In the paper Quantum Observables for continuous control of the Quantum Approximate Optimization Algorithm via Reinforcement Learning, an Hamiltonian is defined in order to solve the MAXCUT problem :
$$ C = \sum_{<i,j>} \frac{1}{2} (I -\sigma_i^z \sigma_j^z) = \sum_{<i,j>} C_{i,j} $$
with $\sigma_j^z$ the pauli matrix $\sigma^z$ applied to the $j^{th}$ qubit. The sum is taken over all adjacent edge in the original graph problem.
The paper indicates the following result :
$$ \lim_{p \rightarrow \infty} [\max_{\beta,\gamma} \langle \beta,\gamma |_p C | \beta,\gamma \rangle_p] = \max C $$.
consider $|\beta,\gamma \rangle_p$ some states produced by the QAOA algorithm and $p$ an integer, their value do not matter for my question.
The quantity $\max_{\beta,\gamma} \langle \beta,\gamma |_p C | \beta,\gamma \rangle_p$ is clearly scalar, whereas I can't make sense of the expression $\max C$.
My question is then, what does $\max C$ means in this context ?
I believe the answer is not given in this paper. I might have the answer from this documentation, where it is said that the Hamiltonian is constructed from the classical function,
$$C(z_1,...,z_n) = \sum_{<i,j>} \frac{1}{2} (1 -z_i z_j)$$
Where the sum is taken over all adjacent edges in the original graph problem, with $z_i = 1$ if $z_i \in S$ or $z_i = -1$ if $z_i \in \bar{S}$ (same for $z_j$) with $S$ and $\bar{S}$ the bipartition of the original graph. I believe the authors of the first paper I linked were referring to this classical function and not to the Hamiltonian when speaking about $\max C$.
My second guess is that it might refers to some matrix norms but none is defined in the article. What do you think ?
