We present a classical algorithm that, for any $D$-dimensional geometrically-local, quantum circuit $C$ of polylogarithmic-depth, and any bit string $x \in {0,1}^n$, can compute the quantity $|<x|C|0^{\otimes n}>|^2$ to within any inverse-polynomial additive error in quasi-polynomial time, for any fixed dimension $D$. This is an extension of the result [CC21], which originally proved this result for $D = 3$. To see why this is interesting, note that, while the $D = 1$ case of this result follows from standard use of Matrix Product States, known for decades, the $D = 2$ case required novel and interesting techniques introduced in [BGM19]. Extending to the case $D = 3$ was even more laborious and required further new techniques introduced in [CC21]. Our work here shows that, while handling each new dimension has historically required a new insight, and fixed algorithmic primitive, based on known techniques for $D \leq 3$, we can now handle any fixed dimension $D > 3$. Our algorithm uses the Divide-and-Conquer framework of [CC21] to approximate the desired quantity via several instantiations of the same problem type, each involving $D$-dimensional circuits on about half the number of qubits as the original. This division step is then applied recursively, until the width of the recursively decomposed circuits in the $D^{th}$ dimension is so small that they can effectively be regarded as $(D-1)$-dimensional problems by absorbing the small width in the $D^{th}$ dimension into the qudit structure at the cost of a moderate increase in runtime. The main technical challenge lies in ensuring that the more involved portions of the recursive circuit decomposition and error analysis from [CC21] still hold in higher dimensions, which requires small modifications to the analysis in some places.
翻译:我们展示了一个经典算法, 对任何以美元维度为单位的地平面, 量电回回路 $C $C 美元, 和任何比特字符串 $x 美元= 美元= 0.1 元美元, 计算数量 $x ⁇ C ⁇ 0 ⁇ otimen n ⁇ 2美元, 计算在准极地平面的任何反球论性添加错误中, 对任何固定维度 $D 来说, 这是结果的延伸 [CC21], 最初证明了这个结果的直径=3美元。 要了解为什么这个结果更高, 注意这个结果的美元比值= 1 美元直径的直流情况, 数十年来, $D = 1 美元 美元 的直径 = 2 案件需要新的技术 。 将这个技术的元维度扩大到[CC21] 。 我们在这里的工作显示, 处理每个新维度的不断的直观, 和固定地平面 直径 d 分析, 直径D 以已知的平面 直径 dal 直方 直方 直方, 直方 直方 显示, 直方平面结构中, 直方 。