Multidimensional Quantum Walks, with Application to $k$-Distinctness

TL;DR

The paper introduces a multidimensional quantum walk framework that extends Belovs’ electric network approach to handle multiple neighbourhoods and variable edge costs, enabling time bounds for $k$-distinctness that match the known query complexity up to polylog factors. By combining phase-estimation techniques with carefully engineered graph constructions and data structures, the authors achieve a time complexity of $ ilde{O}\left(n^{3/4-1/4\left(2^k-1\right)}\right)$ for constant $k\ge3$, advancing beyond prior time bounds. A key demonstrator is welded trees, where the framework yields an $O(n)$ query and $O(n^2)$ time exponential speedup over classical lower bounds, illustrating the framework’s potential for dramatic quantum speedups. The work also provides detailed analyses of positive and negative witnesses, transition subroutines, and data-management strategies within a fully quantum QRAM model, with broad implications for quantum query-to-time tradeoffs in search-type problems.

Abstract

While the quantum query complexity of $k$-distinctness is known to be $O\left(n^{3/4-1/4(2^k-1)}\right)$ for any constant $k \geq 4$, the best previous upper bound on the time complexity was $\widetilde{O}\left(n^{1-1/k}\right)$. We give a new upper bound of $\widetilde{O}\left(n^{3/4-1/4(2^k-1)}\right)$ on the time complexity, matching the query complexity up to polylogarithmic factors. In order to achieve this upper bound, we give a new technique for designing quantum walk search algorithms, which is an extension of the electric network framework. We also show how to solve the welded trees problem in $O(n)$ queries and $O(n^2)$ time using this new technique, showing that the new quantum walk framework can achieve exponential speedups.

Figures (7)

• Figure 4: A sample path from $V_0$ to $V_3$ in our first attempt at a quantum walk for 3-distinctness. The indices shown in blue can be seen to label the edges.
• Figure 5: The data we keep track of for a vertex $v_{R_1,R_2}$. $\diamond$ represents a queried index. $*$ represents an index whose query value is not stored. We only store the query value of an index in $R_2(s)$ if it collides with something in $R_1(\{s\})\cup R_1(\{1,2\})$, shown here by a solid line. If $i_2\in R_2(1)$ collides with some value in $R_1(\{2\})$, shown here by a dashed line, we do not record that, and do not store $x_{i_2}$.
• Figure 6: Example of a graph $G$ with $V_0,V_{\sf M} \subseteq V(G)$ and the induced graph $G'$ that is obtained from $G$ by adding a new vertex $v_0$. This new vertex is connected to all vertices in $V_0$ and only connected to those vertices in $V_{\sf M}$ which are marked (visualised by the red vertices).
• Figure 7: A graph showing the overlap of various sets of states, for an example graph $G$. With the exception of the spaces $\Psi_\star'(u)$ (which we will replace with orthonormal bases in \ref{['sec:fwk-unitary']}), each node represents an orthonormal set. There is an edge between two nodes if and only if the sets contain overlapping vectors.
• Figure 8: The weights of the graph $G'$ (obtained from adding $v_0$ to $G$), and default edge directions.

Theorems & Definitions (74)

• definition 2.1: Network
• definition 2.2: Flow, Circulation
• definition 2.3: Quantum Walk access to $G$
• definition 2.4: Networks with lengths
• definition 2.5: Quantum Subroutine
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8: Hypergeometric Tail Bounds Janson2011RandomGraphs
• corollary 2.9
• definition 2.10