AI 中文总结
研究量子p自旋玻璃态的电路复杂度，证明关闭积态差距所需纠缠不能在浅深度产生，在平均相互作用度随n增长及有界平均度情况下给出电路深度下界，为随机量子自旋玻璃提供类似无低能平凡态问题的阻碍。
AI 中文摘要
量子信息理论中的一个核心问题是标准多体模型产生的态的电路复杂度。我们针对量子p自旋玻璃研究此问题，其为通过泡利串作用于p元量子比特组的随机哈密顿量。Anschuetz等人表明最优能量与积态可达到的最佳能量有差距。我们证明关闭积态差距所需的纠缠不能在浅深度产生。当平均相互作用度随n增长时，对于所有足够大的固定p，任何制备归一化能量在最优值固定正常数内的n量子比特态的电路深度必须为Ω_p(log n)。在有界平均度 regime中，我们证明了固定深度的阻碍。我们的结果为随机量子自旋玻璃给出了类似Freedman和Hastings的无低能平凡态问题的阻碍。
英文摘要
A central question in quantum information theory is the circuit complexity of states arising from standard many-body models. We study this question for quantum $p$-spin glasses, random Hamiltonians whose interactions act on $p$-tuples of qubits through Pauli strings. Anschuetz, Gamarnik, and Kiani (arXiv:2404.07231) showed that the optimum energy is separated from the best energy achievable by product states. This leaves open whether shallow circuits can close the gap, since even depth-one circuits can generate entanglement. We show that the entanglement needed to close the product-state gap cannot be generated at shallow depth. When the average interaction degree grows with $n$, we prove that, for all sufficiently large fixed $p$, any circuit preparing an $n$-qubit state whose normalized energy is within a fixed positive constant of the optimum must have depth $Ω_p(\log n)$. In the bounded-average-degree regime, we prove a fixed-depth obstruction: for every fixed $D$, a sufficiently large degree prefactor rules out depth-$D$ preparation of near-ground states. Both results hold uniformly over circuits with an arbitrary number of ancilla qubits. Our results give an obstruction in the spirit of the No Low-Energy Trivial States problem of Freedman and Hastings (arXiv:1301.1363), but for random quantum spin glasses rather than code-based Hamiltonians such as those of Anshu, Breuckmann, and Nirkhe (arXiv:2206.13228), whose ground states admit polynomial-size preparation circuits. This setting opens a probabilistic route to NLTS-like questions: we recast state-preparation lower bounds for random quantum Hamiltonians as uniform control of Gaussian processes indexed by shallow circuits.
Comments58 pages
