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Multidimensional Quantum Walks, Recursion, and Quantum Divide & Conquer

Stacey Jeffery, Galina Pass

TL;DR

This work introduces subspace graphs as a rigorous framework to blend quantum subroutines with classical reasoning, enabling multidimensional quantum walks to be composed in a principled, recursive manner. By defining switch composition and Boolean formula composition, the authors construct time-efficient quantum divide & conquer algorithms, including a quadratic speedup for Savitch's directed st-connectivity algorithm. The approach yields constructive algorithms, specifies witness-based complexities, and provides detailed examples (OR/AND gadgets) that underpin the general composition theorems. The framework offers a scalable method to design quantum algorithms that preserve classical structure and culminates in concrete improvements for DSTCON with bounded space. Overall, the paper advances a modular, graph-based methodology to reason about and implement complex quantum-classical hybrids.

Abstract

We introduce an object called a \emph{subspace graph} that formalizes the technique of multidimensional quantum walks. Composing subspace graphs allows one to seamlessly combine quantum and classical reasoning, keeping a classical structure in mind, while abstracting quantum parts into subgraphs with simple boundaries as needed. As an example, we show how to combine a \emph{switching network} with arbitrary quantum subroutines, to compute a composed function. As another application, we give a time-efficient implementation of quantum Divide \& Conquer when the sub-problems are combined via a Boolean formula. We use this to quadratically speed up Savitch's algorithm for directed $st$-connectivity.

Multidimensional Quantum Walks, Recursion, and Quantum Divide & Conquer

TL;DR

This work introduces subspace graphs as a rigorous framework to blend quantum subroutines with classical reasoning, enabling multidimensional quantum walks to be composed in a principled, recursive manner. By defining switch composition and Boolean formula composition, the authors construct time-efficient quantum divide & conquer algorithms, including a quadratic speedup for Savitch's directed st-connectivity algorithm. The approach yields constructive algorithms, specifies witness-based complexities, and provides detailed examples (OR/AND gadgets) that underpin the general composition theorems. The framework offers a scalable method to design quantum algorithms that preserve classical structure and culminates in concrete improvements for DSTCON with bounded space. Overall, the paper advances a modular, graph-based methodology to reason about and implement complex quantum-classical hybrids.

Abstract

We introduce an object called a \emph{subspace graph} that formalizes the technique of multidimensional quantum walks. Composing subspace graphs allows one to seamlessly combine quantum and classical reasoning, keeping a classical structure in mind, while abstracting quantum parts into subgraphs with simple boundaries as needed. As an example, we show how to combine a \emph{switching network} with arbitrary quantum subroutines, to compute a composed function. As another application, we give a time-efficient implementation of quantum Divide \& Conquer when the sub-problems are combined via a Boolean formula. We use this to quadratically speed up Savitch's algorithm for directed -connectivity.
Paper Structure (41 sections, 39 theorems, 212 equations, 6 figures, 2 algorithms)

This paper contains 41 sections, 39 theorems, 212 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Subspace Graphs
  3. Time-Efficient Quantum Divide & Conquer
  4. Switching Networks
  5. Application to DSTCON
  6. Model of Computation
  7. Related and Future Work
  8. Preliminaries
  9. Model of Computation
  10. Graph Theory
  11. Function Composition and Boolean Formulas
  12. Phase Estimation Algorithms
  13. Subspace Graphs for Multidimensional Quantum Walks
  14. Subspace Graphs
  15. Phase Estimation Algorithm
  16. ...and 26 more sections

Key Result

Theorem 1.1

Let $\{f_{\ell,n}:D_{\ell,n}\rightarrow\{0,1\}\}_{\ell,n}$ be a family of functions. Let $\varphi$ be a symmetric Boolean formula on $a$ variables, and suppose $f_{\ell,n} = \varphi(f_{\ell/b,n},\dots,f_{\ell/b,n})\vee f_{aux,\ell,n}$, for some $b>1$ and some auxiliary function $f_{aux,\ell,n}$ with

Figures (6)

  • Figure 1: The graph $G_{\textsc{or}}$. The dashed lines represent dangling boundary "edges".
  • Figure 2: The graph $G_{\textsc{and}}$. The dashed lines represent dangling boundary "edges".
  • Figure 3: The graph $G$ for a reversible classical deterministic algorithm with five algorithm states. Between each set $V_t$ and $V_{t+1}$, there is exactly one edge connected to $s=v_{0,z_1}$. The dashed lines represent dangling boundary "edges".
  • Figure 4: The graph $G$. The dashed lines represent dangling boundary "edges".
  • Figure 5: Some perhaps complicated graphs $G_1$, $G_2$ and $G_3$, but we need only consider their boundaries, and can abstract their internal structure. To compose them, we identify points on their boundaries, yielding a new graph, $G^\circ$.
  • ...and 1 more figures

Theorems & Definitions (92)

  • Theorem 1.1: Informal
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: Parameters of a Phase Estimation Algorithm
  • Definition 2.6: Negative Witness
  • Definition 2.7: Positive Witness
  • Theorem 2.8: jeffery2022kDist
  • Definition 2.9: Working Basis Generation
  • ...and 82 more