Efficient tomography of a quantum many-body system

Abstract

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...

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