Tomography of Quantum States from Structured Measurements via quantum-aware transformer

TL;DR

The paper addresses the challenge of reconstructing quantum states from structured measurements by leveraging a quantum-aware transformer (QAT) that explicitly encodes measurement structure. It introduces frequency and operator embeddings, cross-attention, and a loss that combines an approximated Bures distance with Euclidean terms to maximize fidelity between true and reconstructed states. Empirical results show that QAT-QST achieves higher fidelity and robustness across pure and mixed 2–4-qubit states compared to FCN, CNN, and traditional tomography methods, including experiments on IBM quantum hardware. The approach highlights the value of incorporating quantum-specific structures and distance metrics into learnable tomography pipelines, while acknowledging scalability limits and proposing future extensions such as shadow tomography and detector tomography.

Abstract

Quantum state tomography (QST) is the process of reconstructing the state of a quantum system (mathematically described as a density matrix) through a series of different measurements, which can be solved by learning a parameterized function to translate experimentally measured statistics into physical density matrices. However, the specific structure of quantum measurements for characterizing a quantum state has been neglected in previous work. In this paper, we explore the similarity between highly structured sentences in natural language and intrinsically structured measurements in QST. To fully leverage the intrinsic quantum characteristics involved in QST, we design a quantum-aware transformer (QAT) model to capture the complex relationship between measured frequencies and density matrices. In particular, we query quantum operators in the architecture to facilitate informative representations of quantum data and integrate the Bures distance into the loss function to evaluate quantum state fidelity, thereby enabling the reconstruction of quantum states from measured data with high fidelity. Extensive simulations and experiments (on IBM quantum computers) demonstrate the superiority of the QAT in reconstructing quantum states with favorable robustness against experimental noise.

Figures (9)

• Figure 1: Physical process of QST. Given a number of identical copies of a quantum state (denoted as $\rho \in \mathbb{C}^{d\times d}$, with $d$ representing system dimension) (see Section \ref{['sec:preQST']} for detailed information), a set of measurements (with each measurement operator denoted as $\mathbb{O}_i\in \mathbb{C}^{d\times d}$) are performed on quantum states, with the collected outcomes represented into a set of frequencies $\{f_i\in[0,1]\}$.
• Figure 2: Simalirity between languaging model using words and characters, and QST using structured measurements. (a) A sentence is encoded using four words, with each word containing several characters. The character correlations come in two characters in one word or two characters from different words. Although the basic element is a character, words are encoded into vectors for further assessment. (b) To specify a quantum state, several sets of measurements are performed, with each representing a detector that contains a set of positive semi-definite matrices that sum to identity. The measured proabilities from four detectors are represented into four vectors, which can be further utilized for state reconstructions.
• Figure 3: Schematic of the ML-based QST approach. (a) A multi-layer neural network $\Phi(\theta)$ maps the frequencies $\{f_i\}$ to the vector $\hat{\alpha}$; (b) Compute the Cholesky decomposition of the density matrices $\rho$ as the ground-truth vector $\alpha$ for supervised training with a loss function; (c) Obtain the quantum state $\hat{\rho}$ from the network's output $\hat{\alpha}$.
• Figure 4: Framework for the proposed QAT-QST approach. (a) Two embedding modules are combined as the input for the network, namely frequency embedding, and operator embedding. Then, the transformer layers process the mainstream of frequency (Gray solid lines), and query the operator embedding (Orange dash lines), along with a fully connected head to obtain a real vector $\hat{\alpha}$. (b) The transformer layers consist of $d_L$ repeated stacks of normalization, a cross-attention module, normalization, a self-attention module, normalization, and multilayer perception layers. (c) The difference structures between self-attention and cross-attention module, where the cross-attention is used to learn how different elements in operator embedding are related to the mainstream.
• Figure 5: Visualization of three vectors in different criteria. $\boldsymbol{\alpha}$ represents the fixed target vector from the Cholesky decomposition. $\boldsymbol{\hat{\alpha}_1}$, $\boldsymbol{\hat{\alpha}_2}$, $\boldsymbol{\hat{\alpha}_3}$ represent three candidate vectors. If one needs to choose one vector close to $\boldsymbol{\alpha}$, $\boldsymbol{\hat{\alpha}_1}$ is a good candidate regarding the Euclidean distance because $\mu_1 < \mu_2$; $\boldsymbol{\hat{\alpha}_2}$ is a good candidate regarding the Bures distance because $\theta_2 < \theta_1$. When considering both the Bures distance and the Euclidean distance with the target vector $\boldsymbol{\hat{\alpha}}$, $\boldsymbol{\hat{\alpha}_3}$ is a good candidate that achieves good performance on both criteria. As such, $\boldsymbol{\hat{\alpha}_3}$ is a more suitable candidate vector.