Perfect state transfer in Laplacian quantum walk

Abstract

For a graph G and a related symmetric matrix M , the continuous-time quantum walk on G relative to M is defined as the unitary matrix U (t) = exp(−itM ), where t varies over the reals. Perfect state transfer occurs between vertices u and v at time τ if the (u, v)-entry of U (τ ) has unit magnitude. This paper studies quantum walks relative to graph Laplacians. Some main observations include the following closure properties for perfect state transfer:• If a n-vertex graph has perfect state transfer at time τ relative to the Laplacian, then so does its complement if nτ ∈ 2πZ. As a corollary, the double cone over any m-vertex graph has perfect state transfer relative to the Laplacian if and only if m ≡ 2 (mod 4). This was previously known for a double cone over a clique (S. Bose, A. Casaccino, S. Mancini, S. Severini, Int. J. Quant. Inf., 7:11, 2009).• If a graph G has perfect state transfer at time τ relative to the normalized Laplacian, then so does the weak product G × H if for any normalized Laplacian eigenvalues λ of G and µ of H, we have µ(λ − 1)τ ∈ 2πZ. As a corollary, a weak product of P 3 with an even clique or an odd cube has perfect state transfer relative to the normalized Laplacian. It was known earlier that a weak product of a circulant with odd integer eigenvalues and an even cube or a Cartesian power of P 3 has perfect state transfer relative to the adjacency matrix.As for negative results, no path with four vertices or more has antipodal perfect state transfer relative to the normalized Laplacian. This almost matches the state of affairs under the adjacency matrix (C. Godsil, Discrete Math., 312:1, 2011).

Perfect quantum state transfer using Hadamard diagonalizable graphs

“…It turns out that L 1 + L 2 has PST at time π/2 between the pairs (1, 3), (2, 4), (5, 7), (6,8). From the above collection of cases, we find for example that if w 1 ∈ Q and w 2 ∈ Q (or w 1 ∈ Q and w 2 ∈ Q), then G 1 ⊙ w 1 w 2 G 2 has the intriguing property that there is PGST between the pairs (1,8), (1,11), (1,14) (among others).…”

“…If w 1 is even, then (2) simplifies to λ (2) ℓ ≡ (1 − h pℓ hq ℓ ) mod 4, ℓ = 1, · · · , n, so for even w 1 , and odd w 2 and d 2 , G 2 has PST from p toq. If w 1 is odd, then (2) simplifies to λ (1) ℓ + λ (2) ℓ ≡ (1 − h pℓ hq ℓ ) mod 4, ℓ = 1, · · · , n, which shows that the integer-weighted graph with Laplacian L 1 + L 2 has PST from p toq.…”

Laplacian state transfer in coronas

“…Given a graph G and a symmetric matrix M associated with G, the continuous-time quantum walk on G relative to M is given by the unitary matrix U (t) := exp(−itM ). (1) This notion was introduced by Farhi and Gutmann [11] as a paradigm to design efficient quantum algorithms. Physically, this also represents the evolution of a quantum spin system.…”

“…There are several infinite families of graphs known to have perfect state transfer. This includes hypercubes [5], some families of distance-regular graphs [8], complete graphs with a missing edge [4], and some joins [1]. However, recently it has become clear that perfect state transfer is rare.…”

Perfect quantum state transfer in weighted paths with potentials (loops) using orthogonal polynomials

“…From now on, when we mention the eigenvalues as α r , we mean the ones ordered as in (6); and when we mention eigenvalues β r we mean the ones as in (9). If n is even then (8) and (9) yield the fact that β 1 − (−β 1 ) = 2β 1 = (2m 1 + 1), and therefore β 1 = (2m 1 + 1)/2. Using this, we find…”