Manuscript Oai227-A@1 · Oai227-A.tex · section · 7f45f607

QAOA and quantum SK are different dynamical problems

Stage 4: change one hypothesis at a time › QAOA and quantum SK are different dynamical problems · lines 1097–1131

A diagonal quantum cost operator built from the classical energy is

H_{n, J} = - \frac{1}{n} i < j \sum J_{ij} Z_{i} Z_{j} .

It commutes with each $Z_{i}$ . Consequently unitary evolution under this operator leaves every $Z_{i}$ unchanged.

By contrast,

\frac{d}{d ℓ} C_{n, J}^{stat} (ℓ)_{ℓ = 0} = - \frac{1}{n} i \sum π [1 - tanh^{2} h_{i}] < 0.

The strict inequality holds for every finite coupling array. Thus even this elementary diagonal quantum evolution is not the heat-bath dynamics.

QAOA adds prescribed unitary mixing layers and optimizes an energy expectation as a function of its parameters and depth. It does not install independent heat-bath clocks FarhiEtAl. The dossier's Family 281 order-of-limits energy statement is therefore only a catalogue comparison Catalogue,Dossier.

Likewise, a Haldane quantum spectral gap, Anderson localization, and an area-law theorem concern different operators and questions. None identifies the present Markov generator.

Displayed formulas in this passage

unlabeled · line 1101
$H_{n, J} = - \frac{1}{n} i < j \sum J_{ij} Z_{i} Z_{j} .$
unlabeled · line 1110
$\frac{d}{d ℓ} C_{n, J}^{stat} (ℓ)_{ℓ = 0} = - \frac{1}{n} i \sum π [1 - tanh^{2} h_{i}] < 0.$