通过层级辅助量子位控制子空间投影限制等变量子网络中的可训练李代数增长
Restricting Trainable Lie-Algebra Growth in Equivariant Quantum Networks via Hierarchical Ancilla-Controlled Subspace Projections
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中文总结 AI 辅助
本文提出层级辅助控制架构，通过限制等变量子网络的可训练李代数增长，实现多项式上界并改善初始化可训练性，实验验证了梯度优势。
中文摘要 AI 辅助
等变量子网络将对称性编码为归纳偏置，这可以改善泛化能力，并可能有利于优化收敛。然而，仅等变性并不能约束可训练生成器的非交换闭包，而这种闭包在保持对称性的变分电路中仍可能快速增长。我们引入了一种层级辅助量子位控制架构，以解决这种李代数增长机制。数据寄存器上的交换不变扇区投影仪选择共享辅助量子位寄存器上的参数化操作，非交换的可训练动力学被限制在该寄存器中。可训练电路分解为兼容的联合扇区，从而产生扇区概率加权的辅助响应，并清晰地展示跨层级路径的参数共享。对于辅助维度$d_A=2^m$和每层保留的$K_\ell$个控制模式，我们证明了与群无关的界$\dim(\mathfrak g)\le (d_A^2-1)\prod_{\ell=1}^{L}(K_\ell+1)$。当$m$和$K_\ell$在对数深度层级中保持常数时，该界关于数据量子位数是多项式的。粒子数和宇称投影仪说明了通用构造，而固定的Clebsch--Gordan耦合树提供了具体的$SU(2)$实现，具有旋转不变标量输出。有限尺寸态矢量模拟表明，在所研究的系统尺寸上，与通用和常规旋转等变电路相比，梯度方差衰减更慢，初始化梯度更大。相同的实现适用于稀疏旋转不变分类任务和几何相关的海森堡基态能量。这些结果支持将受限的可训练李代数增长作为所考虑体系中初始化可训练性的结构性策略。
英文摘要
Equivariant quantum networks encode symmetry as an inductive bias, which can improve generalization and may also favor optimization convergence. Equivariance alone, however, does not constrain the noncommuting closure of trainable generators, and this closure can still grow rapidly in symmetry-preserving variational circuits. We introduce a hierarchical ancilla-controlled architecture that addresses this Lie-algebra-growth mechanism. Commuting invariant-sector projectors on the data register select parameterized operations on a shared ancilla register, where the noncommuting trainable dynamics is confined. The trainable circuit decomposes into compatible joint sectors, giving a sector-probability-weighted ancilla response and an explicit view of parameter sharing across hierarchical paths. For an ancilla dimension $d_A=2^m$ and $K_\ell$ retained layer-wise control modes, we prove the group-independent bound $\dim(\mathfrak g)\le (d_A^2-1)\prod_{\ell=1}^{L}(K_\ell+1)$. The bound is polynomial in the number of data qubits when $m$ and $K_\ell$ remain constant along a logarithmic-depth hierarchy. Particle-number and parity projectors illustrate the general construction, while a fixed Clebsch--Gordan coupling tree supplies a concrete $SU(2)$ realization with rotation-invariant scalar outputs. Finite-size state-vector simulations exhibit slower gradient-variance decay and larger initialization gradients than generic and conventional rotationally equivariant circuits over the studied system sizes. The same realization fits sparse rotation-invariant classification tasks and geometry-dependent Heisenberg ground-state energies. These results support restricted trainable Lie-algebra growth as a structural strategy for initialization trainability in the regimes considered here.
发表机构
- Nanjing University of Posts and Telecommunications（南京邮电大学）
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