# End-of-year updates on quantum machine learning

*Posted on 31 December 2014*

Over the past two months since I last discussed the latest papers, there has been quite a bit of progress. A Wikipedia page on quantum machine learning was launched in mid-November -- I am hoping to start helping out editing it soon. Many new papers appeared on arXiv -- once there were three in one day -- and I also found some which I missed earlier. Here I quickly jot down a few notes relating to these manuscripts.

## New papers

It is hard not to get excited about a paper with the title "Quantum Deep Learning". The content is even better. The way we train neural networks did not change that much since the perceptron appeared in the sixties: we either do some kind of a gradient descent if the optimization is such, or some heuristic search for the optimum with nonconvex objective functions. This is true for deep learning networks too. This paper bypasses the iterative training all together. The authors suggest a state preparation followed by sampling. The focus is on Boltzmann machines, where the output of the network is based on the Gibbs distribution. The main challenge is the state preparation: this should be such that it approximates the optimal Gibbs distribution. Efficient sampling of the state is comparatively simpler. Apart from having a lower complexity -- especially for fully connected networks -- another advantage is that the quantum state preparation and sampling achieves a better optimum than classical heuristics, leading to better generalization performance.

Sticking with neural networks, to my best knowledge, it has been exactly two decades since the first proposal of quantum perceptrons. Yet, how to train one has been an open question. Now there is one algorithm that teaches a quantum perceptron in *O*(*n*) time (Schuld, Sinayskiy, and Petruccione 2014a). The same authors in a new paper discuss the advantages of applying a quantum associative memory -- essentially a generalized Hopfield network -- to real-world data (Schuld, Sinayskiy, and Petruccione 2014b).

Alex Monràs and Andreas Winter published a work on hidden Markov models where the hidden states are quantum (Monràs and Winder 2014). The idea is to generalize what we know about realization problems: modelling stochastic processes by vector space operations. Roughly speaking, a quasi-representation consists of matrices *D*(**u**) that we assign to elements **u** of the stochastic process. We also take a fixed vector *τ* of the vector space on which the matrices act, and another one *π* from its conjugate space. Then, under some conditions, these should reproduce the probabilities of the stochastic process: *p*(**u**) = *πD*(**u**)*τ*. The quasi-representation is, of course, not unique. The lowest-dimensional quasi-representation of a process is called regular representation. In a regular representation, we want to have objects such that they correspond to a valid quantum physical system. Most notably, the linear maps must be completely positive, and to achieve this, we lift the number of dimensions from a regular representation. The paper is about this generalization and lifting.

A quantum associative memory had several proposed forms before. In an adiabatic setting, Seddiqi and Humble 2014 studied different training strategies and found that the system size and strategy had a great impact on the performance.

Adiabatic quantum optimization is very much relevant to learning theory, so it is worth keeping track of the developments there. While there are alternative proposals, we understand the minimum energy gap during the adiabatic change decides the speed at which the process can complete. This way, the gap defines computational complexity. Finding an analytical solution to the gap is hard in general. A paper titled "Dimensionality Reduction for Adiabatic Quantum Optimizers: Beyond Symmetry Exploitation" suggest a new way of reducing the dimensionality of the Hilbert space of the physical system to be able to calculate the gap (Mandrà, Guerreschi, and Aspuru-Guzik 2014). The title is easily misleading to anyone coming from machine learning -- this is a different dimensionality reduction. Another manuscript argues that the gap will always be exponentially small in a wide range of combinatorial optimization problems (Laumann et al. 2014). In a final bit of adiabatic news, the quest for proving what the D-Wave processor actually does continues. A new manuscript argues that multiple qubit tunnel and this effect is important in the optimization over a nonconvex objective (Boixo et al. 2014).

It is also interesting to note that applying classical machine learning in quantum information theory problems is gain traction. Apart from previous work on quantum metrology and heuristic optimization by particle-swarms and differential evolution, a new manuscript applies nonlinear SVMs to improve the fidelity of measurements (Magesan et al. 2014).

## Papers I missed earlier

I still found some papers that I have not seen earlier, but they are less closely related to machine learning -- subject, of course, to your definition of the domain. Two papers deal with least-squares fitting of data by using quantum algorithms (Wiebe, Braun, and Lloyd 2012,Wang (2014)). The former relies on a trick similar to the one used in least-squares support machines, using a quantum algorithm to solve linear equations. The latter extends the result to the dense case.

I spotted the experimental realization of Seth Lloyd's *k*-means clustering algorithm using a photonic quantum computer at the Hefei National Laboratory before, but I missed an NMR realization of the quantum support vector machines in the same lab (Li et al. 2014). Four qubits deal with a character recognition problem.

While I recently read up on causal structures and Bayesian networks in a quantum setting, I missed this paper: Quantum Inference on Bayesian Networks (Low, Yoder, and Chuang 2014). The paper achieves a quadratic speedup in the probability of evidence.

## References

Boixo, Sergio, Vadim N. Smelyanskiy, Alireza Shabani, Sergei V. Isakov, Mark Dykman, Vasil S. Denchev, Mohammad Amin, Anatoly Smirnov, Masoud Mohseni, and Hartmut Neven. 2014. “Computational Role of Collective Tunneling in a Quantum Annealer.” *arXiv:1411.4036*.

Laumann, Christopher R., Roderich Moessner, Antonello Scardicchio, and S. L. Sondhi. 2014. “Quantum Annealing: The Fastest Route to Quantum Computation?” *arXiv:1411.5710*.

Low, Guang Hao, Theodore J. Yoder, and Isaac L. Chuang. 2014. “Quantum Inference on Bayesian Networks.” *Physics Review A* 89 (6). American Physical Society: 062315. doi:10.1103/PhysRevA.89.062315.

Magesan, Easwar, Jay M. Gambetta, A.D. Córcoles, and Jerry M. Chow. 2014. “Machine Learning for Discriminating Quantum Measurement Trajectories and Improving Readout.” *arXiv:1411.4994*.

Mandrà, Salvatore, Gian Giacomo Guerreschi, and Alán Aspuru-Guzik. 2014. “Dimensionality Reduction for Adiabatic Quantum Optimizers: Beyond Symmetry Exploitation.” *arXiv:1407.8183*.

Monràs, Alex, and Andreas Winder. 2014. “Quantum Learning of Clasiccal Stochastic Processes: The Completely-Positive Realization Problem.” *arXiv:1412:3634*.

Schuld, Maria, Ilya Sinayskiy, and Francesco Petruccione. 2014a. “Simulating a Perceptron on a Quantum Computer.” *arXiv:1412.3635*.

———. 2014b. “Quantum Computing for Pattern Classification.” In *Proceedings of PRICAI-14, 13th Pacific Rim International Conference on Artificial Intelligence*, edited by Duc-Nghia Pham and Seong-Bae Park, 8862:208–20. Lecture Notes in Computer Science. Springer International Publishing. doi:10.1007/978-3-319-13560-1_17.

Seddiqi, Hadayat, and Travis S Humble. 2014. “Adiabatic Quantum Optimization for Associative Memory Recall.” *Frontiers in Physics* 2 (79). doi:10.3389/fphy.2014.00079.

Wang, Guoming. 2014. “Quantum Algorithms for Curve Fitting.” *arXiv:1402.0660*.

Wiebe, Nathan, Daniel Braun, and Seth Lloyd. 2012. “Quantum Algorithm for Data Fitting.” *Physical Review Letters* 109 (5). American Physical Society (APS). doi:10.1103/physrevlett.109.050505.

Li, Zhaokai, Xiaomei Liu, Nanyang Xu, and Jiangfeng Du. 2014. “Experimental Realization of Quantum Artificial Intelligence.” *arXiv:1410.1054*.