There are well developed theoretical tools to analyse how quantum dynamics can solve computational problems by varying Hamiltonian parameters slowly, near the adiabatic limit. On the other hand, there are relatively few tools to understand the opposi
te limit of rapid quenches, as used in quantum annealing and (in the limit of infinitely rapid quenches) in quantum walks. In this paper, we develop several tools which are applicable in the rapid quench regime. Firstly, we analyse the energy expectation value of different elements of the Hamiltonian. From this, we show that monotonic quenches, where the strength of the problem Hamiltonian is consistently increased relative to fluctuation (driver) terms, will yield a better result on average than random guessing. Secondly, we develop methods to determine whether dynamics will occur locally under rapid quench Hamiltonians, and identify cases where a rapid quench will lead to a substantially improved solution. In particular, we find that a technique we refer to as pre-annealing can significantly improve the performance of quantum walks. We also show how these tools can provide efficient heuristic estimates for Hamiltonian parameters, a key requirement for practical application of quantum annealing.
Quantum computation using continuous-time evolution under a natural hardware Hamiltonian is a promising near- and mid-term direction toward powerful quantum computing hardware. We investigate the performance of continuous-time quantum walks as a tool
for finding spin glass ground states, a problem that serves as a useful model for realistic optimization problems. By performing detailed numerics, we uncover significant ways in which solving spin glass problems differs from applying quantum walks to the search problem. Importantly, unlike for the search problem, parameters such as the hopping rate of the quantum walk do not need to be set precisely for the spin glass ground state problem. Heuristic values of the hopping rate determined from the energy scales in the problem Hamiltonian are sufficient for obtaining a better than square-root scaling. This makes it practical to use quantum walks for solving such problems, and opens the door for a range of applications on suitable quantum hardware.
The two degenerate ground states of the anisotropic Heisenberg (XY) spin model of a chain of qubits (pseudo-spins) can encode quantum information, but their degree of protection against local perturbations is known to be only partial. We examine the
properties of the system in the presence of non-local spin-spin interactions, possibly emerging from the quantum electrodynamics of the device. We find a phase distinct from the XY phase admitting two ground states which are highly protected against all local field perturbations, persisting across a range of parameters. In the context of the XY chain we discuss how the coupling between two ground states can be used to observe signatures of topological edge states in a small controlled chain of superconducting transmon qubits.
