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Self-verifying variational quantum simulation of lattice models

An Author Correction to this article was published on 03 April 2020

This article has been updated


Hybrid classical–quantum algorithms aim to variationally solve optimization problems using a feedback loop between a classical computer and a quantum co-processor, while benefiting from quantum resources. Here we present experiments that demonstrate self-verifying, hybrid, variational quantum simulation of lattice models in condensed matter and high-energy physics. In contrast to analogue quantum simulation, this approach forgoes the requirement of realizing the targeted Hamiltonian directly in the laboratory, thus enabling the study of a wide variety of previously intractable target models. We focus on the lattice Schwinger model, a gauge theory of one-dimensional quantum electrodynamics. Our quantum co-processor is a programmable, trapped-ion analogue quantum simulator with up to 20 qubits, capable of generating families of entangled trial states respecting the symmetries of the target Hamiltonian. We determine ground states, energy gaps and additionally, by measuring variances of the Schwinger Hamiltonian, we provide algorithmic errors for the energies, thus taking a step towards verifying quantum simulation.

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Fig. 1: Classical–quantum feedback loop of VQS.
Fig. 2: Schwinger ground state (20 ions).
Fig. 3: Schwinger ground state (eight ions).
Fig. 4: Quantum phase transition.

Data availability

All data generated and analysed in this study are available from the corresponding author on reasonable request.

Code availability

The optimization algorithm developed during this study is available from the corresponding author on reasonable request.

Change history

  • 03 April 2020

    An amendment to this paper has been published and can be accessed via a link at the top of the paper.


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We thank J. Bollinger, A. Elben, K. Holmström, K. Jansen, M. Lukin, E. A. Martinez, S. Montangero, P. Schindler and U. J. Wiese for discussions. The numerical results presented were achieved (in part) using the HPC infrastructure LEO of the University of Innsbruck. The research at Innsbruck is supported by the ERC Synergy Grant UQUAM, by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement number 741541, by the SFB FoQuS (FWF project number F4016-N23) and by QTFLAG—QuantERA. This project (or publication) has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 817482 (PASQuanS). We thank A. Elben and B. Vermersch for the development of the software used for evaluating the Rényi entropies measurement.

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Nature thanks Juergen Berges, Christof Wunderlich and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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Authors and Affiliations



The research topic was developed by C.K., R.v.B., C.A.M., P.S. and C.F.R. (following a suggestion by P.Z.). C.K., R.v.B., P.S. and P.Z. developed the theoretical protocols. Software for the classical-quantum feedback loop and classical simulations was developed by R.v.B. and C.K. C.M., M.K.J., T.B., P.J., C.F.R. and R.B. contributed to the experimental setup. Experimental data were taken by C.M., T.B., P.J. and M.K.J. R.v.B., C.K., C.M., C.A.M., P.S., C.F.R. and P.Z. wrote the manuscript. All authors contributed to the discussion of the results and the manuscript.

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Correspondence to P. Zoller.

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Extended data figures and tables

Extended Data Fig. 1 Energy minimization for 16 ions.

a, Experimentally measured energies \({E}^{(0)}({{\boldsymbol{\theta }}}_{i})\equiv \langle \varPsi ({{\boldsymbol{\theta }}}_{i})|{\hat{H}}_{{\rm{T}}}|\varPsi ({{\boldsymbol{\theta }}}_{i})\rangle \) (upper panel, dots) in the course of a single optimization run for 16 ions, plotted versus iteration number i of the DIRECT optimization algorithm (see text), for \(m=0.6,w= {\bar{g}} =1\). Energy values E(0)(θi) are colour-coded to indicate the Euclidean distance of θi to the final optimized parameter vector θopt, as selected by theoretical fidelity. The solid red line indicates the algorithm’s current estimate of the ground-state energy and its 2σ uncertainty (shaded area), from modelling the energies observed so far as jointly Gaussian distributed random variables (Methods). The inset shows a close-up of a late stage of the optimization, where statistical error bars (Methods) are displayed, and theoretically simulated values are plotted as crosses. The lower panel of a displays the theoretically calculated fidelities \({\mathscr{F}}\) corresponding to the experimentally applied parameters θi. b, Visualization of the energy landscape. Sampled energies are plotted as a function of their distance in the 15-dimensional parameter space \(| \Delta {\boldsymbol{\theta }}| \) relative to the optimal point θopt, and the cell size that each sampling point represents in the DIRECT algorithm (Methods). The algorithm encountered several local minima, appearing as distinct ‘fingers’ with low energies at specific parameter distances and extending towards the direction of smaller representative cell sizes, which is indicative of an increasingly dense sampling around each of the local minima.

Extended Data Fig. 2 Analysis of the algorithmic error bar.

The theoretically expected standard deviations of the target Hamiltonians, obtained from a numerically simulated experiment optimizing the ground state for eight ions, are plotted as a function of the bare mass m around the critical point, and of the circuit depth (with \(w= {\bar{g}} =1\). As expected, algorithmic error bars decrease for increasing circuit depth. This is especially visible in proximity of the critical point (orange dots on peaks in the black curves) where the target states are more entangled, requiring deeper circuits to achieve a required precision.

Extended Data Fig. 3 Protection of target model symmetries.

The left-hand panel shows the Schwinger spin model in the Kogut–Susskind formulation, where matter fields are represented by spin degrees of freedom: \({\hat{H}}_{{\rm{T}}}=2w\mathop{\sum }\limits_{n=1}^{N-1}({\hat{\sigma }}_{n}^{x}{\hat{\sigma }}_{n+1}^{x}+{\hat{\sigma }}_{n}^{y}{\hat{\sigma }}_{n+1}^{y})+\mathop{\sum }\limits_{i=1}^{N}{c}_{i}{\hat{\sigma }}_{i}^{z}+\sum _{i,j > 1}{c}_{ij}{\hat{\sigma }}_{i}^{z}{\hat{\sigma }}_{j}^{z}\). The gauge fields were eliminated using the Gauss law, which results in complicated long-range spin–spin interactions cij (see Supplementary Information section I). The Schwinger Hamiltonian \({ {\hat{H}} }_{{\rm{T}}}\) is block-diagonal with respect to different sectors of \({\sigma }_{{\rm{t}}{\rm{o}}{\rm{t}}}^{z}\). Furthermore, the \({\sigma }_{{\rm{t}}{\rm{o}}{\rm{t}}}^{z}=0\) sector decomposes into two blocks corresponding to quantum numbers \(\hat{{\rm{C}}{\rm{P}}}=+1\) and \(\hat{{\rm{C}}{\rm{P}}}=-1\). We investigate the ground state of \({ {\hat{H}} }_{{\rm{T}}}\) restricted to the symmetry sector with quantum numbers 0 and +1, respectively, for the \({\sigma }_{{\rm{t}}{\rm{o}}{\rm{t}}}^{z}\) and the \(\widehat{{\rm{C}}{\rm{P}}}\) symmetries. The right-hand panel shows that the native resources on an ion trap platform can be exploited to engineer symmetry-preserving quantum circuits specifically tailored to the Schwinger model. Taking \(B\gg {\rm{\max }}\{| {J}_{ij}| \}\) (equation (3) in the main text), results in an approximate protection of the \({\sigma }_{{\rm{t}}{\rm{o}}{\rm{t}}}^{z}\) symmetry (see Supplementary Information). Likewise, single-qubit rotations around the z axis can be forced to be \(\widehat{{\rm{C}}{\rm{P}}}\)-symmetric by linking the rotation angles between the left and the right half of the chain according to \({\theta }_{{\ell }}^{n}=-{\theta }_{{\ell }}^{N-n+1}\) (see Supplementary Information). Such a circuit will thus protect the target symmetries, restricting the variational search to be only within the portion of Hilbert space of interest.

Extended Data Fig. 4 Measurement scheme.

Strategy for measuring the squared Schwinger Hamiltonian \({ {\hat{H}} }_{T}^{2}\) from single-qubit operations and measurements. This diagram shows the scheme for measuring the remaining components of the anticommutator {ΛX, ΛZ} not present in ΛY (see Supplementary Information section IV). They are four-body correlators of the form \({\hat{\sigma }}_{j}^{x}{\hat{\sigma }}_{j+1}^{x}{\hat{\sigma }}_{{j}^{{\rm{{\prime} }}}}^{z}{\hat{\sigma }}_{{j}^{{\rm{{\prime} }}{\rm{{\prime} }}}}^{z}\), with all Pauli operators acting on different sites. On the experiment, all such correlators for a specific j are obtained by rotating sites j and j + 1 under exp(iπσy/4) and then projective measuring in the canonical basis. The procedure has to be repeated for all j, thus N − 1 times.

Extended Data Fig. 5 Qubit chain encoding of the lattice Schwinger model.

Illustration of a specific product state for eight lattice sites in the Schwinger model and its encoding in the corresponding spin configuration. The curves between sites represent the long-range spin–spin interaction pattern, where the thickness of each curve encodes the coupling strength. The inset shows the encoding of particles into spins, with empty sites (vacuum) denoted by ‘vac’.

Extended Data Fig. 6 Theoretical scalability.

Numerical simulation of the VQS for Schwinger ground states with trapped-ion resources. The infidelity of the optimized ground state with respect to the exact ground state is plotted as a function of the number of qubits N and the circuit depth, or number of circuit layers nlayers. Different panels refer to the simulation of the Schwinger model for different values of the bare mass m, reported via the parametric distance δm = (mmc) from the critical point mc (for \(w= {\bar{g}} =1\)). The black dashed line marks the minimal circuit depth required to achieve an optimized infidelity of 5% or less, as a function of the system length N.

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Kokail, C., Maier, C., van Bijnen, R. et al. Self-verifying variational quantum simulation of lattice models. Nature 569, 355–360 (2019).

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