Quantum Variational Learning of the Entanglement Hamiltonian

Learning the structure of the entanglement Hamiltonian (EH) is central to characterizing quantum many-body states in analog quantum simulation. We describe a protocol where spatial deformations of the many-body 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 Fermi-Hubbard models in quasi-1D geometries, finding excellent agreement of the EH with Bisognano-Wichmann predictions. Subsequent on-device spectroscopy enables a direct measurement of the entanglement spectrum, which we illustrate for a Fermi Hubbard model in a topological phase.

Figure 1

Quantum variational learning (QVL) of EH (a) subsystem $A$ is time evolved with the deformed Hamiltonian ${\widehat{H}}_A^{\mathrm{var}}(\mathit{g})$, while measuring observables ${⟨{\widehat{\mathcal{O}}}_A⟩}_t$ at time instances $\{t_n\}$. A classical computer optimizes a cost function $\mathcal{C}(\mathit{g})$ in a feedback loop, minimizing the time variation of observables. (b) Fermi Hubbard model, and subsystem $A$. (c) Time variation of observables indicating convergence in the feedback loop from the initial ${\mathit{g}}^0$ to final ${\mathit{g}}^{\mathrm{opt}}$ parameters. (d) Corresponding cost function vs iteration number of the optimizer. The color map visualizes the distance to the final parameter vector $\mathrm{\Delta }\mathit{g}=|{\mathit{g}}^i-{\mathit{g}}^{\mathrm{opt}}|$. Data plotted in (c) and (d) were obtained in simulated runs for a 2-leg Fermi-Hubbard ladder [see text and Figs. 2 and 2], monitoring double site occupancy by means of a quantum gas microscope.

Figure 2

Quantum variational learning (QVL) of the EH for different geometries of a Hubbard model. Left column: Results for a 5-site subsystem on the right boundary of a 10-site Hubbard chain with $U/J=1$ and $\mu /J=-0.5$. Right column: Results for a single chain in a 2-leg ladder with $U/J_{\parallel }=8$, $J_{\perp }/J_{\parallel }=2$ and $\mu /J=0$. (a),(b) Lattice geometries with highlighted subsystems. (c),(d) Variational parameters $g_j^{\mathrm{opt}}$ obtained from optimization with $6\times {10}^4$ experimental runs (QVL), fixing $g_{j=1}$. The optimal parameters are rescaled (corresponding fidelities shown in Fig. 3) for comparison with the EH parameters obtained by numerically optimizing the relative entropy (black solid lines, see also Supplemental Material [44]). (e),(f) Universal ratios ${\kappa }_{\alpha }$ calculated by diagonalizing the variationally obtained EH ${\widehat{H}}_A^{\mathrm{var}}({\mathit{g}}^{\mathrm{opt}})$ in comparison to exact eigenvalues of the reduced density matrix ${\rho }_A$. All simulations are performed for the ground state in the zero magnetization sector. Error bands are computed by repeating the entire optimization run 10 times and computing the standard error.

Figure 3

Error assessment vs number of experimental runs $N_M$: Maximally achievable Uhlmann fidelity with respect to the exact density matrix ${\widehat{\rho }}_A$ as a function of the total number of measurements for a half-partition of a 10-site Hubbard chain ($N_A=5$) for $U/J=1$ as shown in the inset. For comparison, the blue data points represent a learning protocol based on classical postprocessing of measurement data [19]. The data show the median of the fidelity when the experiment is repeated 100 times. We plot a selection of representative error bars which indicate $2\sigma$ confidence intervals.

Figure 4

Entanglement spectroscopy of the Heisenberg model on a ladder. (a) Schematic of the setup, with subsystem $A$ the right half of a 12-site ladder. We fix the ratio of horizontal to vertical couplings as $J_{\parallel }/J_{\perp }=0.25$ and tune the relative strength of diagonal and vertical couplings with $\lambda \in [0,1]$. (b) Level scheme of the low-lying part of the entanglement spectra, indicating the dominant transitions illustrated here for $\epsilon =0$. (c) Measured entanglement spectrum (see main text), for small values $\epsilon$ of the perturbation, in the trivial phase ($\lambda =0$). (d) Same as (c) at $\lambda =1$. (e) Measured entanglement spectra versus $\lambda$, at $\epsilon =0.05{\overline{J}}_{\perp }$. All spectra are computed assuming 5000 measurements per observable at each time, measured up to $t=1000/{\overline{J}}_{\perp }$.