Mimicking the Probability Distribution of a Two-Dimensional Grover Walk with a Single-Qubit Coin

The nonlocalized case of the spatial density probability of the two-dimensional Grover walk can be obtained using only a two-dimensional coin space and a quantum walk in alternate directions. This significantly reduces the resources necessary for its feasible experimental realization. We present a formal proof of this correspondence and analyze the behavior of the coin-position entanglement as well as the $x\mathrm{\text{−}}y$ spatial entanglement in our scheme with respect to the Grover one. Our scheme allows us to entangle the two orthogonal directions of the walk more efficiently.

Figure 1

(a) Spatial probability distribution after $t=50$ steps of a two-dimensional Grover walk with the initial state of the coin as given in Eq. (4). Only the sites with even $x$ and $y$ are shown, as the probability is zero for all odd sites. (b) Sketch of the proposed scheme, in line with the experiment in Ref. 11. A two-dimensional spin-dependent optical lattice can alternately be shifted into the two orthogonal directions.

Figure 2

(a) Coin-position entanglement against the number of steps $t$ for the Grover walk (blue circles) with the initial conditions as in Eq. (10) and for our alternate walk (purple squares) with the initial conditions as in Eq. (12). (b) Entanglement between $x$ and $y$ position of the particle for the Grover walk and our alternate walk, with the same conditions as in panel (a).