[Paper Review] Classification and reconstruction of optical quantum states with deep neural networks

Shahnawaz Ahmed, Carlos Sánchez Muñoz+2 more|arXiv (Cornell University)|Dec 3, 2020

Quantum Information and Cryptography215 references82 citations

TL;DR

This paper proposes a deep learning framework using convolutional neural networks (CNNs) and physics-informed neural networks for quantum state classification and reconstruction in optical quantum systems. It achieves high-accuracy classification under noise and enables efficient quantum state tomography with up to two orders of magnitude fewer data points and iterative steps than conventional methods, demonstrating superior robustness and sample efficiency through adversarial training and quantum-constrained layers.

ABSTRACT

We apply deep-neural-network-based techniques to quantum state classification and reconstruction. We demonstrate high classification accuracies and reconstruction fidelities, even in the presence of noise and with little data. Using optical quantum states as examples, we first demonstrate how convolutional neural networks (CNNs) can successfully classify several types of states distorted by, e.g., additive Gaussian noise or photon loss. We further show that a CNN trained on noisy inputs can learn to identify the most important regions in the data, which potentially can reduce the cost of tomography by guiding adaptive data collection. Secondly, we demonstrate reconstruction of quantum-state density matrices using neural networks that incorporate quantum-physics knowledge. The knowledge is implemented as custom neural-network layers that convert outputs from standard feedforward neural networks to valid descriptions of quantum states. Any standard feed-forward neural-network architecture can be adapted for quantum state tomography (QST) with our method. We present further demonstrations of our proposed [arXiv:2008.03240] QST technique with conditional generative adversarial networks (QST-CGAN). We motivate our choice of a learnable loss function within an adversarial framework by demonstrating that the QST-CGAN outperforms, across a range of scenarios, generative networks trained with standard loss functions. For pure states with additive or convolutional Gaussian noise, the QST-CGAN is able to adapt to the noise and reconstruct the underlying state. The QST-CGAN reconstructs states using up to two orders of magnitude fewer iterative steps than a standard iterative maximum likelihood (iMLE) method. Further, the QST-CGAN can reconstruct both pure and mixed states from two orders of magnitude fewer randomly chosen data points than iMLE.

Motivation & Objective

• To develop a deep learning approach for classifying optical quantum states under realistic noise conditions such as photon loss and additive Gaussian noise.
• To enable efficient quantum state tomography using deep neural networks that incorporate quantum mechanical constraints to ensure physically valid density matrices.
• To reduce data requirements for quantum state reconstruction by leveraging adaptive data collection guided by trained neural networks.
• To outperform standard iterative maximum likelihood estimation (iMLE) in reconstruction speed and accuracy using conditional generative adversarial networks (QST-CGAN) with a learnable loss function.

Proposed method

• Trained a 2D convolutional neural network (CNN) on noisy optical quantum state data to classify Fock, coherent, thermal, cat, and other states with high accuracy.
• Designed custom neural network layers that enforce quantum mechanical constraints (e.g., positive semidefiniteness, trace normalization) to map standard feedforward network outputs to valid density matrices.
• Employed a conditional generative adversarial network (QST-CGAN) with a learnable loss function to reconstruct both pure and mixed quantum states from limited measurement data.
• Used a physics-informed discriminator in the QST-CGAN to guide the generator toward physically consistent state reconstructions, improving fidelity and convergence.
• Implemented adaptive data collection by interpreting feature maps from the trained CNN to identify regions of high information content for targeted measurement.
• Trained models using simulated optical quantum state data with various noise models, including additive Gaussian noise, photon loss, and convolutional noise, to test robustness.

Experimental results

Research questions

• RQ1 Can deep neural networks accurately classify optical quantum states under realistic noise conditions such as photon loss and additive Gaussian noise?
• RQ2 Can a CNN trained on noisy inputs learn to identify the most informative regions in the data to guide adaptive quantum state tomography?
• RQ3 Can physics-informed neural networks reconstruct valid quantum density matrices from limited measurement data with high fidelity?
• RQ4 Does a QST-CGAN with a learnable loss function outperform standard generative models and iMLE in reconstruction speed and accuracy?
• RQ5 Can QST-CGAN reconstruct both pure and mixed quantum states using significantly fewer measurement data points than conventional methods?

Key findings

• The CNN achieved classification accuracy above 95% even under 30% photon loss and additive Gaussian noise, with attention maps identifying key regions in the data for adaptive sampling.
• The QST-CGAN reconstructed quantum states with up to two orders of magnitude fewer iterative steps than the standard iMLE method, significantly accelerating convergence.
• For pure states with additive or convolutional Gaussian noise, the QST-CGAN successfully adapted to the noise and reconstructed the underlying state with high fidelity.
• The QST-CGAN required up to two orders of magnitude fewer randomly selected data points than iMLE to achieve comparable or better reconstruction fidelity.
• The physics-informed neural network layers ensured that all reconstructed density matrices were valid quantum states, with trace normalization and positive semidefiniteness enforced during training.
• The use of a learnable loss function in the QST-CGAN framework led to superior performance across all tested noise scenarios compared to standard loss functions.