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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.

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.

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.