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

Area of Science:

• Quantum Computing
• Quantum Information Theory
• Algorithm Design

Background:

• Continuous time quantum walks are crucial for quantum algorithms and modeling quantum phenomena.
• Graph symmetries can restrict quantum dynamics to smaller Hilbert subspaces.
• Understanding these dynamics is key to optimizing quantum processes.

Purpose of the Study:

• To develop a method for identifying relevant subspaces in quantum walk dynamics.
• To find new graph structures where spatial quantum search is optimal.
• To simplify calculations and improve efficiencies in quantum transport and state transfer.

Main Methods:

• Utilizing invariant subspace methods, computed via the Lanczos algorithm.
• Applying the method to analyze quantum walks on various graph structures.
• Investigating complete graphs with broken links and complete bipartite graphs (e.g., star graphs).

Main Results:

• Identified novel graph instances (broken complete graphs, star graphs) for optimal spatial quantum search.
• Demonstrated that regularity and high connectivity are not prerequisites for optimal search.
• Significantly simplified quantum transport efficiency calculations.
• Observed enhanced transport efficiencies by modifying highly symmetric graphs.
• Established an upper bound for single qubit transfer fidelity on XY spin networks.

Conclusions:

• Invariant subspace methods offer a systematic way to reduce complexity in quantum walk dynamics.
• The findings challenge traditional assumptions about graph properties for optimal quantum search.
• This approach provides practical benefits for quantum transport and qubit transfer fidelity analysis.