Effect of data encoding on the expressive power of variational quantum-machine-learning models

Quantum computers can be used for supervised learning by treating parametrized quantum circuits as models that map data inputs to predictions. While a lot of work has been done to investigate the practical implications of this approach, many important theoretical properties of these models remain unknown. Here, we investigate how the strategy with which data are encoded into the model influences the expressive power of parametrized quantum circuits as function approximators. We show that one can naturally write a quantum model as a partial Fourier series in the data, where the accessible frequencies are determined by the nature of the data-encoding gates in the circuit. By repeating simple data-encoding gates multiple times, quantum models can access increasingly rich frequency spectra. We show that there exist quantum models which can realize all possible sets of Fourier coefficients, and therefore, if the accessible frequency spectrum is asymptotically rich enough, such models are universal function approximators.

Figure 1

Illustration of the main result of this paper, shown for one-dimensional inputs $x\in \mathbb{R}$: Quantum models consisting of layers of trainable circuit blocks $W=W\left(\mathit{\theta }\right)$ and data-encoding circuit blocks $S\left(x\right)$ can be written as a weighed sum ${\sum }_{\omega }{c}_{\omega }{e}^{i\omega x}$. The data-encoding circuit determines the frequencies $\omega$, and the remainder of the circuit architecture determines the coefficients $c_{\omega }$. If the $\omega$ are integer valued (or integer-valued multiples of a base frequency ${\omega }_0$), the sum becomes a partial Fourier series, which allows us to systematically study properties of the function class a given quantum model can learn.

Figure 2

The general quantum model considered in this paper includes qubit-based circuits where the encoding subroutine consists of a single-qubit gate $\mathcal{G}\left(x\right)$, which is often used in practice. The picture illustrates two special cases investigated in Sec. 3: (a) shows a circuit where the scalar input feature $x$ is encoded by one single-qubit gate, which can be repeated $r=L>1$ times but always acts on the same qubit, and (b) repeats the encoding gate $r$ times in “parallel” using only one layer. Note that the trainable blocks $W$ (purple rectangles) represent arbitrary unitaries, which in practice would be implemented as a sequence of local gates (inset).

Figure 3

A parametrized quantum model is trained with data samples (white circles) to fit a target function $g\left(x\right)={\sum }_{n=-1}^1{c}_n{e}^{-nix}$ or $g^{\prime }\left(x\right)={\sum }_{n=-2}^2{c}_n{e}^{-nix}$ with coefficients $c_0=0.1$, $c_1=c_2=0.15-0.15i$. The variational circuit is of the form $f\left(x\right)=\left.〈0\right|U^{†}\left(x\right){\sigma }_{z}U\left(x\right)\left|0\right.〉$ where $\left|0\right.〉$ is a single qubit, and $U=W^{\left(2\right)}{R}_x\left(x\right)W^{\left(1\right)}$. The $W$ (round blue symbols) are implemented as general rotation gates parametrized by three learnable weights each, and $R_x$ (square blue symbols) is a single Pauli-X rotation. The left panels show the quantum model function $f\left(x\right)$ and target function $g\left(x\right),g^{\prime }\left(x\right)$, while the right panels show the mean-squared error between the data sampled from $g$ and $f$ during a typical training run. Feeding in the input $x$ as is (top row), the quantum model easily fits the target of degree 1. Rescaling the inputs $x\to 2x$ causes a frequency mismatch, and the model can no longer learn the target (middle row). However, even with the correct scaling, the variational circuit cannot fit the target function of degree 2 (bottom row). The experiments in this paper were all performed using the PennyLane software library [35].

Figure 4

Fitting a truncated Fourier series of degree 5, $g\left(x\right)={\sum }_{n=-5}^5{c}_n{e}^{2inx}$ with $c_n=0.05-0.05i$ for $n=1,\cdots ,5$ and $c_0=0$, using a quantum model that repeats the encoding $r=1,3,5$ times in sequence (left) and in parallel (right). Increasing $r$ allows for closer and closer fits until $r=5$ fits the data almost perfectly in both cases—illustrating that parallel and sequential repetitions of Pauli encodings extend the Fourier spectrum in the same manner. All models were trained with at most 200 steps of an Adam optimizer with learning rate 0.3 and batch size 25. For the “parallel” simulations, the $W$ are not arbitrary unitaries but implemented by a smaller ansatz of three layers of parametrized rotations as well as entangling controlled-not (cnot) gates, as per Ref. [29], which is depicted by the open rounded gate symbols. The quantum model still easily fitted the target function, which suggests that the results of this paper are of relevance for realistic quantum models.

Figure 5

Real and imaginary parts of the first six Fourier coefficients sampled from 100 randomly initialized $L=1$ quantum models. The models share the same encoding strategy of parallel Pauli-X rotations (square symbols) but vary in the ansatz and number of layers for the trainable unitaries $W$. Circuit A uses an ansatz of trainable arbitrary single-qubit rotations and layer-dependent entangling structure proposed in Ref. [29] and already used in Fig. 4, while circuit B uses trainable Pauli-X rotations with a simple entangling structure. The plots suggest that the “expressivity” of the trainable circuit block—here represented by increasing the number of times $l$ an ansatz is repeated—has little influence on the distribution of the Fourier coefficients, as opposed to the type of ansatz.

Figure 6

Multivariate $L=1$ quantum model considered for the universality theorem. Here, $S\left(\mathit{x}\right)$ consists of feature-encoding gates acting on different subsystems (green boxes). The Hamiltonians that generate these gates are defined to increase the “richness” of their spectrum with growing dimension of the subsystems (red arrows). Since we assume that the circuit depth and structure of the trainable unitaries $W^{\left(1\right)}$ and $W^{\left(2\right)}$ is sufficient to allow for the realization of arbitrary unitary operations, the trainable circuits grow in dimension along with the total system size.