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So I am preparing for an exam and stumbled upon this question which I am unable to answer:

The problem

Binary Integer Linear Programmingtries to find a binary vector $\vec x=(x_1,x_2, ...,x_n)^T$ with $x \in \mathbb{F_2}$ that maximizes the value of $\vec s * \vec b$ under the constraint that $S\vec x = \vec b$. The values $\vec s \in \mathbb{F_2^n}, b\in \mathbb{F_2^n}$ and $S \in \mathbb{F_2^{n\times m}}$ are given. Construct an Ising-Hamiltonian for the optimization problem.Hint: Construct a Hamiltonian for the constraints, a hamiltonian for the maximization and construct a superposition with the fitting coefficients

While I do understand that the Ising Model can help us with optimization problems and I kinda figured out what my constraints and maximization function would look like, I do not know how I should construct a superposition and what it would be useful for.

Thanks in advance!

1Are you sure that the use of "superposition" in the question statement is really meaning superposition in terms of a state? I wonder if it perhaps meant that you're supposed to add the different Hamiltonians together with some appropriate weights? – DaftWullie – 2019-07-25T07:22:07.263

@DaftWullie Tbh I am not sure, no. Adding them together with appropriate weights would make a ton of sense tough! Do you think that one could interprete the word "superposition" in that way? – beatbrot – 2019-07-25T09:27:25.160

1You shouldn't in normal circumstances! But I struggle to see what else it means. After all, the only state you'll be finding is a ground state, which is a basis state in the case of the Ising model (i.e. not a superposition). Still, someone else might understand things differently. – DaftWullie – 2019-07-25T10:42:18.203

You shouldn't be writing $\mathbb{F}_2$ for this problem. Otherwise $\vec{s} * \vec{b}$ would be an inner product in $\mathbb{F}_2^n$ which would produce an element of $\mathbb{F}_2$ as a result. It would be linear algebra over a finite field. Here, you only want to say the inputs are $0$ and $1$, but still integers, not elements of the finite field. – AHusain – 2019-08-25T00:15:00.873