It has recently been shown that small quantum subsystems generically equilibrate, in the sense that they spend most of the time close to a fixed equilibrium state. This relies on just two assumptions: that the state is spread over many different energies, and that the Hamiltonian has nondegenerate energy gaps. Given the same assumptions, it has also been shown that closed systems equilibrate with respect to realistic measurements. We extend these results in two important ways. Firstly, we prove equilibration over a finite (rather than infinite) time-interval, allowing us to bound the equilibration time. Secondly, we weaken the non-degenerate energy gaps condition, showing that equilibration occurs provided that no energy gap is hugely degenerate. 9 Acknowledgments 10 Appendix A. Proof of equation (13) 10 Appendix B. Proof of theorem 2 11 Appendix C. Proof of theorem 3 111 For a precise definition of these criteria, see the next section. 2 If desired, we could define d E to be the number of distinct energies on which ρ(t) has support. This gives a slight improvement in the results, but has the disadvantage of making d E and N (ε) depend on the state, rather than the Hamiltonian alone.

Neural networks enjoy widespread success in both research and industry and, with the advent of quantum technology, it is a crucial challenge to design quantum neural networks for fully quantum learning tasks. Here we propose a truly quantum analogue of classical neurons, which form quantum feedforward neural networks capable of universal quantum computation. We describe the efficient training of these networks using the fidelity as a cost function, providing both classical and efficient quantum implementations. Our method allows for fast optimisation with reduced memory requirements: the number of qudits required scales with only the width, allowing deep-network optimisation. We benchmark our proposal for the quantum task of learning an unknown unitary and find remarkable generalisation behaviour and a striking robustness to noisy training data.

Considering any Hamiltonian, any initial state, and measurements with a small number of possible outcomes compared to the dimension, we show that most measurements are already equilibrated. To investigate non-trivial equilibration we therefore consider a restricted set of measurements. When the initial state is spread over many energy levels, and we consider the set of observables for which this state is an eigenstate, most observables are initially out of equilibrium yet equilibrate rapidly. Moreover, all two-outcome measurements, where one of the projectors is of low rank, equilibrate rapidly.The topic of equilibration time scales has been of much interest lately [1][2][3][4][5][6][7][8][9]. Given that it has been shown that quantum systems equilibrate under rather general conditions [10][11][12], it is important to understand the time scale for the process. However, attempts to derive an upper bound on equilibration time have resulted in very large time scales. Short and Farrelly [1], for instance, obtain a very general bound, of which we give an improved derivation in appendix A, which scales with the dimension of the system, typically exponentially in the number of particles.A quantum system is said to undergo equilibration when its quantum state spends most of its time almost indistinguishable from a fixed (time-invariant) steadystate. This is not the same as thermalization, in which the steady-state is a Gibbs state. Thus, thermalization is a special case of equilibration, and understanding equilibration times is a key step in understanding thermalization times.When one discusses quantum equilibration, it is common to refer to either subsystem equilibration [10,13], in which a small system equilibrates due to contact with a bath, or observable equilibration, in which a fully closed system appears to equilibrate due to the limited information offered by outcomes of a particular set of observables. The latter was initially shown by Reimann [11,14], as a statement that the expectation values of quantum observables stay predominantly close to a static value, and was later built-on by Short [15], who showed that these results apply even if one considers all the information that can be gathered from the observable, instead of just the expectation value.In this paper we consider any finite-dimensional system and any Hamiltonian, and show that most N -outcome observables are initially in equilibrium (for N small compared to the dimension). To investigate timescales we therefore turn to a natural class of observables which are initially typically out of equilibrium -those with a definite initial value (i.e. observables for which the initial state is an eigenstate). We show that, for pure initial states spread over many energy levels, most of these observables equilibrate in very short times -in fact, most equilibrate essentially as fast as possible. Moreover, in the case of two-outcome observables where one of the projectors is of low rank we show that all observables equilibrate fast (for any initial state spread...