= Input penalty of a history Hamiltonian
{title2=$H_{\rm in}=Q\otimes|0\rangle\langle0|$}
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 $|1\rangle\langle1|$, and plus <ancilla qubits> by $|-\rangle\langle-|$. Their sum $Q$ is positive and has positive integer <eigenvalues>. Together with propagation, its kernel selects histories of valid initial data.
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