Overshifted Parameter-Shift Rules: Optimizing Complex Quantum Systems with Few Measurements

TL;DR

The paper generalizes parameter-shift rules for gradient estimation in variational quantum algorithms to arbitrary gate generators and even infinite-dimensional spaces, by introducing overshifted rules that use more shifts than strictly necessary. It casts the gradient reconstruction as a convex optimization problem that minimizes the L1 norm of the shift coefficients to minimize measurement shots, and it analyzes both symmetric and continuous-limit forms, including a stochastic gradient interpretation. Analytic approximations (triangle, zig-zag, kernel interpolation) and numerical simulations demonstrate that overshifted rules can achieve low-variance gradient estimates with manageable experimental overhead, even for unknown or unbounded spectra. The approach is illustrated across photonic circuits, Gaussian states, many-body Hamiltonians, and Jaynes-Cummings dynamics, showing significant potential to expand the design space and performance of variational quantum algorithms on near-term and future hardware.

Abstract

Gradient-based optimization is a key ingredient of variational quantum algorithms, with applications ranging from quantum machine learning to quantum chemistry and simulation. The parameter-shift rule provides a hardware-friendly method for evaluating gradients of expectation values with respect to circuit parameters, but its applicability is limited to circuits whose gate generators have a particular spectral structure. In this work, we present a generalized framework that, with optimal minimum measurement overhead, extends parameter shift rules beyond this restrictive setting to encompass basically arbitrary gate generator, possibly made of complex multi-qubit interactions with unknown spectrum and, in some settings, even infinite dimensional systems such as those describing photonic devices or qubit-oscillator systems. Our generalization enables the use of more expressive quantum circuits in variational quantum optimization and enlarges its scope by harnessing all the available hardware degrees of freedom.

Figures (9)

• Figure 1: Example circuit with hardware parameter sharing. Three time-bin encoded photonic qubits (red pulses) enter into a beam splitter, with angle $\theta$. The delay line length is tuned according to the time separation between each photons and applies the same beam splitting operation to each neighbouring pair.
• Figure 2: (a) Norm of solutions of \ref{['eq:convex']} vs $P/N$ for different values of $N$. (b) Solutions of \ref{['eq:convex']} vs $\vartheta$ for $N=20$ and two values of $P$, $P=20$ (not overshifted) and $P=40$ (overshifted).
• Figure 3: Discrete Fourier Transform of the solutions of \ref{['eq:convex']} for $N=20$ and $P=40,160$. The real parts are always zero.
• Figure 4: (a) Norm of solutions of \ref{['eq:convex_smooth']} vs $P/N$ for different values of $N$. (b) Solutions of \ref{['eq:convex']} and \ref{['eq:convex_smooth']} vs $\vartheta$ for $N=20$ and $P=95$. For visual clarity, only the values of $c(\vartheta)$ with $|c| >0.05$ are displayed.
• Figure 5: Result of 10000 randomly-generated arbitrary shifts for $N=10$ with an equispaced spectrum. Equispaced shifts are $\{ (-1+\frac{1}{N_{\mathrm{Shifts}}}) \pi, (-1+\frac{3}{N_{\mathrm{Shifts}}}) \pi, \ldots, (1-\frac{1}{N_{\mathrm{Shifts}}}) \pi \}$, encompassing the wierichs2022general and pappalardo2025photonic shifts.