When designing quantum computers, we use undirected, potentially weighted graphs to represent networks of quantum processors. Such a graph is called “S-diagonalizable” if its discrete Laplacian is diagonalizable by some matrix P with all entries from S ⊆ R, and it is said to have an “S-bandwidth” of k if PTP has matrix bandwidth k. It has recently been shown that low {-1,1}- and {-1,0,1}-bandwidths are in many cases indicators of “perfect state transfer”—the reliable transmission of quantum bits, or qubits, between separate nodes. We herein present the first algorithm to determine the minimum {-1,1}- and {-1,0,1}-bandwidths of a given graph in the hopes of advancing the development of quantum information hardware. Particular emphasis is placed on testing for “weak Hadamard diagonalizability,” where a weak Hadamard matrix is some {-1,0,1}-matrix W such that WTW is tridiagonal. Finally, we set forth test results on select unweighted graphs up to order 14 and detail findings on edge weightings (which correspond to different voltages in a quantum circuit) that may induce {-1,1}- and {-1,0,1}-diagonalizability.
(Codebase in progress for Johnston, Plosker, & Varona 2024 summer research.)