通过李代数约束设计量子电路，有效缓解梯度消失问题。
Special-Unitary Parameterization for Trainable Variational Quantum Circuits
- 用特殊酉群子结构限制电路演化空间，降低参数复杂度。
- 梯度方差仅多项式抑制，收敛速度比传统电路快2-3倍。
- 适合近中期量子处理器，无需额外量子比特或数值偏差。
我们提出SUN-VQC，一种变分量子电路架构，其基本层为对称性受限李子群 $[1m\mathrm{SU}(2^{k}) [0m \subset \mathrm{SU}(2^{n})$（$k \ll n$）的单指数形式。将演化限制在该紧致子空间内，可使动力学李代数维度从 $\mathcal{O}(4^{n})$ 降至 $\mathcal{O}(4^{k})$，确保梯度方差仅多项式级抑制，避免硬件高效参数化电路中的荒原峡谷问题。通过广义参数偏移法获得精确且硬件兼容的梯度，无需辅助量子比特或有限差分偏差。在量子自动编码与分类任务上的数值实验表明，SUN-VQC 的梯度信号强度高一个数量级，收敛速度快2–3倍，最终保真度更高。结果表明，李代数工程为抗荒原峡谷的变分量子算法提供了原理性、可扩展的路径，兼容近中期量子处理器。
原文摘要 · Abstract (English)
We propose SUN-VQC, a variational-circuit architecture whose elementary layers are single exponentials of a symmetry-restricted Lie subgroup, $\mathrm{SU}(2^{k}) \subset \mathrm{SU}(2^{n})$ with $k \ll n$. Confining the evolution to this compact subspace reduces the dynamical Lie-algebra dimension from $\mathcal{O}(4^{n})$ to $\mathcal{O}(4^{k})$, ensuring only polynomial suppression of gradient variance and circumventing barren plateaus that plague hardware-efficient ansätze. Exact, hardware-compatible gradients are obtained using a generalized parameter-shift rule, avoiding ancillary qubits and finite-difference bias. Numerical experiments on quantum auto-encoding and classification show that SUN-VQCs sustain order-of-magnitude larger gradient signals, converge 2--3$\times$ faster, and reach higher final fidelities than depth-matched Pauli-rotation or hardware-efficient circuits. These results demonstrate that Lie-subalgebra engineering provides a principled, scalable route to barren-plateau-resilient VQAs compatible with near-term quantum processors.
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