Draft note (v0.1.3). We give a connected simple (unweighted) graph on 12 vertices in which the continuous-time quantum walk exp(itA) has perfect state transfer from the pair state e_1 - e_11 to the plus state e_0 + e_2 at time pi/sqrt(2), with a proof that can be checked by hand, and an infinite family obtained by Cartesian products with powers of the path P3. This concerns the question of Monterde and Pal (arXiv:2505.07982v5, Section 10). The product step is a known construction (Godsil, Kirkland and Monterde, SIAM J. Matrix Anal. Appl. 46 (2025) 2093-2115, Corollary 48 and Example 50 in arXiv:2502.08103v4; see also Monterde and Pal, Section 9, and Kirkland, Monterde and Plosker, arXiv:2609.10713, equation (4.1)); we did not find an example of perfect state transfer from a pair state to a plus state in these papers, and no claim of priority is made. Exhaustive computer searches, with numerical tests at stated tolerances, found no example on at most 10 vertices (adjacency and signless Laplacian matrices); these are finite computations, not proofs. Code and outputs are included. Versions 0.1.2 and 0.1.3 correct attribution and citations only. This is a personal hobby/independent research record in which AI was used for exploration, drafting proof candidates, checking and writing; it has not been peer-reviewed by human experts.
Draft note (v0.1.3). We give a connected simple (unweighted) graph on 12 vertices in which the continuous-time quantum walk exp(itA) has perfect state transfer from the pair state e_1 - e_11 to the plus state e_0 + e_2 at time pi/sqrt(2), with a proof that can be checked by hand, and an infinite family obtained by Cartesian products with powers of the path P3. This concerns the question of Monterde and Pal (arXiv:2505.07982v5, Section 10). The product step is a known construction (Godsil, Kirkland and Monterde, SIAM J. Matrix Anal. Appl. 46 (2025) 2093-2115, Corollary 48 and Example 50 in arXiv:2502.08103v4; see also Monterde and Pal, Section 9, and Kirkland, Monterde and Plosker, arXiv:2609.10713, equation (4.1)); we did not find an example of perfect state transfer from a pair state to a plus state in these papers, and no claim of priority is made. Exhaustive computer searches, with numerical tests at stated tolerances, found no example on at most 10 vertices (adjacency and signless Laplacian matrices); these are finite computations, not proofs. Code and outputs are included. Versions 0.1.2 and 0.1.3 correct attribution and citations only. This is a personal hobby/independent research record in which AI was used for exploration, drafting proof candidates, checking and writing; it has not been peer-reviewed by human experts.
Given a graph with Hermitian adjacency matrix H, perfect state transfer occurs from vertex a to vertex b if the (b,a)-entry of the unitary matrix exp(−iHt) has unit magnitude for some time t. This phenomenon is relevant for information transmission in quantum spin networks and is known to be monogamous under real symmetric matrices. We prove the following results: 1. For oriented graphs (whose nonzero weights are ±i), the oriented 3-cycle and the oriented edge are the only graphs where perfect state transfer occurs between every pair of vertices. This settles a conjecture of Cameron et al. On the other hand, we construct an infinite family of oriented graphs with perfect state transfer between any pair of vertices on a subset of size four. 2. There are infinite families of Hermitian graphs with one-way perfect state transfer, where perfect state transfer occurs without periodicity. In contrast, perfect state transfer implies periodicity whenever the adjacency matrix has algebraic entries (as shown by Godsil). 3. There are infinite families with non-monogamous pretty good state transfer in rooted graph products. In particular, we generalize known results on double stars (due to Fan and Godsil) and on paths with loops (due to Kempton, Lippner and Yau). The latter extends the experimental observation of quantum transport (made by Zimborás et al.) and shows non-monogamous pretty good state transfer can occur amongst distant vertices.
Let X be a graph, A its adjacency matrix, and t a non-negative real number. The matrix exp(i t A) determines the evolution in time of a certain quantum system defined on the graph. It represents a continuous-time quantum walk in X. We say that X admits perfect state transfer from a vertex u to a vertex v if there is a time t such that \n| exp(i t A)_(u,v) | = 1. \n \nThe main problem we study in this thesis is that of determining which simple graphs admit perfect state transfer. For some classes of graphs the problem is solved. For example, a path on n vertices admits perfect state transfer if and only if n=2 or n=3. However, the general problem of determining all graphs that admit perfect state transfer is substantially hard. \n \nIn this thesis, we focus on some special cases. We provide necessary and sufficient conditions for a distance-regular graph to admit perfect state transfer. In particular, we provide a detailed account of which distance-regular graphs of diameter three do so. \n \nA graph is said to be spectrally extremal if the number of distinct eigenvalues is equal to the diameter plus one. Distance-regular graphs are examples of such graphs. We study perfect state transfer in spectrally extremal graphs and explore rich connections to the topic of orthogonal polynomials. We characterize perfect state transfer in such graphs. \n \nWe also provide a general framework in which perfect state transfer in graph products can be studied. We use this to determine when direct products and double covers of graphs admit perfect state transfer. As a consequence, we provide many new examples of simple graphs admitting perfect state transfer. We also provide some advances in the understanding of perfect state transfer in Cayley graphs for the groups (Z_2)^d and Z_n. \n \nFinally, we consider the problem of determining which trees admit perfect state transfer. We show more generally that, except for the path on two vertices, if a connected bipartite graph contains a unique perfect matching, then it cannot admit perfect state transfer. We also consider this problem in the context of another model of quantum walks determined by the matrix exp(i t L), where L is the Laplacian matrix of the graph. In particular, we show that no tree on an odd number of vertices admits perfect state transfer according to this model.
Let G be a graph and let t be a positive real number. Then the evolution of the continuous quantum walk defined on G is described by the transition matrix U(t)=exp(itH).The matrix H is called Hamiltonian. So far the most studied quantum walks are the ones whose Hamiltonians are the adjacency matrices of the underlying graphs and initial states are vertex states e_a, with e_a being the characteristic vector of vertex a. \n \nThis thesis focuses on Laplacian edge state transfer, that is, the quantum walks whose initial states are edge states e_a-e_b and Hamiltonians are the Laplacians of the underlying graphs. So far the research about perfect state transfer only involves vertex states and Laplacian edge state transfer has not been studied before. We extend the known results of perfect vertex state transfer and periodicity of vertex states to Laplacian edge state transfer. \n \nWe prove two useful closure properties for perfect Laplacian edge state transfer. One is that complementation preserves perfect edge state transfer. The other is that if G has perfect Laplacian edge state transfer at time τ and H has perfect Laplacian vertex state transfer also at time τ, then with some mild assumption on the pairs of vertex states and edge states that have perfect state transfer, the Cartesian product G □ H also admits perfect edge state transfer. Those two properties provide us new ways to construct graphs with perfect Laplacian edge state transfer. We also observe one phenomenon that happens in Laplacian edge state transfer which never happens in vertex state transfer: if there is perfect state transfer from e_a-e_b to e_α-e_β and also from e_b-e_c to e_β-e_γ at the same time t in G, then G admits perfect state transfer from e_a-e_c to e_α-e_γ at time t. \n \nWe give characterizations of perfect Laplacian edge state transfer in cycles, paths and complete bipartite graphs K_{2,4n}. We study perfect state transfer and periodicity on edge states with special spectral features. We also consider the case when the unsigned Laplacian is Hamiltonian and initial states are plus states of the form e_a+e_b. In this case, we characterize perfect state transfer in paths, cycles and bipartite graphs. \nWe close this thesis by a list of open questions.
A continuous-time quantum walk on a graph is represented by the complex matrix , where is the adjacency matrix of and is a non-negative time. If the graph models a network of interacting qubits, transfer of state among such qubits throughout time can be formalized as the action of the continuous-time quantum walk operator in the characteristic vectors of the vertices. Here, we are concerned with the problem of determining which graphs admit a perfect transfer of state. More specifically, we will study graphs whose adjacency matrix is a sum of tensor products of -matrices, focusing on the case where a graph is the tensor product of two other graphs. As a result, we will construct many new examples of perfect state transfer.
We prove new results on perfect state transfer of quantum walks on quotient graphs. Since a graph G has perfect state transfer if and only if its quotient G/\pi, under any equitable partition \pi, has perfect state transfer, we exhibit graphs with perfect state transfer between two vertices but which lack automorphism swapping them. This answers a question of Godsil (Discrete Mathematics 312(1):129-147, 2011). We also show that the Cartesian product of quotient graphs \Box_{k} G_{k}/\pi_{k} is isomorphic to the quotient graph \Box_{k} G_{k}/\pi, for some equitable partition \pi. This provides an algebraic description of a construction due to Feder (Physical Review Letters 97, 180502, 2006) which is based on many-boson quantum walk.
This record contains a short preliminary note (TeX source and PDF), Python code and its output, on the question posed in Section 11 of H. Monterde, H. Pal and S. Kirkland, "Laplacian quantum walks on blow-up graphs" (Linear Algebra Appl., 2025, doi:10.1016/j.laa.2025.10.001; numbering of arXiv:2504.11585v1): whether their Theorem 10 (Laplacian perfect state transfer after adding a matching inside the copies of a vertex in the blow-up of n copies of a graph, n divisible by 4) holds for the adjacency matrix. The same question is Problem 10 in H. Monterde's PhD thesis "Quantum pure state transfer" (University of Manitoba, 2025). For matchings whose edges join twin vertices of the blow-up and even n, the note proves that adjacency perfect state transfer between the ends a, b of an added edge forces the matching to cover every neighbour of a, and gives parity conditions on the uncovered vertices. Consequently a nonempty matching inside the copies of a single non-isolated vertex never gives adjacency perfect state transfer, for every graph and every even n. These results, and exact computations on small graphs supporting a conjecture that all sets of copies must be perfectly matched (finite computations only), are the parts claimed as new; a literature search in October 2026 found no earlier treatment of them, which does not establish that none exists. When every set of copies carries a perfect matching (a lexicographic product G[(n/2)K_2]), the note shows that for n divisible by 4 adjacency perfect state transfer between the ends of an added edge occurs if and only if the adjacency eigenvalue support of the vertex is integral, at time pi/2. This is not claimed as new: since G[(n/2)K_2] is isomorphic to Z[K_2], where Z is the blow-up of n/2 copies of G, it follows in a few lines from Proposition 6.3 of A. S. Árnadóttir and K. Guo (arXiv:2608.20476), and its sufficiency for integral graphs is a special case of a lemma of Ge, Greenberg, Perez and Tamon (arXiv:1009.1340). The criterion used for perfect state transfer between adjacent twins is a special case of a theorem of Kirkland, Monterde and Plosker (arXiv:2201.02720). When the matching lies inside the sets of copies, the graphs are edge-perturbed blow-ups in the sense of Monterde and Pal (arXiv:2505.07982v5, Section 9); their Theorem 20 concerns states orthogonal to the all-ones vector of a set of copies and does not cover vertex states. This is a personal hobby/independent research record in which AI was used for exploration, drafting proof candidates, checking and writing; it has not been peer-reviewed by human experts.