Feynman-Kitaev Hamiltonian ($H_{\mathrm{in}}+H_{\mathrm{prop}}+H_{\mathrm{out}}$)

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 prescribed ancilla qubits at clock time zero while leaving the quantum witness unrestricted. A zero ancilla qubit is tested by the bad-state projector $|1⟩⟨1|$; a plus ancilla qubit is tested by $|-⟩⟨-|$.

Let $Q={\sum }_{j=1}^m{\Pi }_j$, where each ${\Pi }_j$ tests a different ancilla qubit. These orthogonal projections commute because they act on separate tensor product factors. Thus $Q$ is a positive operator with integer eigenvalues between zero and $m$. Zero is essential: its kernel contains exactly the valid ancillary input subspace, tensored with arbitrary quantum witness states. Only the nonzero eigenvalues are positive integers.

The clock-conditioned term is $H_{\mathrm{in}}=Q\otimes |c_0⟩⟨c_0|$. For a normalized computational history state $|\eta ⟩=(T+1{)}^{-1/2}{\sum }_t|{\psi }_t⟩|c_t⟩$ with clock labels forming an orthonormal basis,Together with correct propagation, zero input penalty selects histories with valid initial data. The displayed expectation concerns this fixed history state; a bound on the full Hamiltonian's lowest eigenvalue also has to account for propagation penalties.