Research

Quantum Computation

Quantum Algorithms, Quantum Error Correcting Codes, as well as Topological Quantum Computing.

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Quantum Simulation

Investigate the possibilities of this scheme and application to the field of Strongly Correlated Systems.

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Strongly Correlated Systems

Investigate SCS using the framework of Quantum Information, as well as numerical techniques.

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Quantum Information

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Quantum Computation

Over the last decades we have experienced the marvellous and sometimes unbelievable development of computers. Since the invention of Silicon transistors, computer power has increased at a staggering pace whilst computer size has shrank at a similar rate. Indeed this fact has taken the form of a law, Moore's law, which states that the number of transitors on a chip doubles every year and a half.

Although this exponential growth has held true for the last 30 years, it will break down whenever transistors reach the atomic scale and the laws describing the relevant phenomena become quantum instead of classical. Feynman considered this possibility as soon as 1985 [Fey85] realizing that beyond the technical difficulties this problem may pose, it also opens up amazing and novel possibilities in computation. The weirdness of the quantum world may enhance the power of any computer built on the basis of quantum hardware, a quantum computer.

A key breakthrough in Quantum Computation was made in 1994 when Peter Shor showed for the first time that a quantum computer could efficiently perform tasks which are believed to be intractable for its classical analogue [Sho94]. He realized that a quantum computer could factorize a large number in polynomial time, whilst any classical machine would spend exponential time. Two years later, Seth LLoyd showed that the simulation of quantum systems can be implemented efficiently in a quantum rather than classical computer [Llo96] (see section on Quantum Simulation).

In 1995 Grover found another situation where a quantum computer might outperform a classical one; namely, the search of a certain element in an unsorted data base[Gro95]. All these methods take advantage of a common property that has no classical counterpart, quantum entanglement. This property describes correlations between particles at the quantum level, which may exist even when such particles are far apart and which allows Quantum Teleportation [Ben93] . Even though famous physicists as A. Einstein dubbed entanglement as a "spooky action at a distance", it has turned out to be miraculous rather than spooky, since it endows quantum computers with an unimaginable power compared to classical machines.

In order to perform any operation on a quantum computer, we must be able to manipulate the states of quantum bits (qubits) and control their interactions with incredible precision. Unfortunately any quantum system is coupled to the environment and this prevents a perfect experimental control of qubits to be achieved. For instance, environmental thermal fluctuations may couple to our quantum register and flip a certain qubit state leading to a loss of information. This process is coined as Decoherence and is unavoidable in any experimental realization of a quantum computer.

A particular strategy to overcome this problem is to develop quantum error correcting algorithms [Sho95], which use extra qubits to store information and check the fidelity of the processed data. A completely different approach is that of Topological Quantum Computation, where the information is stored in topological properties of certain quasiparticles named anyons, which are resistant to decoherence [Kit97].

One of the research interests in GICC is to investigate quantum algorithms, quantum error correcting codes, as well as Topological Quantum computing proposals.


Bibliography

[Fey85] R.P.Feynman, Opt.News 11, 11(1985).

[Sho94] Shor,P.W., "Polynomial-timr Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer", Proceedings of the 35th Annual Simposium on Foundations of Computer Science. IEEE Computer Society Press, 1994.

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

[Gro95] Grover, L.K., "Quantum Mechanics Helps in Searching for a Needle in a Haystack", Phys.Rev.Lett. 79, 325, 1995.

[Ben93] C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. Wootters, Phys. Rev. Lett. vol.70, 1895,1993.

[Sho95] P.W.Shor, Phys. Rev. A 52, R2493, 1995.

[Kit97] A. Yu. Kitaev, "Fault-Tolerant Quantum Computation by Anyons" ,1997,quantph/9707021.


GICC Publications on this topic

"Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates"

H. Bombin, M.A. Martin-Delgado, Cond-mat/0607736 [1]

"Topological Quantum Distillation"

H. Bombin, M.A. Martin-Delgado, Quant-ph/0605138 [2]

"Homological Error Correction: Classical and Quantum Codes"

H. Bombin, M.A. Martin-Delgado, Quant-ph/0605094 [3]

"Topological Quantum Error Correction with Optimal Encoding Rate"

H. Bombin, M.A. Martin-Delgado, Phys. Rev. A 73, 062303 (2006) [4]

"Relativity and Lorentz Invariance of Entanglement Distillability"

L. Lamata, M.A. Martin-Delgado, E. Solano, quant-ph/0512081 [5]

"Entanglement Distillation Protocols and Number Theory"

H. Bombin, M.A. Martin-Delgado, Phys. Rev. A72, 032313 (2005) [6]

"Distillation Protocols for Mixed States of Multilevel Qubits and the Quantum Renormalization Group"
M.A. Martin-Delgado, M. Navascues, Eur.Phys.J. D27 (2003) 169-180 [7]

"Single-step distillation protocol with generalized beam splitters"

M.A. Martin-Delgado, M. Navascues, Physical Review A68, 012322 (2003) [8]

"The Majorization Arrow in Quantum Algorithm Design"

J.I. Latorre, M.A. Martin-Delgado, Phys. Rev. A66, 022305 (2002) [9]

"Information and Computation: Classical and Quantum Aspects"

A. Galindo, M.A. Martin-Delgado, Rev. Mod. Phys. 74, 347-423 (2002) [10]

"A Family of Grover's Quantum Searching Algorithms"

Alberto Galindo, Miguel A. Martin-Delgado, Phys. Rev. A 62, 62303 (2000) [11]


Quantum Simulation

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]


Strongly Correlated Systems

A wide variety of interesting phenomena in physics can be described by a collection of particles that interact among each other, ranging from Celestial Mechanics to Elementary Particle Physics. These interactions are responsible for the complexity of the phenomena, making a theoretical description both challenging and interesting.

In some cases these interactions can be neglected, which allows an easier treatment of the problem that still describes the physical phenomena faithfully. This is the case of simple metals like copper where the conduction electrons are modeled in such a way that they can wander freely through the sample, and seldom interact with each other.

In some other cases the interaction must be taken into account to describe the system, but it is small enough to be treated approximately. This would be the case of dilute atomic gases and Bose-Einstein Condensation, where a Mean Field Theory can describe appropriately most of the observable effects [Pit03].

However in most of the cases these interactions are strong enough to establish certain correlations between the particles that are responsible of novel phenomena such as Magnetism, High Temperature Superconductivity, Quantum Phase Transitions, Fractional Quantum Hall effect, etc to name just a few [Aue94]. Strongly Correlated Systems is one of the major research areas in contemporary condensed-matter physics. However these high correlations are extremely difficult to deal with by theoretical means.

These correlations are profoundly related to the concept of Quantum Entanglement, which has turned out to be the central resource in quantum information processing responsible of phenomena like Quantum Teleportation, Superdense Coding, etc. The Quantum Information community has made an enormous effort to understand multiparticle entanglement , and as Preskill points out in [Pre00] we can benefit from the tools developed in order to get a better grasp of Strongly Correlated phenomena.

On the other hand the experimental realization of a device capable of outperforming classical computers (i.e. a quantum computer), sometimes relies on strongly correlated systems such as the Mott Insulating phase in an optical lattice, spin chains as quantum channels, etc.

The understanding of these systems is sometimes hindered by the inefficiency of numerical simulations in classical computers. A future solution shall be the use of a Universal Quantum Simulator (see section on Quantum simulation). However its experimental realization requires previous understanding of the particular system that may implement it.

It seems that we have come to a dead end; nevertheless we may use the tools developed by the Strongly Correlated community in order to find a way out. The key fact that prevents an efficient simulation of strongly correlated many-body systems is the huge number of degrees of freedom that must be treated. Using renormalization group techniques, we can get rid of the irrelevant degrees of freedom which are not responsible for the phenomena studied. In this case, we may get rid of high energy variables, since we are interested in low temperature physics. A numerical scheme capable of performing this renormalization preserving the strong correlations in the system was developed by Steven R. White [Whi92], and dubbed the Density Matrix Renormalization Group (DMRG).

One of the research interests in GICC is to investigate Strongly Correlated systems using the theoretical framework of Quantum Information science, as well as numerical techniques for classical computers based on the DMRG scheme.


Bibiliography

[Pit03] L.P. Pitaevskii, S. Stringari, " Bose-Einstein Condensation", Clarendon Press, Oxford, 2003.

[Aue94] A.Auerbach. "Interacting electrons and Magnetism", Springer, New York (1994)

[Pre00] J. Preskill. "Quantum information and physics: Some future directions". J. Mod. Opt., 47:127-137, 2000.

[Whi92] S.R.White,"Density Matrix ormulation for Quantum Renormalization Groups", Phys.Rev.Lett 69, 2863 , 1992.


GICC publications on this topic

"Numerical Computation of Localizable Entanglement in Spin Chains"

Popp, M.; Verstraete, F.; Martin-Delgado, M. A.; Cirac, I., Applied Physics B, Volume 82, Issue 2, pp.225-235, (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]

"Localizable Entanglement"

M. Popp, F. Verstraete, M. A. Martin-Delgado, J. I. Cirac, Phys. Rev. A 71, 042306 (2005) [3]

"Universality Classes of Diagonal Quantum Spin Ladders"

M.A. Martin-Delgado, J. Rodriguez-Laguna, G. Sierra, Phys. Rev. B 72, 104435 (2005) [4]

"Diverging Entanglement Length in Gapped Quantum Spin Systems"

F. Verstraete, M.A. Martin-Delgado, J.I. Cirac, Phys. Rev. Lett. 92, 087201 (2004) [5]

"Matrix Product Density Operators: Simulation of finite-T and dissipative systems"

F. Verstraete, J. J. García-Ripoll, J. I. Cirac , Phys. Rev. Lett. 93, 207204 (2004) [6]

"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) [7]

``Distillation Protocols for Mixed States of Multilevel Qubits and the Quantum Renormalization Group"

M.A. Martin-Delgado, M. Navascues, Eur.Phys.J. D27 (2003) 169-180[8]

"Anderson transition in low-dimensional disordered systems driven by nonrandom long-range hopping"

A. Rodriguez, V.A. Malyshev, G. Sierra, M.A. Martin-Delgado, J. Rodriguez-Laguna, F. Dominguez-Adame, Phys. Rev. Lett. 90, 2, 027404 (1-4) (2003) [9]

"A Density Matrix Renormalization Group study of Excitons in Dendrimers"

M.A. Martin-Delgado, J. Rodriguez-Laguna, G. Sierra, Phys. Rev. B 65, 155116 (2002) [10]

"Exact diagonalisation study of charge order in the quarter-filled two-leg ladder system NaV2O5"

A. Langari, M. A. Martin-Delgado, P. Thalmeier, Phys. Rev. B 63, 144420 (2001) [11]

"Low Energy Properties of Ferrimagnetic 2-leg Ladders: a Lanczos study"

A. Langari, M.A. Martin-Delgado, Phys. Rev. B 63, 054432 (2001).

"Stripe Ansatzs from Exactly Solved Models"

M.A. Martin-Delgado, M. Roncaglia, G. Sierra, Phys. Rev. B 64, 075117 (2001).

"Single-Block Renormalization Group: Quantum Mechanical Problems"

M.A. Martin-Delgado, J. Rodriguez-Laguna, G. Sierra, Nuc. Phys. B 601, 569-590 (2001) [12]

"Matrix Product Approach to Conjugated Polymers "

M. A. Martin-Delgado, G. Sierra, S. Pleutin, E. Jeckelmann, Phys. Rev. B61, 1841, (2000) [13]

"The Density Matrix Renormalization Group applied to single particle Quantum Mechanics"

M.A. Martin-Delgado, G. Sierra and R.M. Noack, J. of Phys. A: Math. and Gen. 32, 6079 (1999) [14]

"The Matrix Product Approach to Quantum Spin Ladders"

J. M. Roman, G. Sierra, J. Dukelsky, M. A. Martin-Delgado, J. Phys. A : Math. Gen. 31, 9729-9759 (1998) [15]

"Equivalence of the Variational Matrix Product Method and the Density Matrix Renormalization Group applied to Spin Chains"

J. Dukelsky, M. A. Martin-Delgado, T. Nishino, G. Sierra, Europhysics Lett. 43 457, (1998) [16]


Quantum Information

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