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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.

Because each term in the history-subspace propagation Hamiltonian is positive semidefinite, the uniform history vector has zero energy and is a ground state:Indeed every edge term annihilates the uniform coefficients. Conversely, a zero-energy vector must have equal coefficients across every edge of the connected path, so this computational history state is the unique ground state in $\mathcal{L}$. It coherently records every intermediate state of the quantum circuit.

The history-subspace propagation Hamiltonian has diagonal entries $1/2$ at the endpoints and $1$ in the interior, and nearest-neighbour entries $-1/2$. The spectrum of a path propagation Hamiltonian isTo see the indexing directly, set $q_k=\pi k/(T+1)$ and $f_t=\cos [q_k(t+1/2)]$. The interior difference equation gives $Ef=(1-\cos q_k)f$. The endpoint equations correspond to $f_{-1}=f_0$ and $f_{T+1}=f_T$, both satisfied by these $q_k$. The $T+1$ distinct eigenvalues and their nonzero eigenvectors exhaust the matrix.

The zero eigenvalue is simple, soThe inequality uses $\sin x\ge 2x/\pi$ on $[0,\pi /2]$ and $1-\cos q=2{\sin }^2(q/2)$.

The supplied path-matrix hint sums through $i=N$ while listing a basis ending at $N$. That last summand would introduce an extra vertex. The matrix derived in (a) has exactly $T$ edges and $T+1$ vertices, as used here; this also proves the needed spectrum without relying on the misindexed summation.

A unary quantum clock represents step $t$ of a $T$-gate circuit by $|1{⟩}^{\otimes t}|0{⟩}^{\otimes (T-t)}$. Clock strings are orthogonal, and adjacent legal strings differ at one qubit. Local patterns around that change permit a history-subspace propagation Hamiltonian with bounded locality.