Integrating Neural Networks with a Quantum Simulator for State Reconstruction

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Integrating Neural Networks with a Quantum Simulator for State Reconstruction

Giacomo Torlai, Brian Timar, Evert P. L. van Nieuwenburg, Harry Levine, Ahmed Omran, Alexander Keesling, Hannes Bernien, Markus Greiner, Vladan Vuletić, Mikhail D. Lukin, Roger G. Melko, and Manuel Endres

Phys. Rev. Lett. 123, 230504 – Published 6 December 2019

Abstract

We demonstrate quantum many-body state reconstruction from experimental data generated by a programmable quantum simulator by means of a neural-network model incorporating known experimental errors. Specifically, we extract restricted Boltzmann machine wave functions from data produced by a Rydberg quantum simulator with eight and nine atoms in a single measurement basis and apply a novel regularization technique to mitigate the effects of measurement errors in the training data. Reconstructions of modest complexity are able to capture one- and two-body observables not accessible to experimentalists, as well as more sophisticated observables such as the Rényi mutual information. Our results open the door to integration of machine learning architectures with intermediate-scale quantum hardware.

Received 29 April 2019
Revised 15 September 2019

DOI:https://doi.org/10.1103/PhysRevLett.123.230504

Physics Subject Headings (PhySH)

Quantum information with atoms & light Quantum tomography

Artificial neural networks Rydberg atoms & molecules

Machine learning

Quantum Information, Science & TechnologyAtomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Giacomo Torlai ^1,2,3, Brian Timar⁴, Evert P. L. van Nieuwenburg⁴, Harry Levine⁵, Ahmed Omran⁵, Alexander Keesling⁵, Hannes Bernien⁶, Markus Greiner⁵, Vladan Vuletić⁷, Mikhail D. Lukin⁵, Roger G. Melko^2,3, and Manuel Endres⁴

¹Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, USA
²Department of Physics and Astronomy, University of Waterloo, Ontario N2L 3G1, Canada
³Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
⁴Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, California 91125, USA
⁵Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
⁶Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA
⁷Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

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Vol. 123, Iss. 23 — 6 December 2019

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Images

Figure 1
Experiment and reconstruction. (a) Model of the reconstruction process. Individual ${}^{87}{Rb}$ atoms (gray circles) are trapped in an array of optical tweezers and coupled to a Rydberg state with Rabi frequency $Ω$ . Site-resolved fluorescence imaging provides imperfect measurement in the ${\hat{σ}}^{z}$ basis. Our neural-network model describes the true quantum state as a RBM (blue and green neurons), while the binary data $τ$ accessible to the experimentalist are included as an auxiliary “noise” layer (red neurons). By training on this data, the network learns parameters $λ$ describing the experimental quantum state, which are subsequently used to compute observables $⟨ \hat{O} ⟩$ . (b) Representation of the ordered state at the end of the adiabatic sweep; see Eq. (2). Darker circles represent a higher probability of Rydberg excitation, and the shading indicates quantum fluctuations localized at bonds (3,4) and (5,6). (c) The effective laser detuning $Δ$ and Rabi frequency $Ω$ as a function of sweep time $t$ . Circular markers indicate the times at which the sweep was halted to collect data. Vertical line: Approximate transition to ordering in the finite system. The nearest-neighbor interaction is $V_{NN} = 30 MHz$ , the final detuning is 10 MHz, and the peak Rabi frequency is 2 MHz; the total sweep time is $T_{ev} = 3.4 μ s$ .
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Figure 2
Benchmarking RBM reconstruction. (a) Fidelity of reconstruction. We sample synthetic datasets from states obtained by exact time evolution under the Hamiltonian (1) without decoherence. The exact quantum state fidelity $F$ between the true state $\hat{ρ}$ and the reconstruction ${\hat{ρ}}_{λ} = | ψ_{λ} ⟩ ⟨ ψ_{λ} |$ is plotted as a function of detuning $Δ$ . Training standard RBMs on datasets without measurement noise (green dashed line), we achieve uniformly high fidelities, demonstrating that the RBM wave function Ansatz is capable of representing states relevant to our experiment. Training on datasets with measurement noise with (red solid line) and without (green solid line) noise-layer regularization shows how the modified training improves reconstruction. Inset: Same data for time evolution including a realistic decoherence model. (b) Model size. Here we compare the number of parameters $N_{p}$ required to specify a RBM wave function with $N$ hidden units with the size of the frequency-distribution (FD) model required to perform direct inference (i.e., number of different configurations in the dataset) for a typical Rydberg ground state, as a function of system size $N$ and for several dataset sizes $N_{s}$ . Note that the FD model size depends on $N_{s}$ , while the RBM size does not. For further discussion, see Ref. [25].
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Figure 3
Few-body observables. Comparison of the RBM reconstruction (squares) with the experiment results (crosses) and the predictions from the Lindbladian master equation (circles) [25]. In order to facilitate comparison with experiment, the values reported in (a) and (b) for the RBM and Lindbladian observables are computed including the known measurement error rates $p (0 | 1) = 0.04$ , $p (1 | 0) = 0.01$ . (a) Nearest-neighbor correlations ${\bar{g}}^{z z} (1)$ in the $z$ basis, spatially averaged (see text for definition). (b) Average correlation ${\bar{g}}^{z z} (s)$ as a function of distance $s$ for $Δ = 10 MHz$ . (c) Spatial average $\bar{x}$ of the transverse field $⟨ {\hat{σ}}_{i}^{x} ⟩$ . (d) Nearest-neighbor correlation $⟨ {\hat{σ}}_{i}^{x} {\hat{σ}}_{i + 1}^{x} ⟩_{c}$ as a function of position $i$ for $Δ = 10 MHz$ . The two peaks correspond to the bonds highlighted in Fig. 1.
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Figure 4
Rényi mutual information. The quantum (Rényi) mutual information $I_{2}$ defined as $I_{2} (s) = S_{2} ({\hat{ρ}}_{s}^{A}) + S_{2} ({\hat{ρ}}_{s}^{B}) - S_{2} (\hat{ρ})$ , where $S_{2} (\hat{ρ}) = - \log Tr {\hat{ρ}}^{2}$ is the second-order Rényi entropy, $\hat{ρ}$ is the (mixed) state of the whole system, and ${\hat{ρ}}_{s}^{A}, {\hat{ρ}}_{s}^{B}$ are the reduced density matrices for the subsystems $A_{s} = {1, \dots, s}$ , $B_{s} = {s + 1, \dots, N}$ , respectively, defined by a partitioning of the system at bond $(s, s + 1)$ . The mutual information is plotted for a partition at bond (3,4), as a function of detuning. Inset: The mutual information $I_{2} (s)$ as a function of the cut bond $s$ for $Δ = 10 MHz$ .
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