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Density functional theory (DFT) is an extremely popular approach to electronic structure problems in both materials science and chemistry and many other fields. Over the past several years, often in collaboration with Klaus Mueller at TU Berlin, we have explored using machine-learning to find the density functionals that must be approximated in DFT calculations. I will summarize our results so far, and report on two new works.

Quantum mechanics is down to earth - quite literally - since the electrons within the tiny crystals found in a handful of dirt manifest a dizzying world of quantum motion. Each crystal has it’s own unique choreography, with the electrons entangled in a myriad of quantum dances. Quantum entanglement also holds the promise of futuristic Quantum Computers - which might be comprised of electron and nuclear spins inside diamond, or of atoms confined in traps, or of small superconducting grains, among a plethora of suggested platforms. In this talk I will describe ongoing efforts to elucidate the mysteries of Quantum Crystals, to design and assemble Quantum Computers, before ruminating about “Quantum Cognition” - the proposal that our brains are capable of quantum processing.

An increase in the efficiency of sampling from Boltzmann distributions would have a significant impact in deep learning and other machine learning applications. Recently, quantum annealers have been proposed as a potential candidate to speed up this task, but several limitations still bar these state-of-the-art technologies from being used effectively. One of the main limitations is that, while the device may indeed sample from a Boltzmann-like distribution, quantum dynamical arguments suggests it will do so with an instance-dependent effective temperature, different from the physical temperature of the device. Unless this unknown temperature can be unveiled, it might not be possible to effectively use a quantum annealer for Boltzmann sampling. In this talk, we present a strategy to overcome this challenge with a simple effective-temperature estimation algorithm. We provide a systematic study assessing the impact of the effective temperatures in the learning of a kind of restricted Boltzmann machine embedded on quantum hardware, which can serve as a building block for deep learning architectures. We also provide a comparison to k-step contrastive divergence (CD-k) with k up to 100. Although assuming a suitable fixed effective temperature also allows to outperform one step contrastive divergence (CD-1), only when using an instance-dependent effective temperature we find a performance close to that of CD-100 for the case studied here. We discuss generalizations of the algorithm to other more expressive generative models, beyond restricted Boltzmann machines.

Machine learning is a rapidly growing field in computer science with applications in computer vision, voice recognition, medical diagnosis, spam filtering, search engines, etc. In this presentation, I will introduce a new machine learning approach based on quantum Boltzmann distribution of a transverse-field Ising Model. Due to the non-commutative nature of quantum mechanics, the training process of the Quantum Boltzmann Machine (QBM) can become nontrivial.  I will show how to circumvent this problem by introducing bounds on the quantum probabilities. This allows training the QBM efficiently by sampling. I will then show examples of QBM training with and without the bound, using exact diagonalization, and compare the results with classical Boltzmann training. Finally, after a brief introduction to D-Wave quantum annealing processors, I will discuss the possibility of using such processors for QBM training and application.

Can quantum computers outperform classical computers on any computational problem in the near future? We study the problem of sampling from the output distribution of random quantum circuits. Sampling from this distribution requires an exponential amount of classical computational resources. We argue that quantum supremacy can be achieved in the near future with approximately fifty superconducting qubits and without error correction despite the fact that quantum random circuits are extremely sensitive to errors.

We propose a family of models with an exponential number of parameters, but which are approximated by a tensor network. Tensor networks are used to represent quantum wavefunctions, and powerful methods for optimizing them can be extended to machine learning applications as well. We use a matrix product state to classify images, and find that a surprisingly small bond dimension yields state-of-the-art results. Tensor networks offer many advantages for machine learning, such as better scaling for existing machine learning approaches and the ability to adapt hyperparameters during training. We will also propose a generative interpretation of the trained models.