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Multidimensional Electrical Networks and their Application to Exponential Speedups for Graph Problems

Jianqiang Li, Sebastian Zur

TL;DR

This work introduces a multidimensional electrical network framework that integrates Alternative Kirchhoff's Law and Alternative Ohm's Law with multidimensional quantum walks to generate and sample quantum-derived electrical flows. By extending the incidence structure to an alternative incidence matrix, the authors establish existence and uniqueness results for s-t alternative flows and their potentials, enabling phase-estimation-based sampling of alt flow states. They apply this framework to one-dimensional random hierarchical graphs and welded-tree circuit graphs, deriving polynomial-time quantum sampling for marked-vertex search and exponential speedups for s-t path finding respectively. The results suggest a broad potential for quantum- and classical-graph algorithm design through alternative electrical-flow states and motivate further exploration in isogeny graphs and quantum linear-system applications. Overall, the paper presents a novel bridge between multidimensional quantum walks and electrical network theory with concrete exponential-speedup demonstrations for key graph problems.

Abstract

Recently, Apers and Piddock [TQC '23] strengthened the connection between quantum walks and electrical networks via Kirchhoff's Law and Ohm's Law. In this work, we develop a new multidimensional electrical network by defining Alternative Kirchhoff's Law and Alternative Ohm's Law based on the multidimensional quantum walk framework by Jeffery and Zur [STOC '23]. In analogy to the connection between the incidence matrix of a graph and Kirchhoff's Law and Ohm's Law in an electrical network, we rebuild the connection between the alternative incidence matrix and Alternative Kirchhoff's Law and Alternative Ohm's Law. This new framework enables generating an alternative electrical flow over the edges on graphs, which has the potential to be applied to a broader range of graph problems, benefiting both quantum and classical algorithm design. We first use this framework to generate quantum alternative electrical flow states and use it 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 exponential quantum-classical separation of the welded tree graph 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, this framework also allows us to demonstrate an exponential quantum speedup for the pathfinding problem in a type of regular graph, which we name the welded tree circuit graph. The exponential quantum advantage is obtained by efficiently generating quantum alternative electrical flow states and then sampling from them to find an s-t path in the welded tree circuit graph. By comparison, Li [arXiv '23] constructed a non-regular graph based on welded trees and used the degree information to achieve a similar speedup.

Multidimensional Electrical Networks and their Application to Exponential Speedups for Graph Problems

TL;DR

This work introduces a multidimensional electrical network framework that integrates Alternative Kirchhoff's Law and Alternative Ohm's Law with multidimensional quantum walks to generate and sample quantum-derived electrical flows. By extending the incidence structure to an alternative incidence matrix, the authors establish existence and uniqueness results for s-t alternative flows and their potentials, enabling phase-estimation-based sampling of alt flow states. They apply this framework to one-dimensional random hierarchical graphs and welded-tree circuit graphs, deriving polynomial-time quantum sampling for marked-vertex search and exponential speedups for s-t path finding respectively. The results suggest a broad potential for quantum- and classical-graph algorithm design through alternative electrical-flow states and motivate further exploration in isogeny graphs and quantum linear-system applications. Overall, the paper presents a novel bridge between multidimensional quantum walks and electrical network theory with concrete exponential-speedup demonstrations for key graph problems.

Abstract

Recently, Apers and Piddock [TQC '23] strengthened the connection between quantum walks and electrical networks via Kirchhoff's Law and Ohm's Law. In this work, we develop a new multidimensional electrical network by defining Alternative Kirchhoff's Law and Alternative Ohm's Law based on the multidimensional quantum walk framework by Jeffery and Zur [STOC '23]. In analogy to the connection between the incidence matrix of a graph and Kirchhoff's Law and Ohm's Law in an electrical network, we rebuild the connection between the alternative incidence matrix and Alternative Kirchhoff's Law and Alternative Ohm's Law. This new framework enables generating an alternative electrical flow over the edges on graphs, which has the potential to be applied to a broader range of graph problems, benefiting both quantum and classical algorithm design. We first use this framework to generate quantum alternative electrical flow states and use it 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 exponential quantum-classical separation of the welded tree graph 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, this framework also allows us to demonstrate an exponential quantum speedup for the pathfinding problem in a type of regular graph, which we name the welded tree circuit graph. The exponential quantum advantage is obtained by efficiently generating quantum alternative electrical flow states and then sampling from them to find an s-t path in the welded tree circuit graph. By comparison, Li [arXiv '23] constructed a non-regular graph based on welded trees and used the degree information to achieve a similar speedup.
Paper Structure (31 sections, 18 theorems, 137 equations, 13 figures, 1 table, 2 algorithms)

This paper contains 31 sections, 18 theorems, 137 equations, 13 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Multidimensional Electrical Network
  3. Multidimensional quantum walks and alternative electrical flows
  4. Alternative incidence matrix and alternative electrical flow
  5. Applications
  6. Finding a marked vertex in random hierarchical graphs
  7. Finding an $s$-$t$ path
  8. Future Outlook
  9. Organization:
  10. Preliminaries
  11. Graph theory and electrical networks
  12. The incidence matrix, Kirchhoff's Law and Ohm's Law
  13. Quantum walks and electrical flow
  14. Multidimensional electrical networks and the alternative incidence matrix
  15. Alternative neighbourhoods
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $\Psi_{\star}$ be a collection of alternative neighborhoods on a network $G = (V,E,{\sf w})$ and let $U_{{\cal A}{\cal B}^{\sf alt}}$ be the quantum walk operator with respect to $\Psi_{\star}$ as defined in eq:walk-alt. Then by performing phase estimation on the initial state ${\lvert}\psi_s^+\

Figures (13)

  • Figure 1: The Welded Tree Circuit Graph
  • Figure 2: The graph $G_1$ with corresponding edge directions where the blue vertices have an additional alternative neighbour as defined in \ref{['eq:G-star-alt']}. For each $(u,v) \in \overrightarrow{E}$, the weights ${\sf w}_{u,v}$ are denoted in black and the flow values $\theta^{\sf alt} _{u,v}$ in red for any valid unit $s$-$t$ alternative flow are parametrised by $x$ and $y = 1-x$.
  • Figure 3: Graph $G$ with its $s$-$t$ electrical flow $\theta$ and corresponding potential ${\sf p}$ at each vertex.
  • Figure 4: Graph $G$ where the blue vertex $x$ has an additional alternative neighbourhood. The $s$-$t$ alternative electrical flow $\theta^{\sf alt}$ with respect to this extra alternative neighbourhood is displayed, as well as the corresponding potential vector ${\sf p}^{\sf alt}$.
  • Figure 5: Graph $G$ where the blue vertex $x$ has an additional alternative neighbourhood ${\lvert}\psi_x^{\textrm{alt}}\rangle$. There is no unit flow from $s$ to $t$ satisfying Alternative Kirchhoff's Law possible in this graph.
  • ...and 8 more figures

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Network
  • Definition 2.2: Flow, Circulation
  • Definition 2.3: Potential
  • Definition 2.4: Electrical Network
  • Definition 2.5: Kirchhoff's Law
  • Definition 2.6: Ohm's Law
  • Definition 2.7: The edge-vertex incidence matrix
  • ...and 42 more