量子电路切割的性能受限于纠缠几何，三者难以兼顾。
Entanglement geometry separates circuit cutting, classical hardness, and trainability
- 通过控制纠缠结构，构造出可低成本切割的电路
- 该电路需超多项式张量网络维数，难以经典模拟
- 用魔术门替代纠缠可避免训练与硬度冲突
电路切割有望突破现有硬件限制实现更大规模量子计算，但实现变分量子优势需同时满足低切割开销、经典难模拟性和可训练性。我们发现这些特性强烈受纠缠几何约束。在缝合区纠缠维数恒定的矩阵乘积态（MPS）和树状张量网络（TTN）电路中，采样开销为 $O(1/ heta^2)$，但可被经典高效模拟，排除了此类家族中的渐近量子优势。通过独立调控缝合区与块内纠缠，我们构建了一类双区块电路，在保持低切割成本的同时，全局MPS纠缠维数需超多项式增长（数值验证至 $n=100$）。然而，MPS的难模拟性要求深度 $d=ω( ext{log} n)$，而可训练性要求 $d=O( ext{log} n)$，二者矛盾。若以魔术门而非纠缠作为难模拟资源，则浅层Clifford+T电路仍可切割与训练，其稳定子模拟代价随T门数量指数增长。
原文摘要 · Abstract (English)
Circuit cutting promises to scale quantum computations beyond current hardware, but variational quantum advantage also requires low cutting overhead, classical hardness, and trainability. We show that these properties are strongly constrained by entanglement geometry. Matrix product state (MPS) and tree tensor network (TTN) circuits with constant seam bond dimension can be cut with \(O(1/\varepsilon^2)\) sampling overhead, but remain efficiently classically simulable, ruling out asymptotic quantum advantage within these families. By independently controlling seam and intra-block entanglement, we construct a two-block circuit family that remains cheaply cuttable while requiring a super-polynomial global MPS bond dimension, as supported numerically up to \(n=100\). However, MPS hardness and trainability require incompatible depth regimes, \(d=ω(\log n)\) and \(d=O(\log n)\), respectively. Using magic rather than entanglement as the hardness resource avoids this conflict: shallow Clifford+\(T\) circuits remain cuttable and trainable while their stabiliser-simulation cost grows exponentially with the \(T\)-count.
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