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The maximum relaxation time of a random walk on regular graphs
Haoran Zhu
We establish sharp quadratic bounds on the relaxation time of simple random walk on connected regular graphs, with leading constants and for even and odd orders,…
Universal Plücker positivity and the Octopus inequality
Haoran Zhu
We prove a positivity theorem for the universal Plücker coordinates of Karp and Purbhoo when exactly one parameter is negative, with a sharp uniform threshold governed by the large…
Random walks on wreath products and spectral gaps for coloured interchange processes
Haoran Zhu
We introduce group-valued coloured interchange processes, a class of continuous-time random walks on wreath products generated by transpositions and arbitrary symmetric base-group…
iSWAP maximises the second-moment spectral gap in random quantum circuits
Yanying Liang, Haoran Zhu
We prove that the gate maximises the spectral gap of the Hermitian second-moment operator on every connected graph with at least three vertices, among all two-loca…
Spectral gaps of ironed two-qubit gadgets matching the iSWAP gap
Yanying Liang, Haoran Zhu
We prove that every ironed two-qubit gadget whose KAK-derived parameter satisfies has, on the complete graph with , the same second-moment spectral gap…
The Hurwitz sum-of-squares problem depends on the base field
Chi Zhang, Haoran Zhu
We show that the Hurwitz problem for sums of squares can depend on the base field. More precisely, we construct an explicit formula of type over every field of charact…