Error-mitigated quantum state tomography using neural networks

TL;DR

This paper addresses the susceptibility of quantum state tomography to experimental noise by introducing a supervised neural-network approach that learns a mapping from measurement data to a physically valid density matrix. The method enforces positivity and unit trace via a Cholesky-based parameterization $ ho=RR^ op$ with normalization to $ ho'$, and employs a multilayer perceptron with a one-hot-inspired encoding for numerical stability. Training uses synthetic data spanning various noise channels and Pauli-type measurements, enabling noise-agnostic mitigation for both pure and mixed states. Results show near-perfect reconstructions for structured pure states up to ten qubits and substantial fidelity gains for two-qubit mixed states, with improvements scaling with measurement repetitions; the approach is data-driven and scalable to larger systems, though it assumes stationary noise and suggests extensions to time-varying noise via recurrent or physics-informed networks.

Abstract

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.

Figures (6)

• Figure 1: Schematic illustration of the neural-network-based quantum state tomography. Starting from a noise-free state $\rho$, two parallel procedures are involved: (1) The blue path represents the physical processing, where noisy channels transform $\rho$ into $\rho_n$ followed by measurements yielding the data vector $\vecfont{D}$; (2) The red path denotes the numerical processing, where a Cholesky decomposition produces a lower triangular matrix $R$ which is subsequently reshaped and encoded into the parameter vector $\boldsymbol{\alpha}$.
• Figure 2: Neural network structures: (a) Perceptron model with direct input-output connections, but limited to linear mappings. (b) Multilayer perceptron (MLP) with hidden layers, enabling approximation of complex functions. The data is feedforward from input to output, while nodes in the middle do not serve as output, turning out to be hidden layers. Properly designed MLPs can universally approximate the mappings from noisy measurements to noise-free states.
• Figure 3: Tomography of six- to ten-qubit mixtures of GHZ-like and Dicke states. Among one thousand samples in each case, the fidelity of the noisy state satisfies $0.75\leq F_{\text{noise}} \leq1$. Histogram width indicates sample counts at each infidelity $1-F_{\text{rec}}$. Most reconstructed states achieve infidelities below $10^{-3}$, with the worst-case infidelity below $0.025$.
• Figure 4: Fidelity comparison for random two-qubit mixed states. The diagonal line $F_{\mathrm{rec}} = F_{\mathrm{noisy}}$ indicates the absence of noise mitigation. Data points above this line demonstrate successful noise suppression by the neural network. The results confirm noise mitigation for almost all the cases tested. For most states, the reconstructed fidelity exceeds the noisy fidelity, with only a small fraction exhibiting slight degradation due to statistical learning effects.
• Figure 5: Deviations of negativity and purity for the reconstructed two-qubit random mixed states. They remain moderate over a wide range of reconstruction infidelities, which confirms that the noise mitigation is also effective in terms of purity and negativity.