固定半径邻居搜索的量子算法
Quantum Algorithm for the Fixed-Radius Neighbor Search
AI总结：
本文提出了一种基于固定点Grover算法的量子算法，用于解决固定半径邻居搜索问题，实现了线性查询复杂度并优化了计算效率。
摘要：
邻居搜索是一个计算上具有挑战性的问题，通常既耗时又耗内存。此类算法的主要问题在于由于缓存缺失导致的执行时间过长。在本工作中，我们提出了一种基于固定点Grover算法变体的固定半径邻居搜索（FRANS）的量子算法。我们提出了一种高效的电路，用于以线性查询复杂度解决FRANS问题，查询复杂度与粒子数量$N$成线性关系。量子电路返回所有在固定半径内的邻居对及其距离，避免了缓存缺失带来的延迟。我们分析了电路的门和查询复杂度。我们的FRANS算法的查询复杂度为$\mathcal{O}(N/\sqrt{M})$，其中$M$是解的数量，达到了Grover算法的最优下界。我们提出了不同的oracle实现方式，这些实现必须根据数据库的精确结构进行选择。其中，我们提出了一种使用切比雪夫距离的实现，其深度为$\mathcal{O}(q_1)$，其中$2^{q_1}$是用于离散化空间维度的网格点数量。最先进的状态准备算法允许在深度和宽度之间进行权衡，其体积（深度×宽度）为$\mathcal{O}(N\log(N))$。这种不利的扩展性在结构化数据集的情况下可以降低到$\mathcal{O}(\text{poly}(\log N))$。我们提出了一种基于贝叶斯干涉的停止准则，并在$1D$模拟中测试了其有效性。最后，我们考虑了读出复杂性，并评估了模型对读出误差的鲁棒性，建议了一种无需纠错策略来检查结果的准确性。
英文摘要：
Neighbor search is a computationally demanding problem, usually both time- and memory-consuming. The main problem of this kind of algorithms is the long execution time due to cache misses. In this work, we propose a quantum algorithm for the Fixed RAdius Neighbor Search problem (FRANS) based on the fixed-point version of Grover's algorithm. We propose an efficient circuit for solving the FRANS with linear query complexity with the number of particles $N$. The quantum circuit returns the list of all the neighbors' pairs within the fixed radius, together with their distance, avoiding the slow down given by cache miss. We analyzed the gate and the query complexity of the circuit. Our FRANS algorithm presents a query complexity of $\mathcal{O}(N/\sqrt{M})$, where $M$ is the number of solutions, reaching the optimal lower bound of the Grover's algorithm. We propose different implementations of the oracle, which must be chosen depending on the precise structure of the database. Among these, we present an implementation using the Chebyshev distance with depth $\mathcal{O}(q_1)$, where $2^{q_1}$ is the number of grid points used to discretize a spatial dimension. State-of-the-art algorithms for state preparation allow for a trade-off between depth and width of the circuit, with a volume (depth$\times$ width) of $\mathcal{O}(N\log(N))$. This unfavorable scaling can be brought down to $\mathcal{O}(\text{poly}(\log N))$ in case of structured datasets. We proposed a stopping criterion based on Bayes interference and tested its validity on $1D$ simulations. Finally, we accounted for the readout complexity and assessed the resilience of the model to the readout error, suggesting an error correction-free strategy to check the accuracy of the results.
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