
A. Elben, J. Yu, G. Zhu, M. Hafezi, F. Pollmann, P. Zoller, B. Vermersch Manybody topological invariants from randomized measurements,
Sci. Adv. 6 (20200410),
http://dx.doi.org/10.1126/sciadv.aaz3666 doi:10.1126/sciadv.aaz3666 (ID: 720289)
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The classification of symmetryprotected topological (SPT) phases in one dimension has been recently achieved, and had a fundamental impact in our understanding of quantum phases in condensed matter physics. In this framework, SPT phases can be identified by manybody topological invariants, which are quantized nonlocal correlators for the manybody wavefunction. While SPT phases can now be realized in interacting synthethic quantum systems, the direct measurement of quantized manybody topological invariants has remained so far elusive. Here, we propose measurement protocols for manybody topological invariants for all types of protecting symmetries of onedimensional interacting bosonic systems. Our approach relies on randomized measurements implemented with local random unitaries, and can be applied to any spin system with singlesite addressability and readout. Our scheme thus provides a versatile toolbox to experimentally classify interacting SPT phases.

M. K. Joshi, A. Elben, B. Vermersch, T. Brydges, C. Maier, P. Zoller, R. Blatt, C. F. Roos Quantum information scrambling in a trappedion quantum simulator with tunable range interactions,
Phys. Rev. Lett. 124 240505 (20200107),
http://dx.doi.org/10.1103/PhysRevLett.124.240505 doi:10.1103/PhysRevLett.124.240505 (ID: 720436)
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In ergodic manybody quantum systems, locally encoded quantum information becomes, in the course of time evolution, inaccessible to local measurements. This concept of "scrambling" is currently of intense research interest, entailing a deep understanding of manybody dynamics such as the processes of chaos and thermalization. Here, we present first experimental demonstrations of quantum information scrambling on a 10qubit trappedion quantum simulator representing a tunable longrange interacting spin system, by estimating outoftime ordered correlators (OTOCs) through randomized measurements. We also analyze the role of decoherence in our system by comparing our measurements to numerical simulations and by measuring Rényi entanglement entropies.

A. Elben, B. Vermersch, R. van Bijnen, C. Kokail, T. Brydges, C. Maier, M. K. Joshi, R. Blatt, C. F. Roos, P. Zoller CrossPlatform Verification of Intermediate Scale Quantum Devices,
Phys. Rev. Lett. 124 10504 (20200106),
http://dx.doi.org/10.1103/PhysRevLett.124.010504 doi:10.1103/PhysRevLett.124.010504 (ID: 720357)
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We describe a protocol for crossplatform verification of quantum simulators and quantum computers. We show how to measure directly the overlap Tr[ρ1ρ2] and the purities Tr[ρ21,2], and thus a (mixedstate) fidelity, of two quantum states ρ1 and ρ2 prepared in separate experimental platforms. We require only local measurements in randomized product bases, which are communicated classically. As a proofofprinciple, we present the measurement of experimenttheory fidelities for entangled 10qubit quantum states in a trapped ion quantum simulator.

L. Sieberer, T. Olsacher, A. Elben, M. Heyl, P. Hauke, F. Haake, P. Zoller Digital Quantum Simulation, Trotter Errors, and Quantum Chaos of the Kicked Top,
npj Quantum Information 5 (20190920),
http://dx.doi.org/10.1038/s4153401901925 doi:10.1038/s4153401901925 (ID: 720105)
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This work aims at giving Trotter errors in digital quantum simulation (DQS) of collective spin systems an interpretation in terms of quantum chaos of the kicked top. In particular, for DQS of such systems, regular dynamics of the kicked top ensures convergence of the Trotterized time evolution, while chaos in the top, which sets in above a sharp threshold value of the Trotter step size, corresponds to the proliferation of Trotter errors. We show the possibility to analyze this phenomenology in a wide variety of experimental realizations of the kicked top, ranging from single atomic spins to trappedion quantum simulators which implement DQS of alltoall interacting spin1/2 systems. These platforms thus enable indepth studies of Trotter errors and their relation to signatures of quantum chaos, including the growth of outoftimeordered correlators.

B. Vermersch, A. Elben, L. Sieberer, N. Y. Yao, P. Zoller Probing scrambling using statistical correlations between randomized measurements,
Phys. Rev. X 9 21061 (20190627),
http://dx.doi.org/10.1103/PhysRevX.9.021061 doi:10.1103/PhysRevX.9.021061 (ID: 720042)
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We present a protocol to study scrambling using statistical correlations between measurements, performed after evolving a quantum system from random initial states. We show that the resulting statistical correlations are directly related to OTOCs and can be used to probe scrambling in manybody systems. Our protocol, which does not require reversing time evolution or auxiliary degrees of freedom, can be realized in stateoftheart quantum simulation experiments.

A. Elben, B. Vermersch, C. F. Roos, P. Zoller Statistical correlations between locally randomized measurements: a toolbox for probing entanglement in manybody quantum states,
Phys. Rev. A 99 52323 (20190515),
http://dx.doi.org/10.1103/PhysRevA.99.052323 doi:10.1103/PhysRevA.99.052323 (ID: 720100)
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We develop a general theoretical framework for measurement protocols employing statistical correlations of randomized measurements. We focus on locally randomized measurements implemented with local random unitaries in quantum lattice models. In particular, we discuss the theoretical details underlying the recent measurement of the second Rényi entropy of highly mixed quantum states consisting of up to 10 qubits in a trappedion quantum simulator [Brydges et al., arXiv:1806.05747]. We generalize the protocol to access the overlap of quantum states, prepared sequentially in an experiment. Furthermore, we discuss proposals for quantum state tomography based on randomized measurements within our framework and the respective scaling of statistical errors with system size.

T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, C. F. Roos Probing Renyi entanglement entropy via randomized measurements,
Science 364 260 (20190419),
http://dx.doi.org/10.1126/science.aau4963 doi:10.1126/science.aau4963 (ID: 720034)
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Entanglement is the key feature of manybody quantum systems, and the development of new tools to probe it in the laboratory is an outstanding challenge. Measuring the entropy of different partitions of a quantum system provides a way to probe its entanglement structure. Here, we present and experimentally demonstrate a new protocol for measuring entropy, based on statistical correlations between randomized measurements. Our experiments, carried out with a trappedion quantum simulator, prove the overall coherent character of the system dynamics and reveal the growth of entanglement between its parts  both in the absence and presence of disorder. Our protocol represents a universal tool for probing and characterizing engineered quantum systems in the laboratory, applicable to arbitrary quantum states of up to several tens of qubits.