Irreconcilable difference between quantum walks and adiabatic quantum computing

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 Schrödinger'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.

Quantum search with hybrid adiabatic–quantum-walk algorithms and realistic noise

“…On the other hand, adiabatic quantum searching alters the Hamiltonian over time, turning on the term for the marked state slowly enough to keep the quantum system in its ground state throughout. On the face of it, these are quite different dynamics, as has been highlighted in [28]. However, both use the same Hamiltonians and initial states, and we argue here that both are best viewed as extreme cases of possible quantum annealing schedules.…”

Resonant transition-based quantum computation

“…To complete the analysis, now we would like to compare the underlying quantum dynamics of the two models and gain a different perspective on their connections. We follow closely a method of analysis presented by Wong and Myer [20] in which one can make AQC to emulate other unitary dynamics based quantum computation models. The idea is to project complicated many-qubit quantum dynamics into the dynamics of an effective qubit.…”

“…In Ref. [20], it was shown that for AQC to emulate the quantum walk search algorithm, it must interpolate between three fixed Hamiltonians (H B ,H E ,H C ). That implies in order to emulate the behavior of a quantum walk via the use of AQC, the corresponding Hamiltonian for AQC is structurally beyond a linear interpolation between the initial Hamiltonian,H B , and the final Hamiltonian,H C .…”

“…with eigenvalue λ ⊥ 0 . Similar to the analysis by Wong and Meyer, to avoid complex number, in [20] with eigenvalues λ 0 = −λ ⊥ 0 , the Hamiltonian is…”

“…Unlike the case presented in Ref. [20], the AQC emulates the RT dynamics via a non-linear and closed adiabatic path which starts with −H C at s = 0 and returns toH C at s = 1. Note the sign difference in the beginning and the end of the adiabatic path.…”