Hypercube continuous-time state transfer

Input: An integer $n\ge 0$, an ordered $n$-qubit register initialized to $e_{0^n}$, and a Hamiltonian-access contract for $A_n:={\sum }_{j=0}^{n-1}{X}_j,$ where $X=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ and $X_j:=I_2^{\otimes j}\otimes X\otimes I_2^{\otimes (n-j-1)}$ is its tensor-map extension acting on factor $j$ and as the identity on every other factor, by Tensor products of finitely many linear maps. The contract must explicitly supply either the whole operation $U_{A_n}(\pi \mathrm{/}2)$ or the $n$ local operations $U_{X_j}(\pi \mathrm{/}2)$ in the declared ordered tensor factors. The local-operation option requests each one-qubit evolution separately; its equivalence to the whole evolution is proved by the correctness argument and is not assumed here. This contract does not make arbitrary matrix exponentiation free. Every supplied query, gate, description-bit, and synthesis cost is recorded under Continuous-time Hamiltonian access cost model and Circuit-family and computation cost model.

Procedure: If the whole operation is supplied, apply $U_{A_n}(\pi \mathrm{/}2)$ once. If the local operations are supplied, apply $U_{X_j}(\pi \mathrm{/}2)$ once for each $j=0,\dots ,n-1$ in increasing order. Then perform the computational-basis projective measurement and return the observed $n$-bit word. For $n=0$, the sum is empty, the register is the one-dimensional empty tensor product, the evolution is the identity, and the output word is empty. For every $n$, the operation and readout costs come from the declared access and circuit models; the matrix formula alone supplies no implementation or cost.