Irreconcilable Difference Between Quantum Walks and Adiabatic Quantum Computing

TL;DR

This paper demonstrates that continuous-time quantum walks and adiabatic quantum computing are fundamentally incompatible methods, each requiring complex structures beyond standard unstructured search oracles to mimic each other's evolution.

Contribution

It establishes a fundamental difference between quantum walks and adiabatic quantum computing, showing they cannot be made to simulate each other without introducing stronger-than-oracle structures.

Findings

Quantum walks require a chiral, weighted, directed star graph structure.

Adiabatic evolution must interpolate between three fixed Hamiltonians, including a complex one.

The two methods compute via fundamentally irreconcilable means.

Abstract

Continuous-time quantum walks and adiabatic quantum evolution are two general techniques for quantum computing, both of which are described by Hamiltonians that govern their evolutions by Schr\"odinger's equation. In the former, the Hamiltonian is fixed, while in the latter, the Hamiltonian varies with time. As a result, their formulations of Grover's algorithm evolve differently through Hilbert space. We show that this difference is fundamental; they cannot be made to evolve along each other's path without introducing structure more powerful than the standard oracle for unstructured search. For an adiabatic quantum evolution to evolve like the quantum walk search algorithm, it must interpolate between three fixed Hamiltonians, one of which is complex and introduces structure that is stronger than the oracle for unstructured search. Conversely, for a quantum walk to evolve along the path of the adiabatic search algorithm, it must be a chiral quantum walk on a weighted, directed star graph with structure that is also stronger than the oracle for unstructured search. Thus the two techniques, although similar in being described by Hamiltonians that govern their evolution, compute by fundamentally irreconcilable means.