Extended parameter shift rules with minimal derivative variance for parameterized quantum circuits

TL;DR

This work introduces Extended Parameter Shift Rules (EPSR), a unifying framework that generalizes traditional parameter shift rules to arbitrary Hermitian generators in parameterized quantum circuits, enabling exact computation of derivatives of any order from shifted evaluations. EPSR leverages a trigonometric representation of the univariate cost function and derives a main theorem that expresses derivatives as linear combinations of function values at chosen shifts, with coefficients determined by inverting small real matrices; optimal shifts are characterized under two-shot schemes, with equidistant shifts globally optimal under a weighted-shot allocation. The authors rigorously compare EPSR to existing PSRs, prove robustness to non-equidistant spectra, and provide detailed derivations, variance minimization strategies, and practical guidance. Through numerical experiments on XXZ/HVA circuits using the Qiskit Aer simulator, they demonstrate that optimal shifts significantly reduce derivative variance and improve gradient estimates, highlighting EPSR’s potential to enhance quantum-classical optimization in PQCs. Overall, EPSR broadens PSR applicability, reduces measurement overhead, and offers a principled path to more reliable quantum-gradient-based optimization.

Abstract

Parameter shift rules (PSRs) are useful methods for computing arbitrary-order derivatives of the cost function in parameterized quantum circuits. The basic idea of PSRs is to evaluate the cost function at different parameter shifts, then use specific coefficients to combine them linearly to obtain the exact derivatives. In this work, we propose an extended parameter shift rule (EPSR) which generalizes a broad range of existing PSRs and has the following two advantages. First, EPSR offers an infinite number of possible parameter shifts, allowing the selection of the optimal parameter shifts to minimize the final derivative variance and thereby obtaining the more accurate derivative estimates with limited quantum resources. Second, EPSR extends the scope of the PSRs in the sense that EPSR can handle arbitrary Hermitian operator $H$ in gate $U(x) = \exp (iHx)$ in the parameterized quantum circuits, while existing PSRs are valid only for simple Hermitian generators $H$ such as simple Pauli words. Additionally, we show that the widely used ``general PSR'', introduced by Wierichs et al. (2022), is a special case of our EPSR, and we prove that it yields globally optimal shifts for minimizing the derivative variance under the weighted-shot scheme. Finally, through numerical simulations, we demonstrate the effectiveness of EPSR and show that the usage of the optimal parameter shifts indeed leads to more accurate derivative estimates.

Figures (6)

• Figure 1: Landscapes of $F_{\mathrm{wgt}}(\mathbf{x})$ for equidistant frequency set $\{1,2\}$ with $r=2$, across derivative orders $d=1$ to $6$. Panels (a), (c), and (e) show odd orders $d=1,3,5$ with $\mathbf{x} \in \mathbb{R}^2$, where the global minimizers are the equidistant PSR shifts $\left(\frac{\pi}{4}, \frac{3\pi}{4}\right)$. Panels (b), (d), and (f) show even orders $d=2,4,6$ with $\mathbf{x} \in \mathbb{R}^3$ (fixing $x_0 \equiv 0$), where the global minimizers are the equidistant PSR shifts $\left(0, \frac{\pi}{2}, \pi\right)$.
• Figure 2: Maximum error between the numerical global minimizer obtained by differential evolution (DE), $\mathbf{x}_{\mathrm{DE}}$, and the equidistant PSR nodes, $\mathbf{x}_{\mathrm{equi}}$, for combinations of sizes $r = 1$ to $8$ and orders $d = 1$ to $8$, categorized by parity. Each color block indicates $\max_i |(\mathbf{x}_{\mathrm{DE}})_i - (\mathbf{x}_{\mathrm{equi}})_i|$.
• Figure 3: The HVA quantum circuit for the XXZ model with $q = 5$ qubits and $p = 1$ layer.
• Figure 4: Error of EPSR derivatives compared to numdifftools derivatives for different orders in a noise-free ideal setting.
• Figure 5: Comparison of PDF distributions under different $r$ values and different shot allocation schemes (uniform vs. weighted) using equidistant PSR nodes.

Theorems & Definitions (25)

• Remark 1
• Theorem 1: Extended parameter shift rules (EPSR)
• Remark 2
• Theorem 2
• Lemma 1
• Remark 3
• Remark 4
• Lemma 2: Optimal shot allocation scheme
• proof
• Lemma 3: (Sub)gradients of $F_{\mathrm{unif}}$ and $F_{\mathrm{wgt}}$ for odd $d$-order derivatives