arXiv subjects

Xingkun Song

Publications and source records attributed to Xingkun Song.

5 recordsLinked to original sources

arXiv2026

Spectral width and polynomial degree in perfect state transfer

We study the minimum time for perfect state transfer under polynomial Hamiltonians with bounded degree and spectral width. For a strongly cospectral pair and width bound $W$, the optimum, when finite, is an integer multiple of $π/W$, determined by integer interpolation with prescribed parities. For equally spaced supported eigenvalues with alternating signs, we give degree bounds under which every minimizer is affine, and sharp asymptotics for each fixed exact degree. Near-minimizing phase polynomials satisfy a quantitative Chebyshev stability estimate. We determine the optimal transfer time for every degree bound on hypercubes of odd prime dimension. For complementary vertices of $J(2m,m)$, the optimal time at fixed spectral width grows exponentially in $m$ throughout an interval of feasible degrees. We also construct polynomial Hamiltonians showing that every feasible degree $m-t$ with $t=o(m)$ admits subexponential transfer time.

quant-ph↗

arXiv2026

Perfect state transfer under matrix powers: parity and spectral arithmetic

For a real symmetric matrix $H$ and distinct vertices $a,b$, we classify exponents $k$ for which $H^k$ has perfect state transfer (PST) from $a$ to $b$. If their supported eigenvalues are integer multiples of a common positive number, every odd exponent reduces to $H$ and every positive even exponent reduces to $H^2$. We determine the minimum transfer times using a greatest common divisor of supported spectral differences. For rational symmetric matrices, symmetry of the source vertex support about zero implies the odd-power equivalence without a commensurability assumption; this includes all bipartite graphs. If the source vertex supports zero, PST under one positive even power implies PST under every positive even power. For a symmetric three-point quadratic spectrum whose outer projection signs agree and differ from the central sign, a nonzero rational shift leaves exactly one PST exponent. We classify all adjacency powers of hypercubes, cycles, and Johnson graphs, and all adjacency squares of paths. In particular, the adjacency matrix of $P_7$ has PST from vertex $2$ to vertex $6$ only at exponent $2$.

math.CO↗

arXiv2026

An Extremal Spectral Problem for Triangle-Free Graphs Arising from Quantum Transport

For a graph $G$ of order $n$ with adjacency matrix $A$, let $F_G(t)$ be the average of $|(\exp(-\ii tA))_{vu}|^2$ over distinct ordered vertex pairs. Under the dense scaling $t=τ/n$, the quantities $n^2F_G(τ/n)$ lead to a graphon functional $Φ_τ$ whose leading term is $τ^2$ times the edge density and whose remaining terms form a weighted alternating series of even cycle densities. For $0\leτ\leτ_{\mathrm c}$, we determine the exact maximum of $Φ_τ$ over all triangle-free graphons. The balanced complete bipartite graphon $B_1$ is the unique maximizer, up to weak isomorphism, when $0<τ\leτ_{\mathrm c}$, where $τ_{\mathrm c}$ is the unique positive solution of \[ τ_{\mathrm c}=4\sin(τ_{\mathrm c}/2), \qquad τ_{\mathrm c}\approx3.79099, \] and the maximum equals $4(1-\cos(τ/2))$. This threshold is sharp: $B_1$ is not globally optimal for $τ>τ_{\mathrm c}$. For $τ>τ_{\mathrm c}$, the unique maximizer within the bipartite class, up to weak isomorphism, is the balanced bipartite graphon $B_{q_τ}$, where $q_τ\in(0,1)$; the unrestricted maximization problem beyond $τ_{\mathrm c}$ remains open. We also prove an explicit edge density deficit bound and quantitative cut distance stability, uniform for $τ$ in compact subintervals of $(0,τ_{\mathrm c})$, together with qualitative cut distance stability on compact subintervals of $(0,τ_{\mathrm c}]$. The corresponding finite triangle-free extremal values converge locally uniformly to the graphon maximum, with an $O(n^{-1})$ error uniformly on $[0,τ_{\mathrm c}]$. The proof uses a coefficient criterion for spectral graphon functionals and combines a sixth-degree spectral minorant with a four-vertex inequality and a six-vertex moment inequality; the latter is established by an exact rational flag algebra certificate.

math.CO↗

arXiv2026

Perfect State Transfer on Oriented Circulant Graphs: A Complete Classification

The continuous-time quantum walk on an oriented circulant graph is determined by the Fourier eigenvalues of its Hermitian adjacency matrix. We classify perfect state transfer (PST) between distinct vertices in every nonempty oriented circulant graph. We show that each such graph is described by an odd primitive quadratic Dirichlet character of conductor $Δ$, a set of gcd-classes, and a choice between the two orientations of each selected class. For a graph of order $n$, we derive an explicit formula for every Fourier eigenvalue without assuming that $n/Δ$ is coprime to $Δ$. We prove that PST occurs only for $Δ\in\{3,4,8\}$ and give necessary and sufficient conditions on the connection set for each conductor. Equivalently, the square-free radicands of oriented circulant graphs with PST are exactly $1$, $2$, and $3$. More generally, when $λ_j=\sqrt{D}η_j$ with $η_j\in\mathbb{Z}$, congruences satisfied by the integers $η_j$ determine all PST pairs and times, the minimum period, and the largest vertex sets supporting multiple state transfer (MST). In this class, pretty good state transfer is equivalent to PST. We also determine the connected orders and enumerate the resulting graphs.

quant-ph↗

arXiv2026

Zero transfer on mixed graphs

In this paper, we investigate zero transfer on mixed graphs. Zero transfer is a quantum walk phenomenon in which the transition amplitude between two vertices is identically zero for all times, so that no quantum state transfer occurs between them. Using the Hermitian adjacency matrix, we derive necessary and sufficient conditions for zero transfer in mixed graphs. We then specialize these criteria to oriented circulant graphs, obtaining nonexistence results for prime order, structural restrictions for even order, and exhaustive computational classifications for small orders.

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