Using recurrent neural networks to optimize dynamical decoupling for quantum memory

We utilize machine learning models that are based on recurrent neural networks to optimize dynamical decoupling (DD) sequences. Dynamical decoupling is a relatively simple technique for suppressing the errors in quantum memory for certain noise models. In numerical simulations, we show that with minimum use of prior knowledge and starting from random sequences, the models are able to improve over time and eventually output DD sequences with performance better than that of the well known DD families. Furthermore, our algorithm is easy to implement in experiments to find solutions tailored to the specific hardware, as it treats the figure of merit as a black box.

Figure 1

Convergence of the algorithm in (a) $E3$ compared to the case where LSTMs are replaced by 5/6-gram models and (b) $E1$ compared to $E4$ as both consider the same problem setting. In (a) it is clearly visible that LSTMs outperform the $n\text{-gram}$ models, while (b) reflects the physical knowledge that the Pauli unitaries are a better choice than random gates. Note that the red (dark) crosses in (a) are almost covered by the red (dark) circles. As a reference, we show the score of the best DD sequence obtained from the known DD classes.

Figure 2

The 3- and 5/6-gram experiments without data reusage. Otherwise, the experiments are done in the same way as in Fig. 1.

Figure 3

Plot of the function (C1).

Figure 4

Outline of our algorithm: (a) demonstrates that if we can model the distribution correctly, then we will be able to sample from good solutions more efficiently [red (darker) points correspond to smaller $f(x,y)$] and (b) illustrates the idea of concatenating the step performed in (a) in order to achieve an exponential speed-up compared to random search.

Figure 5

Standard model of a recurrent neural network shown for three time steps.

Figure 6

Illustration of an RNN with three hidden layers.

Figure 7

Long short-term memory model illustrated in a schematic way. In addition to the diagram, the input gate $i$, the forget gate $f$, and the output gate $o$ all depend on the current input $x_t$ and previous state of the network $s_{t-1}$, as described in (D1).

Figure 8

Illustration of how we truncate the gradient computation for long sequences. Here we divide the sequences into two halves. As the first step, we compute the gradient of the error function $\mathcal{E}({\vec{x}}_1,{\vec{o}}_1)$ with respect to the parameters $U,V,W$, while ignoring the other half of the network. In the second step, we compute the gradient of $\mathcal{E}({\vec{x}}_2,{\vec{o}}_2)$, while treating the final state of the network $s_t$ of the first half as a constant. The final gradients are approximated by the sums of these two constituents. Thus, we are able to avoid the instability of computing gradients, but still capture the correlation between two halves, since we feed the final network state $s_t$ into the second half.