标题： 通过量子计数的Benincasa-Dowker因果集作用量 显示英文标题

标题： Benincasa-Dowker causal set actions by quantum counting

摘要： 因果集理论是一种量子引力的方法，其中时空在根本上是离散的，同时保留了局部洛伦兹不变性。 Benincasa--Dowker作用量是因果集理论中对应于爱因斯坦广义相对论中的爱因斯坦--希尔伯特作用量的等价物。 我们提出了一种运行时间为$\tilde{O}(n^{2})$的量子算法，用于计算任意时空维度下具有$n$个元素的因果集的 Benincasa--Dowker 作用量，该算法渐近最优，并且相比于所有已知的经典或量子算法提供了多项式加速。 为此，我们在计算基态的一个大小为$O(n^{2})$的任意子集上的均匀叠加态，该子集编码了感兴趣因果集的经典描述。 然后构建$\tilde{O}(n)$深度的oracle电路，用于测试因果集元素对之间不同的离散体积。 通过重复使用这些oracle执行量子计数的两阶段变体，得到所需的算法。 显示英文摘要

摘要： Causal set theory is an approach to quantum gravity in which spacetime is fundamentally discrete while retaining local Lorentz invariance. The Benincasa--Dowker action is the causal set equivalent to the Einstein--Hilbert action underpinning Einstein's general theory of relativity. We present a $\tilde{O}(n^{2})$ running-time quantum algorithm to compute the Benincasa--Dowker action in arbitrary spacetime dimensions for causal sets with $n$ elements which is asymptotically optimal and offers a polynomial speed up compared to all known classical or quantum algorithms. To do this, we prepare a uniform superposition over an $O(n^{2})$-size arbitrary subset of computational basis states encoding the classical description of a causal set of interest. We then construct $\tilde{O}(n)$-depth oracle circuits testing for different discrete volumes between pairs of causal set elements. Repeatedly performing a two-stage variant of quantum counting using these oracles yields the desired algorithm.