Abstract
After the discussion of classical-quantum interfaces, we are now ready to dive into techniques that can bee used to construct quantum machine learning algorithms. As laid out in the introduction, there are two strategies to solve learning task with quantum computers. First, one can try to translate a classical machine learning method into the language of quantum computing. The challenge here is to combine quantum routines in a clever way so that the overall quantum algorithm reproduces the results of the classical model. The second strategy is more exploratory and creates new models that are tailor-made for the working principles of a quantum device. Here, the numerical analysis of the model performance can be much more important than theoretical promises of asymptotic speedups. In the remaining chapters we will look at methods that are useful for both approaches.
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- 1.
Such an angle encoded qubit has been called a quron [10] in the context of quantum neural networks.
- 2.
Thanks to Gian Giacomo Guerreschi for this simplified presentation.
- 3.
The form of Eq. (6.10) is a so called non-parametric model: The number of (potentially zero) parameters \(\nu _m \) grows with the size M of the training set.
- 4.
In fact, the Hilbert space of some quantum systems can easily be constructed as a reproducing kernel Hilbert space [14].
- 5.
Strictly speaking, according to the definition in Eq. (6.8) a feature map maps an input to a function, and not to a vector. However, a quantum state is also called a wave function, and a more general definition of a feature map is a map from \(\mathcal {X}\) to a general Hilbert space. We can therefore overlook this subtlety here and refer to [14] for more details.
- 6.
In statistical physics, a Gibbs distribution is a state of a system that does not change under the evolution of the system.
References
Cleve, R., Ekert, A., Macchiavello, C., Mosca, M.: Quantum algorithms revisited. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 454, pp. 339–354. The Royal Society (1998)
Kobayashi, H., Matsumoto, K., Yamakami, T.: Quantum Merlin-Arthur proof systems: Are multiple Merlins more helpful to Arthur? In: Algorithms and Computation, pp. 189–198. Springer (2003)
Zhao, Z., Fitzsimons, J.K., Fitzsimons, J.F.: Quantum assisted Gaussian process regression (2015). arXiv:1512.03929
Romero, J., Olson, J.P., Aspuru-Guzik, A.: Quantum autoencoders for efficient compression of quantum data. Quantum Sci. Technol. 2(4), 045001 (2017)
Schuld, M., Sinayskiy, I., Petruccione, F.: How to simulate a perceptron using quantum circuits. Phys. Lett. A 379, 660–663 (2015)
Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2010)
Tommiska, M.T.: Efficient digital implementation of the sigmoid function for reprogrammable logic. IEE Proc. Comput. Digit. Tech. 150(6), 403–411 (2003)
Quine, V.W.: A way to simplify truth functions. Am. Math. Mon. 62(9), 627–631 (1955)
McCluskey, J.E.: Minimization of boolean functions. Bell Labs Techn. J. 35(6), 1417–1444 (1956)
Schuld, M., Sinayskiy, I., Petruccione, F.: The quest for a quantum neural network. Quantum Inf. Process. 13(11), 2567–2586 (2014)
Cao, Y., Guerreschi, G.G., Aspuru-Guzik, A.: Quantum neuron: an elementary building block for machine learning on quantum computers (2017). arXiv:1711.11240
Paetznick, A., Svore, K.M.: Repeat-until-success: non-deterministic decomposition of single-qubit unitaries. Quantum Inf. Comput. 14, 1277–1301 (2013)
Wiebe, N., Kliuchnikov, V.: Floating point representations in quantum circuit synthesis. New J. Phys. 15(9), 093041 (2013)
Schuld, M., Killoran, N.: Quantum machine learning in feature Hilbert spaces (2018). arXiv:1803.07128v1
Schölkopf, B., Herbrich, R., Smola, A.: A generalized representer theorem. In: Computational Learning Theory, pp. 416–426. Springer (2001)
Schölkopf, B., Smola, A.J.: Learning with kernels: support vector machines, regularization, optimization, and beyond. MIT Press (2002)
Rebentrost, P., Mohseni, M., Lloyd, S.: Quantum support vector machine for big data classification. Phys. Rev. Lett. 113, 130503 (2014)
Schuld, M., Bocharov, A., Svore, K., Wiebe, N.: Circuit-centric quantum classifiers (2018). arXiv:1804.00633
Stoudenmire, E., Schwab, D.J.: Supervised learning with tensor networks. In: Advances in Neural Information Processing Systems, pp. 4799–4807 (2016)
Chatterjee, R., Ting, Y.: Generalized coherent states, reproducing kernels, and quantum support vector machines. Quantum Inf. Commun. 17(15&16), 1292 (2017)
Klauder, J.R., Bo-Sture S.: Coherent States: Applications in Physics and Mathematical Physics. World Scientific (1985)
Trugenberger, C.A.: Quantum pattern recognition. Quantum Inf. Process. 1(6), 471–493 (2002)
Schuld, M., Sinayskiy, I., Petruccione, F.: Quantum Computing for Pattern Classification. Lecture Notes in Computer Science, vol. 8862, pp. 208–220. Springer (2014)
Schuld, M., Fingerhuth, M., Petruccione, F.: Implementing a distance-based classifier with a quantum interference circuit. EPL (Europhys. Lett.) 119(6), 60002 (2017)
Yoder, T.J., Low, G.H., Chuang, I.L.: Quantum inference on Bayesian networks. Phys. Rev. A 89, 062315 (2014)
Ozols, M., Roetteler, M., Roland, J.: Quantum rejection sampling. ACM Trans. Comput. Theory (TOCT) 5(3), 11 (2013)
Wiebe, N., Kapoor, A., Svore,K.M.: Quantum deep learning (2014). arXiv: 1412.3489v1
Wittek, P., Gogolin, C.: Quantum enhanced inference in markov logic networks. Sci. Rep. 7 (2017)
Brandão, F.G.S.L., Svore, K.M.: Quantum speed-ups for solving semidefinite programs. In: 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pp. 415–426. IEEE (2017)
Poulin, D., Wocjan, P.: Sampling from the thermal quantum gibbs state and evaluating partition functions with a quantum computer. Phys. Rev. Lett. 103(22), 220502 (2009)
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Schuld, M., Petruccione, F. (2018). Quantum Computing for Inference. In: Supervised Learning with Quantum Computers. Quantum Science and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-96424-9_6
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