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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 ?

1I suspect your first guess is the right one. I've not looked at this in-depth, but my impression is that the paper you give is rather too brief on the setup and the connection to the actual problem it's trying to solve. I think the use of the $C_{i,j}$ in your first equation adds weight to this as those are probably supposed to be classical variables of whether or not a particular edge is cut. – DaftWullie – 2020-04-06T14:33:36.623

1Of course, there is

alsoa connection to matrix norms: basically what you're interested in is the maximum eigenvalue of the matrix. – DaftWullie – 2020-04-06T14:35:40.767ok thank you @DaftWullie, I'll go with that definition for now for $\max C$. Would you be kind enough to explain me this sentence ? "Of course, there is also a connection to matrix norms: basically what you're interested in is the maximum eigenvalue of the matrix". I fail to see the link between the maximum eigenvalue of the Hamiltonian operator and the solution to the MAXCUT problem. – nathan raynal – 2020-04-07T14:22:33.670

That is the whole purpose of this scheme - to make the connection between the classical problem and a quantum problem! – DaftWullie – 2020-04-07T15:03:02.343