Unary quantum clock ($|c_t⟩=|1{⟩}^{\otimes t}|0{⟩}^{\otimes (T-t)}$)

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.

In the orthonormal history basis $|e_t⟩=|{\psi }_t⟩|c_t⟩$, circuit propagation restricts to $E=\frac{1}{2}{\sum }_{t=1}^T(|e_t⟩-|e_{t-1}⟩)(⟨e_t|-⟨e_{t-1}|)$. This is a Graph Laplacian on a path. Its uniform zero-energy vector is the computational history state, while initialization restricts to $D=I-|e_0⟩⟨e_0|$.

For $M(s)=(1-s)D+sE$ in the unary history subspace, the first Gershgorin disc lies in $[0,s]$ and every other disc has real part at least $1-s$. For $s<1/2$, the first disc is isolated and contains the unique ground state energy. Thus the spectral gap is at least $1-2s$, in particular at least $1/3$ for $s\le 1/3$.

For a path with $T+1$ vertices and edge weight $1/2$, the eigenvalues are $1-\cos [\pi k/(T+1)]$, $k=0,\dots ,T$. The corresponding coefficients are proportional to $\cos [\pi k(t+1/2)/(T+1)]$. The zero mode is uniform, and the spectral gap is at least $2/(T+1{)}^2$. The endpoint difference equations use reflecting ghost values, not periodic ones.