Scrambling ability of quantum neural network architectures

In this Letter, we propose a guiding principle for how to design the architecture of a quantum neural network in order to achieve a high learning efficiency. This principle is inspired by the equivalence between extracting information from the input state to the readout qubit and scrambling information from the readout qubit to input qubits. We characterize the quantum information scrambling by operator size growth. By Haar random averaging over operator sizes, we propose an averaged operator size to describe the information scrambling ability of a given quantum neural network architecture. The key conjecture of this Letter is that this quantity is positively correlated with the learning efficiency of this architecture. To support this conjecture, we consider several different architectures, and we also consider two typical learning tasks. One is a regression task of a quantum problem, and the other is a classification task on classical images. In both cases, we find that, for the architecture with a larger averaged operator size, the loss function decreases faster or the prediction accuracy increases faster as the training epoch increases, which means higher learning efficiency. Our results can be generalized to more complicated quantum versions of machine learning algorithms.

Figure 1

(a) The global structure of the QNN. Each ${\widehat{U}}_l$ is called a unit in this Letter, which is chosen among one of the building blocks shown in (b)–(f). (b)–(f) Various typical building blocks for constructing the QNN, which are respectively called the “brick wall”($B$), “chain”($C$), “lambda” ($\mathrm{\Lambda }$), “hyperbolic”($H$) and “supercube”($S$) in this Letter.

Figure 2

The Haar-random-averaged operator size $\overline{\mathrm{size}}$ defined in Eq. (7) for different architectures. For each architecture, all units share the same structure chosen as one of the cases shown in Figs. 1 and labeled by the same label introduced in Figs. 1. The horizontal axis is the number of units. Given the number of units, the numbers of two-qubit gates are the same for different cases under comparison [55]. Cross markers with different labels are obtained by numerical simulations of sampling ${10}^3$ different parameters, and the solid lines with empty circles are obtained by the analytical formula assuming infinite sampling. Here we have taken the number of qubits $N=8$.

Figure 3

Performance of different architectures for two different tasks. (a) and (b) Loss as a function of a training epoch for the information recovering task on the quantum spin problem. (c) and (d) Prediction accuracy as a function of training epoch for the second classification problem of red-green-blue (RGB) images of numbers. The number of units $N_{\mathrm{unit}}=4$ for (a) and (c) and $N_{\mathrm{unit}}=7$ for (b) and (d). In both cases, the results have been averaged over ten different initializations. The shaded area indicates the standard deviation of averaging over ten different initializations.