References and further reading


[R1]Diederik P Kingma and Jimmy Ba. Adam: a method for stochastic optimization. 2014. arXiv:1804.03159.
[R2]Maria Schuld, Alex Bocharov, Krysta Svore, and Nathan Wiebe. Circuit-centric quantum classifiers. 2018. arXiv:1804.00633.
[R3]Nathan Killoran, Thomas R Bromley, Juan Miguel Arrazola, Maria Schuld, Nicolás Quesada, and Seth Lloyd. Continuous-variable quantum neural networks. 2018. arXiv:1806.06871.
[R4]Maria Schuld and Nathan Killoran. Quantum machine learning in feature hilbert spaces. Physical review letters, 122(4):040504, 2019. arXiv:1803.07128.
[R5]William R. Clements, Peter C. Humphreys, Benjamin J. Metcalf, W. Steven Kolthammer, and Ian A. Walmsley. An optimal design for universal multiport interferometers. Optica, 3(12):1460–1465, 2016. doi:10.1364/OPTICA.3.001460.
[R6]Michael Reck, Anton Zeilinger, Herbert J Bernstein, and Philip Bertani. Experimental realization of any discrete unitary operator. Physical review letters, 73(1):58, 1994. doi:10.1103/PhysRevLett.73.58.
[R7]Kosuke Mitarai, Makoto Negoro, Masahiro Kitagawa, and Keisuke Fujii. Quantum circuit learning. 2018. arXiv:1803.00745.
[R8]Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O’Brien. A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 2014. doi:10.1038/ncomms5213.
[R9]Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, and Dacheng Tao. The expressive power of parameterized quantum circuits. 2018. arXiv:1810.11922.
[R10]Jonathan Romero, Jonathan P Olson, and Alan Aspuru-Guzik. Quantum autoencoders for efficient compression of quantum data. Quantum Science and Technology, 2(4):045001, 2017. doi:10.1088/2058-9565/aa8072.
[R11]D. Shepherd and M. J. Bremner. Temporally unstructured quantum computation. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 465(2105):1413–1439, 2009. doi:10.1098/rspa.2008.0443.
[R12]Vojtech Havlicek, Antonio D Córcoles, Kristan Temme, Aram W Harrow, Jerry M Chow, and Jay M Gambetta. Supervised learning with quantum enhanced feature spaces. 2018. arXiv:1804.11326.
[R13]Juan Miguel Arrazola, Patrick Rebentrost, and Christian Weedbrook. Quantum supremacy and high-dimensional integration. 2017. arXiv:1712.07288.
[R14]Gregory R Steinbrecher, Jonathan P Olson, Dirk Englund, and Jacques Carolan. Quantum optical neural networks. arXiv:1808.10047.
[R15]Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm. 2014. arXiv:1411.4028.
[R16]Guillaume Verdon, Michael Broughton, and Jacob Biamonte. A quantum algorithm to train neural networks using low-depth circuits. 2017. arXiv:1712.05304.
[R17]Mark Fingerhuth, Tomáš Babej, and others. A quantum alternating operator ansatz with hard and soft constraints for lattice protein folding. 2018. arXiv:1810.13411.
[R18]William Huggins, Piyush Patel, K Birgitta Whaley, and E Miles Stoudenmire. Towards quantum machine learning with tensor networks. 2018. arXiv:1803.11537.
[R19]Edwin Stoudenmire and David J Schwab. Supervised learning with tensor networks. In D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett, editors, Advances in Neural Information Processing Systems 29, 4799–4807. Curran Associates, Inc., 2016. URL:
[R20]Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Hartmut Neven. Quantum algorithms for fixed qubit architectures. 2017. arXiv:1703.06199.
[R21]Marcello Benedetti, Delfina Garcia-Pintos, Yunseong Nam, and Alejandro Perdomo-Ortiz. A generative modeling approach for benchmarking and training shallow quantum circuits. 2018. arXiv:1801.07686.
[R22]Pierre-Luc Dallaire-Demers and Nathan Killoran. Quantum generative adversarial networks. 2018. arXiv:1804.08641.
[R23]Seth Lloyd and Christian Weedbrook. Quantum generative adversarial learning. Physical Review Letters, 2018. doi:10.1103/physrevlett.121.040502.

Further reading

For more details and further information on quantum machine learning and quantum computation, please see the following:

[FR1]Dave Wecker, Matthew B. Hastings, and Matthias Troyer. Progress towards practical quantum variational algorithms. Physical Review A, 2015. doi:10.1103/physreva.92.042303.
[FR2]Nathan Killoran, Josh Izaac, Nicolás Quesada, Ville Bergholm, Matthew Amy, and Christian Weedbrook. Strawberry Fields: A software platform for photonic quantum computing. 2018. arXiv:1804.03159.
[FR3]Dougal Maclaurin, David Duvenaud, and Ryan P Adams. Autograd: effortless gradients in numpy. In ICML 2015 AutoML Workshop. 2015. URL:
[FR4]Atılım Güneş Baydin, Barak A. Pearlmutter, Alexey Andreyevich Radul, and Jeffrey Mark Siskind. Automatic differentiation in machine learning: a survey. Journal of Machine Learning Research, 18(153):1–43, 2018. URL:, arXiv:1502.05767.
[FR5]JS Otterbach, R Manenti, N Alidoust, A Bestwick, M Block, B Bloom, S Caldwell, N Didier, E Schuyler Fried, S Hong, and others. Unsupervised machine learning on a hybrid quantum computer. 2017. arXiv:1712.05771.
[FR6]Andrea Rocchetto, Edward Grant, Sergii Strelchuk, Giuseppe Carleo, and Simone Severini. Learning hard quantum distributions with variational autoencoders. npj Quantum Information, 2018. doi:10.1038/s41534-018-0077-z.
[FR7]Gian Giacomo Guerreschi and Mikhail Smelyanskiy. Practical optimization for hybrid quantum-classical algorithms. 2017. arXiv:1701.01450.
[FR8]Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics, 18(2):023023, 2016.
[FR9]Edward Farhi and Hartmut Neven. Classification with quantum neural networks on near term processors. 2018. arXiv:1802.06002.
[FR10]Edward Grant, Marcello Benedetti, Shuxiang Cao, Andrew Hallam, Joshua Lockhart, Vid Stojevic, Andrew G Green, and Simone Severini. Hierarchical quantum classifiers. 2018. arXiv:1804.03680.
[FR11]Marcello Benedetti, John Realpe-Gómez, and Alejandro Perdomo-Ortiz. Quantum-assisted helmholtz machines: a quantum–classical deep learning framework for industrial datasets in near-term devices. Quantum Science and Technology, 3(3):034007, 2018. doi:10.1088/2058-9565/aabd98.
[FR12]Marcello Benedetti, John Realpe-Gómez, Rupak Biswas, and Alejandro Perdomo-Ortiz. Quantum-assisted learning of hardware-embedded probabilistic graphical models. Physical Review X, 2017. doi:10.1103/physrevx.7.041052.
[FR13]Sergio Boixo, Sergei V Isakov, Vadim N Smelyanskiy, and Hartmut Neven. Simulation of low-depth quantum circuits as complex undirected graphical models. 2017. arXiv:1712.05384.
[FR14]Jin-Guo Liu and Lei Wang. Differentiable learning of quantum circuit born machine. 2018. arXiv:1804.04168.
[FR15]Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven. Barren plateaus in quantum neural network training landscapes. 2018. arXiv:1803.11173.