Quantum state tomography is the standard technique for estimating the quantum state of small systems 1 . But its application to larger systems soon becomes impractical as the required resources scale exponentially with the size. Therefore, considerable e ort is dedicated to the development of new characterization tools for quantum many-body states 2-11 . Here we demonstrate matrix product state tomography 2 , which is theoretically proven to allow for the e cient and accurate estimation of a broad class of quantum states. We use this technique to reconstruct the dynamical state of a trapped-ion quantum simulator comprising up to 14 entangled and individually controlled spins: a size far beyond the practical limits of quantum state tomography. Our results reveal the dynamical growth of entanglement and describe its complexity as correlations spread out during a quench: a necessary condition for future demonstrations of better-than-classical performance. Matrix product state tomography should therefore find widespread use in the study of large quantum many-body systems and the benchmarking and verification of quantum simulators and computers.A matrix product state (MPS) is a way of expressing a manyparticle wavefunction which, for a broad class of physical states, offers a compact and accurate description with a number of parameters that increases only polynomially (that is, efficiently) in system components 12 . MPS tomography recognizes that the information required to identify the compact MPS is typically accessible locally; that is, by making measurements only on subsets of particles that lie in the same neighbourhood 2 . In such cases, the total effort required to obtain a reliable estimate for the state in the laboratory increases at most polynomially in system components 2,6 . States suited to MPS tomography and its generalizations to higher dimensions 2 include those with a maximum distance over which significant quantum correlations exist between constituents (locally correlated states): for example, the 2D cluster states-universal resource states for quantum computing-and the ground states of a broad class of one-dimensional (1D) systems [13][14][15] . We find that MPS tomography is also well-suited to characterize the states generated during the dynamical evolution of systems with shortranged interactions, as found in many physical systems.Consider an N -component quantum system initially in a product state (or other locally correlated state) in which interactions are abruptly turned on. In the presence of finite-range interactions, information and correlations spread out in the system with a strict maximum group velocity [16][17][18] . Therefore, after a finite evolution time, there is a maximum distance over which correlations extend in the system (the correlation length, L), beyond which correlations decay exponentially in distance. The information required to describe the state is largely contained in the local reductions: the reduced states (density matrices) of all groups of neighbouring particles...

We investigate the dynamics following a global parameter quench for two 1D models with variablerange power-law interactions: a long-range transverse Ising model, which has recently been realised in chains of trapped ions, and a long-range lattice model for spinless fermions with long-range tunnelling. For the transverse Ising model, the spreading of correlations and growth of entanglement are computed using numerical matrix product state techniques, and are compared with exact solutions for the fermionic tunnelling model. We identify transitions between regimes with and without an apparent linear light cone for correlations, which correspond closely between the two models. For long-range interactions (in terms of separation distance r, decaying slower than 1/r), we find that despite the lack of a light-cone, correlations grow slowly as a power law at short times, and that -depending on the structure of the initial state -the growth of entanglement can also be sublinear. These results are understood through analytical calculations, and should be measurable in experiments with trapped ions.

We propose an experimentally realizable quantum spin model that exhibits fast scrambling, based on non-local interactions which couple sites whose separation is a power of 2. By controlling the relative strengths of deterministic, non-random couplings, we can continuously tune from the linear geometry of a nearest-neighbor spin chain to an ultrametric geometry in which the effective distance between spins is governed by their positions on a tree graph. The transition in geometry can be observed in quench dynamics, and is furthermore manifest in calculations of the entanglement entropy. Between the linear and treelike regimes, we find a peak in entanglement and exponentially fast spreading of quantum information across the system. Our proposed implementation, harnessing photon-mediated interactions among cold atoms in an optical cavity, offers a test case for experimentally observing the emergent geometry of a quantum many-body system.

In an optical lattice entropy and mass transport by first-order tunneling is much faster than spin transport via superexchange. Here we show that adding a constant force (tilt) suppresses first-order tunneling, but not spin transport, realizing new features for spin Hamiltonians. Suppression of the superfluid transition can stabilize larger systems with faster spin dynamics. For the first time in a many-body spin system, we vary superexchange rates by over a factor of 100 and tune spin-spin interactions via the tilt. In a tilted lattice, defects are immobile and pure spin dynamics can be studied.arXiv:1908.09870v1 [cond-mat.quant-gas]

We show that atoms in tilted optical superlattices provide a platform for exploring coupled spin chains of forms that are not present in other systems. In particular, using a period-2 superlattice in 1D, we show that coupled Ising spin chains with XZ and ZZ spin coupling terms can be engineered. We use optimized tensor network techniques to explore the criticality and non-equilibrium dynamics in these models, finding a tricritical Ising point in regimes that are accessible in current experiments. These setups are ideal for studying low-entropy physics, as initial entropy is "frozen-out" in realizing the spin models, and provide an example of the complex critical behaviour that can arise from interaction-projected models.