The combination of the compactness of networks, featuring small diameters, and their
complex architectures results in a variety of critical effects dramatically different from those in …

Among various algorithms designed to exploit the specific properties of quantum computers
with respect to classical ones, the quantum adiabatic algorithm is a versatile proposition to …

The field of quantum algorithms aims to find ways to speed up the solution of computational
problems by using a quantum computer. A key milestone in this field will be when a …

One of the most prominent application areas for quantum computers is solving hard
constraint satisfaction and optimization problems. However, detailed analyses of the …

An instance of a random constraint satisfaction problem defines a random subset 𝒮 (the set
of solutions) of a large product space XN (the set of assignments). We consider two …

We establish the satisfiability threshold for random k-SAT for all k≥ k0. That is, there exists a
limiting density αs (k) such that a random k-SAT formula of clause density α is with high …

Quantum algorithms can deliver asymptotic speedups over their classical counterparts.
However, there are few cases where a substantial quantum speedup has been worked out …

It is widely believed that for many optimization problems, no algorithm is substantially more
efficient than exhaustive search. This means that finding optimal solutions for many practical …

Using elementary rigorous methods we prove the existence of a clustered phase in the
random K-SAT problem, for K≥ 8. In this phase the solutions are grouped into clusters …

Since the early 2000s physicists have developed an ingenious but non-rigorous formalism
called the cavity method to put forward precise conjectures as to the phase transitions in …