Fourier-Parameterized Schedules

• Fourier-Parameterized Schedules are mathematical constructs that use truncated Fourier series to represent control parameters with periodic, oscillatory, and universal approximation properties.
• They enable efficient schedule evaluation, low-variance gradient estimates, and analytical tractability across applications like reinforcement learning, quantum computing, and time series forecasting.
• Optimizing a small set of Fourier coefficients provides discretization invariance and improved interpretability, balancing expressive power with robustness against overfitting.

Fourier-Parameterized Schedules are mathematical and algorithmic constructs in which the trajectory or schedule of a control parameter (such as a reinforcement learning policy update, quantum circuit parameterization, annealing path, or neural activation) is expressed explicitly in terms of Fourier basis functions—most commonly as a truncated sum of sinusoids or complex exponentials. This parameterization harnesses the analytical tractability of Fourier analysis, enabling efficient manipulation, optimization, and generalization of schedules that are periodic, cyclical, or exhibit frequency-specific structure. Fourier-parameterized schedules underpin advancements across reinforcement learning, quantum algorithms, time series forecasting, scientific computing, and deep neural architectures by conferring beneficial properties such as universal approximation, low-variance gradient estimates, discretization-invariance, and increased interpretability.

1. Mathematical Foundations and Parameterization

Fourier-parameterized schedules formalize the time evolution or path of a control parameter via an expansion in Fourier basis functions. Canonical forms include:

• Continuous-time functions: $$s(t) = a_0 + \sum_{n=1}^{M}\, [a_n \cos(n \omega t) + b_n \sin(n \omega t)]$$
• Discrete schedules: For $K$ steps, the schedule is often written $$s[i] = a_0 + \sum_{n=1}^{M}\, [a_n \cos(2 \pi n i/K) + b_n \sin(2 \pi n i/K)]$$

The coefficients $(a_n, b_n)$, or complex coefficients $c_k$ in $f(\theta) = \sum_k c_k e^{-i k \cdot \theta}$, encode the schedule’s amplitude and phase per frequency. The truncation index $M$ (number of harmonics) governs complexity. Schedules may be optimized directly over these coefficients, subject to physical or task constraints (e.g., $|s(t)| \in [0,1]$ for hardware limits).

Key frameworks leveraging this formalism include:

• Quantum annealing schedules: $$s(t) = (t/T) + \sum_{m=1}^{M}\, \theta_m \sin(m\pi t/T)$$ (Jeong et al., 17 Oct 2025).
• Policy schedules in RL: Convolutions of critic and policy expanded via $Q * \tilde{\beta}$ with analytic gradients in Fourier space (Fellows et al., 2018).
• Neural architectures: Schedules embedded as weighted sums of periodic basis functions for projection or embedding layers (Yang et al., 13 Jul 2025, Kong et al., 2 Aug 2025).

2. Analytical and Computational Advantages

Fourier parameterization yields tractable analytic and computational properties:

• Convolution as Multiplication: Integrals of expected values (e.g., $E_\beta[Q]$ in RL) become convolutions in the spatial domain, which are pointwise multiplications in the frequency domain: $$F(Q * \tilde{\beta})(\omega) = F(Q)(\omega) \cdot F(\tilde{\beta})(\omega)$$ (Fellows et al., 2018).
• Efficient Evaluation: Once Fourier coefficients are obtained, schedules can be evaluated at arbitrary times with $O(M)$ cost. Fast Partial Fourier Transform (PFT) algorithms further provide $O(N + M \log M)$ complexity for sparse coefficient computation (Park et al., 2020).
• Low-Variance Gradient Estimates: Analytical gradients can be derived, reducing variance relative to explicit Monte Carlo sampling. For instance, policy gradient updates are written as $$I_\theta(s) = \nabla_\theta F^{-1}[F(Q)(\omega) \cdot F(\tilde{\beta})(\omega)](\mu)$$ (Fellows et al., 2018).
• Universal Approximation: Trigonometric and radial basis functions offer dense approximations for continuous target schedules; the selection of harmonics can be tailored to encode prior knowledge or constraints (Fellows et al., 2018).

These properties establish the Fourier basis as a natural and powerful substrate for schedule parameterization whenever periodicity, oscillation, or spectral structure are salient.

3. Design and Optimization Strategies

Optimization of Fourier-parameterized schedules typically involves:

• Coefficient Optimization: The schedule is encoded by a small vector of coefficients (e.g., $\{\theta_m\}$), drastically reducing dimensionality and enabling tractable search (Jeong et al., 17 Oct 2025).
• Bayesian Surrogates: Gaussian process models, often with Matérn kernels and automatic relevance determination, are constructed over the schedule coefficient space to balance exploration and exploitation in noisy hardware evaluation settings. Acquisition functions such as expected improvement guide candidate schedule selection, with trust regions adapting focus and step size (Jeong et al., 17 Oct 2025).
• Orthogonal Expansion Approaches: For quantum circuit depth scheduling, iterative interpolation leverages the observed smoothness of optimal schedules to expand them in Chebyshev, Legendre, or Fourier bases, adjusting both circuit depth and basis size until convergence (Apte et al., 2 Apr 2025).

Table: Contrasting Schedule Optimization Methods

Method	Variables Optimized	Dimensionality Reduction	Typical Application
Direct param./full layer	2p (layers)	None	QAOA, RL
Fourier/Basis expansion	M (harmonics)	Yes	Annealing, QAOA
Bayesian surrogate	$\{\theta_m\}, T$	Yes	Quantum annealing
Iterative interpolation	$C$ (basis coeffs)	Yes/adaptive	QAOA depth scheduling

4. Practical Implementations and Domain-Specific Applications

Fourier-parameterized schedules are applied across several domains:

• Quantum Computing: Schedules govern the smooth transition between Hamiltonians in quantum annealing, improving fidelity and solution quality under hardware constraints (Jeong et al., 17 Oct 2025, Finžgar et al., 2023). In quantum circuit design, Fourier expansions are embedded as control gates to synthesize periodic functions (Li et al., 2021).
• Machine Learning and Neural Operators: Time-frequency basis expansions are used for time series forecasting (FBM, FBM-S) (Yang et al., 13 Jul 2025), as frequency-constrained embeddings in Transformers (Kong et al., 2 Aug 2025), and as operator parameterizations in FNO architectures, efficiently learning quantum Floquet dynamics (Qi et al., 8 Sep 2025). In convolution, parameterization in the Fourier basis enables equivariant filters for rotation and scale (Sun et al., 2023).
• Scientific Computing: The Taylor-Fourier method combines truncated Fourier and Taylor series, producing closed-form, uniformly-accurate approximations for highly oscillatory ODEs—and by extension, closed-form scheduled control (Calvo et al., 5 Jun 2024).
• Reinforcement Learning: Policy updates are analytically tractable via convolution/Fourier formalism, and gradient estimators unify canonical approaches (score-function, reparameterization, EPGs) (Fellows et al., 2018).

5. Tradeoffs, Limitations, and Mitigation of Pathologies

Fourier-parameterized schedules offer a tradeoff between expressive capacity and optimization efficiency:

• Expressiveness vs Overfitting: Increasing harmonic count ($M$) increases representational power but risks overfitting, especially in noisy or nonperiodic data (Fitzpatrick et al., 2021).
• Hardware Constraints: Schedules must be clipped and discretized to comply with device amplitude and time grids; Fourier coefficients are box-constrained for feasibility (Jeong et al., 17 Oct 2025).
• Spectral Suppression and Barren Plateaus: In parameterized quantum circuits, excessive expressibility (e.g., approaching a 2-design) suppresses all Fourier coefficients exponentially, causing barren plateaus ($\sum_k |c_k|^2 = 1/(2^n+1)$). Thus, maintaining significant harmonics is vital for trainability (Okumura et al., 2023).

A plausible implication is that controlling basis selection, regularizing harmonics, and employing explicit frequency-constrained learning schedules are important for mitigating such effects and ensuring robust optimization and generalization.

6. Interpretability and Extensions

Fourier parameterization inherently provides interpretability:

• Coefficients map directly to frequency content—allowing identification of dominant cycles, trends, or physical mechanisms (e.g., extracted driving frequencies in time series or quantum Hamiltonians) (Kong et al., 2 Aug 2025, Qi et al., 8 Sep 2025).
• Schedules are directly interpretable in the context of their physics (as in adiabatic preparation, STAs, or quantum simulation) (Finžgar et al., 2023).
• Explicit time-frequency mappings facilitate diagnostic analyses and debugging of learned schedules, with ability to recompose, extend, and transfer between domains and discretizations (Qi et al., 8 Sep 2025).

A plausible implication is that Fourier-parameterized schedules will continue to underpin advancements in interpretable scheduling and control in hybrid quantum-classical domains, reinforcement learning, and scientific computation.

7. Summary and Outlook

Fourier-parameterized schedules formalize control functions as finite sums of harmonics, yielding analytic tractability, universality, and computational efficiency. Their adoption in quantum computing, machine learning, and dynamical systems has enabled new capabilities in low-variance optimization, equivariant representation learning, scalable surrogate modeling, and interpretable forecasting. Key challenges remain in the balance between expressiveness and generalization, managing hardware constraints, and avoiding suppression pathologies such as barren plateaus. Current research is actively developing adaptive basis selection, Bayesian optimization frameworks, hybrid analytic-numeric estimation, and schedule transfer learning across domains, ensuring Fourier parameterization remains central to both practical and theoretical progress.