Direct parameter estimations from machine-learning enhanced quantum state tomography (2203.16385v1)

Abstract: With the capability to find the best fit to arbitrarily complicated data patterns, machine-learning (ML) enhanced quantum state tomography (QST) has demonstrated its advantages in extracting complete information about the quantum states. Instead of using the reconstruction model in training a truncated density matrix, we develop a high-performance, lightweight, and easy-to-install supervised characteristic model by generating the target parameters directly. Such a characteristic model-based ML-QST can avoid the problem of dealing with large Hilbert space, but keep feature extractions with high precision. With the experimentally measured data generated from the balanced homodyne detectors, we compare the degradation information about quantum noise squeezed states predicted by the reconstruction and characteristic models, both give agreement to the empirically fitting curves obtained from the covariance method. Such a ML-QST with direct parameter estimations illustrates a crucial diagnostic toolbox for applications with squeezed states, including advanced gravitational wave detectors, quantum metrology, macroscopic quantum state generation, and quantum information process.

• The paper introduces a supervised ML method that directly estimates quantum state parameters, reducing the computational burden of traditional density matrix reconstruction.
• The approach employs CNN-based analysis of quadrature data to accurately predict squeezing parameters and average photon numbers from experimental measurements.
• The method demonstrates robust, real-time QST potential, paving the way for applications in gravitational wave detection, quantum metrology, and quantum information processing.

The paper introduces a supervised machine learning (ML) approach to quantum state tomography (QST) that directly estimates parameters, offering a lightweight alternative to traditional reconstruction methods. This characteristic model-based ML-QST bypasses the need to reconstruct truncated density matrices in large Hilbert spaces, maintaining high precision in feature extraction. The method is validated using experimental data from balanced homodyne detectors, demonstrating its ability to accurately predict degradation information about quantum noise squeezed states.

The authors detail the limitations of maximum likelihood estimation (MLE) in QST, particularly its overestimation issues when dealing with an increasing number of modes. Alternative algorithms, including permutationally invariant tomography, quantum compressed sensing, tensor networks, generative models, and restricted Boltzmann machines, have been proposed to address these limitations. ML-enhanced QST has emerged as a fast, robust, and precise method for continuous variables.

The core of the work involves a characteristic model-based ML-QST that skips training on the truncated density matrix. This approach is designed to mitigate the challenges associated with large Hilbert spaces while preserving feature extraction accuracy. The model directly estimates parameters such as average photon numbers in pure squeezed states, squeezed thermal states, and thermal reservoirs, aligning with results from reconstruction models.

Key components and methods include:

• Supervised ML with Convolutional Neural Networks (CNNs): The method uses CNNs to analyze quadrature sequence data $X_{\theta}$ obtained via quantum homodyne tomography. The CNN architecture consists of 17 convolutional layers with 5 convolution blocks, each containing 2 convolution layers of varying sizes. Shortcuts are incorporated to mitigate the gradient vanishing problem.
• Reconstruction Model: This model predicts a truncated density matrix, mapping the estimated function $f_{\text{est}: X_{\theta}\rightarrow \hat{\rho}_{m \times m}$. To ensure physical validity, a positive semi-definite constraint is imposed using Cholesky decomposition, $$\hat{\rho}_{m \times m} \equiv L_{m \times m}\, L_{m \times m}^{\ast}$$. The training set consists of quadrature data and corresponding lower triangular matrices, $$\left \{X_\theta^{j}, L_{m \times m}^{j} \vert \text{dim}(X^j_\theta) = 4096, \theta \in [0,2 \pi], j = 1,2,3 \dots N\right \}$$.
• Characteristic Model: This model directly predicts physical parameters $(r, \theta, n_{th})$, where $r$ is the squeezing ratio, $\theta$ is the squeezing angle, and $n_{th}$ is the average photon number. This avoids dealing with high-dimensional Hilbert spaces. The mapping function is $f_{\text{est}: X_{\theta}\rightarrow (r, \theta, n_{th})$.

The experimental setup involves generating squeezed states using a bow-tie optical parametric oscillator cavity with a periodically poled KTiOPO$_4$ (PPKTP) crystal, operated below the threshold at 1064 nm. Quantum homodyne tomography is performed by collecting quadrature sequences using a spectrum analyzer at 2.5 MHz with specific resolution and video bandwidths.

The paper presents a comparison between the reconstruction and characteristic models, focusing on the predicted average photon number as a function of pump power. Both models show agreement in predicting the average photon numbers for measured data $\langle n\rangle_{\text{total}}$, pure squeezed states $\langle n\rangle_{\text{sq}}$, and non-pure components $\langle n\rangle_{\text{other}}$. The models also capture the cross-over point where non-pure components become dominant, leading to degraded quantum noises.

The degradation in squeezed states is analyzed by comparing the squeezing level versus the anti-squeezing level. Experimental data are fitted using orthogonal distance regression, taking into account optical loss ($L$) and phase noise ($\theta$). The measured squeezing $V^{\text{SQZ}$ and anti-squeezing $V^{\text{ASQZ}$ levels are modeled as:

$V^{\text{SQZ} = (1-L) [V^{\text{SQZ}_{id}\times \cos^2 \theta + V^{\text{ASQZ}_{id}\times \sin^2\theta]+L$

$V^{\text{ASQZ} = (1-L)[V^{\text{ASQZ}_{id}\times \cos^2\theta + V^{\text{SQZ}_{id}\times \sin^2\theta]+L$

where $V^{\text{SQZ}_{id}$ and $V^{\text{ASQZ}_{id}$ are the ideal squeezing and anti-squeezing levels. The ML-QST, based on both reconstruction and characteristic models, aligns with experimental data, demonstrating its ability to extract degradation information precisely and quickly.

The authors conclude by highlighting the potential of the characteristic model for real-time QST and its applicability in advanced gravitational wave detectors, quantum metrology, macroscopic quantum state generation, and quantum information processing.