Quantum Metropolis sampling.

The original motivation to build a quantum computer came from Feynman, who imagined a machine capable of simulating generic quantum mechanical systems--a task that is believed to be intractable for classical computers. Such a machine could have far-reaching applications in the simulation of many-body quantum physics in condensed-matter, chemical and high-energy systems. Part of Feynman's challenge was met by Lloyd, who showed how to approximately decompose the time evolution operator of interacting quantum particles into a short sequence of elementary gates, suitable for operation on a quantum computer. However, this left open the problem of how to simulate the equilibrium and static properties of quantum systems. This requires the preparation of ground and Gibbs states on a quantum computer. For classical systems, this problem is solved by the ubiquitous Metropolis algorithm, a method that has basically acquired a monopoly on the simulation of interacting particles. Here we demonstrate how to implement a quantum version of the Metropolis algorithm. This algorithm permits sampling directly from the eigenstates of the Hamiltonian, and thus evades the sign problem present in classical simulations. A small-scale implementation of this algorithm should be achievable with today's technology.

Quantum simulators, continuous-time automata, and translationally invariant systems.

The general problem of finding the ground state energy of lattice Hamiltonians is known to be very hard, even for a quantum computer. We show here that this is the case even for translationally invariant systems in 1D. We also show that a quantum computer can be built in a 1D chain with a fixed, translationally invariant Hamitonian consisting of nearest-neighbor interactions only. The result of the computation is obtained after a prescribed time with high probability.

Delocalized entanglement of atoms in optical lattices.

We show how to detect and quantify entanglement of atoms in optical lattices in terms of correlation functions of the momentum distribution. These distributions can be measured directly in the experiments. We introduce two kinds of entanglement measures related to the position and the spin of the atoms.

Ensemble quantum computation with atoms in periodic potentials.

We show how to perform universal quantum computation with atoms confined in optical lattices which works both in the presence of defects and without individual addressing. The method is based on using the defects in the lattice, wherever they are, both to "mark" different copies on which ensemble quantum computation is carried out and to define pointer atoms which perform the quantum gates. We also show how to overcome the problem of scalability in this system.

Distillability via protocols respecting the positivity of partial transpose.

We show that all quantum states that do not have a positive partial transpose are distillable via channels, which preserve the positivity of the partial transpose. The question whether bound entangled states with non-positive partial transpose exist is therefore closely related to the connection between the set of separable superoperators and positive partial transpose-preserving maps.