Efficient Gradient Estimation of Variational Quantum Circuits with Lie Algebraic Symmetries

TL;DR

This work develops a scalable framework for efficiently estimating gradients in variational quantum circuits by exploiting Lie algebraic symmetries and Pauli-string structures. It reformulates the gradient as a linear combination of Hadamard-test observables and leverages subgroup and DLA (dynamical Lie algebra) structures to achieve poly$(d)$-time algorithms with dramatically reduced measurement shots. By combining binary Pauli-string encodings, joint-measurability, and shadow tomography (including accelerated CST via a QCQC scheme), the approach yields $O(p)$ Hadamard tests and $O(\log p)$ shot scaling under mild assumptions, and extends to general DLAs with poly$(d)$ resources. The results offer a near-term practical route to efficient gradient-based optimization in quantum-classical hybrid systems, potentially mitigating barren-plateau issues and reducing experimental overhead.

Abstract

Hybrid quantum-classical optimization and learning strategies are among the most promising approaches to harnessing quantum information or gaining a quantum advantage over classical methods. However, efficient estimation of the gradient of the objective function in such models remains a challenge due to several factors including the exponential dimensionality of the Hilbert spaces, and information loss of quantum measurements. In this work, we developed an efficient framework that makes the Hadamard test efficiently applicable to gradient estimation for a broad range of quantum systems, an advance that had been wanting from the outset. Under certain mild structural assumptions, the gradient is estimated with the measurement shots that scale logarithmically with the number of parameters and with polynomial classical and quantum time. This is an exponential reduction in the measurement cost and polynomial speed up in time compared to existing works. The structural assumptions are (1) the dimension of the dynamical Lie algebra is polynomial in the number of qubits, and (2) the observable has a bounded Hilbert-Schmidt norm.

Figures (3)

• Figure 1: Hadamard test with backpropagation for measuring the partial derivative of product ansätze with respect to a parameter $a_{\mathbf{s}_l}$ appearing at layer $l$. Here $U_{\leq l}$ corresponds to the first $l$ layers of the ansatz , and $U_{>l}$ to the remaining layers. in addition , $X$ is the X-gate , $H$ is the Hadamard gate , and $R_{\mathbf{s}_l}$ is the controlled rotation around Pauli $\sigma^{\mathbf{s}_l}$.
• Figure 2: This figure illustrates the concept behind the subgroup gradient estimation algorithm. The expression in Theorem \ref{['thm:subgroup']} points to an alternative representation of the problem. Instead of picturing the gradient as a function in the space of the parameters (the left picture) , we can view it as a vector in the landscape of Hadamard tests $D_\mathbf{s}$.
• Figure 3: Estimation error for gradient approximation as a function of the number of terms for various Hamiltonians with different numbers of qubits.

Theorems & Definitions (29)

• Theorem 1: abbreviated
• Theorem 2: abbreviated
• Theorem 3: abbreviated
• Example 1
• Corollary 4
• Example 2
• Lemma 5
• Lemma 6
• Theorem 7
• Example 3