AI 中文总结
本文研究变分量子电路中离散CNOT路由对容量与局部可达性的影响，提出路由条件化插入以在保持学习电路的同时开启新下降方向，并通过数值实验验证其有效性。
AI 中文摘要
寻找有用的变分量子电路不仅需要表示能力：训练还必须能够获得改进目标的方向。我们研究了离散的CNOT路由如何塑造容量和局部可达性。对于由CNOT和任意单量子比特门构成的电路，我们将二进制路由矩阵与连续旋转分离，以将全局容量与局部切向可达性联系起来。一个关于CNOT块算子施密特秩的精确有限域公式和累积多块界限产生了容量约束和贝尔对保真度障碍。在局部，路由将生成元传播到依赖于路由的切空间，因此即使环境目标梯度非零，所有参数梯度也可能消失；框架界限区分了梯度覆盖的丧失与不良条件。一个路由条件化的恒等插入保留了当前酉算子、状态、损失和现有梯度，同时添加了新的切向方向。候选仅和增量投影分数为插入排序提供了局部标准。精确态矢量测试在受控贝尔任务上显示出强的短视界分数-下降相关性，并在多达十二个量子比特的连通海森堡链上实现了有效的等成本选择，并辅以互补的TFIM选择器控制。一个更广泛的TFIM路由集成表现出依赖于路由的近停滞。总之，这些结果通过一个共同的离散路由描述将表示容量、局部可达性和同点修复联系起来。架构在训练期间不必保持固定：任务相关的几何信息可以指导新下降方向的开启，同时在插入时保留已学习的电路。这为更广泛的问题提供了一种具体的局部方法：如何从任务提供的信息中发现有用的量子电路。
英文摘要
Finding useful variational quantum circuits requires more than representational capacity: training must also have access to directions that improve the objective. We study how discrete CNOT routing shapes both capacity and local accessibility. For circuits built from CNOT and arbitrary one-qubit gates, we separate binary routing matrices from continuous rotations to relate global capacity to local tangent accessibility. An exact finite-field formula for the operator Schmidt rank of a CNOT block and cumulative multiblock bounds yield capacity constraints and a Bell-pair fidelity obstruction. Locally, routing propagates generators into a routing-dependent tangent space, so all parameter gradients can vanish even when the ambient objective gradient is nonzero; frame bounds distinguish loss of gradient coverage from poor conditioning. A routing-conditioned identity insertion preserves the current unitary, state, loss, and existing gradients while adding new tangent directions. Candidate-only and incremental projection scores provide local criteria for ranking insertions. Exact-statevector tests show strong short-horizon score-descent correlations on controlled Bell tasks and effective equal-cost selection on connected Heisenberg chains up to twelve qubits, with complementary TFIM selector controls. A broader TFIM routing ensemble exhibits routing-dependent near-stagnation. Together, these results connect representational capacity, local accessibility, and same-point repair through a common discrete routing description. An architecture need not remain fixed during training: task-dependent geometric information can guide the opening of new descent directions while preserving the learned circuit at insertion. This provides a concrete local approach to a broader problem: how useful quantum circuits can be discovered from the information a task supplies.
Comments55 pages, 8 figures. Code and data: https://github.com/wang-yu-quantum/VQC-routing-geometry
