Neural networks for quantum state tomography with constrained measurements

TL;DR

The paper tackles quantum state tomography under constrained measurements by proposing a neural-network-based framework (DNN-QST) that maps observed measurement frequencies to a physical density matrix through a Cholesky-based parameterization. By training on simulated data, the approach achieves higher fidelity than traditional estimators and CNNs in scenarios with limited copies, incomplete measurements, or noisy operators, while offering fast inference once trained. Key contributions include a unified NN architecture that guarantees physical states via $ ho = ho_L ho_L^ op / \mathrm{Tr}( ho_L ho_L^ op)$, effective handling of cube and MUB measurement settings, and demonstrated robustness across noise models. This method reduces resource requirements for QST in practical quantum technologies and opens avenues for transfer learning and noise-aware measurement optimization.

Abstract

Quantum state tomography (QST) aiming at reconstructing the density matrix of a quantum state plays an important role in various emerging quantum technologies. Recognizing the challenges posed by imperfect measurement data, we develop a unified neural network(NN)-based approach for QST under constrained measurement scenarios, including limited measurement copies, incomplete measurements, and noisy measurements. Through comprehensive comparison with other estimation methods, we demonstrate that our method improves the estimation accuracy in scenarios with limited measurement resources, showcasing notable robustness in noisy measurement settings. These findings highlight the capability of NNs to enhance QST with constrained measurements.

Figures (12)

• Figure 1: Schematic of the DNN-QST approach. (a) Obtain the measured data and compute the Cholesky decomposition of the density matrices as the target vectors; (b) A multi-layer NN maps the frequencies to the $\bold{\alpha}$-vector; (c) Obtain the quantum states from the networks' output.
• Figure 2: Comparison of several methods on 2-qubit states using cube measurements. a. Infidelity vs copies of each measurement for pure states; b. Infidelity vs copies of each measurement for mixed states.
• Figure 3: The performance for 2-qubit states with different purities using few measurement copies under the cube measurement. a. Infidelity vs purity for cube basis with $S=10$; b. Infidelity vs purity for cube basis with $S=100$.
• Figure 4: The performance for 2-qubit states with different purities using few measurement copies under the mub measurement. a. Infidelity vs purity for MUB basis with $S=10$; b. Infidelity vs purity for MUB basis with $S=100$.
• Figure 5: Results for 3-qubit random pure states when the cube measurement has few copies.

Theorems & Definitions (1)

• Remark 1