. Support, B. We-thank-ryszard-michalczyk, and . Mac-donald, Cliff Unkefer, and David Cory for their help. *Electronic address: knill@lanl.gov ? Electronic address: laflamme@lanl.gov

D. G. Cory, A. F. Fahmy, and T. F. Havel, Ensemble quantum computing by NMR spectroscopy, Proceedings of the National Academy of Sciences, vol.275, issue.5278, p.1634, 1997.
DOI : 10.1126/science.273.5278.1073

W. S. Warren, The Usefulness of NMR Quantum Computing, Science, vol.277, issue.5332, p.1688, 1997.
DOI : 10.1126/science.277.5332.1688

C. H. Bennett, D. P. Divincenzo, J. A. Smolin, and W. K. Wootters, Mixed-state entanglement and quantum error correction, Physical Review A, vol.54, issue.5, p.3824, 1996.
DOI : 10.1103/PhysRevA.54.3824

R. Laflamme, C. Miquel, J. Paz, and W. H. Zurek, Perfect Quantum Error Correcting Code, Physical Review Letters, vol.77, issue.1, p.198, 1996.
DOI : 10.1103/PhysRevLett.77.198

D. Gottesman, Class of quantum error-correcting codes saturating the quantum Hamming bound, Physical Review A, vol.54, issue.3, p.1862, 1996.
DOI : 10.1103/PhysRevA.54.1862

A. Calderbank, E. Rains, P. Shor, and N. Sloane, Quantum Error Correction and Orthogonal Geometry, Physical Review Letters, vol.78, issue.3, p.405, 1997.
DOI : 10.1103/PhysRevLett.78.405

URL : http://arxiv.org/abs/quant-ph/9605005

B. Schumacher, Sending entanglement through noisy quantum channels, Physical Review A, vol.54, issue.4, p.2614, 1996.
DOI : 10.1103/PhysRevA.54.2614

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.123.8322

E. Knill, R. Laflamme, R. Martinez, and C. Tseng, An algorithmic benchmark for quantum information processing, Nature, vol.404, issue.6776, p.368, 2000.
DOI : 10.1038/35006012

D. G. Cory, M. Price, W. Maas, E. Knill, R. Laflamme et al., Experimental Quantum Error Correction, Physical Review Letters, vol.81, issue.10, p.2152, 1998.
DOI : 10.1103/PhysRevLett.81.2152

I. L. Chuang, N. Gershenfeld, and M. Kubinec, Experimental Implementation of Fast Quantum Searching, Physical Review Letters, vol.80, issue.15, p.3408, 1998.
DOI : 10.1103/PhysRevLett.80.3408

L. M. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, R. Cleve et al., Experimental Realization of an Order-Finding Algorithm with an NMR Quantum Computer, Physical Review Letters, vol.85, issue.25, p.5452, 2000.
DOI : 10.1103/PhysRevLett.85.5452

D. Leung, L. Vandersypen, X. L. Zhou, M. Sherwood, C. Yannoni et al., Experimental realization of a two-bit phase damping quantum code, Physical Review A, vol.60, issue.3, p.1924, 1999.
DOI : 10.1103/PhysRevA.60.1924

T. Oka, A Search for Interstellar HFormula, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.303, issue.1480, pp.543-549, 1981.
DOI : 10.1098/rsta.1981.0223

J. P. Maillard, P. Drossart, J. K. Watson, S. J. Kim, and J. Caldwell, H3(+) fundamental band in Jupiter's auroral zones at high resolution from 2400 to 2900 inverse centimeters, The Astrophysical Journal, vol.363, pp.37-41, 1990.
DOI : 10.1086/185859

P. Drossart, Detection of H3+ on Jupiter, Nature, vol.340, issue.6234, pp.539-541, 1989.
DOI : 10.1038/340539a0

R. S. Fisher, C. M. Telesco, R. K. Piña, R. F. Knacke, and M. C. Wyatt, Detection of Extended Thermal Infrared Emission around the Vega-like Source HD 141569, The Astrophysical Journal, vol.532, issue.2, pp.141-144, 2000.
DOI : 10.1086/312575

A. J. Weinberger, The Circumstellar Disk of HD 141569 Imaged with NICMOS, The Astrophysical Journal, vol.525, issue.1, pp.53-56, 1999.
DOI : 10.1086/312334

B. Zuckerman, T. Forveille, and J. H. Kastner, Inhibition of giant-planet formation by rapid gas depletion around young stars, Nature, vol.373, issue.6514, pp.494-496, 1995.
DOI : 10.1038/373494a0

K. Malfait, E. Bogaert, and C. Waelkens, An ultraviolet, optical and infrared study of Herbig Ae/Be stars, Astron. Astrophys, vol.331, pp.211-223, 1998.

A. J. Weinberger, R. M. Rich, E. E. Becklin, B. Zuckerman, and K. Matthews, Stellar Companions and the Age of HD 141569 and Its Circumstellar Disk, The Astrophysical Journal, vol.544, issue.2, pp.937-943, 2000.
DOI : 10.1086/317243

W. F. Thi, and CO Emission from Disks around T Tauri and Herbig Ae Pre???Main???Sequence Stars and from Debris Disks around Young Stars: Warm and Cold Circumstellar Gas, The Astrophysical Journal, vol.561, issue.2, pp.1074-1094, 2001.
DOI : 10.1086/323361

S. P. Ruden and J. B. Pollack, The dynamical evolution of the protosolar nebula, The Astrophysical Journal, vol.375, pp.740-760, 1991.
DOI : 10.1086/170239

J. J. Lissauer, Planet Formation, Annual Review of Astronomy and Astrophysics, vol.31, issue.1, pp.129-174, 1993.
DOI : 10.1146/annurev.aa.31.090193.001021

S. Miller, J. Tennyson, and R. D. Joseph, Infrared emission of H 3 þ in the atmosphere of Jupiter in the 2

T. Oka and T. R. Geballe, Observations of the 4 micron fundamental band of H3(+) in Jupiter, The Astrophysical Journal, vol.351, pp.53-56, 1990.
DOI : 10.1086/185678

B. J. Mccall, K. H. Hinkle, T. R. Geballe, and T. Oka, H3+ in dense and diffuse clouds, Faraday Discussions, vol.109, pp.267-280, 1998.
DOI : 10.1039/a800655e

A. P. Boss and . Symp, Planetary Systems in the Universe, pp.202-205, 2000.

A. P. Boss, Formation of extrasolar giant planets: core accretion or disk instability?, Earth, Moon, and Planets, vol.81, issue.1, pp.19-26, 1998.
DOI : 10.1023/A:1006370021545

V. G. Kunde and W. C. Maguire, Direct integration transmittance model, Journal of Quantitative Spectroscopy and Radiative Transfer, vol.14, issue.8, pp.803-817, 1974.
DOI : 10.1016/0022-4073(74)90124-1

L. S. Rothman, The HITRAN molecular database: Editions of 1991 and 1992, Journal of Quantitative Spectroscopy and Radiative Transfer, vol.48, issue.5-6, pp.469-507, 1992.
DOI : 10.1016/0022-4073(92)90115-K

S. D. Dunkin, M. J. Barlow, and S. G. Ryan, High-resolution spectroscopy of Vega-like stars -- I. Effective temperatures, gravities and photospheric abundances, Monthly Notices of the Royal Astronomical Society, vol.286, issue.3, pp.604-616, 1997.
DOI : 10.1093/mnras/286.3.604

P. C. Frisch, Radial velocities in selected B-G stars, The Astrophysical Journal Supplement Series, vol.65, pp.313-317, 1987.
DOI : 10.1086/191227

D. T. Smithey, M. Beck, M. G. Reymer, and A. Farydani, Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum, Physical Review Letters, vol.70, issue.9, pp.1244-1247, 1993.
DOI : 10.1103/PhysRevLett.70.1244

A. Kitaev and . Yu, Quantum measurements and the Abelian stabilizer problem, 1995.

S. Lloyd, Universal Quantum Simulators, Science, vol.273, issue.5278, pp.1073-1078, 1996.
DOI : 10.1126/science.273.5278.1073

P. W. Shor, Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, SIAM Journal on Computing, vol.26, issue.5, pp.1484-1509, 1997.
DOI : 10.1137/S0097539795293172

R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, Quantum algorithms revisited, Proc. R. Soc. Lond. A 454, pp.339-354, 1998.
DOI : 10.1098/rspa.1998.0164

URL : http://arxiv.org/abs/quant-ph/9708016

A. Barenco, Elementary gates for quantum computation, Physical Review A, vol.52, issue.5, pp.3457-3467, 1995.
DOI : 10.1103/PhysRevA.52.3457

R. P. Feynman, Simulating physics with computers, International Journal of Theoretical Physics, vol.21, issue.6-7, pp.467-488, 1962.
DOI : 10.1007/BF02650179

D. Abraham and S. Lloyd, Quantum Algorithm Providing Exponential Speed Increase for Finding Eigenvalues and Eigenvectors, Physical Review Letters, vol.83, issue.24, pp.5162-5165, 1999.
DOI : 10.1103/PhysRevLett.83.5162

E. Knill and R. Laflamme, Power of One Bit of Quantum Information, Physical Review Letters, vol.81, issue.25, pp.5672-5675, 1999.
DOI : 10.1103/PhysRevLett.81.5672

M. Hillery, R. F. O-'connell, M. O. Scully, and E. P. Wigner, Distribution functions in physics: Fundamentals, Physics Reports, vol.106, issue.3, pp.121-167, 1984.
DOI : 10.1016/0370-1573(84)90160-1

W. K. Wootters, A Wigner-function formulation of finite-state quantum mechanics, Annals of Physics, vol.176, issue.1, pp.1-21, 1987.
DOI : 10.1016/0003-4916(87)90176-X

U. Leonhardt, Discrete Wigner function and quantum-state tomography, Physical Review A, vol.53, issue.5, pp.2998-3013, 1996.
DOI : 10.1103/PhysRevA.53.2998

C. Miquel, J. P. Paz, and M. Saraceno, Quantum computers in phase space A (in the press); preprint quant-ph, Phys. Rev, 2002.

C. Miquel, J. P. Paz, and R. Perazzo, Factoring in a dissipative quantum computer, Physical Review A, vol.54, issue.4, pp.2605-2613, 1996.
DOI : 10.1103/PhysRevA.54.2605

W. Zurek, Sub-Planck structure in phase space and its relevance for quantum decoherence, Nature, vol.95, issue.6848, pp.712-717, 2001.
DOI : 10.1038/35089017

D. G. Cory, Experimental Quantum Error Correction, Physical Review Letters, vol.81, issue.10, pp.2152-2155, 1998.
DOI : 10.1103/PhysRevLett.81.2152

E. Knill, I. Chuang, and R. Laflamme, Effective pure states for bulk quantum computation, Physical Review A, vol.57, issue.5, pp.3348-3363, 1998.
DOI : 10.1103/PhysRevA.57.3348

L. G. Lutterbach and L. Davidovich, Method for Direct Measurement of the Wigner Function in Cavity QED and Ion Traps, Physical Review Letters, vol.78, issue.13, pp.2547-2550, 1997.
DOI : 10.1103/PhysRevLett.78.2547

L. Lvovsky, Quantum State Reconstruction of the Single-Photon Fock State, Physical Review Letters, vol.87, issue.5, pp.402-405, 2001.
DOI : 10.1103/PhysRevLett.87.050402

G. Nogues, Measurement of a negative value for the Wigner function of radiation, Physical Review A, vol.62, issue.5, pp.4101-4104, 2000.
DOI : 10.1103/PhysRevA.62.054101

B. Aharonov and D. Bohm, Significance of Electromagnetic Potentials in the Quantum Theory, Physical Review, vol.115, issue.3, 1959.
DOI : 10.1103/PhysRev.115.485

C. Batista and G. Ortiz, Generalized Jordan-Wigner Transformations, Physical Review Letters, vol.86, issue.6, p.1082, 2001.
DOI : 10.1103/PhysRevLett.86.1082

URL : http://arxiv.org/abs/cond-mat/0008374

C. Bennett, Logical Reversibility of Computation, IBM Journal of Research and Development, vol.17, issue.6, pp.525-557, 1973.
DOI : 10.1147/rd.176.0525

D. Chang, L. Vandersapien, and M. Stephen, NMR implementation of a building block for scalable quantum computation, Chemical Physics Letters, vol.338, issue.4-6, pp.337-344, 2001.
DOI : 10.1016/S0009-2614(01)00280-9

R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, Quantum algorithms revisited, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.454, issue.1969, p.339, 1998.
DOI : 10.1098/rspa.1998.0164

URL : http://arxiv.org/abs/quant-ph/9708016

D. Cory, A. Fahmy, and T. Havel, Ensemble quantum computing by NMR spectroscopy, Proc. Nat. Acad. Sci. USA, pp.1634-1639, 1997.
DOI : 10.1126/science.273.5278.1073

D. Cory, R. Laflamme, E. Knill, L. Viola, T. Havel et al., NMR Based Quantum Information Processing: Achievements and Prospects, Fortschritte der Physik, vol.48, issue.9-11, pp.875-907, 2000.
DOI : 10.1002/1521-3978(200009)48:9/11<875::AID-PROP875>3.0.CO;2-V

URL : http://arxiv.org/abs/quant-ph/0004104

D. Cory, W. Maas, M. Price, E. Knill, R. Laflamme et al., Experimental Quantum Error Correction, Physical Review Letters, vol.81, issue.10, pp.2152-2155, 1998.
DOI : 10.1103/PhysRevLett.81.2152

D. Cory, M. Price, and T. Havel, Nuclear magnetic resonance spectroscopy: An experimentally accessible paradigm for quantum computing, Physica D: Nonlinear Phenomena, vol.120, issue.1-2, pp.82-101, 1998.
DOI : 10.1016/S0167-2789(98)00046-3

D. Deutsch, Quantum theory, the Church-Turing principle and the universal quqntum computer, Proc.R.Soc.Lond.A, pp.97-117, 1985.

D. and D. Vincenzo, Two-bit gates are universal for quantum computation, Physical Review A, vol.51, issue.2, pp.1015-1022, 1995.
DOI : 10.1103/PhysRevA.51.1015

D. and D. Vincenzo, The Physical Implementation of Quantum Computation, Fortschritte der Physik, vol.48, issue.9-11, pp.771-783, 2000.
DOI : 10.1002/1521-3978(200009)48:9/11<771::AID-PROP771>3.0.CO;2-E

R. Feynman, Simulating physics with computers, International Journal of Theoretical Physics, vol.21, issue.6-7, p.467, 1982.
DOI : 10.1007/BF02650179

E. Fortunato, N. Pravia, M. Boulant, G. Teklemariam, D. Havel et al., Design of strongly modulating pulses to implement precise effective Hamiltonians for quantum information processing, The Journal of Chemical Physics, vol.116, issue.17, pp.7599-7606, 2002.
DOI : 10.1063/1.1465412

R. Freeman, Spin choreography. Spektrum, 1997.

N. Gershenfeld and I. Chuang, Bulk Spin-Resonance Quantum Computation, Science, vol.58, issue.1907, pp.350-356, 1997.
DOI : 10.1098/rspa.1992.0167

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.28.8877

D. Gottesman, Theory of fault-tolerant quantum computation, Physical Review A, vol.57, issue.1, pp.127-137, 1998.
DOI : 10.1103/PhysRevA.57.127

J. Jones and E. Knill, Efficient Refocusing of One-Spin and Two-Spin Interactions for NMR Quantum Computation, Journal of Magnetic Resonance, vol.141, issue.2, pp.322-325, 1999.
DOI : 10.1006/jmre.1999.1890

E. Knill, I. Chuang, and R. Laflamme, Effective pure states for bulk quantum computation, Physical Review A, vol.57, issue.5, pp.3348-3363, 1998.
DOI : 10.1103/PhysRevA.57.3348

URL : http://arxiv.org/abs/quant-ph/9706053

E. Knill, R. Laflamme, R. Martinez, and C. Negrevergne, Benchmarking Quantum Computers: The Five-Qubit Error Correcting Code, Physical Review Letters, vol.86, issue.25, pp.5811-5814, 2001.
DOI : 10.1103/PhysRevLett.86.5811

E. Knill, R. Laflamme, R. Martinez, and C. Tseng, An algorithmic benchmark for quantum information processing, Nature, vol.404, issue.6776, pp.368-370, 2000.
DOI : 10.1038/35006012

E. Knill, R. Laflamme, and W. Zurek, Resilient Quantum Computation, Science, vol.279, issue.5349, pp.342-345, 1998.
DOI : 10.1126/science.279.5349.342

E. Knill and R. Laflamme, Power of One Bit of Quantum Information, Physical Review Letters, vol.81, issue.25, pp.5672-5675, 1998.
DOI : 10.1103/PhysRevLett.81.5672

R. Laflamme, C. Miquel, J. Paz, and W. Zurek, Perfect Quantum Error Correcting Code, Physical Review Letters, vol.77, issue.1, p.198, 1996.
DOI : 10.1103/PhysRevLett.77.198

R. Laflamme, D. Cory, C. Négrevergne, and L. Viola, NMR quantum information processing and entanglement, Quantum Information and Computation, pp.166-176, 2002.

S. Loyd, Universal Quantum Simulators, Science, vol.273, issue.5278, pp.1073-1078, 1996.
DOI : 10.1126/science.273.5278.1073

G. Milburn, R. Laflamme, B. Sanders, and E. Knill, Quantum dynamics of two coupled qubits, Physical Review A, vol.65, issue.3, 2002.
DOI : 10.1103/PhysRevA.65.032316

C. Miquel, J. Paz, P. , M. Saraceno, R. Knill et al., Interpretation of tomography and spectroscopy as dual forms of quantum computation, Nature, vol.176, issue.6893, pp.59-62, 2002.
DOI : 10.1103/PHYSREVA.62.054101

D. Monroe, C. Meekhof, W. King, B. Itano, and W. , Demonstration of a Fundamental Quantum Logic Gate, Physical Review Letters, vol.75, issue.25, p.4714, 1995.
DOI : 10.1103/PhysRevLett.75.4714

C. Négrevergne, R. Somma, G. Ortiz, E. Knill, and R. Laflamme, Anyonic quantum simulations. en préparation

M. Nielsen, A. , E. Knill, and R. Laflamme, Complete quantum teleportation, Nature, vol.396, issue.6706, pp.52-55, 1998.
DOI : 10.1038/23891

G. Ortiz, J. Gubernatis, E. , E. Knill, and R. Laflamme, Quantum algorithms for fermionic simulations, Physical Review A, vol.64, issue.2, 2001.
DOI : 10.1103/PhysRevA.64.022319

A. Raushenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune et al., Step-by-Step Engineered Multiparticle Entanglement, Science, vol.288, issue.5473, pp.2024-2028, 2000.
DOI : 10.1126/science.288.5473.2024

C. Sackett, D. Kielpinsky, B. King, B. Langer, V. Meyer et al., Experimental entanglement of four particules, Nature, vol.404, pp.256-258, 2000.

L. Schuman and U. Vazirani, Scalable NMR quantum computation, Proc. 31 st ACM Symp. on Theory of Computing, pp.322-329, 1998.

P. Shor, Algorithms for quantum computation: discrete logarithms and factoring, Proceedings 35th Annual Symposium on Foundations of Computer Science, pp.116-123, 1994.
DOI : 10.1109/SFCS.1994.365700

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.3862

P. Shor, Scheme for reducing decoherence in quantum computer memory, Physical Review A, vol.52, issue.4, p.2493, 1995.
DOI : 10.1103/PhysRevA.52.R2493

P. Shor, Scheme for reducing decoherence in quantum computer memory, Physical Review A, vol.52, issue.4, p.2493, 1995.
DOI : 10.1103/PhysRevA.52.R2493

S. Somaroo, C. Tseng, T. Havel, R. Laflamme, and D. Cory, Quantum Simulations on a Quantum Computer, Physical Review Letters, vol.82, issue.26, pp.5381-5384, 1999.
DOI : 10.1103/PhysRevLett.82.5381

R. Somma, G. Ortiz, J. Gubernatis, E. , E. Knill et al., Simulating physical phenomena by quantum networks, Physical Review A, vol.65, issue.4, pp.1-17, 2002.
DOI : 10.1103/PhysRevA.65.042323

O. Sorensen, G. Eich, M. Levitt, G. Bodenhaussen, and R. Ernst, Product operator formalism for the description of NMR pulse experiments, Progess in NMR spectroscopy, pp.163-192, 1983.
DOI : 10.1016/0079-6565(84)80005-9

A. Steane, Error Correcting Codes in Quantum Theory, Physical Review Letters, vol.77, issue.5, p.793, 1996.
DOI : 10.1103/PhysRevLett.77.793

A. Steane and M. , Introduction to quantum error correction. phil, Trans. R. Lond. A, vol.356, pp.1739-1758, 1998.

Q. Truchette, C. Hood, W. Lange, H. Mabuchi, and H. Kimble, Measurement of Conditional Phase Shifts for Quantum Logic, Physical Review Letters, vol.75, issue.25, p.4710, 1995.
DOI : 10.1103/PhysRevLett.75.4710

C. Tseng, S. Somaroo, Y. Sharf, E. Knill, R. Laflamme et al., Quantum simulation of a three-body-interaction Hamiltonian on an NMR quantum computer, Physical Review A, vol.61, issue.1, pp.1-6, 2000.
DOI : 10.1103/PhysRevA.61.012302

A. Turing, On computable numbers, with an application to the entscheidungsproblem, Proc. Lond, p.230, 1936.

L. Vandersypen, M. Steffen, G. Breyta, S. Yannoni, I. Sherwood et al., Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance, Nature, vol.61, issue.3, pp.883-887, 2001.
DOI : 10.1038/414883a

L. Viola, E. Fortunato, E. Knill, R. Laflamme, and D. Cory, Experimental Realization of Noiseless Subsystems for Quantum Information Processing, Science, vol.293, issue.5537, pp.2059-2063, 2001.
DOI : 10.1126/science.1064460

L. Viola, E. Knill, and R. Laflamme, Constructing qubits in physical systems, Journal of Physics A: Mathematical and General, vol.34, issue.35, pp.7067-7080, 2001.
DOI : 10.1088/0305-4470/34/35/331

W. Zurek, Environment-induced superselection rules, Physical Review D, vol.26, issue.8, pp.1862-1880, 1982.
DOI : 10.1103/PhysRevD.26.1862

W. Zurek, Decoherence and the Transition from Quantum to Classical, Physics Today, vol.44, issue.10, pp.36-44, 1991.
DOI : 10.1063/1.881293