超越光锥：量子自旋玻璃中的态制备复杂性

中文摘要 AI 辅助

我们提出了一种研究稠密量子 $p$-自旋哈密顿量在 $n$ 个量子比特上的态制备复杂性的方法，超越了仅基于电路光锥的界限。关键输入是该类算符的有效轮廓复杂性，它由其泡利轮廓的度量熵导出。这些轮廓记录了所有恰好支持在 $p$ 个量子比特上的泡利算符的期望值。对于足够大的固定 $p$，具有一致有界二次有效轮廓复杂性的类与基态能量之间保持一个正的 $\n√n$ 倍数。在亚二次有效轮廓复杂性下，该类在前导阶上无法超越合适的基准类，其中乘积态提供了通用基准。证明结合了 Berta 等人（arXiv:1810.12197）的非对称量子 de Finetti 定理的改编版本与高斯过程熵界。应用此框架，我们表明达到近基态能量需要 $\nΩ(n^2/\nlog n)$ 个单量子比特和双量子比特门，即使有任意可丢弃的辅助量子比特。我们还获得了深度-宽度权衡、纠缠深度和矩阵乘积态键维下界，以及在 Parham 魔法层级（arXiv:2504.19966）的每个固定级别上对两个方向的阻碍，总电路宽度为 $O(n)$。在第一级反向魔法中，浅电路后跟一个无限制的 Clifford 电路。后者可以将局部可观测量扩散到整个系统，阻止直接应用小光锥界限。对于这个第一级类，我们的界限还允许在固定浅电路深度下任意多的干净辅助量子比特。一个更锐利的基准表明，具有 $o(n)$ 个 $T$ 门的 Clifford+$T$ 电路在前导阶上相对于乘积稳定子态没有能量优势，即使有无限制的 Clifford 操作和任意可丢弃的辅助量子比特。

英文摘要

We introduce a method for studying state preparation complexity in dense quantum $p$-spin Hamiltonians on $n$ qubits, going beyond bounds based only on circuit lightcones. The key input is the class's effective profile complexity, which is derived from the metric entropy of its Pauli profiles. These profiles record expectations of all Pauli operators supported on exactly $p$ qubits. Classes with uniformly bounded quadratic effective profile complexity remain separated from the ground-state energy by a positive multiple of $\sqrt n$ for sufficiently large fixed $p$. At subquadratic effective profile complexity, the class cannot outperform a suitable benchmark class at leading order, with product states providing a universal benchmark. The proof combines an adaptation of a nonsymmetric quantum de Finetti theorem of Berta et al. (arXiv:1810.12197) with Gaussian process entropy bounds. Applying this framework, we show that attaining near-ground-state energy requires $Ω(n^2/\log n)$ one- and two-qubit gates, even with arbitrary discardable ancillas. We also obtain depth-width tradeoffs, entanglement-depth and matrix product state bond-dimension lower bounds, and obstructions for both orientations at every fixed level of Parham's magic hierarchy (arXiv:2504.19966), with total circuit width $O(n)$. In first-level reverse magic, a shallow circuit is followed by an unrestricted Clifford circuit. The latter can spread local observables across the system, preventing a direct application of small-lightcone bounds. For this first-level class, our bounds also allow arbitrarily many clean ancillas at fixed shallow-circuit depth. A sharper benchmark shows that Clifford+$T$ circuits with $o(n)$ $T$-gates have no leading-order energy advantage over product stabilizer states, even with unrestricted Clifford operations and arbitrary discardable ancillas.

发表机构

• Massachusetts Institute of Technology（麻省理工学院）
• Bowdoin College（鲍登学院）

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86 pages