Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 67 / 2 / a / Solution

Define the orthonormal history basiswhere ${\psi }_t=U_t\cdots U_1{\psi }_0$. Orthogonality follows from the distinct unary quantum clock strings, regardless of overlaps between the computational states.

The initialization term acts diagonally:Each propagation term only couples the two clock strings adjacent to its step. The gate and its adjoint giveand $H_t|e_j⟩=0$ for $j\notin \{t-1,t\}$. The same calculation covers the two endpoint terms.

Thus both Hamiltonians preserve $\mathcal{L}$, and in the history basis their restrictions areThis history-subspace propagation Hamiltonian is a path graph Laplacian with a factor $1/2$. The specified two-clock endpoint formulas assume $T\ge 2$; a shorter circuit can first be padded with identity gates.