bibkey: bauschcrosson2018modifiedcircuits
authors: Johannes Bausch; Elizabeth Crosson
year: 2018
title: Analysis and limitations of modified circuit-to-Hamiltonian constructions
doi: null
url: https://arxiv.org/abs/1609.08571
claim: Controlled removal of the circuit gates makes the modified propagation Hamiltonian unitarily equivalent to a clock Hamiltonian tensored with the register identity.
strata_touched: []
license: citation-only
triage: anchor

Propagation constraints do not determine a physical clock scale

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The author version of Analysis and limitations of modified circuit-to-Hamiltonian constructions, Lemma 2 (PDF p. 5), supplies the controlled gate-removal equivalence ${\mathcal{W}}^{†}{H}_{\mathrm{prop}}\mathcal{W}=H_{\mathrm{clock}}\otimes I$. Its proof is on PDF p. 10. The source considers a broad class of weighted clocks; the positive scalar weights below multiply fixed constraint terms and do not change the constraints themselves.

Positive weights on fixed constraints

For finite-dimensional Hilbert spaces and fixed linear maps $A_j$, let

$$H_{\lambda }=\sum _{j=0}^{N-1}{\lambda }_j{A}_j^{†}{A}_j,{\lambda }_j>0.$$

For every $v$, $⟨v,H_{\lambda }v⟩={\sum }_j{\lambda }_j\Vert A_{j}v{\Vert }^2$. Consequently

$$\ker H_{\lambda }=\bigcap _j\ker A_j.$$

Indeed, membership in the kernel makes the nonnegative quadratic form zero, so each $A_{j}v=0$; the reverse implication follows by substitution. This is the standard positive-sum kernel argument, used as supporting linear algebra, not a new theorem of the project. Strictly positive weights are essential to this argument for retaining every constraint; a zero weight may remove a constraint.

For fixed gates and $A_j=⟨j+1|\otimes I-⟨j|\otimes U_{j+1}$, the resulting history space is therefore unchanged by these weights. With $\mathcal{W}={\sum }_j|j⟩⟨j|\otimes W_j$, $W_j=U_j\cdots U_1$, direct conjugation gives ${\mathcal{W}}^{†}{H}_{\lambda }\mathcal{W}=L_{\lambda }\otimes I$, where $L_{\lambda }$ is the path Laplacian with edge weights ${\lambda }_j$. The project-specific finite instrument correspondence is already in Collective oscillators, Definition 10.1 and Theorem 10.2.

For every $c>0$, $H_{c\lambda }=c{H}_{\lambda }$ has the same kernel, whereas its specified physical evolution obeys

$$e^{-it{H}_{c\lambda }/\hslash }=e^{-i(ct)H_{\lambda }/\hslash }.$$

Thus a constraint kernel does not specify the energy scale. A vector in that kernel is stationary under this propagation Hamiltonian and does not automatically execute the encoded sequence. Actual gate control, state preparation, repeated clock measurement and physical calibration remain separate implementation assumptions.