稀疏随机哈密顿量在量子计算下是易解的
Sparse random Hamiltonians are quantumly easy
AI总结：
该研究针对量子模拟中低能试探态高效制备的理论难题，证明对两类随机稀疏哈密顿量，最大混合态可作为有效试探态，结合相位估计能高效制备近基态，且这类低能态经典计算非平凡、相位估计是最优制备方法。
摘要：
量子计算机的一个潜在应用是模拟量子系统的低温特性。针对这一任务，已有一种被广泛研究的量子算法：对与低能态存在不可忽略重叠的初始试探态执行量子相位估计。然而，要从理论上保证这类试探态可被高效制备，是出了名的困难。此外，目前可用的启发式方案（例如绝热泵浦制备方案）在实际场景中似乎并不够用。本文证明，对于大多数随机稀疏哈密顿量，最大混合态是足够好的试探态，相位估计可高效制备出能量任意接近基态能量的量子态。此外，任何低能态都必然具有不可忽略的量子线路复杂度，这表明低能态在经典计算下是非平凡的，且相位估计是制备这类态的最优方法（至多差多项式因子）。这些结论适用于两种随机哈密顿量模型：（i）随机带号泡利串之和模型，（ii）随机带号$d$-稀疏哈密顿量模型。核心技术论证基于非渐近随机矩阵理论的若干新结果，其中需要用到一种改进的谱密度集中界，来为这些随机哈密顿量提供复杂度保证。
英文摘要：
A candidate application for quantum computers is to simulate the low-temperature properties of quantum systems. For this task, there is a well-studied quantum algorithm that performs quantum phase estimation on an initial trial state that has a nonnegligible overlap with a low-energy state. However, it is notoriously hard to give theoretical guarantees that such a trial state can be prepared efficiently. Moreover, the heuristic proposals that are currently available, such as with adiabatic state preparation, appear insufficient in practical cases. This paper shows that, for most random sparse Hamiltonians, the maximally mixed state is a sufficiently good trial state, and phase estimation efficiently prepares states with energy arbitrarily close to the ground energy. Furthermore, any low-energy state must have nonnegligible quantum circuit complexity, suggesting that low-energy states are classically nontrivial and phase estimation is the optimal method for preparing such states (up to polynomial factors). These statements hold for two models of random Hamiltonians: (i) a sum of random signed Pauli strings and (ii) a random signed $d$-sparse Hamiltonian. The main technical argument is based on some new results in nonasymptotic random matrix theory. In particular, a refined concentration bound for the spectral density is required to obtain complexity guarantees for these random Hamiltonians.
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