Learning Informative Latent Representation for Quantum State Tomography

TL;DR

This work tackles quantum state tomography under imperfect measurement data, where frequencies $f_i$ may be under-sampled or measurements may be missing, rendering the reconstruction of the density matrix $ ho$ ill-posed. It introduces a transformer-based autoencoder that learns an informative latent representation (ILR) from imperfect data, aided by a pre-training task that recovers high-quality probabilities $ ilde{ extbf{p}}$ from masked frequencies. The approach uses operator embedding, a large expressive encoder, an asymmetrical decoder, and a post-processing module to enforce density-matrix constraints, achieving superior density matrix reconstructions and direct quantum-property predictions compared to traditional and neural baselines. Exhaustive simulations and IBM quantum-device experiments show ILR’s robustness to noise and incomplete data, with demonstrated scalability to at least 4 qubits and promising extensions to higher-dimensional states. Overall, ILR offers a practical, scalable pathway for accurate partial QST and quantum-property inference under realistic experimental constraints.

Abstract

Quantum state tomography (QST) is the process of reconstructing the complete state of a quantum system (mathematically described as a density matrix) through a series of different measurements. These measurements are performed on a number of identical copies of the quantum system, with outcomes gathered as frequencies. QST aims to recover the density matrix or the properties of the quantum state from the measured frequencies. Although an informationally complete set of measurements can specify the quantum state accurately in an ideal scenario with a large number of identical copies, both the measurements and identical copies are restricted and imperfect in practical scenarios, making QST highly ill-posed. The conventional QST methods usually assume accurate measured frequencies or rely on manually designed regularizers to handle the ill-posed reconstruction problem, suffering from limited applications in realistic scenarios. Recent advances in deep neural networks (DNN) led to the emergence of deep learning in QST. However, existing DL-based QST approaches often employ generic DNN models that are not optimized for imperfect conditions of QST. In this paper, we propose a transformer-based autoencoder architecture tailored for QST with imperfect measurement data. Our method leverages a transformer-based encoder to extract an informative latent representation (ILR) from imperfect measurement data and employs a decoder to predict the quantum states based on the ILR. We anticipate that the high-dimensional ILR will capture more comprehensive information about the quantum states. To achieve this, we conduct pre-training of the encoder using a pretext task that involves reconstructing high-quality frequencies from measured frequencies. Extensive simulations and experiments demonstrate the remarkable ability of the informative latent representation to deal with imperfect measurement data in QST.

Figures (16)

• Figure 1: Schematic of QST. (a) Perform different measurements on quantum states and obtain frequencies from collected outcomes. (b) Infer quantum representations, including density matrix (full description) and quantum properties (partial description) from measured frequencies.
• Figure 2: The ill-posed phenomena of QST. $m\#$ denotes the number of missed measurements. The point with the lower value of the log of infidelity is better.
• Figure 3: General schematic of learning an informative representation for quantum state tomography with imperfect measurement data. (a) The QST process aims to approximate a map from imperfect inputs (inaccurate and incomplete frequency vectors) to desired outputs (density matrices or quantum properties) through a transformer-based autoencoder and a state decoder. (b) The pre-training process aims to retrieve high-quality probabilities $\hat{\mathbf{p}}$ from masked frequency vector $\widetilde{\mathbf{f}}$ through a combination of a large encoder (share with (a)) and a small frequency decoder.
• Figure 4: Comparison of different training strategies on 2-qubit pure states.
• Figure 5: Comparison results of reconstructing density matrices for pure states ($N_t=100$) with other methods including LRE, MLE, CNN, GAN, and FCN. "ILR w/o P" indicates ILR without the pre-training strategy.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3