Systematic Dimensionality Reduction for Quantum Walks: Optimal Spatial Search and Transport on Non-Regular Graphs

TL;DR

This paper introduces a systematic dimensionality reduction method for continuous-time quantum walks using invariant subspace techniques, enabling optimal spatial search and improved quantum transport analysis on non-regular graphs.

Contribution

The authors develop a Lanczos-based invariant subspace method to simplify quantum walk analysis, revealing optimal search on non-regular graphs and providing bounds for quantum state transfer.

Findings

Optimal spatial search on non-regular graphs like star and bipartite graphs.

Simplified calculation of quantum transport efficiencies.

Improved quantum transport by removing links in symmetric graphs.

Abstract

Continuous time quantum walks provide an important framework for designing new algorithms and modelling quantum transport and state transfer problems. Often, the graph representing the structure of a problem contains certain symmetries that confine the dynamics to a smaller subspace of the full Hilbert space. In this work, we use invariant subspace methods, that can be computed systematically using Lanczos algorithm, to obtain the reduced set of states that encompass the dynamics of the problem at hand without the specific knowledge of underlying symmetries. First, we apply this method to obtain new instances of graphs where the spatial quantum search algorithm is optimal: complete graphs with broken links and complete bipartite graphs, in particular, the star graph. These examples show that regularity and high-connectivity are not needed to achieve optimal spatial search. We also show that this method considerably simplifies the calculation of quantum transport efficiencies. Furthermore, we observe improved efficiencies by removing a few links from highly symmetric graphs. Finally, we show that this reduction method also allows us to obtain an upper bound for the fidelity of a single qubit transfer on an XY spin network.