# Integer Eigenvalues leading to periodicity

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 ?

Cross-posted on physics.SE

• could you link the paper that these equations come from? Apr 25 at 22:08
• What are $U_B$ and $U_C$? Apr 27 at 16:37
• To confirm: when eigenvalues are non-integer rational numbers then the claim is that $F_p$ is not periodic? Apr 27 at 16:40
• Please refrain from vandalising your own post. If for whatever reason you need to hide some information, like references to papers, you can just delete the question altogether
– glS
May 11 at 16:30

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.
• What about the other direction? Also, the claim appears to be that if $\lambda_i$ are non-integer rational numbers then $F_p$ is not periodic, but this proof suggests otherwise (TBH, my bet is your proof is correct and the claim is incorrect, but the question isn't clear enough to say with certainty). Apr 27 at 16:40
• @AdamZalcman I don't think you can make this claim. If you take $a H_c$ where $a$ non-integer and $H_c$ has integer eigenvalues, you have non-integer eigenvalues and a periodic $F_p$ Apr 27 at 17:29