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 ?