Volatility time series modeling by single-qubit quantum circuit learning

We employ single-qubit quantum circuit learning (QCL) to model the dynamics of volatility time series. To assess its effectiveness, we generate synthetic data using the Rational GARCH model, which is specifically designed to capture volatility asymmetry. Our results show that QCL-based volatility predictions preserve the negative return-volatility correlation, a hallmark of asymmetric volatility dynamics. Moreover, analysis of the Hurst exponent and multifractal characteristics indicates that the predicted series, like the original synthetic data, exhibits anti-persistent behavior and retains its multifractal structure.

💡 Research Summary

This paper investigates the feasibility of modeling financial volatility time series using Quantum Circuit Learning (QCL), a hybrid classical‑quantum algorithm that employs a parameterized quantum circuit to approximate nonlinear functions. The authors focus on asymmetric volatility, a stylized fact in finance where negative returns tend to increase future volatility more than positive returns—a phenomenon commonly referred to as the leverage effect. To generate a controlled testbed, they employ the Rational GARCH (R‑GARCH) model, which extends the standard GARCH framework by incorporating an exponential term that captures asymmetry through a parameter γ. With parameters (α=0.11, β=0.85, ω=0.005, γ=0.1) they simulate 1,095 daily observations (approximately three years) of returns rₜ and conditional variances σ²ₜ.

The QCL architecture consists of a single‑qubit parameterized circuit. Input data are encoded via angle encoding: the return rₜ is mapped to rotation angles using arcsin(rₜ) for an RY gate and arccos(rₜ²) for an RZ gate, while the variance σ²ₜ is encoded with arcsin(2σ²ₜ−1) and arccos(σ⁴ₜ) respectively. After these rotations, a trainable unitary U(θ) =