3 research outputs found
Exact simulation of coined quantum walks with the continuous-time model
The connection between coined and continuous-time quantum walk models has
been addressed in a number of papers. In most of those studies, the
continuous-time model is derived from coined quantum walks by employing
dimensional reduction and taking appropriate limits. In this work, we produce
the evolution of a coined quantum walk on a generic graph using a
continuous-time quantum walk on a larger graph. In addition to expanding the
underlying structure, we also have to switch on and off edges during the
continuous-time evolution to accommodate the alternation between the shift and
coin operators from the coined model. In one particular case, the connection is
very natural, and the continuous-time quantum walk that simulates the coined
quantum walk is driven by the graph Laplacian on the dynamically changing
expanded graph
Eigenbasis of the Evolution Operator of 2-Tessellable Quantum Walks
Staggered quantum walks on graphs are based on the concept of graph
tessellation and generalize some well-known discrete-time quantum walk models.
In this work, we address the class of 2-tessellable quantum walks with the goal
of obtaining an eigenbasis of the evolution operator. By interpreting the
evolution operator as a quantum Markov chain on an underlying multigraph, we
define the concept of quantum detailed balance, which helps to obtain the
eigenbasis. A subset of the eigenvectors is obtained from the eigenvectors of
the double discriminant matrix of the quantum Markov chain. To obtain the
remaining eigenvectors, we have to use the quantum detailed balance conditions.
If the quantum Markov chain has a quantum detailed balance, there is an
eigenvector for each fundamental cycle of the underlying multigraph. If the
quantum Markov chain does not have a quantum detailed balance, we have to use
two fundamental cycles linked by a path in order to find the remaining
eigenvectors. We exemplify the process of obtaining the eigenbasis of the
evolution operator using the kagome lattice (the line graph of the hexagonal
lattice), which has symmetry properties that help in the calculation process.Comment: 21 pages, 3 figure
Quantum Walks via Quantum Cellular Automata
Very much as its classical counterpart, quantum cellular automata are
expected to be a great tool for simulating complex quantum systems. Here we
introduce a partitioned model of quantum cellular automata and show how it can
simulate, with the same amount of resources (in terms of effective Hilbert
space dimension), various models of quantum walks. All the algorithms developed
within quantum walk models are thus directly inherited by the quantum cellular
automata. The latter, however, has its structure based on local interactions
between qubits, and as such it can be more suitable for present (and future)
experimental implementations.Comment: 10 pages, 3 figures. Comments are welcom