Quantum walks have been extensively studied recently, mainly due to their
vast difference in behavior to classical random walks. This paper is concerned
with discrete time and space quantum walks of particles that propagate through
a one-dimensional line. This line can be either a lattice or a graph or any
other form of mathematical structure that can be viewed as a one-dimensional
line. First is defined a concrete way to describe the unitary evolution of a
quantum walk through a balanced coin operator and a shift operator. Then
follows the implementation of the quantum walk on an
8-cycle, i.e a cycle
graph with
8 nodes, which is then run locally as a simulation and on IBM's
quantum computer. The paper explores two implementations of the quantum walk as
a quantum circuit: the first one consists of generalised controlled inversions,
as introduced in \cite{EffWalk}, whereas the second one tries to replace them
with rotation operators around the basis states. The main aim is to find a way
around the caveat resulting from the large amount of ancilla qubits required to
carry out the computation in the case where only generalised inverters are
used. Next, another three experiments are computed, involving cycles with a
larger state space, more specifically
16,
32 and
64 possible positions.
In order to measure the magnitude of the error of the circuit we use the cross
entropy benchmarking method, calculated through the Hellinger distance.
Finally, a derivation of the variance of the quantum walk is provided along
with a calculation of the variance for our experiment.
Abstract We mainly study exceptional configuration for coined quantum walk search. For searching on a two-dimensional grid by AKR algorithm, we find some new classes of exceptional configurations that cannot be found by the AKR algorithm effectively and the known diagonal configuration can be regarded as its special case. Meanwhile, we give two modified quantum walk models that can improve the success probability in the exceptional configurations by numerical simulation. Furthermore, we introduce the concept of generalized exceptional configuration and consider search by quantum walk on a cycle with Grover coin. We find that the most common coin combination model (G, −), where G is a Grover diffusion transformation, is a generalized exceptional configuration when just searching one marked vertex on the cycle. In the end, we find generalized exceptional configuration has a different evolution of quantum coherence from exceptional configuration. These extend largely the range of exceptional configuration of quantum walk search in some sense.
The recent paper entitled “Generalized teleportation by means of discrete-time quantum walks on [Formula: see text]-lines and [Formula: see text]-cycles” by Yang et al. [Mod. Phys. Lett. B 33(6) (2019) 1950069] proposed the quantum teleportation by means of discrete-time quantum walks on [Formula: see text]-lines and [Formula: see text]-cycles. However, further investigation shows that the quantum walk over the one-dimensional infinite line can be based over the [Formula: see text]-cycles and cannot be based on [Formula: see text]-lines. The proofs of our claims on quantum walks based on finite lines are also provided in detail.