One of the most challenging problems in modern theoretical physics is the so-called many-body problem. Typical many-body systems are composed of a large number of strongly interacting particles. Few such systems are amenable to exact mathematical treatment and numerical techniques are needed to make progress. However, since the resources required to specify a generic many-body quantum state depend exponentially on the number of particles in the system (more precisely, on the number of degrees of freedom), even today's best supercomputers lack sufficient power to exactly encode such states (they can handle only relatively small systems, with less than 45 particles). As we shall see, recent applications of machine learning techniques (artificial neural networks in particular) have been shown to provide highly efficient representations of such complex states, making their overwhelming complexity computationally tractable.

The complexity of interacting quantum many-body systems leads to exceedingly long recurrence times of the initial quantum state for all but the smallest systems. Thus, experimentally, recurrences can only be determined on the level of the accessible observables. Realizing a commensurate spectrum of collective excitations in one-dimensional superfluids, we demonstrate recurrences of coherence and long-range order in an interacting quantum many-body system containing thousands of particles. Our findings will enable the study of the coherent dynamics of large quantum systems even after they have reached a transient thermal-like state.

A quantum walk is the quantum mechanical analog of a classical random walk, describing the propagation of quantum walkers (photons) through an optical circuit. Because quantum walks generate large-scale quantum superposed states, they can be used for simulating many-body quantum systems and the development of algorithms for quantum computation. Nejadsattari et al. describe the photonic simulation with cyclic quantum systems. With the ability to simulate a variety of different quantum operations and gates, they claim that the versatility of the approach should allow the study of more complex many-body systems.

The same type of artificial intelligence that mastered the ancient game of Go could help wrestle with the amazing complexity of quantum systems containing billions of particles. Google's AlphaGo artificial neural network made headlines last year when it bested a world champion at Go. After marvelling at this feat, Giuseppe Carleo of ETH Zurich in Switzerland thought it might be possible to build a similar machine-learning tool to crack one of the knottiest problems in quantum physics. Now, he has built just such a neural network – which could turn out to be a game changer in understanding quantum systems. Go is far more complex than chess, in that the number of possible positions on a Go board could exceed the number of atoms in the universe.

In this talk I will discuss some of the long-term challenges emerging with the effort of making deep learning a relevant tool for controlled scientific discovery in many-body quantum physics. The current state of the art of deep neural quantum states and learning tools will be discussed in connection with open challenging problems in condensed matter physics, including frustrated magnetism and quantum dynamics. Variational algorithms for a gate-based quantum computer, like the QAOA, prescribe a fixed circuit ansatz --- up to a set of continuous parameters --- that is designed to find a low-energy state of a given target Hamiltonian. After reviewing the relevant aspects of the QAOA, I will describe attempts to make the algorithm more efficient. The strategies I will explore are 1) tuning the variational objective function away from the energy expectation value, 2) analytical estimates that allow elimination of some of the gates in the QAOA circuit, and 3) using methods of machine learning to search the design space of nearby circuits for improvements to the original ansatz.