenberger, Fred Glover, Mark Lewis, Zhipeng Lü, Haibo Wang, and Yang Wang, ``The unconstrained binary quadratic programming problem: a survey'' Journal of Combinatorial Optimization 28, 58–81 (2014).

[70] Ehsan Zahedinejadand Arman Zaribafiyan ``Combinatorial Optimization on Gate Model Quantum Computers: A Survey'' (2017).

[71] C. E. Shannon ``A mathematical theory of communication'' The Bell System Technical Journal 27, 379–423 (1948).

[72] T.M. Coverand J.A. Thomas ``Elements of Information Theory'' Wiley (2012).

[73] Nisha Saini ``Review of Selection Methods in Genetic Algorithms'' International Journal of Engineering and Computer Science 6, 22261–22263 (2017).

[74] G. Cowan ``Statistics'' Oxford University Press (2020).

[75] Jerzy Neyman ``A selection of early statistical papers'' University of California Press (1967).

[76] Niek J. Boumanand Serge Fehr ``Sampling in a Quantum Population, and Applications'' (2012).

[77] Lawrence D. Brown, T. Tony Cai, and Anirban DasGupta, ``Interval Estimation for a Binomial Proportion'' Statistical Science 16, 101–133 (2001).

[78] Abdullah Ash Saki, Mahabubul Alam, and Swaroop Ghosh, ``Study of Decoherence in Quantum Computers: A Circuit-Design Perspective'' (2019).

[79] G. Lindblad ``On the generators of quantum dynamical semigroups'' Communications in Mathematical Physics 48, 119–130 (1976).

[80] B. M. Villegas-Martínez, F. Soto-Eguibar, and H. M. Moya-Cessa, ``Application of Perturbation Theory to a Master Equation'' Advances in Mathematical Physics 2016, 9265039 (2016).

[81] Kamil Khadievand Liliya Safina ``The Quantum Version of Random Forest Model for Binary Classification Problem'' YRID-2020: International Workshop on Data Mining and Knowledge Engineering 30–35 (2020).

[82] Maria Schuldand Francesco Petruccione ``Quantum ensembles of quantum classifiers'' Scientific Reports 8, 2772 (2018).

[83] Zeke Xieand Issei Sato ``A Quantum-Inspired Ensemble Method and Quantum-Inspired Forest Regressors'' (2017).

[84] Mohsen Shahhosseiniand Guiping Hu ``Improved Weighted Random Forest for Classification Problems'' Progress in Intelligent Decision Science 42–56 (2021).

Cited by

[1] Weikang Li and Dong-Ling Deng, "Recent advances for quantum classifiers", Science China Physics, Mechanics, and Astronomy 65 2, 220301 (2022).

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