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...

Discretizing spacetime is often a natural step towards modelling physical systems. For quantum systems, if we also demand a strict bound on the speed of information propagation, we get quantum cellular automata (QCAs). These originally arose as an alternative paradigm for quantum computation, though more recently they have found application in understanding topological phases of matter and have} been proposed as models of periodically driven (Floquet) quantum systems, where QCA methods were used to classify their phases. QCAs have also been used as a natural discretization of quantum field theory, and some interesting examples of QCAs have been introduced that become interacting quantum field theories in the continuum limit. This review discusses all of these applications, as well as some other interesting results on the structure of quantum cellular automata, including the tensor-network unitary approach, the index theory and higher dimensional classifications of QCAs.

Thermal states are the bedrock of statistical physics. Nevertheless, when and how they actually arise in closed quantum systems is not fully understood. We consider this question for systems with local Hamiltonians on finite quantum lattices. In a first step, we show that states with exponentially decaying correlations equilibrate after a quantum quench. Then, we show that the equilibrium state is locally equivalent to a thermal state, provided that the free energy of the equilibrium state is sufficiently small and the thermal state has exponentially decaying correlations. As an application, we look at a related important question: When are thermal states stable against noise? In other words, if we locally disturb a closed quantum system in a thermal state, will it return to thermal equilibrium? We rigorously show that this occurs when the correlations in the thermal state are exponentially decaying. All our results come with finite-size bounds, which are crucial for the growing field of quantum thermodynamics and other physical applications. DOI: 10.1103/PhysRevLett.118.140601 To understand the strengths and limitations of statistical physics, it makes sense to derive it from physical principles, without ad hoc assumptions. Along these lines, over the past twenty years, ideas from quantum information have given new insights into the foundations of statistical physics [1][2][3]. In particular, some progress was made towards understanding how and when thermalization occurs [4][5][6]. A large class of states of systems with weak intensive interactions (e.g., one dimensional systems) were shown to thermalize [5]. In [6], thermalization was also proved for a large class of states, in the thermodynamic limit. (We will compare the results of [6] to our results in detail below.) More recently, the equivalence of the microcanonical and canonical ensemble (i.e., thermal state) was proved for finite quantum lattice systems, when correlations in the thermal state decay sufficiently quickly [7] (see, also, [6,8]).Here, we prove thermalization results for closed quantum systems in two parts. First, we build upon previous equilibration results (e.g., Refs. [9,10]). A requirement for equilibration is that the effective dimension, defined below, is large. While there are physical arguments for this in some cases [11] (and it is true for most states drawn from the Haar measure on large subspaces [10]), there are no techniques for deciding whether a given initial state will equilibrate under a given Hamiltonian. Here, we prove that a large effective dimension is guaranteed for local Hamiltonian systems if the correlations in the initial state decay sufficiently quickly and the energy variance is sufficiently large. The latter is known for thermal states with intensive specific heat capacity and may, for large classes of states, be computed straightforwardly. The second part of thermalization is to show that the equilibrium state is locally indistinguishable from a thermal state. We prove that this occurs if the correlations in...

We propose a discrete spacetime formulation of quantum electrodynamics in one-dimension (a.k.a the Schwinger model) in terms of quantum cellular automata, i.e. translationally invariant circuits of local quantum gates. These have exact gauge covariance and a maximum speed of information propagation. In this picture, the interacting quantum field theory is defined as a "convergent" sequence of quantum cellular automata, parameterized by the spacetime lattice spacing-encompassing the notions of continuum limit and renormalization, and at the same time providing a quantum simulation algorithm for the dynamics.

Even after almost a century, the foundations of quantum statistical mechanics are still not completely understood. In this work, we provide a precise account on these foundations for a class of systems of paradigmatic importance that appear frequently as mean-field models in condensed matter physics, namely non-interacting lattice models of fermions (with straightforward extension to bosons). We demonstrate that already the translation invariance of the Hamiltonian governing the dynamics and a finite correlation length of the possibly non-Gaussian initial state provide sufficient structure to make mathematically precise statements about the equilibration of the system towards a generalized Gibbs ensemble, even for highly non-translation invariant initial states far from ground states of non-interacting models. Whenever these are given, the system will equilibrate rapidly according to a power-law in time as long as there are no longwavelength dislocations in the initial second moments that would render the system resilient to relaxation. Our proof technique is rooted in the machinery of Kusmin-Landau bounds. Subsequently, we numerically illustrate our analytical findings by discussing quench scenarios with an initial state corresponding to an Anderson insulator observing power-law equilibration. We discuss the implications of the results for the understanding of current quantum simulators, both in how one can understand the behaviour of equilibration in time, as well as concerning perspectives for realizing distinct instances of generalized Gibbs ensembles in optical lattice-based architectures.

We explore the different meanings of "quantum uncertainty" contained in Heisenberg's seminal paper from 1927, and also some of the precise definitions that were explored later. We recount the controversy about "Anschaulichkeit", visualizability of the theory, which Heisenberg claims to resolve. Moreover, we consider Heisenberg's programme of operational analysis of concepts, in which he sees himself as following Einstein. Heisenberg's work is marked by the tensions between semiclassical arguments and the emerging modern quantum theory, between intuition and rigour, and between shaky arguments and overarching claims. Nevertheless, the main message can be taken into the new quantum theory, and can be brought into the form of general theorems. They come in two kinds, not distinguished by Heisenberg. These are, on one hand, constraints on preparations, like the usual textbook uncertainty relation, and, on the other, constraints on joint measurability, including trade-offs between accuracy and disturbance.