Approximation rates of quantum neural networks for periodic functions via Jackson's inequality

TL;DR

The paper develops a Jackson-inequality-based framework for approximating $2\pi$-periodic functions with quantum neural networks. By translating truncated Fourier (trigonometric) polynomials into QNN outputs, it achieves explicit univariate and multivariate approximation rates, with a quadratic parameter reduction for periodic targets and rates governed by function smoothness. The results quantify parameter and qubit requirements: univariate cases need $\mathcal{O}(\epsilon^{-1})$ parameters with one qubit, while $d$-variate cases with $K_j$ derivatives use $\mathcal{O}(\epsilon^{-(d+1)/k})$ parameters and $\mathcal{O}(d/k\log(d\epsilon^{-1}) + d\log k)$ ancilla qubits, where $k_j=K_j+1$. Numerical experiments on univariate functions and the heat equation solution corroborate the theoretical rates and illustrate how smoother functions admit faster convergence in practice.

Abstract

Quantum neural networks (QNNs) are an analog of classical neural networks in the world of quantum computing, which are represented by a unitary matrix with trainable parameters. Inspired by the universal approximation property of classical neural networks, ensuring that every continuous function can be arbitrarily well approximated uniformly on a compact set of a Euclidean space, some recent works have established analogous results for QNNs, ranging from single-qubit to multi-qubit QNNs, and even hybrid classical-quantum models. In this paper, we study the approximation capabilities of QNNs for periodic functions with respect to the supremum norm. We use the Jackson inequality to approximate a given function by implementing its approximating trigonometric polynomial via a suitable QNN. In particular, we see that by restricting to the class of periodic functions, one can achieve a quadratic reduction of the number of parameters, producing better approximation results than in the literature. Moreover, the smoother the function, the fewer parameters are needed to construct a QNN to approximate the function.

Figures (4)

• Figure 1: Approximating the continuous but non-differentiable function $f_1(x) := \vert \sin(x) \vert$ by a quantum neural network (QNN) $U^{2L}_{\theta,\phi}$ with $L := \lceil \frac{K+3}{2} \rceil \lfloor \frac{N}{2} \rfloor$. In (a), the approximation error $\Vert f_1 - f^{2L}_{\theta,\phi} \Vert_\infty$ is displayed against $N \in \lbrace 1,\dots,20 \rbrace$, for $K \in \lbrace 0,\dots,5 \rbrace$. In (b), the function $f_1$ and its QNN-based approximation $f^{2L}_{\theta,\phi}$ are shown, for $N = 20$ and $K \in \lbrace 0,\dots,5 \rbrace$.
• Figure 2: Approximating the twice but not three times differentiable function $f_{2.5}(x) := \vert \sin(x) \vert^{2.5}$ by a quantum neural network (QNN) $U^{2L}_{\theta,\phi}$ with $L := \lceil \frac{K+3}{2} \rceil \lfloor \frac{N}{2} \rfloor$. In (a), the approximation error $\Vert f_{2.5} - f^{2L}_{\theta,\phi} \Vert_\infty$ is displayed against $N \in \lbrace 1,\dots,20 \rbrace$, for $K \in \lbrace 0,\dots,5 \rbrace$. In (b), the function $f_{2.5}$ and its QNN-based approximation $f^{2L}_{\theta,\phi}$ are shown, for $N = 20$ and $K \in \lbrace 0,\dots,5 \rbrace$.
• Figure 3: Learning the solution of the heat equation $\mathbf{x} \mapsto u(0.5,\mathbf{x})$ in \ref{['eqheat']} by a quantum neural network (QNN) $U^{\mathbf{L}_{\mathbf{N},\mathbf{K}}}_{\boldsymbol\theta,\boldsymbol\phi}$, with $d = 2$ and $g(\mathbf{x}) := \prod_{j=1}^d g(x_j)$, $g_j$ defined in \ref{['eq:heat:init_cond']}. In (a), the approximation error $\Vert u(0.5,\mathbf{x}) - f^{\mathbf{L}_{\mathbf{N},\mathbf{K}}}_{\boldsymbol\theta,\boldsymbol\phi} \Vert_\infty$ is displayed against $N := N_1 = N_2 \in \lbrace 2,\dots,7 \rbrace$, for different $K := K_1 = K_2 \in \lbrace 0, 1, 2 \rbrace$. In (b), the function $\mathbf{x} \mapsto u(0.5,\mathbf{x})$ (wireframe) and its QNN-based approximation $\mathbf{x} \mapsto f^{\mathbf{L}_{\mathbf{N},\mathbf{K}}}_{\boldsymbol\theta,\boldsymbol\phi}(\mathbf{x})$ (colormap) are shown, for $N := N_1 = N_2 \in \lbrace 2,\dots,7 \rbrace$ and $K := K_1 = K_2 \in \lbrace 0, 1, 2 \rbrace$.
• Figure 4: Learning the solution of the heat equation $\mathbf{x} \mapsto u(1,\mathbf{x})$ in \ref{['eqheat']} by a quantum neural network (QNN) $U^{\mathbf{L}_{\mathbf{N},\mathbf{K}}}_{\boldsymbol\theta,\boldsymbol\phi}$, with $d = 2$ and $g(\mathbf{x}) := \prod_{j=1}^d g(x_j)$, $g_j$ defined in \ref{['eq:heat:init_cond']}. In (a), the approximation error $\Vert u(1,\mathbf{x}) - f^{\mathbf{L}_{\mathbf{N},\mathbf{K}}}_{\boldsymbol\theta,\boldsymbol\phi} \Vert_\infty$ is displayed against $N := N_1 = N_2 \in \lbrace 2,\dots,7 \rbrace$, for different $K := K_1 = K_2 \in \lbrace 0, 1, 2 \rbrace$. In (b), the function $\mathbf{x} \mapsto u(1,\mathbf{x})$ (wireframe) and its QNN-based approximation $\mathbf{x} \mapsto f^{\mathbf{L}_{\mathbf{N},\mathbf{K}}}_{\boldsymbol\theta,\boldsymbol\phi}(\mathbf{x})$ (colormap) are shown, for $N := N_1 = N_2 \in \lbrace 2,\dots,7 \rbrace$ and $K := K_1 = K_2 \in \lbrace 0, 1, 2 \rbrace$.

Theorems & Definitions (20)

• Remark 2.1: Ancilla qubits
• Theorem 3.1
• Remark 3.2: Complexity analysis
• Remark 3.3: General periodic functions
• Theorem 3.4
• Remark 3.5: Complexity analysis
• Remark 3.6: Comparison with previous works
• Lemma 5.1
• proof
• Proposition 5.2