Integer Eigenvalues leading to periodicity

Question

I've been reading a number of papers about approximate algorithms and they mention that when the standard Ising Hamiltonian of the form

$$ H_{c}=\sum_{} c_{i} Z_{i} +\sum_{} J_{i j} Z_{i} Z_{j}, $$

has integer eigenvalues, the value of $$ F_{p}(\boldsymbol{\gamma}, {\boldsymbol{\beta}})= \left\langle\psi_{p}(\boldsymbol{\gamma}, {\boldsymbol{\beta}})\left|H_{C}\right| \psi_{p}(\boldsymbol{\gamma}, {\boldsymbol{\beta}})\right\rangle, $$

where, $$ |\psi_{p}(\boldsymbol{\gamma}, \boldsymbol{\beta})\rangle=U_{B}\left(\beta_{p}\right) U_{C}\left(\gamma_{p}\right) \cdots U_{B}\left(\beta_{1}\right) U_{C}\left(\gamma_{1}\right)|+\rangle^{\otimes N} $$

is periodic, but the value when $H_{c}$ is not integer eigenvalued isn't: why is this ?

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Cross-posted on physics.SE

Answer 1

If you diagonalize $ H_c = \sum \lambda_i |u_i \rangle \langle u_i| $ and the eigenvalues $\lambda_i$ are integers, then: $$ e^{- i (\gamma + 2\pi) H_c} = \sum e^{- i (\gamma + 2\pi) \lambda_i} |u_i \rangle \langle u_i| = \sum e^{- i \gamma \lambda_i} |u_i \rangle \langle u_i| = e^{- i \gamma H_c} $$

since $ e^{- i 2\pi k} = 1, \forall k \text{ integer}$.

This means $ |\psi_p(\gamma, \beta) \rangle = |\psi_p(\gamma + 2\pi, \beta) \rangle \implies F_p(\gamma, \beta) = (\gamma + 2\pi, \beta)$.

A similar argument holds for $ \beta $ if the mixer Hamiltonian $H_B$ has integer eigenvalues.