Non-asymptotic Approximation Error Bounds of Parameterized Quantum Circuits

TL;DR

This paper establishes the first non-asymptotic approximation error bounds for these functions in terms of the number of qubits, quantum circuit depth, and number of trainable parameters and demonstrates that for approximating functions that satisfy specific smoothness criteria, the quantum circuit size and number of trainable parameters of the proposed PQCs can be smaller than those of deep ReLU neural networks.

Abstract

Parameterized quantum circuits (PQCs) have emerged as a promising approach for quantum neural networks. However, understanding their expressive power in accomplishing machine learning tasks remains a crucial question. This paper investigates the expressivity of PQCs for approximating general multivariate function classes. Unlike previous Universal Approximation Theorems for PQCs, which are either nonconstructive or rely on parameterized classical data processing, we explicitly construct data re-uploading PQCs for approximating multivariate polynomials and smooth functions. We establish the first non-asymptotic approximation error bounds for these functions in terms of the number of qubits, quantum circuit depth, and number of trainable parameters. Notably, we demonstrate that for approximating functions that satisfy specific smoothness criteria, the quantum circuit size and number of trainable parameters of our proposed PQCs can be smaller than those of deep ReLU neural networks. We further validate the approximation capability of PQCs through numerical experiments. Our results provide a theoretical foundation for designing practical PQCs and quantum neural networks for machine learning tasks that can be implemented on near-term quantum devices, paving the way for the advancement of quantum machine learning.

Figures (4)

• Figure 1: Overview of PQCs for approximating continuous functions. (a) Flowchart illustrating the strategy for using PQCs to approximate continuous functions via implementing Bernstein polynomials. The input data $x$ is encoded into the PQC through $S(x)$, with the PQC (blue background) capable of representing parity-constrained polynomials up to degree $3$ (as $x$ is encoded three times). The technique of linear combination of unitaries (LCU) is used to aggregate these polynomials together. The output of PQC derives from measurement with a specific observable. Fine-tuning trainable parameters in $R_Z$ gates yields a polynomial output depicted in the right panel. (b) Flowchart illustrating the strategy of approximation via local Taylor expansions. We first apply a PQC to localize the input domain into $K=5$ regions. For example, for input $x\in [0.8, 1]$, PQC outputs $x^{\prime}=0.8$ as a fixed point. Then $x-x^{\prime}$ will be fed into a new PQC for implementing the local Taylor expansions at the fixed point $x'$, forming a nesting architecture. Control gates with pink backgrounds implement the Taylor coefficients. Fine-tuning trainable parameters in $R_X$ and $R_Z$ gates yields a piecewise polynomial with degree $3$ that approximates the target function.
• Figure 2: An illustration of localization. The left panel demonstrates the localization $\bigcup_{\bm{\eta}} Q_{\bm{\eta}}$ for $K=5$ and $d=1$. The right panel shows the case of localization for $K=5$ and $d=2$. The "volume" of the trifling region $\Lambda(d, K, \Delta)$ is no more than $dK\Delta$.
• Figure 3: Simulation results of localization. We use single-qubit PQCs to approximate the localization function $D(x)$ for $K=2$ and $K=10$ respectively.
• Figure 4: Simulation results for learning $f(x, y)$. The left two panels are derived by interpolating and smoothing the output values of PQC on 100 test data points.

Theorems & Definitions (39)

• Theorem 1
• Theorem 2: The Universal Approximation Theorem of PQC
• Theorem 3
• Theorem 4
• Lemma S1: gilyen2019quantum
• Corollary S2: gilyen2019quantum
• Remark S1
• Lemma S3: yu2022power
• Corollary S4: yu2022powerwang2023quantum
• Lemma S5