39 problems

Coutinho–Liu's conjecture. No tree with at least four vertices admits perfect state transfer in the adjacency matrix model.

Riemann hypothesis for $\Lambda_G^{QW}$. If $\rho\in\operatorname{Zero}(\Lambda_G^{QW})$, then $\operatorname{Re}(\rho)=1/2$. This is the analogue of the Riemann hypothesis for the…

Emms–Hancock–Severini–Wilson conjecture.

Let $X$ be a graph, and suppose that $X$ admits uniform mixing at time $t$. Mullin's conjecture. Then $e^{it}$ is a root of unity. This conjecture concerns an arithmetic constraint…

Consider the network of nodes arising from the three-player quantum walk, with board size $N$, and suppose that the nodes are partitioned into subnetworks according to connectivity…

Let $G$ be a graph, let ${{\mathrm{HL}}'_2}^r(G)$ denote its $r$-fold iterated lift, and let the relative entropy of coherence be measured for the corresponding lifted quantum syst…

Let a $p$-adic Schrödinger equation be obtained from a $p$-adic heat equation by a Wick rotation, and let a CTQW denote a continuous-time quantum walk on a graph. The p-adic quantu…

Let $n\geq 5$ be odd. Let $P_n$ be the path on $n$ vertices, let $u=1$ be an end vertex, and let $P_n'$ be obtained from $P_n$ by adding a vertex $v:=n+1$ such that $u$ and $v$ are…

Let $X$ be a simple connected unweighted graph on an odd number of vertices. Odd-order Laplacian PST conjecture. Laplacian perfect state transfer does not occur in $X$. This extend…

Let $P_n$ be the path on $n$ vertices, and let $\overset{2}{\uplus}~P_n$ denote its double blow-up, with corresponding vertices $(0,u)$ and $(1,u)$ for each vertex $u$ of $P_n$. Le…

Let $P_{-M}$ be the transition matrix restricted to the non-marked vertices, and let $\mathcal{D}_{-M}D_r^{1/2}=V\Lambda V^{-1}$, where…

Let $X$ be an oriented graph, meaning that its adjacency matrix is Hermitian with nonzero entries in $7i$; an oriented graph has universal perfect state transfer if perfect state t…

Let $n,m \geq 3$ with $(n,m) \neq (4,4)$. A toroidal $(n,m)$-grid is the toroidal grid with parameters $n$ and $m$, and periodic means periodic under the discrete-time quantum walk…

Let a directed network be represented by the Hamiltonian matrix used in the continuous-time quantum-walk ranking methods, and let its eigenvectors have localization properties. Loc…

Let $x$ and $t$ be integers satisfying … Let $a(x,t)$ be the complex amplitude under consideration. Simultaneous nonvanishing conjecture. Then … unless $(x,t)\in\{(-3,11),(5,11)\}$…

Normalized-Laplacian tree state-transfer conjecture. Apart from the paths on $2$ and $3$ vertices, no tree admits perfect state transfer according to the normalized Laplacian.

Let $\mu_n$ be the probability distribution of the quantum-walk-related random walk (QWRW), and let $\rho_n$ be the probability distribution of the quantum-state-related random wal…

Let $Z$ be the graph shown in Figure 6: a longer path $X$ joins the relevant subgraphs, with vertices $a,b,c,d$ labelled as in that figure. No-perfect-state-transfer conjecture. Pe…

No state transfer in trees conjecture. No tree on four or more vertices admits perfect state transfer.

Let a graph carry a Hamiltonian whose eigenstates include localized states, and consider the return probability of a quantum walk to its starting point. High-energy localized-state…

Consider spatial search in a $d$-dimensional lattice with $N$ sites, and let $t_2$ be the number of times the oracle is called. Patel–Raghunathan lower-bound conjecture. The search…

Let $b_1(x,t)$ denote the first component of the modified Feynman checker amplitude at position $x$ and time $t$. Modified Feynman checker limiting-probability conjecture. The tota…

Let $a_1(x,t,u)$ denote the first component of the Feynman checker amplitude at position $x$ and time $t$ in the external field $u$, and define $u(x+1/2,t+1/2)=-1$ if both $x$ and…

Nonanalyticity conjecture. These limiting “probabilities” are nonanalytic at

Godsil's integrality conjecture. If an abelian Cayley graph has uniform mixing, then the graph is integral; equivalently, all eigenvalues of its adjacency matrix are integers.