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arXiv:2405.10052 · Physica Scripta

Deep neural network-assisted improvement of quantum compressed sensing tomography

Adriano Macarone Palmieri, Leonardo Zambrano, Maciej Lewenstein, Antonio Acín, Donato Farina

Published Nov 1, 2025• 2 citations
arXivHTMLPDF

Abstract

Abstract Quantum compressed sensing is a fundamental technique for tomographic reconstruction of low-rank density matrices in informationally incomplete scenarios. However, when the available measurement data is insufficient, standard compressed sensing, like other algorithmic estimators such as maximum-likelihood, can yield imprecise or noisy estimates. To mitigate this, we propose a deep neural network-based post-processing to enhance the initial reconstruction. By treating the compressed sensing-estimated quantum state as a noisy input, the network performs a supervised denoising task. After the network is applied, a projection onto the space of feasible density matrices is performed to obtain an improved final state estimation. We demonstrate the effectiveness of our method through numerical experiments. Additionally, we explore iterative application of the inference process to further enhance performance, and we analyze the network’s generalization ability by testing its performance on states and noise models not seen during training.

DOI
10.1088/1402-4896/ae1ada

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Abstract Quantum compressed sensing is a fundamental technique for tomographic reconstruction of low-rank density matrices in informationally incomplete scenarios. However, when the available measurement data is insufficient, standard compressed sensing, like other algorithmic estimators such as maximum-likelihood, can yield imprecise or noisy estimates. To mitigate this, we propose a deep neural network-based post-processing to enhance the initial reconstruction. By treating the compressed sensing-estimated quantum state as a noisy input, the network performs a supervised denoising task. After the network is applied, a projection onto the space of feasible density matrices is performed to obtain an improved final state estimation. We demonstrate the effectiveness of our method through numerical experiments. Additionally, we explore iterative application of the inference process to further enhance performance, and we analyze the network’s generalization ability by testing its performance on states and noise models not seen during training.
We revisit quantum tomography in an informationally incomplete scenario and propose improved state reconstruction methods using deep neural networks. In the first approach, the trained network predicts an optimal linear or quadratic reconstructor with coefficients depending only on the collection of (already taken) measurement operators. This effectively refines the undercomplete tomographic reconstructor based on pseudoinverse operation. The second, based on an LSTM recurrent network performs state reconstruction sequentially. It can also optimize the measurement sequence, which suggests a no-free-lunch theorem for tomography: by narrowing the state space, we gain the possibility of more efficient tomography by learning the optimal sequence of measurements. Numerical experiments for a 2-qubit system show that both methods outperform standard maximum likelihood estimation and also scale to larger 3- and 4-qubit systems. Our results demonstrate that neural networks can effectively learn the underlying geometry of multi-qubit states and use it for their reconstruction.
Abstract 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-measurement (SPAM) apparatus. However, physical models suffer from intrinsic limitations as actual measurement operators and trial states cannot be known precisely. This scenario inevitably leads to SPAM errors degrading reconstruction performance. Here we develop a framework based on machine learning which generally applies to both the tomography and SPAM mitigation problem. We experimentally implement our method. 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 an 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.
We apply deep-neural-network-based techniques to quantum state classification and reconstruction. Our methods 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 feed-forward 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 QST technique with conditional generative adversarial networks (QST-CGAN) [Ahmed et al., Phys. Rev. Lett. 127, 140502 (2021)]. 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 iterative and accelerated projected-gradient-based maximum-likelihood estimation (MLE) methods. We also demonstrate that the QST-CGAN can reconstruct both pure and mixed states from two orders of magnitude fewer randomly chosen data points than these MLE methods. Our paper opens possibilities to use state-of-the-art deep-learning methods for quantum state classification and reconstruction under various types of noise.
The reliable characterization of quantum states is a fundamental task in quantum information science. For this purpose, quantum state tomography provides a standard framework for reconstructing quantum states from measurement data, yet it is often degraded by experimental noise. Mitigating such noise is therefore essential for the accurate estimation of the states in realistic settings. In this work, we propose a scalable tomography method based on multilayer perceptron networks that mitigate unknown noise through supervised learning. This approach is data-driven and thus does not rely on explicit assumptions about the noise model or measurement, making it readily extendable to general quantum systems. Numerical simulations, ranging from special pure states to random mixed states, demonstrate that the proposed method effectively mitigates noise across a broad range of scenarios, compared with the case without mitigation.
In this paper, we propose a method for denoising experimental density matrices that combines standard quantum state tomography with an attention-based neural network architecture. The algorithm learns the noise from the data itself, without knowledge of its sources. Firstly, we show how the proposed protocol can improve the averaged fidelity of reconstruction over linear inversion and maximum likelihood estimation in the finite-statistics regime, reducing at least by an order of magnitude the amount of necessary training data. Next, we demonstrate its use for out-of-distribution data in realistic scenarios. In particular, we consider squeezed states of few spins in the presence of depolarizing noise and measurement/calibration errors and certify its metrologically useful entanglement content. The protocol introduced here targets experiments involving few degrees of freedom and afflicted by a significant amount of unspecified noise. These include NISQ devices and platforms such as trapped ions or photonic qudits. Published by the American Physical Society 2024
Intuitively, if a density operator has small rank, then it should be easier to estimate from experimental data, since in this case only a few eigenvectors need to be learned. We prove two complementary results that confirm this intuition. Firstly, we show that a low-rank density matrix can be estimated using fewer copies of the state, i.e. the sample complexity of tomography decreases with the rank. Secondly, we show that unknown low-rank states can be reconstructed from an incomplete set of measurements, using techniques from compressed sensing and matrix completion. These techniques use simple Pauli measurements, and their output can be certified without making any assumptions about the unknown state. In this paper, we present a new theoretical analysis of compressed tomography, based on the restricted isometry property for low-rank matrices. Using these tools, we obtain near-optimal error bounds for the realistic situation where the data contain noise due to finite statistics, and the density matrix is full-rank with decaying eigenvalues. We also obtain upper bounds on the sample complexity of compressed tomography, and almost-matching lower bounds on the sample complexity of any procedure using adaptive sequences of Pauli measurements. Using numerical simulations, we compare the performance of two compressed sensing estimators—the matrix Dantzig selector and the matrix Lasso—with standard maximum-likelihood estimation (MLE). We find that, given comparable experimental resources, the compressed sensing estimators consistently produce higher fidelity state reconstructions than MLE. In addition, the use of an incomplete set of measurements leads to faster classical processing with no loss of accuracy. Finally, we show how to certify the accuracy of a low-rank estimate using direct fidelity estimation, and describe a method for compressed quantum process tomography that works for processes with small Kraus rank and requires only Pauli eigenstate preparations and Pauli measurements.