This paper is a sequel to the work of Bhattacharjya et al. (2024) [2] on quantum state transfer on blow-up graphs, where instead of the adjacency matrix, we take the Laplacian matrix as the time-independent Hamiltonian associated with a blow-up graph. We characterize Laplacian strong cospectrality, periodicity, perfect state transfer (LPST) and pretty good state transfer (LPGST) on blow-up graphs. We present several constructions of blow-up graphs with LPST and produce new infinite families of regular graphs where each vertex is involved in LPST. We also determine LPST and LPGST in blow-ups of classes of trees. Finally, if n ≡ 0 (mod 4), then the blow-up of n copies of a graph has no LPST, but we show that under certain conditions, the addition of an appropriate matching to this blow-up graph results in LPST.
This paper is a sequel to the work of Bhattacharjya et al. (2024) [2] on quantum state transfer on blow-up graphs, where instead of the adjacency matrix, we take the Laplacian matrix as the time-independent Hamiltonian associated with a blow-up graph. We characterize Laplacian strong cospectrality, periodicity, perfect state transfer (LPST) and pretty good state transfer (LPGST) on blow-up graphs. We present several constructions of blow-up graphs with LPST and produce new infinite families of regular graphs where each vertex is involved in LPST. We also determine LPST and LPGST in blow-ups of classes of trees. Finally, if n ≡ 0 (mod 4), then the blow-up of n copies of a graph has no LPST, but we show that under certain conditions, the addition of an appropriate matching to this blow-up graph results in LPST.
Abstract A blow-up of n copies of a graph G is the graph obtained by replacing every vertex of G by an independent set of size n, where the copies of two vertices in G are adjacent in the blow-up if and only if they are adjacent in G. In this work, we characterize strong cospectrality, periodicity, perfect state transfer (PST) and pretty good state transfer (PGST) in blow-up graphs. We prove that if a blow-up admits PST or PGST, then n = 2. In particular, if G has an invertible adjacency matrix, then each vertex in a blow of two copies of G pairs up with a unique vertex to exhibit strong cospectrality. Under mild conditions, we show that periodicity (resp., almost periodicity) of a vertex in G guarantees PST (resp. PGST) between the two copies of the vertex in the blow-up. This allows us to construct new families of graphs with PST from graphs that do not admit PST. We also characterize PST and PGST in the blow-ups of complete graphs, paths, cycles and cones. Finally, while trees in general do not admit PST, we provide infinite families of stars and subdivided stars whose blow-ups admit PST.
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.
Quantum spin systems carry information in the form of quantum states, and the propagation of these quantum states is described by a quantum walk. In order to construct an operational quantum computer, the task of reliably transmitting quantum states from one part of a quantum computer to another must be accomplished. Quantum walks are known to be powerful tools in achieving this task. Pure states are quantum states represented by unit complex vectors. In this thesis, we use algebraic and combinatorial techniques to develop the theory of perfect state transfer on pure states in quantum walks, with emphasis on the adjacency and Laplacian matrices as Hamiltonians of a graph representing a spin network. We prove basic results about eigenvalue supports, periodicity, and strong cospectrality on pure states. We investigate a special type of strong cospectrality called m-strong cospectrality, and show that m-strongly cospectral pure states admit desirable quantum state transfer properties. Several characterisations of perfect state transfer are also given, one for general pure states, one for m-strongly pure states, and another one for real pure states. These characterisations give rise to algebraic and analytic properties of pure states admitting perfect state transfer. Moreover, we determine all complete graphs, complete bipartite graphs, cycles, and paths that admit perfect state transfer between m-strongly pure states. Constructions of infinite families of graphs admitting perfect state transfer are also given. We also characterise perfect state transfer between vertex states in joins and blowup graphs. We determine when the join operation preserves or induces perfect state transfer in the resulting graph. We also use blow-up graphs to construct new families of regular graphs with perfect state transfer. We use these two graph operations to demonstrate that graphs with perfect state transfer can be constructed from graphs that do not exhibit such a property. Finally, we investigate perfect state transfer on s-pair states – pure states that represent entanglement between two vertices. We establish combinatorial and spectral properties of s-pair states with perfect state transfer, and characterise its existence in complete graphs, complete bipartite graphs, paths and distance-regular graphs. Further, we construct infinite families of graphs with s-pair state transfer.
The join X ∨ Y of two graphs X and Y is the graph obtained by joining each vertex of X to each vertex of Y . We explore the behaviour of a continuous quantum walk on a join graph with positive edge weights having the adjacency matrix or Laplacian matrix as its associated Hamiltonian, where the underlying graphs are assumed to be regular when dealing with the adjacency matrix. We characterize strong cospectrality, periodicity and perfect state transfer (PST) in a join graph. We also determine conditions in which strong cospectrality, periodicity and PST are preserved in the join. Under certain conditions, we show that there are graphs with no PST that exhibit PST when joined with another graph. We also show that | | U M ( X ∨ Y , t ) u , v | − | U M ( X , t ) u , v | | ≤ 2 | V ( X ) | for all vertices u and v of X , where U M ( X ∨ Y , t ) and U M ( X , t ) denote the transition matrices of X ∨ Y and X respectively relative to the adjacency or Laplacian matrix. We demonstrate that the bound 2 | V ( X ) | is tight for infinite families of graphs.
Abstract We establish the theory for pretty good state transfer in discrete-time quantum walks. For a class of walks, we show that pretty good state transfer is characterized by the spectrum of certain Hermitian adjacency matrix of the graph; more specifically, the vertices involved in pretty good state transfer must be strongly cospectral relative to this matrix, and the arccosines of its eigenvalues must satisfy some number theoretic conditions. Using normalized adjacency matrices, cyclic covers, and the theory on linear relations between geodetic angles, we construct several infinite families of walks that exhibits this phenomenon.
Let UX(t) be the transition matrix of a quantum walk on a graph X relative to its adjacency matrix A or the Laplacian matrix L. This paper investigates the behavior of quantum walks under Cartesian products, joins, and graph complements. We have two main goals. First, we characterize the conditions such that peak state transfer and pretty good state transfer are preserved under these operations, allowing us to construct new families of graphs admitting these properties. Our second goal is to analyze the relationship between the quantum walks on a graph and its complement. We provide bounds for fu,v(t)=UXc(t)u,v−eitδUX(−t)u,v and gu,v(t)=∣UX(t)u,v∣−∣UXc(t)u,v∣, where δ=−1 when dealing with A and δ=n otherwise. Note that fu,v(t) and gu,v(t) both measure the difference between the behavior of quantum state transfer between vertices u and v in a graph and its complement. If X is regular or M=L, then fu,v(t) is bounded above by ∣V(X)∣2. If X is non-regular and M=A, then we utilize the main eigenvalues of a graph to obtain an upper bound for fu,v(t) which depends only on A. We also use the bounding matrix of the graph to give bounds for the Nordhaus-Gaddum type relations ∣UX(t)u,v∣+∣UXc(t)u,v∣ and ∣UX(t)u,v∣⋅∣UXc(t)u,v∣. Finally, we demonstrate that most of our bounds are sharp for certain families of graphs.