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arXiv:1801.00999 · Physical Review A

Unitary n -designs via random quenches in atomic Hubbard and spin models: Application to the measurement of Rényi entropies

Benoît Vermersch, Andreas Elben, Marcello Dalmonte, J. I. Cirac, P. Zoller

Published Feb 2, 2018• 107 citations
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Abstract

We present a general framework for the generation of random unitaries based on random quenches in atomic Hubbard and spin models, forming approximate unitary $n$-designs, and their application to the measurement of R\'enyi entropies. We generalize our protocol presented in Elben et al. [Phys. Rev. Lett. 120, 050406 (2018)] to a broad class of atomic and spin-lattice models. We further present an in-depth numerical and analytical study of experimental imperfections, including the effect of decoherence and statistical errors, and discuss connections of our approach with many-body quantum chaos.

DOI
10.1103/physreva.97.023604

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We present a scheme for measuring Rényi entropies in generic atomic Hubbard and spin models using single copies of a quantum state and for partitions in arbitrary spatial dimensions. Our approach is based on the generation of random unitaries from random quenches, implemented using engineered time-dependent disorder potentials, and standard projective measurements, as realized by quantum gas microscopes. By analyzing the properties of the generated unitaries and the role of statistical errors, with respect to the size of the partition, we show that the protocol can be realized in existing quantum simulators and used to measure, for instance, area law scaling of entanglement in two-dimensional spin models or the entanglement growth in many-body localized systems.
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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 R\'enyi entropy of highly mixed quantum states consisting of up to ten qubits in a trapped-ion quantum simulator [Brydges et al., Science 364, 260 (2019)]. We generalize the protocol to access the overlap of quantum states, prepared sequentially in one 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.
Three Hamiltonians are Sufficient for Unitary kkk-Design in Temporal Ensemble
Unitary kkk-designs are central to quantum information and quantum many-body physics, as they provide efficient proxies for Haar-random dynamics. We study how chaotic Hamiltonian evolution can generate unitary kkk-designs using the frame potential (FP). Unlike standard approaches based on independently sampled Hamiltonians or fine-tuned evolution times, we consider a quenched temporal ensemble in which the Hamiltonians are sampled once and then held fixed, while randomness enters only through the evolution times. We compare a two-step protocol (2SP), with evolution under two fixed Hamiltonians, to a three-step protocol (3SP) with one additional quench, with three evolution times sampled independently. Analytically and numerically, we show that the 2SP fails to realize general unitary kkk-designs, whereas the 3SP realizes them for arbitrary kkk. The random phases in the 3SP impose stronger index-matching constraints in the FP, eliminating the independent permutation degrees of freedom that still remain in the 2SP. Even with imperfect time averaging, the 3SP reaches a given accuracy within a parametrically shorter time window. These results hold universally across different symmetry classes and persist in both all-to-all and local models with random interactions.
We present a single-quench protocol that generates unitary k-designs with minimal control. A system first evolves under a random Hamiltonian H_{1}; at a switch time t_{s}≥t_{Th} (the Thouless time), it is quenched to an independently drawn H_{2} from the same ensemble and then evolves under H_{2}. This single quench breaks residual spectral correlations that prevent strictly time-independent chaotic dynamics from forming higher-order designs. The resulting ensemble approaches a unitary k-design using only a single control operation-far simpler than Brownian schemes with continuously randomized couplings or protocols that apply random quenches at short time intervals. Our analytical results are derived for Gaussian unitary random matrices, while our numerical demonstrations are carried out primarily in the Sachdev-Ye-Kitaev model. Beyond offering a direct route to Haar-like randomness, the protocol yields an operational, measurement-friendly definition of t_{Th} and provides a quantitative diagnostic of chaoticity. It further suggests symmetry-resolved and open-system extensions, circuit-level single-quench analogs, and applications to randomized measurements, benchmarking, and tomography.