Quantum state tomography (QST) aims to estimate the density matrix of an unknown quantum state through repeated measurements. It's crucial for characterizing quantum states and processes, but resource-intensive due to the large number of measurements needed for accurate estimation of the d²−1 real parameters describing a d-dimensional quantum system. Research focuses on minimizing resource costs for target precision. Adaptive measurements, adjusting the next measurement based on previous information, offer a promising approach. Adaptive Bayesian quantum tomography (ABQT) is one such method, using Bayesian principles and a particle filter algorithm to update the prior distribution. However, ABQT suffers from computational cost due to particle weight decay, necessitating frequent, expensive resampling. This paper proposes a faster alternative, Neural Adaptive Quantum Tomography (NAQT), using a custom-built recurrent neural network (RNN) architecture to learn an efficient approximation of the Bayesian update rule, eliminating the need for computationally expensive resampling and achieving significant speedups while retaining high accuracy.

Existing QST methods, including adaptive ones like ABQT, often face challenges in computational efficiency. ABQT, while effective, uses a particle filter with resampling to handle Bayesian updates. This approach, however, is hampered by the decay of particle weights, requiring repeated computationally intensive resampling steps, making it slow for a large number of measurements. This paper builds upon the foundation of adaptive QST methods, addressing the limitations of existing techniques by leveraging the power of neural networks for efficient state estimation and measurement adaptation.

NAQT employs a custom-built RNN architecture to perform adaptive QST. The network's input is the measurement data from a chosen Positive Operator-Valued Measure (POVM), and its output is an estimate of the density matrix, ρ. The RNN iteratively refines its estimate with each new batch of data, converging toward the true state. The network maintains a bank B of particles (candidate quantum states) with associated weights. Unlike ABQT, where weights are updated according to Bayes' rule, NAQT's neural network learns to update both weights and particles to efficiently move toward the true state. The measurement adaptation step in NAQT uses the information-gain quantity I(Π, D) = H(P(Π|D)) – ⟨H(P(Π|ρ))⟩, maximizing information gain by choosing the POVM Π that maximizes I(Π, D). The NAQT algorithm is trained using simulated data. The training process involves iteratively simulating measurements, updating the particle bank based on the network's learned rules, and adapting the POVM to maximize information gain. The network learns the optimal rules for updating the particle bank and choosing the next measurement, effectively learning an approximation of Bayes' rule without the computationally expensive resampling of ABQT. The algorithm's architecture includes a differentiable implementation of quantum mechanics in TensorFlow to simulate POVM measurements.

NAQT demonstrates comparable reconstruction accuracy to ABQT but achieves orders-of-magnitude faster computation times. In a two-qubit tomography experiment with basis POVMs, NAQT using 100 particles completed a reconstruction with 100,000 measurements in 2 seconds, while ABQT with 100 particles required 24,000 seconds (6.7 hours). The runtime scaling for NAQT appears logarithmic with the number of measurements, while ABQT's scaling appears linear. The speedup stems from the neural network's ability to learn an effective heuristic for state perturbations, avoiding ABQT's computationally expensive resampling steps. NAQT's performance is robust across different POVMs, as shown by similar runtime behavior for product tetrahedron POVMs. The key advantage lies in the neural network's ability to learn an effective bank update strategy, avoiding the computationally expensive resampling step caused by particle weight decay in ABQT. This is illustrated in the logarithmic vs. linear runtime comparisons, showing a huge speedup advantage for NAQT. The use of a resampling schedule that scales logarithmically with the number of copies measured is shown to contribute significantly to the logarithmic scaling of the runtime for NAQT.

NAQT addresses the computational bottleneck of adaptive QST by leveraging the power of neural networks. The neural network effectively learns an approximate Bayesian update rule, eliminating the need for the computationally expensive resampling inherent in ABQT. The algorithm's flexibility allows for easy adaptation to different measurement choices and types of states. The NAQT algorithm shows a significant speed improvement over the ABQT algorithm, potentially pushing the boundary of tractable QST to higher dimensions. The ease of retraining for different POVMs and state classes demonstrates the versatility of the NAQT approach. This is a significant advantage over methods that require specific estimators or ad hoc fixes depending on the POVM type. The ability to avoid an initial ansatz for the quantum state further enhances the algorithm's generality.

NAQT provides a significant advancement in quantum state tomography by achieving orders-of-magnitude speedups compared to existing adaptive methods like ABQT without sacrificing accuracy. The use of neural networks to learn an effective heuristic for state updates allows for efficient and flexible tomography. Future research could explore the application of NAQT to higher-dimensional systems and noisy measurements, potentially pushing the boundaries of practical quantum state tomography.

The current implementation of NAQT has been primarily tested on two-qubit systems. While the authors suggest scalability to higher dimensions, further experimentation is needed to confirm its performance and efficiency for larger quantum systems. The training of the neural network requires simulated data; the performance on real experimental data might be impacted by noise and other real-world factors. The choice of the specific neural network architecture and hyperparameters could also influence the algorithm's performance, requiring careful optimization for specific applications.