Recently, Apers and Piddock [TQC '23] strengthened the natural connection between quantum walks and electrical networks by considering Kirchhoff's Law and Ohm's Law. In this work, we develop the multidimensional electrical network by defining Kirchhoff's Alternative Law and Ohm's Alternative Law based on the novel multidimensional quantum walk framework by Jeffery and Zur [STOC '23]. This multidimensional electrical network allows us to sample from the electrical flow obtained via a multidimensional quantum walk algorithm and achieve exponential quantum-classical separations for certain graph problems. We first use this framework to find a marked vertex in one-dimensional random hierarchical graphs as defined by Balasubramanian, Li, and Harrow [arXiv '23]. In this work, they generalised the well known exponential quantum-classical separation of the welded tree problem by Childs, Cleve, Deotto, Farhi, Gutmann, and Spielman [STOC '03] to random hierarchical graphs. Our result partially recovers their results with an arguably simpler analysis. Furthermore, by constructing a $3$-regular graph based on welded trees, this framework also allows us to show an exponential speedup for the pathfinding problem. This solves one of the open problems by Li [arXiv '23], where they construct a non-regular graph and use the degree information to achieve a similar speedup. In analogy to the connection between the (edge-vertex) incidence matrix of a graph and Kirchhoff's Law and Ohm's Law in an electrical network, we also rebuild the connection between the alternative incidence matrix and Kirchhoff's Alternative Law and Ohm's Alternative Law. By establishing this connection, we expect that the multidimensional electrical network could have more applications beyond quantum walks.

翻译：近期，Apers与Piddock [TQC '23] 通过引入基尔霍夫定律和欧姆定律，加强了量子游走与电路网络之间的自然联系。本文基于Jeffery与Zur [STOC '23] 提出的新型多维量子游走框架，通过定义基尔霍夫替代定律与欧姆替代定律，建立了多维电路网络。该多维电路网络使我们能够从多维量子游走算法获得的电流分布中采样，并在特定图问题上实现指数级量子-经典分离。首先，我们利用该框架在Balasubramanian、Li与Harrow [arXiv '23] 定义的一维随机分层图中寻找标记顶点。他们在此工作中将Childs、Cleve、Deotto、Farhi、Gutmann与Spielman [STOC '03] 提出的焊接树问题中著名的指数级量子-经典分离扩展到随机分层图。我们的结果以更简洁的分析部分复现了他们的结论。此外，通过构建基于焊接树的3-正则图，该框架还使我们能够证明寻径问题的指数加速。这解决了Li [arXiv '23] 提出的开放问题之一——他们通过构建非正则图并利用度数信息实现了类似的加速。类比于图的（边-顶点）关联矩阵与电路网络中基尔霍夫定律和欧姆定律之间的联系，我们重新建立了替代关联矩阵与基尔霍夫替代定律及欧姆替代定律之间的对应关系。通过建立这种联系，我们预期多维电路网络在量子游走之外将具有更广泛的应用前景。