State transfer in discrete-time quantum walks via projected transition matrices

Abstract

In this paper, we analyze state transfer in quantum walks by using combinatorial methods. We generalize perfect state transfer in two-reflection discrete-time quantum walks to a notion that we call 'peak state transfer'; we define peak state transfer as the highest state transfer that can be achieved between an initial and a target state under unitary evolution, even when perfect state transfer is unattainable. We give a spectral characterization of peak state transfer that allows us to fully characterize peak state transfer in the arc-reversal (Grover) walk on various families of graphs, including strongly regular graphs and incidence graphs of block designs (assuming that the walk starts at a point of the design). In addition, we provide many examples of peak state transfer, including an infinite family where the amount of peak state transfer tends to $1$ as the number of vertices grows. We further demonstrate that peak state transfer properties extend to infinite families of graphs generated by vertex blow-ups, and we characterize periodicity in the vertex-face walk on toroidal grids. In our analysis, we make extensive use of the spectral decomposition of a matrix that is obtained by projecting the transition matrix down onto a subspace. Though we are motivated by a problem in quantum computing, we identify several open problems that are purely combinatorial, arising from the spectral conditions required for peak state transfer in discrete-time quantum walks.

Figures (11)

• Figure 1: The pictures in the top row show two quantum states; each state $\phi_i$ represents a vector indexed by a set of $96$ arcs. Its value is $0$ on every arc, except for the $4$ red-colored arcs, on which it has value $0.5$. The second row depicts the first four steps of a discrete-time quantum walk, where red (resp. blue) arcs represent positive (resp. negative) entries of the corresponding states. The level of opaqueness of each arc reflects the magnitude of the corresponding entry, making the arc invisible whenever the entry is equal to zero.
• Figure 2: The arc-reversal walk on this $7$-vertex graph is $12$-periodic at the vertex $u$ an not periodic at any other vertex. For the vertices $u$ and $v$, the value of $m(t,\gamma)$ is lower bounded by $2 - \sqrt{3} \approx 0.267949$, a bound that is reached for $t = 6$ and $\gamma = 1$. It turns out that this is an instance of peak state transfer, the definition of which is given in Section \ref{['sec:peakst']}.
• Figure 3: This embedding of the complete graph $K_4$ in the torus (genus $1$) has two faces $f_1$ and $f_2$. (The pairs of opposite sides of the square are identified to form the torus.) The corresponding clockwise walks in black and gray have lengths $4$ and $8$ respectively so that $d(f_1) = 4$ and $d(f_2) = 8$. Since $v$ appears on the clockwise walk of $f_2$ twice, so that the $(v,f_1)$-entry of $N^TM$ is $1/\sqrt{12}$ and the $(v,f_2)$-entry is $1/\sqrt{6}$.
• Figure 4: Pictured are the first few steps of three different two-reflection walks on the arcs of $K_4$; the arc-reversal walk (in row 1), the vertex-face walk on $K_4$ embedded in the plane (in row 2) and the vertex-face walk on $K_4$ embedded in the torus with a faces of size $4$ and $8$(in row 3). For each walk, we initialize the system at the out-going arcs at vertex $v$ and the picture depict the amplitudes at times $0,1,\ldots, 4$. Like in Figure \ref{['fig:grid_vxfwalk_peakST']}, red (resp. blue) arcs represent positive (resp. negative) entries of the corresponding states and the magnitudes are reflected by the opaqueness.
• Figure 5: The 'signed' arc-reversal walk has zero state transfer between the antipodal pairs of vertices, even though the underlying graph is connected.

Theorems & Definitions (45)

• Remark 2.1
• Lemma 2.2
• Lemma 2.3: ConJon1976, Theorem 7
• Lemma 3.1
• proof
• Remark 3.2
• Lemma 3.3
• Remark 3.4
• Theorem 3.5
• proof