# Quantum Simulation

### From GICC

In 1982, Richard Feynman wondered how a Universal Computer could be used to simulate a certain physical system [Fey82]. He soon realized that the number of variables that must be taken into account to simulate a quantum system increases exponentially with the number of particles simulated. He finally claimed that *the problem of simulating quantum physics was not tractable in any classical computer*, and conjectured how a new kind of computer built at the quantum level should be able to carry out this task efficiently.

He realized that certain phenomena in Quantum Field Theory are well imitated by certain Condensed Matter systems (e.g. spin waves in a solid mimicking Bose particles in a field theory). Building on these ideas, he thought that there should be a certain class of quantum mechanical systems which would simulate any other system, a **Universal Quantum Simulator** (UQS).

Fourteen years later, Seth LLoyd showed that Feynman was right once more. *Quantum Computers can simulate efficiently any system at the quantum level* as soon as the interactions have some degree of locality [LLo96]. Furthermore, he also realized that the experimental requirements to achieve an efficient simulation of interesting many-body systems were less demanding as compared to the implementation of Shor's and Grover's algorithm. Quantum simulation requires a quantum computer with a few tens or hundreds of qubits which may be available in decades from now. Therefore it stands as a midpoint between the current technology that builds computers with a few quantum bits, and the large scale quantum computer that might need millions of qubits.

With the advent of the UQS we shall be able to deepen our knowledge in many Condensed Matter phenomena, as intriguing as High Temperature Superconductivity, which rely on a many-body Hamiltonian that cannot be treated either analytically or numerically with classical computers. Therefore a *UQS will serve as a quantum laboratory where the validity of several theoretical models may be tested*, and where important phenomena in Physics, Chemistry and even Biology shall be understood.

One of the most promising proposals for the experimental realization of this simulator is based on **ultracold atoms stored in optical lattices**. These lattices are artificial crystals made of laser light where ultracold atoms are stored in arrays of microscopic potentials, and can be manipulated externally by means of optical techniques [Gre02]. The high degree of control that the experimenter has over the atoms makes this system best suited in order to perform a quantum simulation. Actually it has been shown by Jané et al. that this system can simulate a great variety of phenomena ranging from Magnetism to Quantum Phase Transitions [Cir03].

One of the research interests in GICC is to investigate the possibilities of this scheme and its application to the field of Strongly Correlated Systems.

## Bibliography

**[Fey82]** R.P.Feynman, "Simulating physics with computers" , Int.J.Theor.Phys. 21,467 (1982).

**[LLo96]** S.LLoyd, "Universal Quantum Simulators", Science 273,1073 (1996).

**[Gre02]** Greiner M, Mandel O, Esslinger T, H¨ansch T W, Bloch I, "Quantum phase transition from a superfluid to a Mott incuslator in an ultracold gas of atoms", Nature 415 39 (2002).

**[Cir03]** E.Jané, G.Vidal, W.Dür, P.Zoller, J.I.Cirac, "Simulation of quantum dynamics with quantum optical systems", Quant. Inf. Comp. 3, 15 (2003)

## GICC Publications on this topic

**"Cooling toolbox for atoms in optical lattices"**

M. Popp, J. J. García-Ripoll, K. G. H. Vollbrecht, J. I. Cirac, New J. Phys. 8 164 (2006) [1]

**"Ground state cooling of atoms in optical lattices"**

M. Popp, J. J. García-Ripoll, K. G. H. Vollbrecht, J. I. Cirac, Phys. Rev. A 74, 013622 (2006) [2]

**"Coherent control of trapped ions using off-resonant lasers"**

J. J. García-Ripoll, P. Zoller, J. I. Cirac , Phys. Rev. A 71, 062309 (2005) [3]

**"Quantum information processing with cold atoms and trapped ions"**

J. J. García-Ripoll, P. Zoller and J. I. Cirac , J. Phys. B 38 S567-S578 (2005) [4]

**"Implementation of Spin Hamiltonians in Optical Lattices"**

J. J. Garcia-Ripoll, M. A. Martin-Delgado, J. I. Cirac, Phys. Rev. Lett. 93, 250405 (2004) [5]

**"Variational ansatz for the superfluid Mott-insulator transition in optical lattices"**

J. J. García-Ripoll, C. Kollath, U. Schollwoeck, P. Zoller, J. von Delft, and J. I. Cirac, Optics Express 12, 42 (2004) [6]

**"Quantum computation with cold bosonic atoms in an optical lattice"**

J. J. García-Ripoll, and J. I. Cirac , Phil. Trans. R. Soc. Lond. A 361, 1537-1548 (2003).

**"Spin dynamics for bosons in an optical lattice"**

J. J. García-Ripoll, and J. I. Cirac, New J. Phys. 5, 76 (2003) [7]

**"Quantum computation with unknown parameters"**

J. J. García-Ripoll, and J. I. Cirac , Phys. Rev. Lett. 90, 127902 (2003) [8]

**"Split vortices in optically coupled Bose-Einstein condensates"**

J. J. García-Ripoll, V. M. Pérez-García, and F. Sols , Phys. Rev. A 66, 021602 (2002) [9]