Quantum State Tomography using Quantum Machine Learning

TL;DR

The paper surveys classical and quantum machine learning approaches to Quantum State Tomography (QST), addressing the critical challenge of measurement overhead in reconstructing quantum states. It compares traditional methods (Linear Inversion, MLE, Least Squares, covariance-based estimators, Bayesian) with several QML strategies, including Variational Quantum Circuits (VQC), Quantum PCA (qPCA), Bayesian QST, and Quantum Variational with Classical Statistics (QVCS). Empirical results on simulated and experimental data show high fidelity for QST on small to moderate systems; notably, GHZ-state tomography achieves ≈$98\%$ fidelity, VQC depth optimization reaches favorable performance around depth 13, and qPCA can yield near-90% fidelity for low-rank states, illustrating potential speedups and reduced measurements. The study highlights practical benefits for near-term quantum information processing while acknowledging current limitations due to noise and scalability, and outlines future directions toward hybrid and more advanced QML-QST methods to extend these gains. $\rho$ and $F$ are central quantities throughout, with fidelity and positive semidefinite, unit-trace constraints ensuring physical density matrices.

Abstract

Quantum State Tomography (QST) is a fundamental technique in Quantum Information Processing (QIP) for reconstructing unknown quantum states. However, the conventional QST methods are limited by the number of measurements required, which makes them impractical for large-scale quantum systems. To overcome this challenge, we propose the integration of Quantum Machine Learning (QML) techniques to enhance the efficiency of QST. In this paper, we conduct a comprehensive investigation into various approaches for QST, encompassing both classical and quantum methodologies; We also implement different QML approaches for QST and demonstrate their effectiveness on various simulated and experimental quantum systems, including multi-qubit networks. Our results show that our QML-based QST approach can achieve high fidelity (98%) with significantly fewer measurements than conventional methods, making it a promising tool for practical QIP applications.

Figures (19)

• Figure 1: Flowchart of the Variational Quantum Circuit for QST Algorithm.
• Figure 2: Variational Quantum Circuit diagram. The VQC initiates with $n$ input qubits that undergo a feature map before being processed through the circuit itself. After this transformation, all qubits are measured, and their outcomes are subsequently assessed by a classical model to evaluate the cost function.
• Figure 3: Variational Quantum Circuit used for QST. The depicted quantum circuit is constructed using the Construct VariationalCirc algorithm, employing rotational Rx, Ry, and CNOT gates to represent generic quantum states. Each layer consists of CNOT gates, defining the circuit depth $s$, which is set to 10 layers for this representation.
• Figure 4: A schematic diagram of Bayesian Inference for Quantum State Tomography. $q$ defines the $Quantum Registers$ for $n$-qubits. While $\theta_1...\theta_n$ are prior parameters. $P(\mathcal{D}|\theta)$ represents a Maximum Likelihood function. $\theta_1'$ are posterior parameters used for Posterior State probability distribution, and then after measurements, the sampling procedure is performed.
• Figure 5: Perceptron model architecture to represent $p_\lambda$, and $\phi_\mu$ for the pure state $\ket{\phi}$, according to the parameterisation defined in Eq.\ref{['quantum state']}. Here, the two networks, one for the probabilities and one for the phases, have been represented in a single diagram.