分治 Van Vleck 灾难
Dividing and Conquering the Van Vleck Catastrophe
AI总结：
针对 Van Vleck 灾难导致的基态制备指数级困难问题，提出一种分治策略，通过反复使用相位估计合并子系统基态，将 N 个相互作用系统的基态制备查询复杂度降至准多项式级别。
摘要：
通过量子相位估计确定基态能量的量子计算成本，取决于易于制备的初始态与目标基态之间的重叠度。Van Vleck 正交性灾难常被援引，以表明量子计算机可能无法高效制备大型系统的基态，因为即使对于非相互作用系统，与初始态的重叠度也往往随系统规模呈指数级下降。我们证明这一直觉并不一定正确。具体而言，我们引入了一种分治策略，反复使用相位估计来合并逐渐增大的子系统的基态。我们为该方法提供了严格的边界，并表明如果每次合并的最小成功概率以一个常数为下界，那么制备 N 个相互作用系统的基态的查询复杂度为 O(N^{log log(N)} poly(N))，这是关于 N 的准多项式，与 Van Vleck 灾难预期的指数级缩放形成对比。我们还讨论了哈密顿量的充分条件，以确保准多项式运行时间。
英文摘要：
The quantum-computational cost of determining ground state energies through quantum phase estimation depends on the overlap between an easily preparable initial state and the targeted ground state. The Van Vleck orthogonality catastrophe has frequently been invoked to suggest that quantum computers may not be able to efficiently prepare ground states of large systems because the overlap with the initial state tends to decrease exponentially with the system size, even for non-interacting systems. We show that this intuition is not necessarily true. Specifically, we introduce a divide-and-conquer strategy that repeatedly uses phase estimation to merge ground states of increasingly larger subsystems. We provide rigorous bounds for this approach and show that if the minimum success probability of each merge is lower bounded by a constant, then the query complexity of preparing the ground state of $N$ interacting systems is in $O(N^{\log\log(N)} {\rm poly}(N))$, which is quasi-polynomial in $N$, in contrast to the exponential scaling anticipated by the Van Vleck catastrophe. We also discuss sufficient conditions on the Hamiltonian that ensure a quasi-polynomial running time.
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