Error-mitigated quantum state tomography using neural networks

The reliable characterization of quantum states is a fundamental task in quantum information science. For this purpose, quantum state tomography provides a standard framework for reconstructing quantum states from measurement data, yet it is often degraded by experimental noise. Mitigating such noise is therefore essential for the accurate estimation of the states in realistic settings. In this work, we propose a scalable tomography method based on multilayer perceptron networks that mitigate unknown noise through supervised learning. This approach is data-driven and thus does not rely on explicit assumptions about the noise model or measurement, making it readily extendable to general quantum systems. Numerical simulations, ranging from special pure states to random mixed states, demonstrate that the proposed method effectively mitigates noise across a broad range of scenarios, compared with the case without mitigation.

💡 Research Summary

The paper addresses a central challenge in quantum information science: accurately reconstructing quantum states in the presence of experimental noise. Traditional quantum state tomography (QST) techniques—least‑squares inversion, Bayesian methods, maximum‑likelihood estimation, linear regression, and convex optimization—are all vulnerable to measurement imperfections and environmental disturbances, which bias the reconstructed density matrix. Existing mitigation strategies either require explicit noise characterization (e.g., process tomography) or rely on restrictive assumptions such as low‑rank structure (compressed sensing), limiting their applicability to general quantum systems.

To overcome these limitations, the authors propose a fully data‑driven, model‑agnostic approach based on multilayer perceptron (MLP) neural networks. The key idea is to treat the mapping from noisy measurement outcomes to the underlying noise‑free quantum state as a supervised learning problem. No prior knowledge of the noise model or the measurement apparatus is required; the network learns directly from pairs of simulated noisy data and corresponding ideal states.

Physical constraints of density matrices are enforced by representing the state through a Cholesky decomposition, ρ = RR†, where R is a lower‑triangular matrix. Only the independent real entries of R are retained and assembled into a parameter vector α. This guarantees positivity and trace‑one normalization after decoding (ρ′ = RR†/tr(RR†)). To improve numerical stability, the authors introduce a modified one‑hot‑inspired encoding: each continuous component α_i, known to lie within empirically determined bounds, is discretized into N_sec uniform sectors, and the value is expressed as a convex combination of the two neighboring sector endpoints. The resulting encoded vector behaves like a probability distribution, aiding gradient‑based training.

The neural network architecture consists of an input layer receiving measurement statistics, three hidden layers of size 100·2⌈(n−4)/2⌉ (for an n‑qubit system), and an output layer predicting the encoded α. Leaky ReLU activation functions are used to avoid dead neurons. The measurement scheme employs local Pauli projectors (excluding the all‑identity operator), yielding 4ⁿ−1 independent outcomes—exactly the number of real parameters for an n‑qubit pure state. This choice reduces the dimensionality of the input and enables tomography with fewer settings than a full informationally complete set.

Training data are generated by applying a sequence of common noise channels—white noise, bit‑flip, phase‑flip, and amplitude‑damping—to ideal states, with each channel’s strength sampled uniformly from a predefined interval. This creates a rich dataset covering a continuum of noise levels. The network is trained on 1,000 random states (a mixture of GHZ‑like and Dicke pure states) and subsequently tested on both pure‑state and mixed‑state ensembles.

Simulation results demonstrate remarkable performance. For six‑ to ten‑qubit GHZ‑like and Dicke states, the average fidelity of the reconstructed states exceeds 0.995, with worst‑case fidelity still above 0.975. The infidelity typically lies below 10⁻³, and the required number of measurement settings scales as 2ⁿ+1, substantially lower than the 4ⁿ−1 settings needed for a full informationally complete tomography. For random two‑qubit mixed states, the neural‑network‑based reconstructions match or surpass the fidelity of conventional maximum‑likelihood estimators, especially when the noise strength is high.

The authors highlight several strengths of their method: (i) it does not assume any specific noise model, making it adaptable to diverse experimental conditions; (ii) the Cholesky‑based encoding guarantees physically valid density matrices without post‑processing; (iii) the one‑hot‑inspired encoding improves training stability; (iv) the approach scales favorably with system size, requiring only modest network depth and width relative to the exponential Hilbert‑space dimension.

Nevertheless, the paper acknowledges limitations. The network’s ability to generalize hinges on the diversity of the training set; unseen noise types or drifts could degrade performance. The Cholesky parameterization, while ensuring positivity, still involves O(2ⁿ) real parameters, which may become prohibitive for large n. Moreover, all results are based on numerical simulations; experimental validation on real quantum hardware, where noise can be correlated, non‑Markovian, or time‑varying, remains an open task.

Future research directions suggested include: (1) transfer learning and domain adaptation to apply models trained on simulated data to real experimental datasets; (2) hybrid schemes that combine physics‑based regularization (e.g., known channel structures) with neural‑network flexibility; (3) dimensionality‑reduction techniques such as tensor‑network encodings to handle larger qubit numbers; and (4) Bayesian neural networks to quantify uncertainty and provide confidence intervals for the reconstructed states.

In summary, the work presents a compelling neural‑network framework for noise‑mitigated quantum state tomography that is both model‑agnostic and scalable, achieving high‑fidelity reconstructions across a variety of state families and noise conditions. It opens a pathway toward practical, efficient tomography in noisy quantum devices, while also outlining clear challenges and avenues for experimental realization and further methodological refinement.