Quantum Monte Carlo simulations, while being efficient for bosons, suffer from the "negative sign problem" when applied to fermions -causing an exponential increase of the computing time with the number of particles. A polynomial time solution to the sign problem is highly desired since it would provide an unbiased and numerically exact method to simulate correlated quantum systems. Here we show, that such a solution is almost certainly unattainable by proving that the sign problem is NP-hard, implying that a generic solution of the sign problem would also solve all problems in the complexity class NP (nondeterministic polynomial) in polynomial time.Half a century after the seminal paper of Metropolis et al.[1] the Monte Carlo method has widely been established as one of the most important numerical methods and as a key to the simulation of many-body problems. Its main advantage is that it allows phase space integrals for many-particle problems, such as thermal averages, to be evaluated in a time that scales only polynomially with the particle number N although the configuration space grows exponentially with N . This enables the accurate simulation of large systems with millions of particles.Monte Carlo simulations of quantum systems, such as fermions, bosons, or quantum spins, can be performed after mapping the quantum system to an equivalent classical system. For fermionic or frustrated models this mapping may yield configurations with negative Boltzmann weights, resulting in an exponential growth of the statistical error and hence the simulation time with the number of particles, defeating the advantage of the Monte Carlo method. A polynomial time solution of this "sign problem" of negative weights would revolutionize electronic structure calculations by providing an unbiased and approximation-free method to study correlated fermionic systems. This would be of invaluable help, for example, in finding the mechanism for high-temperature superconductivity or in determining the properties of dense nuclear matter and quark matter.The difficulties in finding polynomial time solutions to the sign problem are reminiscent of the apparent impossibility to find polynomial time algorithms for nondeterministic polynomial (NP)-complete decision problems, which could be solved in polynomial time on a hypothetical non-deterministic machine, but for which no polynomial time algorithm is known for deterministic classical computers. A hypothetical non-deterministic machine can always follow both branches of an if-statement simultaneously, but can never merge the branches again. It can, equivalently, be viewed as having exponentially many processors, but without any communication between them. In addition, it must be possible to check a positive answer to a problem in NP on a classical computer in polynomial time.Many important computational problems in the complexity class NP, including the traveling salesman problem and the problem of finding ground states of spin glasses have the additional property of being NP-hard, forming...

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Abelian and non-Abelian gauge theories are of central importance in many areas of physics. In condensed matter physics, Abelian U(1) lattice gauge theories arise in the description of certain quantum spin liquids. In quantum information theory, Kitaev's toric code is a Z(2) lattice gauge theory. In particle physics, Quantum Chromodynamics (QCD), the non-Abelian SU(3) gauge theory of the strong interactions between quarks and gluons, is nonperturbatively regularized on a lattice. Quantum link models extend the concept of lattice gauge theories beyond the Wilson formulation, and are well suited for both digital and analog quantum simulation using ultracold atomic gases in optical lattices. Since quantum simulators do not suffer from the notorious sign problem, they open the door to studies of the real-time evolution of strongly coupled quantum systems, which are impossible with classical simulation methods. A plethora of interesting lattice gauge theories suggests itself for quantum simulation, which should allow us to address very challenging problems, ranging from confinement and deconfinement, or chiral symmetry breaking and its restoration at finite baryon density, to color superconductivity and the real-time evolution of heavy-ion collisions, first in simpler model gauge theories and ultimately in QCD.