Quantum Machine Learning for State Tomography Using Classical Data

TL;DR

The paper tackles the scalability barrier in quantum state tomography by introducing a QML-based protocol that reconstructs quantum states using only classical measurement data, enabling practical deployment on NISQ devices. It employs a variational quantum circuit whose parameters are trained to match the measurement distributions of a target state across Pauli bases, using a symmetric KL-divergence loss and SPSA optimization. The approach yields high-fidelity reconstructions for a variety of states (3- and 6-qubit GHZ, spin-chain ground states, and random circuits) in simulations and demonstrates feasibility on IBM and IonQ hardware, including scenarios with incomplete measurement bases. This work demonstrates a scalable pathway for practical quantum state reconstruction and provides a foundation for further improvements in trainability, symmetry-guided measurements, and mixed-state tomography.

Abstract

Reconstructing quantum states from measurement data represents a formidable challenge in quantum information science, especially as system sizes grow beyond the reach of traditional tomography methods. While recent studies have explored quantum machine learning (QML) for quantum state tomography (QST), nearly all rely on idealized assumptions, such as direct access to the unknown quantum state as quantum data input, which are incompatible with current hardware constraints. In this work, we present a QML-based tomography protocol that operates entirely on classical measurement data and is fully executable on noisy intermediate-scale quantum (NISQ) devices. Our approach employs a variational quantum circuit trained to reconstruct quantum states based solely on measurement outcomes. We test the method in simulation, achieving high-fidelity reconstructions of diverse quantum states, including GHZ states, spin chain ground states, and states generated by random circuits. The protocol is then validated on quantum hardware from IBM and IonQ. Additionally, we demonstrate accurate tomography is possible using incomplete measurement bases, a crucial step towards scaling up our protocol. Our results in various scenarios illustrate successful state reconstruction with fidelity reaching 90% or higher. To our knowledge, this is the first QML-based tomography scheme that has been implemented on real quantum processors using exclusively classical measurements. This work establishes the feasibility of QML-based tomography on current quantum platforms and offers a scalable pathway for practical quantum state reconstruction.

Figures (15)

• Figure 1: Example of a simple quantum circuit.
• Figure 2:
• Figure 5: The circuit ansatz. The trainable layers are the alternating $R_x$ and $R_y$ layers. Between the $R_x$ and $R_y$ layers, CNOT gates are used to create entanglement. The plot shows an ansatz with 5 layers.
• Figure 6: Reconstruction of the 3-qubit GHZ state. \ref{['fig:q3_ghz_state_city_spsa']} shows the visualization of the target state, while the \ref{['fig:q3_reconstructed_state_city_spsa']} shows the reconstructed state in a typical trial. The difference between them are calculated $\rho_\text{target}-\rho_\text{ansatz}$, and \ref{['fig:dm_diff_q3_ghz']} visualizes the magnitude of its matrix elements.
• Figure 7: Reconstruction of the 3-qubit spin chain ground state. \ref{['fig:q3_spch_state_city_spsa']} shows the visualization of the target state, while the \ref{['fig:q3_spch_reconstructed_state_city_spsa']} shows the reconstructed state in a typical trial. The difference between them are calculated $\rho_\text{target}-\rho_\text{ansatz}$, and \ref{['fig:dm_diff_q3_spch']} visualizes the magnitude of its matrix elements.