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For an arbitrary vector $|\Psi ⟩={\sum }_{t=0}^T|{\eta }_t⟩|t⟩$, the preceding positive quadratic forms show that zero energy requiresThus all its clock components are generated by one correctly initialized input. Ifthen $\ker Q$ consists of $|\zeta ⟩\otimes |a⟩$, with unrestricted quantum witness $\zeta$. Normalization gives the computational history stateConversely, every such computational history state annihilates every propagation term and the input term, hence has zero energy. These states are exactly the ground space, not merely examples of its vectors. In particular, histories of computational basis quantum witnesses span the ground space; their coherent superpositions are also zero-energy states.

Another way to see the propagation constraint is to conjugate by $W={\sum }_t{V}_t\otimes |t⟩⟨t|$. Then $W^{†}{H}_{\mathrm{prop}}W=I\otimes E$, whereThe spectrum of a path propagation Hamiltonian has a unique uniform clock zero mode. This is the path with $T+1$ vertices. The upper summation index in the printed auxiliary formula must be $T-1$, rather than $T$, to agree with its stated dimension and spectrum; using $T$ would introduce an extra vertex.

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.