Error-mitigated quantum state tomography using neural networks

1. Problem Statement

Quantum State Tomography (QST) is the standard framework for reconstructing unknown quantum states from measurement data. However, in realistic experimental settings, measurement imperfections and environmental noise inevitably contaminate data, leading to biased or inaccurate state reconstructions.

• Limitations of Existing Methods:
  • Explicit Noise Characterization: Requires additional measurements (process tomography) and assumes noise stability.
  • Compressed Sensing: Relies on low-rank assumptions, making it unsuitable for general mixed states.
  • Traditional ML Approaches: Often require specific parametric noise models or prior knowledge about the target state.
• The Challenge: There is a need for a scalable, data-driven method that can mitigate unknown, non-parametric noise without requiring explicit assumptions about the noise model or the measurement apparatus, while remaining applicable to general quantum systems (including mixed states).

2. Methodology

The authors propose a supervised learning framework based on Multilayer Perceptrons (MLPs) to map noisy measurement data directly to the ideal quantum state.

A. General Framework

The approach treats QST as a function approximation problem: learning a mapping $f: D \to \rho$, where $D$ is the vector of measurement outcomes and $\rho$ is the target density matrix.

• Data-Driven Nature: The network learns noise patterns directly from training data. It does not require an analytical model of the noise channel or the measurement device.
• Training Data Generation: Ideal states are subjected to various noise channels (depolarizing, bit-flip, phase-flip, amplitude-damping) with randomly sampled strengths. The resulting noisy states are measured to generate the input-output pairs for training.

B. Encoding and Decoding Schemes

To ensure the output satisfies physical constraints (Hermiticity, positive semi-definiteness, unit trace) and to improve training efficiency, the authors employ a specific encoding-decoding architecture:

1. Cholesky Decomposition (Decoding): Instead of outputting the density matrix directly, the network outputs a parameter vector $\alpha$ representing the independent elements of a lower triangular matrix $R$. The density matrix is reconstructed as $\rho = RR^\dagger / \text{tr}(RR^\dagger)$. This guarantees $\rho$ is a valid density matrix.
2. One-Hot Inspired Encoding (Input/Hidden): To handle continuous parameter values efficiently, the authors propose a modified "one-hot" encoding. Continuous parameters are discretized into sectors, and the value is represented as a convex combination of two neighboring sector endpoints. This improves numerical stability and training convergence.

C. Network Architecture

• Model: Multilayer Perceptrons (MLPs) with fully connected layers.
• Activation: Leaky Rectified Linear Unit (Leaky ReLU).
• Optimization: Trained using Mean Squared Error (MSE) loss and the Adam optimizer.
• Measurements: The method supports both informationally complete measurements (e.g., SIC-POVM) and informationally incomplete measurements (e.g., random subsets of local Pauli measurements), particularly for pure states.

3. Key Contributions

1. Model-Agnostic Noise Mitigation: The method mitigates noise without explicit knowledge of the noise model. It learns to recognize and compensate for noise patterns purely from data statistics.
2. Scalability: The approach demonstrates favorable scaling with the number of qubits. It successfully reconstructs states for systems ranging from 6 to 10 qubits.
3. Handling Mixed States: Unlike many ML-QST methods focused on pure states, this framework is validated on general random mixed states, preserving key physical properties like purity and entanglement negativity.
4. Reduced Measurement Overhead: For structured pure states, the method achieves high fidelity using informationally incomplete measurements (fewer settings than required for full tomography), significantly reducing experimental complexity.

4. Results

Numerical simulations were conducted on two-qubit mixed states and multi-qubit (6–10 qubit) pure states (GHZ-like and Dicke states).

• Pure States (6–10 Qubits):

  • The method achieved an average fidelity $\bar{F} > 0.995$ across a wide range of parameters.
  • No systematic degradation in performance was observed as the number of qubits increased.
  • Even in worst-case scenarios, infidelity remained below 0.025.
  • Successful reconstruction was achieved using fewer measurement settings than required for full informationally complete tomography.

• Mixed States (Two-Qubit):

  • Fidelity: For the vast majority of 1,000 random samples, the reconstructed fidelity ($F_{rec}$) exceeded the noisy input fidelity ($F_{noisy}$), confirming effective noise suppression.
  • Physical Properties: The method preserved key physical quantities. Deviations in purity ($D_\Gamma$) and entanglement negativity ($D_N$) remained moderate (e.g., $<0.06$) even when infidelity was up to 0.1.
  • Noise Strength Dependence: Interestingly, the network performed best at moderate noise levels. For nearly noise-free states, the network applied "conservative denoising," slightly reducing fidelity, which is a trade-off of the learned statistical regularization.

• Statistical Robustness: Increasing the number of measurement repetitions ($m$) systematically improved reconstruction fidelity and reduced variance, aligning with theoretical expectations.

5. Significance and Future Outlook

• Practicality: This work provides a robust tool for experimentalists working in noisy intermediate-scale quantum (NISQ) devices where noise is complex and difficult to model analytically.
• Scalability: The ability to scale to larger qubit numbers while maintaining high fidelity suggests this is a viable path for characterizing large-scale quantum circuits.
• Future Directions:
  • Extending the framework to time-dependent or correlated noise (potentially using Recurrent Neural Networks).
  • Incorporating Physics-Informed Neural Networks (PINNs) to further constrain the solution space and improve accuracy.
  • Experimental validation on real quantum hardware.

In conclusion, the paper establishes that neural networks, specifically MLPs with tailored encoding schemes, offer a powerful, flexible, and scalable solution for error-mitigated quantum state tomography, overcoming the limitations of traditional model-based approaches.