Quantum machine learning in 2015

Posted on 04 January 2016

It is getting really difficult to keep track of the developments of the field. I counted nearly seventy relevant papers on arXiv alone, and my scope is rather narrow. The first QML workshop happened at NIPS in December. More workshops are coming: there will be a workshop on Physics and Machine Learning: Emerging Paradigms at ESANN in April, and another QML workshop in South Africa in July, followed by a summer school in 2017.

The year certainly revolved around adiabatic quantum optimization and D-Wave. Apart from the big guys investing in it, a new company started to offer consulting, and they also do genuine research on the side.

The whole exercise of writing up the year's advances is somewhat redundant, as an overview article appeared less than a month ago (Adcock et al. 2015). Nevertheless, I like to collect references and links for my own benefit, so here they are.

A year of annealing

Most of the activity focus on D-Wave and much progress has been made understanding its limits and advantages. There was also a scattering of papers talking about quantum annealing and optimization with no reference to implementation.


A few papers dealt with understanding the implementation issues that one faces when using a D-Wave system. For instance, the couplings might have persistent biases, and if we are able to recalibrate them, optimization performance is bound to improve (Perdomo-Ortiz et al. 2015). A similar problem appears in large optimization problems: they are annealed at higher temperatures due analogue control errors, which can be addressed, yielding better scaling (King 2015).

Apparently, a small number very hard instances disproportionally increase the total time to solve a set of random input instance of an optimization problem, which we can call "heavy tails". It is possible to suppress such degenerate cases in the D-Wave annealer, leading to a performance that is magnitudes better (A. D. King et al. 2015)

Finally, a manuscript studied the vulnerability of quantum annealers to qubit and coupler failures on Chimera topologies (Melchert, Katzgraber, and Novotny 2015).

Benchmarking and quantum advantage

Great progress has been made to verify the advantages of using a quantum annealer. To start of, we need rigorous verifiable benchmarks in place of arguments over the physical properties (Katzgraber et al. 2015). The Time-to-target metric is such, identifying low-cost target solutions found by the D-Wave processor within very short time limits, and then asking how much time competing software solvers need to find solution energies of matching or better quality (J. King et al. 2015)

A wave of papers suggest comparing simulated (thermal) annealing, simulated quantum annealing, and physical quantum annealing to understand the limits and advantages of the adiabatic method (Sowa et al. 2015). Simulated quantum annealing itself has clear advantages over simulated annealing in a physics problem (Zanca and Santoro 2015), but the relationship between simulated and physical quantum annealing is more interesting. Simulated quantum annealing relies on different flavours of quantum Monte Carlo simulations. The path-integral quantum Monte Carlo algorithm was found to succeed in the same regimes where quantum adiabatic optimization succeeds (Brady and Dam 2015, Isakov et al. (2015)), while the projective variant seems to be more efficient in simulations (Inack and Pilati 2015).

What type of objective functions work well? Not all that surprisingly, tall and narrow energy barriers are best (Denchev, Boixo, et al. 2015). In a more surprising turn, in some cases, short and narrow barriers can also give a quantum advantage, and we might even relax the adiabatic dynamics (Muthukrishnan, Albash, and Lidar 2015). Non-adiabatic annealing schedules apparently also help with the "heavy tails" (Steiger, Rønnow, and Troyer 2015). These are indications that a pure, idealistic adiabatic quantum optimization might actually be overrated.

In any case, the annealing schedule depends on the gap between the ground state and the lowest excited state throughout the adiabatic change. Identifying this analytically is a hard problem in general. There is a new analytical result for the case of tall and narrow cost functions (Kong and Crosson 2015). Then introducing intermediate Hamiltonians can help achieving better gaps (Zeng, Zhang, and Sarovar 2015). It seems, however, that these Hamiltonians have to be carefully crafted, as randomly constructed ones improve the gap only by a small margin. The relationship between the gap and the annealing time is not entirely well understood either, and sometimes we end up with intractable problems while respecting the scaling law (Knysh 2015).

Sampling a Boltzmann distribution

It is tempting to be able to a sample a Boltzmann (Gibbs) distribution for a variety of applications. With the D-Wave hardware, the distribution of the ground state and excited states roughly follows the Boltzmann distribution, which can be used for studying spin glasses and probing the performance of quantum annealing (Chancellor et al. 2015).

A few papers before suggested training reduced or full Boltzmann machines this way, because then we can entirely bypass the contrastive divergence algorithm that is normally used to train these networks. Using the D-Wave machine, we can obtain some tangible results (Adachi and Henderson 2015), even if we can only estimate the effective temperature (Benedetti et al. 2015). Three talks at the NIPS workshop were also about this approach.

Other applications

Structural risk minimization pivots on regularization, which ensures the sparsity of a learned model. We usually cheat on the regularization on relax the optimization problem underlying learning to end up with a convex objective function, which in turn we can solve efficiently. This is exactly where quantum adiabatic optimization could be huge from the perspective of statistical learning theory: theoretically we can solve nonconvex objectives fast. Totally corrective boosting aims to do just that, either by simulated quantum annealing or by the real physical process (Denchev, Ding, et al. 2015).

Last year we have seen a paper on applying quantum resources to a topological learning method called persistent homology. Keeping the topological theme, a paper proposes using quantum annealing to calculate the homology of point clouds (Dridi and Alghassi 2015).

Moving on to more generic optimization problems, we can solve larger problems on the D-Wave annealer by iteratively finding the local optimum of smaller chunks (Rosenberg et al. 2015). Inequality constraints are always trouble because they cannot be simply added with Lagrange multipliers to the objective function, but there is a new way of dealing with them using the quantum adiabatic approach (Ronagh, Woods, and Iranmanesh 2015). A curious application of the D-Wave hardware is job-shop scheduling (Venturelli, Marchand, and Rojo 2015).

More neural networks

Returning for a moment to Boltzmann sampling, following last years proposal for training a Boltzmann machine with state preparation and subsequent sampling, a paper by the same authors follows the same protocol with a classical simulation. The resulting algorithm is called quantum-inspired deep learning (Wiebe et al. 2015). It is an interesting way how insights from quantum computing can give hints to improve classical algorithms.

A new training algorithm for a quantum perceptron can compute the XOR function (Seow, Behrman, and Steck 2015). This XOR problem with the perceptron has been around for about fifty years.

The great thing about quantum neural networks that they are the closest to physical implementations, apart from adiabatic quantum optimization. In 2015, there was a proposal for a classical optical system for recurrent neural networks (Hermans et al. 2015), a nano-optical perceptron (Tezak and Mabuchi 2015), and an actual implementation using quantum dots (M. V. Altaisky et al. 2015, Mikhail V. Altaisky et al. (2015)).

Causal networks

Probabilistic inference, Bayesian or causal networks lie closer to symbolic AI, and naturally the progress continues to generalize results to the quantum case. For a starter, we can use quantum conditional operators to talk about probabilistic inference, with an immediate application to the Monty Hall problem (Glos and Kurzyk 2015).

Nevertheless, the task in general is hard, primarily due to the entanglement and nonlocal correlations. Even defining causality is difficult, and we need to extend the notion (Oreshkov and Giarmatzi 2015). Then, the original Reichenbach Common Cause Principle assumed that the arrow between cause and effect and time direction are the same. A new manuscript goes beyond this, giving algorithmic rather than statistical foundations to causal inference (Janzing, Chaves, and Schoelkopf 2015). Another generalization uses open quantum systems and quantum channels, and avoids Bayesian inference (Costa and Shrapnel 2015). Despite the progress, it is still an open problem how causal models, nonlocality, and the nonsignalling principle fit together (Evans 2015).


On a good day, quantum cryptography cannot be broken. For this reason, the UK government will probably ban it soon enough. Until then, there is a series of proposals to perform quantum machine learning either remotely or by slicing up the learning problem to small chunks (Bang, Lee, and Jeong 2015, Ying, Ying, and Feng (2015), Sheng and Zhou (2015)). It is a fun idea to play with, especially when the learning algorithm itself cannot compromise privacy.

Machine learning applied on quantum physics problems

This topic leads out of the domain of quantum machine learning, as here we are interested in applying the principles of learning theory to solve problems in quantum physics. Yet, there is a smooth transition. For instance, we can use a quantum neural network to learn the unknown entanglement of a system in a way that is robust to noise and decoherence (Behrman et al. 2015). Another manuscript derived limits on the learnability of quantum measurements and quantum state (Cheng, Hsieh, and Yeh 2015). In all cases the authors used quantum resources and ideas from learning theory to derive results in quantum physics.

If we abandon quantum resources in learning and uses classical algorithms in quantum physics problems, we encounter a flood of new papers. Two paper suggests learning target quantum gates: in one case, we would like to map the gates to a spin system consisting of target and ancilla qubits (Banchi, Pancotti, and Bose 2015), in the other case, differential evolution approximates three-qubit gates on an architecture of nearest-neighbor-coupled superconducting artificial atoms (Zahedinejad, Ghosh, and Sanders 2015). Refreshingly for a physics paper, the code for the latter is available under an open source licence.

Calibration is core to quantum computers at any scale, and naturally the same stands for quantum simulators. (Wiebe, Granade, and Cory 2015) proposes learning a Hamiltonian model for a larger quantum system using a small quantum simulator using Bayesian updates, then iteratively building up larger and larger systems.

Moving beyond quantum computation, a reinforcement learning scheme helps producing Bose-Einstein condensates (Wigley et al. 2015). The code is also available for this project. An evolutionary learning algorithms helps characterizing decohering quantum systems (Stenberg, Köhn, and Wilhelm 2015). On the more general level of quantum control, a manuscript proposes training on a sample, followed by testing, which eerily resembles supervised learning (Dong et al. 2015).


Departing from the angle of the theory of computation and focusing on a very specific learning problem of a binary-output function, a manuscript demonstrates a quantum advantage in the number oracle queries using a system of five transmon qubits (Ristè et al. 2015).

Proposals for exponential speedup in quantum machine learning tend to stem from the HHL algorithm for solving a linear system. There is an update to this algorithm with exponentially improved dependence on precision (Childs, Kothari, and Somma 2015).

The concept of agency has not been well explored thus far. We can ask what it means to have a quantum agent: what does it learn, how do we know it learns, and how does it interact with its (quantum) environment? (Dunjko, Friis, and Briegel 2015, Dunjko, Taylor, and Briegel (2015), Wiebe and Granade (2015)). These are important questions, but the answers remain elusive.


Adachi, Steven H., and Maxwell P. Henderson. 2015. “Application of Quantum Annealing to Training of Deep Neural Networks.” arXiv:1510.06356.

Adcock, Jeremy, Euan Allen, Matthew Day, Stefan Frick, Janna Hinchliff, Mack Johnson, Sam Morley-Short, Sam Pallister, Alasdair Price, and Stasja Stanisic. 2015. “Advances in Quantum Machine Learning.” arXiv:1512.02900.

Altaisky, M. V., N. N. Zolnikova, N. E. Kaputkina, V. A. Krylov, Yu. E. Lozovik, and N. S. Dattani. 2015. “Towards a Feasible Implementation of Quantum Neural Networks Using Quantum Dots.” arXiv:1503.05125.

Altaisky, Mikhail V., Nadezhda N. Zolnikova, Natalia E. Kaputkina, Victor A. Krylov, Yurii E. Lozovik, and Nikesh S. Dattani. 2015. “Entanglement in a Quantum Neural Network Based on Quantum Dots.” arXiv:1512.01141.

Banchi, Leonardo, Nicola Pancotti, and Sougato Bose. 2015. “Quantum Gate Learning in Engineered Qubit Networks: Toffoli Gate with Always-on Interactions.” arXiv:1509.04298.

Bang, Jeongho, Seung-Woo Lee, and Hyunseok Jeong. 2015. “Protocol for Secure Quantum Machine Learning at a Distant Place.” arXiv:1504.04929.

Behrman, E. C., N. H. Nguyen, J. E. Steck, and M. McCann. 2015. “Quantum Neural Computation of Entanglement Is Robust to Noise and Decoherence.” arXiv:1510.09173.

Benedetti, Marcello, John Realpe-Gómez, Rupak Biswas, and Alejandro Perdomo-Ortiz. 2015. “Estimation of Effective Temperatures in a Quantum Annealer and Its Impact in Sampling Applications: A Case Study Towards Deep Learning Applications.” arXiv:1510.07611.

Brady, Lucas T., and Wim van Dam. 2015. “Quantum Monte Carlo Simulations of Tunneling in Quantum Adiabatic Optimization.” arXiv:1509.02562.

Chancellor, Nicholas, Szilard Szoke, Walter Vinci, Gabriel Aeppli, and Paul A. Warburton. 2015. “Maximum-Entropy Inference with a Programmable Annealer.” arXiv:1506.08140.

Cheng, Hao-Chung, Min-Hsiu Hsieh, and Ping-Cheng Yeh. 2015. “The Learnability of Unknown Quantum Measurements.” arXiv:1501.00559.

Childs, Andrew M., Robin Kothari, and Rolando D. Somma. 2015. “Quantum Linear Systems Algorithm with Exponentially Improved Dependence on Precision.” arXiv:1511.02306.

Costa, Fabio, and Sally Shrapnel. 2015. “Quantum Causal Modelling.” arXiv:1512.07106.

Denchev, Vasil S., Sergio Boixo, Sergei V. Isakov, Nan Ding, Ryan Babbush, Vadim Smelyanskiy, John Martinis, and Hartmut Neven. 2015. “What Is the Computational Value of Finite Range Tunneling?” arXiv:1512.02206.

Denchev, Vasil S., Nan Ding, Shin Matsushima, S. V. N. Vishwanathan, and Hartmut Neven. 2015. “Totally Corrective Boosting with Cardinality Penalization.” arXiv:1504.01446.

Dong, Daoyi, Mohamed A. Mabrok, Ian R. Petersen, Bo Qi, Chunlin Chen, and Herschel Rabitz. 2015. “Sampling-Based Learning Control for Quantum Systems with Uncertainties.” IEEE Transactions on Control Systems Technology. IEEE, 1–1. doi:10.1109/tcst.2015.2404292.

Dridi, Raouf, and Hedayat Alghassi. 2015. “Homology Computation of Large Point Clouds Using Quantum Annealing.” arXiv:1512.09328.

Dunjko, Vedran, Nicolai Friis, and Hans J. Briegel. 2015. “Quantum-Enhanced Deliberation of Learning Agents Using Trapped Ions.” New J. Phys. 17 (2). IOP Publishing: 023006. doi:10.1088/1367-2630/17/2/023006.

Dunjko, Vedran, Jacob M. Taylor, and Hans J. Briegel. 2015. “Framework for Learning Agents in Quantum Environments.” arXiv:1507.08482.

Evans, Peter W. 2015. “Quantum Causal Models, Faithfulness and Retrocausality.” arXiv:1506.08925.

Glos, Adam, and Dariusz Kurzyk. 2015. “Quantum Inferring Acausal Structure.” arXiv:1504.01917.

Hermans, Michiel, Miguel Soriano, Joni Dambre, Peter Bienstman, and Ingo Fischer. 2015. “Photonic Delay Systems as Machine Learning Implementations.” arXiv:1501.02592.

Inack, E. M., and S. Pilati. 2015. “Simulated Quantum Annealing of Double-Well and Multi-Well Potentials.” arXiv:1510.04650.

Isakov, Sergei V., Guglielmo Mazzola, Vadim N. Smelyanskiy, Zhang Jiang, Sergio Boixo, Hartmut Neven, and Matthias Troyer. 2015. “Understanding Quantum Tunneling Through Quantum Monte Carlo Simulations.” arXiv:1510.08057.

Janzing, Dominik, Rafael Chaves, and Bernhard Schoelkopf. 2015. “Algorithmic Independence of Initial Condition and Dynamical Law in Thermodynamics and Causal Inference.” arXiv:1512.02057.

Katzgraber, Helmut G., Firas Hamze, Zheng Zhu, Andrew J. Ochoa, and H. Munoz-Bauza. 2015. “Seeking Quantum Speedup Through Spin Glasses: The Good, the Bad, and the Ugly.” Physical Review X 5 (3). American Physical Society (APS). doi:10.1103/physrevx.5.031026.

King, Andrew D. 2015. “Performance of a Quantum Annealer on Range-Limited Constraint Satisfaction Problems.” arXiv:1502.02098.

King, Andrew D., Emile Hoskinson, Trevor Lanting, Evgeny Andriyash, and Mohammad H. Amin. 2015. “Degeneracy, Degree, and Heavy Tails in Quantum Annealing.” arXiv:1512.07325.

King, James, Sheir Yarkoni, Mayssam M. Nevisi, Jeremy P. Hilton, and Catherine C. McGeoch. 2015. “Benchmarking a Quantum Annealing Processor with the Time-to-Target Metric.” arXiv:1508.05087.

Knysh, Sergey. 2015. “Computational Bottlenecks of Quantum Annealing.” arXiv:1506.08608.

Kong, Linghang, and Elizabeth Crosson. 2015. “The Performance of the Quantum Adiabatic Algorithm on Spike Hamiltonians.” arXiv:1511.06991.

Melchert, O., Helmut G. Katzgraber, and M. A. Novotny. 2015. “Site and Bond Percolation Thresholds in Kn, n-Based Lattices: Vulnerability of Quantum Annealers to Random Qubit and Coupler Failures on Chimera Topologies.” arXiv:1511.07078.

Muthukrishnan, Siddharth, Tameem Albash, and Daniel A. Lidar. 2015. “Tunneling and Speedup in Quantum Optimization for Permutation-Symmetric Problems.” arXiv:1511.03910.

Oreshkov, Ognyan, and Christina Giarmatzi. 2015. “Causal and Causally Separable Processes.” arXiv:1506.05449.

Perdomo-Ortiz, Alejandro, Bryan O’Gorman, Joseph Fluegemann, Rupak Biswas, and Vadim N. Smelyanskiy. 2015. “Determination and Correction of Persistent Biases in Quantum Annealers.” arXiv:1503.05679.

Ristè, D., Marcus P. da Silva, Colm A. Ryan, Andrew W. Cross, John A. Smolin, Jay M. Gambetta, Jerry M. Chow, and Blake R. Johnson. 2015. “Demonstration of Quantum Advantage in Machine Learning.” arXiv:1512.06069.

Ronagh, Pooya, Brad Woods, and Ehsan Iranmanesh. 2015. “Solving Constrained Quadratic Binary Problems via Quantum Adiabatic Evolution.” arXiv:1509.05001.

Rosenberg, Gili, Mohammad Vazifeh, Brad Woods, and Eldad Haber. 2015. “Building an Iterative Heuristic Solver for a Quantum Annealer.” arXiv:1507.07605.

Seow, Kok-Leong, Elizabeth Behrman, and James Steck. 2015. “Efficient Learning Algorithm for Quantum Perceptron Unitary Weights.” arXiv:1512.00522.

Sheng, Yu-Bo, and Lan Zhou. 2015. “Blind Quantum Machine Learning.” arXiv:1507.07195.

Sowa, A P, M J Everitt, J H Samson, S E Savel’ev, A M Zagoskin, S Heidel, and J C Zúñiga-Anaya. 2015. “Recursive Simulation of Quantum Annealing.” Journal of Physics A: Mathematical and Theoretical 48 (41). IOP Publishing: 415301. doi:10.1088/1751-8113/48/41/415301.

Steiger, Damian S., Troels F. Rønnow, and Matthias Troyer. 2015. “Heavy Tails in the Distribution of Time to Solution for Classical and Quantum Annealing.” Physical Review Letters 115 (23). American Physical Society (APS). doi:10.1103/physrevlett.115.230501.

Stenberg, Markku P. V., Oliver Köhn, and Frank K. Wilhelm. 2015. “Characterization of Decohering Quantum Systems: Machine Learning Approach.” arXiv:1510.05655.

Tezak, Nikolas, and Hideo Mabuchi. 2015. “A Coherent Perceptron for All-Optical Learning.” arXiv:1501.01608.

Venturelli, Davide, Dominic J. J. Marchand, and Galo Rojo. 2015. “Quantum Annealing Implementation of Job-Shop Scheduling.” arXiv:1506.08479.

Wiebe, Nathan, and Christopher Granade. 2015. “Can Small Quantum Systems Learn?” arXiv:1512.03145.

Wiebe, Nathan, Christopher Granade, and D G Cory. 2015. “Quantum Bootstrapping via Compressed Quantum Hamiltonian Learning.” New Journal of Physics 17 (2). IOP Publishing: 022005. doi:10.1088/1367-2630/17/2/022005.

Wiebe, Nathan, Ashish Kapoor, Christopher Granade, and Krysta M Svore. 2015. “Quantum Inspired Training for Boltzmann Machines.” arXiv:1507.02642.

Wigley, P. B., P. J. Everitt, A. van den Hengel, J. W. Bastian, M. A. Sooriyabandara, G. D. McDonald, K. S. Hardman, et al. 2015. “Fast Machine-Learning Online Optimization of Ultra-Cold-Atom Experiments.” arXiv:1507.04964.

Ying, Shenggang, Mingsheng Ying, and Yuan Feng. 2015. “Quantum Privacy-Preserving Data Mining.” arXiv:1512.04009.

Zahedinejad, Ehsan, Joydip Ghosh, and Barry C. Sanders. 2015. “Designing High-Fidelity Single-Shot Three-Qubit Gates: A Machine Learning Approach.” arXiv:1511.08862.

Zanca, Tommaso, and Giuseppe E. Santoro. 2015. “Quantum Annealing Speedup over Simulated Annealing on Random Ising Chains.” arXiv:1511.01906.

Zeng, Lishan, Jun Zhang, and Mohan Sarovar. 2015. “Schedule Path Optimization for Quantum Annealing and Adiabatic Quantum Computing.” arXiv:1505.00209.

Tags: Quantum machine learning, Machine learning, Quantum information theory

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