= Solution
Define the orthonormal history basis
$$
|e_t\rangle=|\psi_t\rangle\otimes|c_t\rangle,\qquad
|c_t\rangle=|1\rangle^{\otimes t}|0\rangle^{\otimes(T-t)},
\quad t=0,\ldots,T,
$$
where $\psi_t=U_t\cdots U_1\psi_0$. Orthogonality follows from the distinct <unary quantum clock> strings, regardless of overlaps between the computational states.
The initialization term acts diagonally:
$$
H_{\rm init}|e_0\rangle=0,\qquad
H_{\rm init}|e_t\rangle=|e_t\rangle\quad(t\geq1).
$$
Each propagation term only couples the two clock strings adjacent to its step. The gate and its adjoint give
$$
H_t|e_{t-1}\rangle=\frac12(|e_{t-1}\rangle-|e_t\rangle),\qquad
H_t|e_t\rangle=\frac12(|e_t\rangle-|e_{t-1}\rangle),
$$
and $H_t|e_j\rangle=0$ for $j\notin\{t-1,t\}$. The same calculation covers the two endpoint terms.
Thus \b[both Hamiltonians preserve $\mathcal L$], and in the history basis their restrictions are
$$
\boxed{D:=H_{\rm init}|_{\mathcal L}=I-|e_0\rangle\langle e_0|,
\qquad
E:=H_{\rm final}|_{\mathcal L}
=\frac12\sum_{t=1}^T
(|e_t\rangle-|e_{t-1}\rangle)
(\langle e_t|-\langle e_{t-1}|).}
$$
This <history-subspace propagation Hamiltonian> is a path <graph Laplacian> with a factor $1/2$. The specified two-clock endpoint formulas assume $T\geq2$; a shorter circuit can first be padded with identity gates.
Back to article page
