
T. Olsacher, L. Pastori, C. Kokail, L. Sieberer, Peter Zoller Digital quantum simulation, learning of the Floquet Hamiltonian, and quantum chaos of the kicked top,
J. Phys. A: Math. Gen. 55 (20220819),
http://dx.doi.org/10.1088/17518121/ac8087 doi:10.1088/17518121/ac8087 (ID: 720859)
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The kicked top is one of the paradigmatic models in the study of quantum chaos (Haake et al 2018 Quantum Signatures of Chaos (Springer Series in Synergetics vol 54)). Recently it has been shown that the onset of quantum chaos in the kicked top can be related to the proliferation of Trotter errors in digital quantum simulation (DQS) of collective spin systems. Specifically, the proliferation of Trotter errors becomes manifest in expectation values of fewbody observables strongly deviating from the target dynamics above a critical Trotter step, where the spectral statistics of the Floquet operator of the kicked top can be predicted by random matrix theory. In this work, we study these phenomena in the framework of Hamiltonian learning (HL). We show how a recently developed HL protocol can be employed to reconstruct the generator of the stroboscopic dynamics, i.e., the Floquet Hamiltonian, of the kicked top. We further show how the proliferation of Trotter errors is revealed by HL as the transition to a regime in which the dynamics cannot be approximately described by a loworder truncation of the Floquet–Magnus expansion. This opens up new experimental possibilities for the analysis of Trotter errors on the level of the generator of the implemented dynamics, that can be generalized to the DQS of quantum manybody systems in a scalable way. This paper is in memory of our colleague and friend Fritz Haake.

L. Pastori, T. Olsacher, C. Kokail, Peter Zoller Characterization and Verification of Trotterized Digital Quantum Simulation via Hamiltonian and Liouvillian Learning,
PRX Quantum 3 30324 (20220818),
http://dx.doi.org/10.1103/PRXQuantum.3.030324 doi:10.1103/PRXQuantum.3.030324 (ID: 720828)
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The goal of digital quantum simulation is to approximate the dynamics of a given target Hamiltonian via a sequence of quantum gates, a procedure known as Trotterization. The quality of this approximation can be controlled by the so called Trotter step, that governs the number of required quantum gates per unit simulation time, and is intimately related to the existence of a timeindependent, quasilocal Hamiltonian that governs the stroboscopic dynamics, refered to as the Floquet Hamiltonian of the Trotterization. In this work, we propose a Hamiltonian learning scheme to reconstruct the implemented Floquet Hamiltonian orderbyorder in the Trotter step: this procedure is efficient, i.e., it requires a number of measurements that scales polynomially in the system size, and can be readily implemented in stateoftheart experiments. With numerical examples, we propose several applications of our method in the context of verification of quantum devices, from the characterization of the distinct sources of errors in digital quantum simulators to the design of new types of quantum gates. Furthermore, we show how our approach can be extended to the case of nonunitary dynamics and used to learn Floquet Liouvillians, thereby offering a way of characterizing the dissipative processes present in NISQ quantum devices.

A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, Peter Zoller Practical quantum advantage in quantum simulation,
Nature 607 676 (20220727),
http://dx.doi.org/10.1038/s41586022049406 doi:10.1038/s41586022049406 (ID: 720857)
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The development of quantum computing across several technologies and platforms has reached the point of having an advantage over classical computers for an artificial problem, a point known as ‘quantum advantage’. As a next step along the development of this technology, it is now important to discuss ‘practical quantum advantage’, the point at which quantum devices will solve problems of practical interest that are not tractable for traditional supercomputers. Many of the most promising shortterm applications of quantum computers fall under the umbrella of quantum simulation: modelling the quantum properties of microscopic particles that are directly relevant to modern materials science, highenergy physics and quantum chemistry. This would impact several important realworld applications, such as developing materials for batteries, industrial catalysis or nitrogen fixing. Much as aerodynamics can be studied either through simulations on a digital computer or in a wind tunnel, quantum simulation can be performed not only on future faulttolerant digital quantum computers but also already today through specialpurpose analogue quantum simulators. Here we overview the state of the art and future perspectives for quantum simulation, arguing that a first practical quantum advantage already exists in the case of specialized applications of analogue devices, and that fully digital devices open a full range of applications but require further development of faulttolerant hardware. Hybrid digital–analogue devices that exist today already promise substantial flexibility in nearterm applications.

T. Zache, C. Kokail, B. Sundar, P. Zoller Entanglement Spectroscopy and probing the LiHaldane Conjecture in Topological Quantum Matter,
Quantum 6 702 (20220427),
http://dx.doi.org/10.22331/q20220427702 doi:10.22331/q20220427702 (ID: 720692)
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Topological phases are characterized by their entanglement properties, which is manifest in a direct relation between entanglement spectra and edge states discovered by Li and Haldane. We propose to leverage the power of synthetic quantum systems for measuring entanglement via the Entanglement Hamiltonian to probe this relationship experimentally. This is made possible by exploiting the quasilocal structure of Entanglement Hamiltonians. The feasibility of this proposal is illustrated for two paradigmatic examples realizable with current technology, an integer quantum Hall state of noninteracting fermions on a 2D lattice and a symmetry protected topological state of interacting fermions on a 1D chain. Our results pave the road towards an experimental identification of topological order in strongly correlated quantum manybody systems.

C. Kokail, B. Sundar, T. Zache, A. Elben, B. Vermersch, M. Dalmonte, R. van Bijnen, P. Zoller Quantum Variational Learning of the Entanglement Hamiltonian,
Phys. Rev. Lett. 127 170501 (20211022),
http://dx.doi.org/10.1103/PhysRevLett.127.170501 doi:10.1103/PhysRevLett.127.170501 (ID: 720649)
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Learning the structure of the entanglement Hamiltonian (EH) is central to characterizing quantum manybody states in analog quantum simulation. We describe a protocol where spatial deformations of the manybody Hamiltonian, physically realized on the quantum device, serve as an efficient variational ansatz for a local EH. Optimal variational parameters are determined in a feedback loop, involving quench dynamics with the deformed Hamiltonian as a quantum processing step, and classical optimization. We simulate the protocol for the ground state of FermiHubbard models in quasi1D geometries, finding excellent agreement of the EH with BisognanoWichmann predictions. Subsequent ondevice spectroscopy enables a direct measurement of the entanglement spectrum, which we illustrate for a Fermi Hubbard model in a topological phase.

A. Neven, J. Carrasco, V. Vitale, C. Kokail, A. Elben, M. Dalmonte, P. Calabrese, P. Zoller, B. Vermersch, R. Kueng, B. Kraus Symmetryresolved entanglement detection using partial transpose moments,
npj Quantum Information 7 (20211020),
http://dx.doi.org/10.1038/s4153402100487y doi:10.1038/s4153402100487y (ID: 720635)
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We propose an ordered set of experimentally accessible conditions for detecting entanglement in mixed states. The kth condition involves comparing moments of the partially transposed density operator up to order k. Remarkably, the union of all moment inequalities reproduces the PeresHorodecki criterion for detecting entanglement. Our empirical studies highlight that the first four conditions already detect mixed state entanglement reliably in a variety of quantum architectures. Exploiting symmetries can help to further improve their detection capabilities. We also show how to estimate moment inequalities based on local random measurements of single state copies (classical shadows) and derive statistically sound confidence intervals as a function of the number of performed measurements. Our analysis includes the experimentally relevant situation of drifting sources, i.e. nonidentical, but independent, state copies.

D. Paulson, L. Dellantonio, J. Haase, A. Celi, A. Kan, A. Jena, C. Kokail, R. van Bijnen, K. Jansen, P. Zoller, C. A. Muschik Simulating 2D effects in lattice gauge theories on a quantum computer,
PRX Quantum 2 30334 (20210825),
http://dx.doi.org/10.1103/PRXQuantum.2.030334 doi:10.1103/PRXQuantum.2.030334 (ID: 720526)
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Quantum computing is in its greatest upswing, with socalled noisyintermediatescalequantum devices heralding the computational power to be expected in the near future. While the field is progressing toward quantum advantage, quantum computers already have the potential to tackle classically intractable problems. Here, we consider gauge theories describing fundamentalparticle interactions. On the way to their fullfledged quantum simulations, the challenge of limited resources on nearterm quantum devices has to be overcome. We propose an experimental quantum simulation scheme to study groundstate properties in twodimensional quantum electrodynamics (2D QED) using existing quantum technology. Our protocols can be adapted to larger lattices and offer the perspective to connect the lattice simulation to lowenergy observable quantities, e.g., the hadron spectrum, in the continuum theory. By including both dynamical matter and a nonminimal gaugefield truncation, we provide the novel opportunity to observe 2D effects on presentday quantum hardware. More specifically, we present two variationalquantumeigensolver (VQE) based protocols for the study of magnetic field effects and for taking an important first step toward computing the running coupling of QED. For both instances, we include variational quantum circuits for qubitbased hardware. We simulate the proposed VQE experiments classically to calculate the required measurement budget under realistic conditions. While this feasibility analysis is done for trapped ions, our approach can be directly adapted to other platforms. The techniques presented here, combined with advancements in quantum hardware, pave the way for reaching beyond the capabilities of classical simulations.

C. Kokail, R. van Bijnen, A. Elben, B. Vermersch, P. Zoller Entanglement Hamiltonian Tomography in Quantum Simulation,
Nature Phys. (20210624),
http://dx.doi.org/10.1038/s4156702101260w doi:10.1038/s4156702101260w (ID: 720530)
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Entanglement is the crucial ingredient of quantum manybody physics, and characterizing and quantifying entanglement in closed system dynamics of quantum simulators is an outstanding challenge in today's era of intermediate scale quantum devices. Here we discuss an efficient tomographic protocol for reconstructing reduced density matrices and entanglement spectra for spin systems. The key step is a parametrization of the reduced density matrix in terms of an entanglement Hamiltonian involving only quasi local fewbody terms. This ansatz is fitted to, and can be independently verified from, a small number of randomised measurements. The ansatz is suggested by Conformal Field Theory in quench dynamics, and via the BisognanoWichmann theorem for ground states. Not only does the protocol provide a testbed for these theories in quantum simulators, it is also applicable outside these regimes. We show the validity and efficiency of the protocol for a longrange Ising model in 1D using numerical simulations. Furthermore, by analyzing data from 10 and 20 ion quantum simulators [Brydges \textit{et al.}, Science, 2019], we demonstrate measurement of the evolution of the entanglement spectrum in quench dynamics.

J. Carrasco, A. Elben, C. Kokail, B. Kraus, P. Zoller Theoretical and Experimental Perspectives of Quantum Verification,
PRX Quantum 2 10102 (20210303),
http://dx.doi.org/10.1103/PRXQuantum.2.010102 doi:10.1103/PRXQuantum.2.010102 (ID: 720626)
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In this perspective we discuss verification of quantum devices in the context of specific examples, formulated as proposed experiments. Our first example is verification of analog quantum simulators as Hamiltonian learning, where the input Hamiltonian as design goal is compared with the parent Hamiltonian for the quantum states prepared on the device. The second example discusses crossdevice verification on the quantum level, i.e. by comparing quantum states prepared on different quantum devices. We focus in particular on protocols using randomized measurements, and we propose establishing a central data repository, where existing experimental devices and platforms can be compared. In our final example, we address verification of the output of a quantum device from a computer science perspective, addressing the question of how a user of a quantum processor can be certain about the correctness of its output, and propose minimal demonstrations on present day devices.

C. R. Kaubrügger, P. Silvi, C. Kokail, R. van Bijnen, A. M. Rey, J. Ye, A. Kaufman, P. Zoller Variational spinsqueezing algorithms on programmable quantum sensors,
Phys. Rev. Lett. 123 260505 (20190822),
http://dx.doi.org/10.1103/PhysRevLett.123.260505 doi:10.1103/PhysRevLett.123.260505 (ID: 720356)
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Arrays of atoms trapped in optical tweezers combine features of programmable analog quantum simulators with atomic quantum sensors. Here we propose variational quantum algorithms, tailored for tweezer arrays as programmable quantum sensors, capable of generating entangled states ondemand for precision metrology. The scheme is designed to generate metrological enhancement by optimizing it in a feedback loop on the quantum device itself, thus preparing the best entangled states given the available quantum resources. We apply our ideas to generate spinsqueezed states on Sr atom tweezer arrays, where finiterange interactions are generated through Rydberg dressing. The complexity of experimental variational optimization of our quantum circuits is expected to scale favorably with system size. We numerically show our approach to be robust to noise, and surpassing known protocols.

C. Kokail, C. Maier, R. van Bijnen, T. Brydges, M. K. Joshi, P. Jurcevic, C. A. Muschik, P. Silvi, R. Blatt, C. F. Roos, P. Zoller Selfverifying variational quantum simulation of lattice models,
Nature 569 360 (20190515),
http://dx.doi.org/10.1038/s4158601911774 doi:10.1038/s4158601911774 (ID: 720076)
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Hybrid classicalquantum algorithms aim at variationally solving optimisation problems, using a feedback loop between a classical computer and a quantum coprocessor, while benefitting from quantum resources. Here we present experiments demonstrating selfverifying, hybrid, variational quantum simulation of lattice models in condensed matter and highenergy physics. Contrary to analog quantum simulation, this approach forgoes the requirement of realising the targeted Hamiltonian directly in the laboratory, thus allowing the study of a wide variety of previously intractable target models. Here, we focus on the Lattice Schwinger model, a gauge theory of 1D quantum electrodynamics. Our quantum coprocessor is a programmable, trappedion analog quantum simulator with up to 20 qubits, capable of generating families of entangled trial states respecting symmetries of the target Hamiltonian. We determine ground states, energy gaps and, by measuring variances of the Schwinger Hamiltonian, we provide algorithmic error bars for energies, thus addressing the longstanding challenge of verifying quantum simulation.