93 research outputs found
Solving Set Cover with Pairs Problem using Quantum Annealing
Here we consider using quantum annealing to solve Set Cover with Pairs (SCP), an NP-hard combinatorial optimization problem that plays an important role in networking, computational biology, and biochemistry. We show an explicit construction of Ising Hamiltonians whose ground states encode the solution of SCP instances. We numerically simulate the time-dependent Schrödinger equation in order to test the performance of quantum annealing for random instances and compare with that of simulated annealing. We also discuss explicit embedding strategies for realizing our Hamiltonian construction on the D-wave type restricted Ising Hamiltonian based on Chimera graphs. Our embedding on the Chimera graph preserves the structure of the original SCP instance and in particular, the embedding for general complete bipartite graphs and logical disjunctions may be of broader use than that the specific problem we deal with
Finding a marked node on any graph by continuous-time quantum walk
Spatial search by discrete-time quantum walk can find a marked node on any
ergodic, reversible Markov chain quadratically faster than its classical
counterpart, i.e.\ in a time that is in the square root of the hitting time of
. However, in the framework of continuous-time quantum walks, it was
previously unknown whether such general speed-up is possible. In fact, in this
framework, the widely used quantum algorithm by Childs and Goldstone fails to
achieve such a speedup. Furthermore, it is not clear how to apply this
algorithm for searching any Markov chain . In this article, we aim to
reconcile the apparent differences between the running times of spatial search
algorithms in these two frameworks. We first present a modified version of the
Childs and Goldstone algorithm which can search for a marked element for any
ergodic, reversible by performing a quantum walk on its edges. Although
this approach improves the algorithmic running time for several instances, it
cannot provide a generic quadratic speedup for any . Secondly, using the
framework of interpolated Markov chains, we provide a new spatial search
algorithm by continuous-time quantum walk which can find a marked node on any
in the square root of the classical hitting time. In the scenario where
multiple nodes are marked, the algorithmic running time scales as the square
root of a quantity known as the extended hitting time. Our results establish a
novel connection between discrete-time and continuous-time quantum walks and
can be used to develop a number of Markov chain-based quantum algorithms.Comment: This version deals only with new algorithms for spatial search by
continuous-time quantum walk (CTQW) on ergodic, reversible Markov chains.
Please see arXiv:2004.12686 for results on the necessary and sufficient
conditions for the optimality of the Childs and Goldstone algorithm for
spatial search by CTQ
- …