Let quantum neural networks choose their own frequencies

Parameterized quantum circuits as machine learning models are typically well described by their representation as a partial Fourier series of the input features, with frequencies uniquely determined by the feature map's generator Hamiltonians. Ordinarily, these data-encoding generators are chosen in advance, fixing the space of functions that can be represented. In this work we consider a generalization of quantum models to include a set of trainable parameters in the generator, leading to a trainable-frequency (TF) quantum model. We numerically demonstrate how TF models can learn generators with desirable properties for solving the task at hand, including nonregularly spaced frequencies in their spectra and flexible spectral richness. Finally, we showcase the real-world effectiveness of our approach, demonstrating an improved accuracy in solving the Navier-Stokes equations using a TF model with only a single parameter added to each encoding operation. Since TF models encompass conventional fixed-frequency models, they may offer a sensible default choice for variational quantum machine learning.

Figure 1

An overview of the concepts discussed in this paper. Top: we introduce a parameterized quantum circuit in which the generator of the data-encoding block is a function of trainable parameters ${\vec{\theta }}_F$ alongside the standard trainable variational ansatz. Bottom left: the output of such a model is a Fourier-like sum over different individual modes. Bottom right: in conventional quantum models, tuning the ansatz parameters ${\vec{\theta }}_A$ allows the coefficients of each mode to be changed. By using a trainable-frequency feature map (TFFM), tuning ${\vec{\theta }}_F$ leads to a quantum model in which the frequencies of each mode can also be trained.

Figure 2

Fitting cosine series of different frequencies using fixed-frequency (FF) and trainable-frequency (TF) QNNs. (a) Prediction after training on data with ${\mathrm{\Omega }}_d=\{1,2,3\}$. (b) Prediction after training on data with ${\mathrm{\Omega }}_d=\{1,1.2,3\}$. (c) Spectra of trained models in (b) as obtained using a discrete Fourier transform. The blue dashed lines indicate frequencies of the data.

Figure 3

Prediction MSE of simple, tower, and exponential FFFMs compared to a TFFM when training on multiple datasets with equally spaced frequencies $\overrightarrow{{\omega }_d}\in [1,3]$. For each box, the triangle and orange line denote the mean and median, respectively.

Figure 4

Circuit diagram of the TF QNN architecture used in experiments solving the Navier-Stokes equations. The TFFM (dashed box) contains the generator parameters ${\theta }_F$ that allow training of the underlying model frequencies. The TFFM is followed by $L=8$ ansatz layers, data reuploading, and then a final ansatz layer before the qubits are measured. Here, we study both digital and digital-analog versions of such layered abstraction.

Figure 5

Pressure field at $t=3.5$ of the wake flow of fluid passing a circular cylinder at $x=0$. Left: reference solution obtained with the finite-element method (FEM) in [38]. Cells with red borders indicate the training data accessible to the quantum models (see text). Middle: prediction of TF QNN using generators ${\widehat{G}}_{\theta }$. Right: prediction of FF QNN using generators $\widehat{G}$. The quantum circuits are based on a sliced digital-analog approach [28] suitable for platforms such as neutral atom quantum computers.

Figure 6

A single layer of the different types of hardware-efficient ansätze used in this work. All parameters $\theta$ are separate and independent.

Figure 7

Pressure field at $t=4.0$ of the wake flow of fluid passing a circular cylinder at $x=0$. Left: reference solution obtained by FEM. Middle: prediction of the overparameterized TF QNN model with $L=64$. Right: prediction of the equivalent FF QNN.