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Experimental neural network enhanced quantum tomography
Adriano Macarone Palmieri, Egor Kovlakov, Federico Bianchi, Dmitry Yudin, Stanislav Straupe, Jacob Biamonte, Sergei Kulik
TL;DR
Quantum tomography can be degraded by systematic measurement effects, including unknown Gouy phase shifts. The paper uses a neural network to reduce these effects and reports higher reconstruction fidelity than raw reconstruction, while experiments also expose a data-related limitation.
Problem
Unknown measurement effects, including Gouy phase shifts, can reduce quantum-state reconstruction fidelity when the detector model is not fully known.
Method
A supervised neural network learns to generate measurement probabilities closer to ideal values, providing state reconstruction without explicitly identifying all detector phase shifts.
Results
The neural-network reconstruction achieved average fidelity F(nn) = 0.81 ± 0.19, compared with F(raw) = 0.54 ± 0.12 without phase correction.
Takeaways & Limitations
Neural-network post-processing can improve reconstruction despite unknown Gouy phase shifts, while complex spatial measurements retain a tradeoff between projection quality and detection efficiency.
Takeaways & Limitations
Performance may recover with heralded-photon data for training, suggesting that the available training data were limited by experimental drift and sampling.
Abstract
from arXiv · showhide
Quantum tomography is currently ubiquitous for testing any implementation of a quantum information processing device. Various sophisticated procedures for state and process reconstruction from measured data are well developed and benefit from precise knowledge of the model describing state preparation and the measurement apparatus. However, physical models suffer from intrinsic limitations as actual measurement operators and trial states cannot be known precisely. This scenario inevitably leads to state-preparation-and-measurement (SPAM) errors degrading reconstruction performance. Here we develop and experimentally implement a machine learning based protocol reducing SPAM errors. We trained a supervised neural network to filter the experimental data and hence uncovered salient patterns that characterize the measurement probabilities for the original state and the ideal experimental apparatus free from SPAM errors. We compared the neural network state reconstruction protocol with a protocol treating SPAM errors by process tomography, as well as to a SPAM-agnostic protocol with idealized measurements. The average reconstruction fidelity is shown to be enhanced by 10\% and 27\%, respectively. The presented methods apply to the vast range of quantum experiments which rely on tomography.
1. Experimental setup
The experiment prepares six-dimensional photonic states in Hermite-Gaussian modes and measures them with spatial-light-modulator holograms, while comparing attenuated-laser and heralded-photon sources. Dataset non-uniformity from experimental drift limits transfer of neural-network training between sources.
- State encoding: Six-dimensional states use Hermite-Gaussian modes with n + m ≤2, incorporating the full two-dimensional spatial-mode spectrum.This choice includes radial structure in addition to orbital angular momentum and addresses the poor projective-measurement quality of such modes.
- Optical setup: The preparation stage transforms an initially Gaussian beam into arbitrary Hermite-Gaussian superpositions using phase and amplitude modulation.The detection stage uses a phase-only hologram and single-mode fiber to filter a selected Gaussian mode.
- Light sources: The setup uses both an attenuated 808 nm diode laser and a heralded single-photon source generated by spontaneous parametric down-conversion.Training and evaluation data for the neural network were collected with the attenuated laser because it provided a much higher acquisition rate.
- Cross-source evaluation: Neural-network transfer to heralded-photon data yielded F(nn) = 0.86 ± 0.04 versus F(raw) = 0.81 ± 0.05, and π(nn) = 0.84 ± 0.04 versus π(raw) = 0.75 ± 0.07.The authors attribute the degraded transfer performance most likely to dataset non-uniformity caused by experimental drifts.
2. State generation and detection methods
The detection design balances projection quality against efficiency: phase-only holograms increase counts, while modified mode calculations compensate for Gaussian-fiber effects. The resulting measurement quality and efficiency are quantified through cross-talk and similarity measurements.
- Hologram tradeoff: Phase-only detection holograms sacrifice projection quality for efficiency, whereas amplitude modulation corrects diffraction alterations but significantly lowers diffraction efficiency.The hologram design therefore exposes a direct measurement-quality versus detection-efficiency tradeoff.
- Measurement characterization: The SIC POVM projectors were implemented in a d = 6 Hilbert space, with SPAM-corrupted counterparts used to quantify measurement deviations.Cross-talk probabilities compare experimentally measured and theoretically expected projector responses.
- Experimental result: Switching off detection-hologram amplitude modulation reduced the similarity parameter from 0.99 to 0.96 while increasing observed counts from 6.2 × 10^6 to 40.9 × 10^6.This demonstrates the experimentally observed tradeoff between projection measurement quality and detection efficiency.
- Detection-mode correction: The modified hologram calculation introduces a second width parameter ˜w to compensate for the single-mode-fiber term and restore detection-mode orthogonality.Increasing the detection-mode waist is an alternative, but reduces detection efficiency to a few percent.
3. Gouy phases reconstruction by process tomography
Process tomography reconstructed the spatial evolution between preparation and measurement, revealing mode-dependent Gouy phase shifts that strongly affect state reconstruction. A neural network can compensate for these shifts without explicitly knowing their model.
- Process reconstruction: Process tomography reconstructed the spatial evolution process and found a dominant operator element E1 close to diagonal with phase factors.The process used amplitude-modulation masks for state preparation.
- Gouy phase shifts: 0.92 ± 0.02 and 1.97 ± 0.03 radians were the inferred Gouy phase shifts for mode orders n+m = 1 and n+m = 2, respectively.The phases were nearly equal for modes sharing the same total order n+m.
- Reconstruction performance: Only after incorporating the additional phase shifts did reconstruction without neural-network post-processing reach fidelities above 0.8.The phase shifts therefore represented a substantial systematic measurement effect in this setup.
- Neural-network compensation: The neural network treated unknown Gouy phase shifts as systematic SPAM errors, avoiding the need for full process reconstruction to identify them.The comparison used SIC-POVM measurements whose projectors were modified by the reconstructed process element.
- Reconstruction performance: Without phase correction, raw reconstruction achieved F(raw) = 0.54 ± 0.12, while neural-network processing achieved F(nn) = 0.81 ± 0.19.These results used 2000 test states and a network trained without Gouy-phase information.
4. State Reconstruction
For an exemplary reconstructed spatial state, neural-network processing produced a higher-fidelity reconstruction than using the raw data.
- Example reconstruction: F(nn) = 0.91 exceeded F(raw) = 0.75 ± 0.02 for the exemplary reconstructed spatial state.The figure compares raw-data reconstruction, neural-network predicted probabilities, and the prepared state.
5. Neural Network
The paper uses a two-hidden-layer feed-forward neural network to map input probabilities toward ideal ones, evaluating how dataset size and training procedures affect prediction quality. A small training subset already produces high classical fidelity on unseen test data.
- Neural-network model: A feed-forward neural network with 400 and 200 neurons in its two hidden layers maps input probabilities to ideal probabilities.Dropout with probability 0.2 is applied between the hidden layers to reduce overfitting.
- Dataset-size evaluation: The study varies the training-set fraction from η = 0.1 to η = 1 using a dataset of 10500 states, while evaluating loss and Bhattacharyya distance on test data.The experiment reserves 2000 states for testing and samples training subsets from the remaining 8500 states.
- Dataset-size evaluation: Prediction quality increases as the training-set size increases, according to the dependence of KLD loss and classical fidelity on dataset size.The figure also examines the effect of learning rate on the neural network.
- Dataset-size evaluation: Fidelity for η = 0.1, corresponding to 10% of the available data, is already 0.9720.The reported values are averaged over five repetitions to provide a stable estimate.
- Training and evaluation: An independent validation set enables early stopping when validation error stops decreasing, after which the weights with the best validation loss are restored before testing.The final test set remains completely unseen during model selection, providing a less biased loss estimate.