Experimental demonstration of reconstructing quantum states with generative models

Quantum state tomography, a process that reconstructs a quantum state from measurements on an ensemble of identically prepared copies, plays a crucial role in benchmarking quantum devices. However, brute-force approaches to quantum state tomography would become impractical for large systems, as the required resources scale exponentially with the system size. Here, we explore a machine learning approach and report an experimental demonstration of reconstructing quantum states based on neural network generative models with an array of programmable superconducting transmon qubits. In particular, we experimentally prepare the Greenberger-Horne-Zeilinger states and random states up to five qubits and demonstrate that the machine learning approach can efficiently reconstruct these states with the number of required experimental samples scaling linearly with system size. Our results experimentally showcase the intriguing potential for exploiting machine learning techniques in validating and characterizing complex quantum devices, offering a valuable guide for the future development of quantum technologies.

💡 Research Summary

The paper presents an experimental demonstration of a machine‑learning‑based quantum state tomography (QST) method that leverages a recurrent neural network (RNN) generative model to reconstruct noisy quantum states on a superconducting processor. The authors first prepare a set of target states—Greenberger‑Horne‑Zeilinger (GHZ) states of 2 to 5 qubits and a collection of random many‑body states—using calibrated single‑qubit rotations and controlled‑phase (CZ) gates on a linear chain of five transmon qubits. Each qubit can be individually driven along the X, Y, and Z axes and read out via a dispersive measurement chain equipped with a broadband Josephson parametric amplifier, ensuring simultaneous high‑fidelity readout of all qubits.

For tomography, the authors employ an informationally complete Pauli‑4 positive‑operator‑valued measure (POVM), which consists of projectors onto the eigenstates of σ_x, σ_y, and σ_z (plus a mixed projector). Randomly selecting one of these three measurement bases for each experimental shot yields a dataset of binary strings that encode the outcomes of multi‑qubit measurements. The raw data, which includes readout errors and gate imperfections, is fed directly into an RNN. The network treats each qubit position as a time step, propagating a hidden state through the chain so that non‑local correlations can be captured efficiently. Training minimizes the Kullback‑Leibler (KL) divergence between the empirical measurement distribution P_exp and the model‑generated distribution P_model. The optimization uses the Adam optimizer and converges within a few hundred epochs.

Two performance metrics are reported. Classical fidelity (F_C) quantifies the overlap between P_exp and P_model and reaches >99 % after a modest number of training samples. Quantum fidelity (F_Q) is computed by first reconstructing a density matrix from the learned distribution via maximum‑likelihood estimation (MLE) and then evaluating the trace fidelity with the density matrix obtained from conventional QST. For the four‑qubit GHZ state, F_Q stabilizes around 95 %, reflecting the sensitivity of quantum fidelity to residual experimental noise that the RNN does not fully capture.

A central contribution of the work is the scaling analysis of the required number of measurement samples (N_s). By repeatedly training the RNN on randomly selected subsets of the full dataset, the authors determine the critical sample size N* at which F_C exceeds 99 %. Plotting N* versus the number of qubits N reveals an approximately linear relationship (N* ≈ a N + b), in stark contrast to the exponential scaling observed for standard QST. This linear scaling holds both for GHZ states and for the randomly generated states, indicating that the generative‑model approach is not limited to highly structured states.

Beyond state reconstruction, the trained RNN is used to compute physical observables directly from the learned probability distribution. The authors evaluate two‑body correlators ⟨X_i X_j⟩ and ⟨Z_i Z_j⟩ for all qubit pairs. After applying an MLE‑based correction to enforce physicality of the reconstructed density matrix, the correlators match those obtained from conventional QST, with Z‑Z correlations reproducing the expected GHZ pattern and X‑X correlations remaining near zero (subject to experimental noise). This demonstrates that the generative model can serve as a compact classical representation from which a variety of quantum properties can be extracted without reconstructing the full density matrix.

The paper also discusses practical considerations. While Bayesian post‑processing and MLE can improve the fidelity of the reconstructed state, both procedures scale exponentially and are therefore only applied here for benchmarking. The authors argue that as hardware improves and readout errors diminish, raw measurement data will suffice for training, eliminating the need for costly post‑processing. Moreover, the RNN’s parameter count remains modest (on the order of a few thousand), ensuring that classical computational resources scale linearly with system size.

In summary, the experimental results demonstrate that (i) an RNN‑based generative model can learn the full measurement distribution of noisy multi‑qubit states with high accuracy, (ii) the number of required experimental samples grows only linearly with the number of qubits, and (iii) once trained, the model provides a versatile tool for extracting observables and characterizing quantum devices in the noisy intermediate‑scale quantum (NISQ) regime. This work highlights the promise of integrating modern machine‑learning techniques with quantum information protocols to achieve scalable, efficient quantum state characterization.