Computational history state ($|\phi ⟩=(T+1{)}^{-1/2}{\sum }_t|{\psi }_t⟩|c_t⟩$)

A computational history state coherently records a circuit's intermediate states with orthogonal clock labels: $(T+1{)}^{-1/2}{\sum }_{t=0}^T|{\psi }_t⟩|c_t⟩$. A propagation term penalizes a mismatch between adjacent time labels and the corresponding gate. With correct propagation, uniform coefficients form a zero-energy ground state within the history subspace.

A Feynman-Kitaev Hamiltonian penalizes incorrect input initialization, disagreement between successive quantum circuit steps and clock labels, and optionally a rejecting output. Its propagation quadratic form is a sum of $\frac{1}{2}\Vert {\eta }_t-U_t{\eta }_{t-1}{\Vert }^2$. Without output penalty, its zero-energy space consists of correctly initialized computational history states. A nonlocal quantum clock makes the propagation formula simple but does not supply fixed qubit locality by itself.

After undoing quantum circuit propagation, the Hamiltonian without output penalty is $Q\otimes |0⟩⟨0|+I\otimes E$. The common kernel is the valid input space times the uniform clock vector. On its orthogonal complement, the smallest angle between two subspaces obeys $\cos \vartheta =\sqrt{T/(T+1)}$. The Kitaev geometrical lemma and path gap $2/(T+1{)}^2$ give $\Delta \ge (T+1{)}^{-3}$. Degenerate valid quantum witnesses must be removed together when computing this angle.

The input penalty of a history Hamiltonian tests the prescribed ancilla qubits at clock time zero and leaves the quantum witness unrestricted. Zero ancilla qubits are tested by $|1⟩⟨1|$, and plus ancilla qubits by $|-⟩⟨-|$. Their sum $Q$ is positive and has positive integer eigenvalues. Together with propagation, its kernel selects histories of valid initial data.

A nonlocal quantum clock records quantum circuit time in an abstract $T+1$ dimensional register. A binary encoding uses logarithmically many qubits, but transitions $|t⟩⟨t-1|$ need not have bounded qubit locality. Extra diagonal penalties may exclude unused binary labels. A unary quantum clock provides a different encoding for fixed-locality constructions.

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.