# What does the maximum of a Hamiltonian means (in a particular paper)?

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 :

$$C = \sum_{} \frac{1}{2} (I -\sigma_i^z \sigma_j^z) = \sum_{} C_{i,j}$$

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,

$$C(z_1,...,z_n) = \sum_{} \frac{1}{2} (1 -z_i z_j)$$

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 ?

• I 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. Apr 6, 2020 at 14:33
• Of course, there is also a connection to matrix norms: basically what you're interested in is the maximum eigenvalue of the matrix. Apr 6, 2020 at 14:35
• ok 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. Apr 7, 2020 at 14:22
• That is the whole purpose of this scheme - to make the connection between the classical problem and a quantum problem! Apr 7, 2020 at 15:03

$$C$$ is a diagonal matrix, and $$\max{C}$$ is simply the maximum element (which is also the maximum eigenvalue, since the matrix is already diagonal).