Quantum feedback control with a transformer neural network architecture

TL;DR

This work numerically shows that the bespoke transformer architecture can achieve unit fidelity to a target state in a short time even in the presence of inefficient measurement and Hamiltonian perturbations that were not included in the training set.

Abstract

Attention-based neural networks such as transformers have revolutionized various fields such as natural language processing, genomics, and vision. Here, we demonstrate the use of transformers for quantum feedback control through a supervised learning approach. In particular, due to the transformer's ability to capture long-range temporal correlations and training efficiency, we show that it can surpass some of the limitations of previous control approaches, e.g.~those based on recurrent neural networks trained using a similar approach or reinforcement learning. We numerically show, for the example of state stabilization of a two-level system, that our bespoke transformer architecture can achieve unit fidelity to a target state in a short time even in the presence of inefficient measurement and Hamiltonian perturbations that were not included in the training set. We also demonstrate that this approach generalizes well to the control of non-Markovian systems. Our approach can be used for quantum error correction, fast control of quantum states in the presence of colored noise, as well as real-time tuning, and characterization of quantum devices.

Figures (4)

• Figure 1: Problem and Architecture Overview:a, The two level system (TLS) interacting with a bath in a non-Markovian manner that is embedded with a reaction coordinate (RC) which interacts in a Markovian manner. The measurement record obtained continously is used to predict optimal values of the control parameters by the transformer. b, The transformer's structure consists of an encoder and decoder architecture. During training, the encoder takes the initial state and the measurement record as input (green dotted boxes). The decoder takes the encoder output as part of the cross attention layer and the optimal parameters (blue dotted boxes), to autoregressively predict the optimal parameters for the next time steps. However, during inference, only the initial state and measurement record is given as input to the encoder (solid arrows). The decoder then predicts the optimal next values of the control parameters based on this data.
• Figure 2: Fidelity $\mathcal{F}$ with a target state as a function of time under feedback control. The initial state is $\hat{\rho}_0 = |\psi_0\rangle \langle \psi_0|$, where $\ket{\psi_0} =\alpha|0\rangle+\beta|1\rangle$ with $\alpha = \sqrt{\frac{7}{12}}$ and $\beta= \sqrt{\frac{5}{12}}$. The target state is $\ket{\psi_{\rm targ}}=\frac{1}{\sqrt{2}}(|0\rangle+i|1\rangle)$. The performance of the transformer under imperfect measurement efficiency ($\eta=0.7$) (blue, circles), and an increase in bias ($\epsilon =0.5$) (green, squares) is benchmarked against the fidelity improvement when randomly selecting $\lambda_t$ values (orange, crosses).
• Figure 3: The fidelity $\mathcal{F}$ as a function of time while benchmarking the performance of the transformer (red, crosses) as compared to a vanilla recurrent neural network (green, circles) and a gated-recurrent unit recurrent neural network GRU-RNN (yellow, stars). The context of 2000 measurement record samples is provided in the case of the non-Markovian setting accounted for by the reaction coordinate embedding. The coupling with the bath provided by $g=0.5$. The dimension of the reaction coordinate is truncated to $d=6$.
• Figure 4: a, The magnitude and phase map of the mixed initial state of $\rho_0=0.7|0\rangle\langle 0|+0.3|1\rangle\langle 1|$ and target pure state of $\left|\psi_T\right\rangle=\sqrt{0.3}|0\rangle+i \sqrt{0.7}|1\rangle$. The third plot represents the increase in fidelity towards the target pure state based on control parameters produced by the transformer (blue, circles) when undergoing continuous measurement with a measurement efficiency of 0.8. b, The magnitude and phase map of the mixed initial state of $\rho_0=0.6|0\rangle\langle 0|+(0.2+0.1 i)|0\rangle\langle 1|+(0.2-0.1 i)|1\rangle\langle 0|+0.4|1\rangle\langle 1|$ and target pure state of $\psi_T=\frac{|0\rangle}{\sqrt{2}}+\frac{1+i}{2}|1\rangle$. The third plot represents the increase in fidelity towards the target pure state based on control parameters produced by the transformer (blue, circles) when undergoing continuous measurement with a measurement efficiency of 0.65.