  Quadratic Unconstrained Binary Optimization (QUBO or UBQP) is concerned with
  maximizing/minimizing the quadratic form
  $H(J, \eta) = W \sum_{i,j} J_{i,j} \eta_{i} \eta_{j}$ with $J$ a matrix of
coefficients,
  $\eta \in \{0, 1\}^N$ and $W$ a normalizing constant.
  In the statistical mechanics literature, QUBO is a lattice gas counterpart to
the (generalized) Sherrington--Kirkpatrick spin glass model.
  Finding the optima of $H$ is an NP-hard problem.
  Several problems in combinatorial optimization and data analysis can be
mapped to QUBO in a straightforward manner.
  In the combinatorial optimization literature, random instances of QUBO are
often used to test the effectiveness of heuristic algorithms.
  Here we consider QUBO with random independent coefficients and show that if
  the $J_{i,j}$'s have zero mean and finite variance then, after proper
normalization,
  the minimum and
  maximum \emph{per particle} of $H$ do not depend on the details of the
distribution
  of the couplings and are concentrated around their expected values.
  Further, with the help of numerical simulations, we study the
  minimum and maximum of the objective function and provide some insight
  into the structure of the minimizer and the maximizer of $H$.
  We argue that also this structure is rather robust.
  Our findings hold also in the diluted case where each of the $J_{i,j}$'s is
allowed
  to be zero with probability going to $1$ as $N \to \infty$ in a suitable way.
