CritPt / Challenge_61_main / We replace each of the M/2+1 vertices of an M/2-simplex with

# Problem setup:
We replace each of the M/2+1 vertices of an M/2-simplex with
a complete graph of M/2 vertices, resulting in a total of M/2(M/2+1) vertices.

The Hamiltonian is given by
\begin{equation}
H=-\gamma A-|a\rangle\langle a|,
\end{equation}
where γ is a tunable parameter, A is the adjacency matrix of this graph, and each vertex of the graph corresponds to a basis for this Hamiltonian. The state |a⟩ is the marked vertex on this graph. We choose the initial state |s⟩ to be the equal superposition of all vertices. We want to choose the proper value of γ, and let the system evolve for the appropriate duration of time such that, at the end of the process, the system is maximally concentrated at the state |a⟩.

# Main problem:

Suppose M=200. Find the evolution time T (rounded to the nearest integer) and the achievable probability P (to two decimal places) of getting basis |a⟩.

# Problem setup:
We replace each of the $M/2+1$ vertices of an $M/2$-simplex with
a complete graph of $M/2$ vertices, resulting in a total of $M/2(M/2+1)$ vertices.

The Hamiltonian is given by
\begin{equation}
H=-\gamma A-|a\rangle\langle a|,
\end{equation}
where $\gamma$ is a tunable parameter, $A$ is the adjacency matrix of this graph, and each vertex of the graph corresponds to a basis for this Hamiltonian. The state $|a\rangle$ is the marked vertex on this graph. We choose the initial state $|s\rangle$ to be the equal superposition of all vertices. We want to choose the proper value of $\gamma$, and let the system evolve for the appropriate duration of time such that, at the end of the process, the system is maximally concentrated at the state $\left|a\right\rangle$.

# Main problem:

Suppose $M=200$. Find the evolution time $T$ (rounded to the nearest integer) and the achievable probability $P$ (to two decimal places) of getting basis $\left|a\right\rangle$.