Physics-inspired neural networks as quasi inverse of quantum channels

TL;DR

The paper tackles inverting quantum channels by learning a quasi-inverse with a neural network that remains physically realizable as a CPTP map across arbitrary channel parameters $\mathcal{E}$. It introduces a physics-inspired MSMTD loss based on the scaled Bloch-vector length $r'$, guiding the network to recover a rotated, scaled original state while preserving trace and positivity. A simple 3-32-3 feedforward network, trained over 1000 samples for four qubit channels, learns the inverse from $\mathbf{r}'$ to $\mathbf{r}''$ and is validated by quantum process tomography to extract Kraus operators and Chi matrices, with CPTP verification $\sum_i E_i^{\dagger} E_i = I + \delta$ and $\delta \le 10^{-15}$. Empirically, the approach reduces the mean square modified trace distance (MSMTD) in about 80% of test cases, suggesting a practical framework for noise mitigation in quantum devices, while highlighting limitations for high-purity states where original SMTD is already small. This work provides a principled route to learn physically valid quasi-inverses of quantum channels and a methodology to extract their Kraus representations.

Abstract

Quantum channels are not invertible in general. A quasi-inverse allows for a partial recovery of the input state, but its analytical results are found only in a restricted space of its parameters. This work explores the potential of neural networks to find the quasi-inverse of qubit channels for any values of the channel parameters while keeping the quasi-inverse as a physically realizable quantum operation. We introduce a physics-inspired loss function based on the mean of the square of the modified trace distance (MSMTD). The scaled trace distance is used so that the neural network does not increase the length of the Bloch vector of the quantum states, which ensures that the network behaves as a completely positive and trace-preserving (CPTP) quantum channel. The Kraus operators of the quasi-inverse channel were obtained by performing quantum process tomography on the trained neural network.

Figures (4)

• Figure 1: A schematic representation of the quantum channel and the inversion process using a neural network. The quantum channel transforms the original Bloch vector $\textbf{r}$ into $\textbf{r}^{\,\prime}$. The neural network predicts an approximation of the original Bloch vector denoted as $\textbf{r}^{\,\prime \prime}$.
• Figure 2: A schematic representation of our fully connected neural network.
• Figure 3: Training loss history of neural networks trained to invert quantum channels.
• Figure 4: Mean of the square of the modified trace distance (MSMTD) of single qubit (a) bit flip, (b) phase flip and (c) bit-phase flip channels for varying noise probability $p$ and (d) amplitude damping channel for varying damping parameter $\gamma$.