Nuclear magnetic resonance (NMR) is a direct macroscopic manifestation of the quantum mechanics of the intrinsic angular momentum of atomic nuclei. It is best known for its extraordinary range of applications, which include molecular structure determination, medical imaging, and measurements of flow and diffusion rates. Most recently, liquid-state NMR spectroscopy has been found to provide a powerful experimental tool for the development and evaluation of the coherent control techniques needed for quantum information processing. This burgeoning new interdisciplinary field has the potential to achieve cryptographic, communications, and computational feats far beyond what is possible with known classical physics. Indeed, NMR has made the demonstration of many of these feats sufficiently simple to be carried out by high school summer interns working in our laboratory (see the last two authors). In this paper the basic principles of quantum information processing by NMR spectroscopy are described, along with several illustrative experiments suitable for incorporation into the undergraduate physics curriculum. These experiments are spin–spin interferometry, an implementation of the quantum Fourier transform, and the quantum simulation of a harmonic oscillator.

1.
D. Z. Albert, Quantum Mechanics and Experience (Harvard U.P., Cambridge, MA, 1992).
2.
I. L.
Chuang
,
N.
Gershenfeld
,
M. G.
Kubinec
, and
D. W.
Leung
, “
Bulk quantum computation with nuclear magnetic resonance: Theory and experiment
,”
Proc. R. Soc. London, Ser. A
454
,
447
467
(
1998
).
3.
D. G.
Cory
,
M. D.
Price
, and
T. F.
Havel
, “
Nuclear magnetic resonance spectroscopy: An experimentally accessible paradigm for quantum computing
,”
Physica D
120
,
82
101
(
1998
).
4.
J. A.
Jones
,
M.
Mosca
, and
R. H.
Hansen
, “
Implementation of a quantum search algorithm on a quantum computer
,”
Nature (London)
393
,
344
346
(
1998
).
5.
Phys. World, special issue on quantum information (March, 1998); available on-line at http://physicsweb.org/article/world/11/3/8.
6.
C. H.
Bennett
and
D. P.
DiVincenzo
, “
Quantum information and computation
,”
Nature (London)
404
,
247
255
(
2000
).
7.
S.
Haroche
and
J.-M.
Raimond
, “
Quantum computing: Dream or nightmare?
,”
Phys. Today
49
(
8
),
51
52
(
1996
).
8.
S.
Lloyd
, “
Quantum-mechanical computers
,”
Sci. Am.
273
,
140
145
(
October
, 1995).
9.
C. P. Williams and S. H. Clearwater, Ultimate Zero and One: Computing at the Quantum Frontier (Copernicus Books, Springer-Verlag, New York, 1999).
10.
W. S.
Warren
, “
The usefulness of NMR quantum computing
,”
Science
277
,
1688
1689
(
1997
);
see also response by
N.
Gershenfeld
and
I.
Chuang
,
Science
277
,
1689
1690
(
1997
).
11.
S. L.
Braunstein
,
C. M.
Caves
,
R.
Jozsa
,
N.
Linden
,
S.
Popescu
, and
R.
Schack
, “
Separability of very noisy mixed states and implications for NMR quantum computing
,”
Phys. Rev. Lett.
83
,
1054
1057
(
1999
).
12.
R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison–Wesley, Reading, MA, 1965), Vols. I–III.
13.
D. G.
Cory
et al., “
NMR based quantum information processing
,”
Prog. Phys.
48
,
875
907
(
2000
).
14.
H. K.
Cummins
and
J. A.
Jones
, “
Nuclear magnetic resonance: A quantum technology for computation and spectroscopy
,”
Contemp. Phys.
41
,
383
399
(
2000
).
15.
N. A.
Gershenfeld
and
I. L.
Chuang
, “
Quantum computing with molecules
,”
Sci. Am.
278
,
66
71
(June,
1998
).
16.
T. F.
Havel
,
S. S.
Somaroo
,
C.-H.
Tseng
, and
D. G.
Cory
, “Principles and demonstrations of quantum information processing by NMR spectroscopy,” in Applicable Algebra in Engineering, Communications and Computing, edited by T. Beth and M. Grassl (Springer-Verlag, Berlin, 2000), Vol. 10, pp. 339–374. See also
T. F.
Havel
,
S. S.
Somaroo
,
C.-H.
Tseng
, and
D. G.
Cory
, quant-ph/9812086.
17.
C. H.
Bennett
and
R.
Landauer
, “
The fundamental physical limits of computation
,”
Sci. Am.
253
,
38
46
(July,
1985
).
18.
D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, Berlin, 2001).
19.
D. Giulini, E. Joos, C. Kiefer, J. Kupsch, I.-O. Stamatescu, and H. D. Zeh, Decoherence and the Appearance of a Classical World in Quantum Theory (Springer-Verlag, Berlin, 1996).
20.
G. Mahler and V. A. Weberruss, Quantum Networks (Springer-Verlag, Berlin, 1998), 2nd ed.
21.
A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, Amsterdam, 1993).
22.
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U.P., Cambridge, 2000).
23.
K. Beauchamp, Applications of Walsh and Related Functions: With an Introduction to Sequency Theory (Academic, Englewood Cliffs, NJ, 1984).
24.
D.
Deutsch
and
R.
Jozsa
, “
Rapid solution of problems by quantum computation
,”
Proc. R. Soc. London, Ser. A
439
,
553
558
(
1992
).
25.
A.
Ekert
and
R.
Jozsa
, “
Quantum algorithms: Entanglement enhanced information processing
,”
Philos. Trans. R. Soc. London, Ser. A
356
,
1769
1782
(
1998
).
26.
R.
Jozsa
, “
Quantum algorithms and the Fourier transform
,”
Proc. R. Soc. London, Ser. A
454
,
323
337
(
1998
).
27.
R.
Jozsa
, “
Quantum factoring, discrete logarithms, and the hidden subgroup problem
,”
Comput. Sci. Eng.
3
,
34
43
(
2001
).
28.
R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford U.P., Oxford, 1987).
29.
S. S.
Somaroo
,
D. G.
Cory
, and
T. F.
Havel
, “
Expressing the operations of quantum computing in multiparticle geometric algebra
,”
Phys. Lett. A
240
,
1
7
(
1998
).
30.
O. W.
Sörensen
,
G. W.
Eich
,
M. H.
Levitt
,
G.
Bodenhausen
, and
R. R.
Ernst
, “
Product operator formalism for the description of NMR pulse experiments
,”
Prog. Nucl. Magn. Reson. Spectrosc.
16
,
163
192
(
1983
).
31.
T. F. Havel, D. G. Cory, S. S. Somaroo, and C.-H. Tseng, “Geometric algebra methods in quantum information processing by NMR spectroscopy,” in Advances in Geometric Algebra with Applications, edited by E. Corrochano and G. Sobczyk (Birkhauser, Boston, 2000).
32.
T. F.
Havel
and
C.
Doran
, “Geometric algebra in quantum information processing,” Contemporary Math. (in press)
T. F.
Havel
and
C.
Doran
, (see LANL preprint quant-ph/0004031).
33.
M. D.
Price
,
S. S.
Somaroo
,
A. E.
Dunlop
,
T. F.
Havel
, and
D. G.
Cory
, “
Generalized (controlled)n-NOT quantum logic gates
,”
Phys. Rev. A
60
,
2777
2780
(
1999
).
34.
J. J. Sakurai, Modern Quantum Mechanics (Addison–Wesley, Reading, MA, 1994), revised edition.
35.
W. H.
Zurek
, “
Decoherence and the transition from quantum to classical
,”
Phys. Today
44
(
10
),
36
44
(
1991
).
36.
Y.
Sharf
,
T. F.
Havel
, and
D. G.
Cory
, “
Spatially encoded psuedo-pure states for NMR quantum information processing
,”
Phys. Rev. A
62
,
052314
(
2000
).
37.
R.
Cleve
,
A.
Ekert
,
C.
Macchiavello
, and
M.
Mosca
, “
Quantum algorithms revisited
,”
Proc. R. Soc. London, Ser. A
454
,
339
354
(
1998
).
38.
M. Mosca and A. Ekert, “The hidden subgroup problem and eigenvalue estimation on a quantum computer,” in Proceedings of the NASA International Conference on Quantum Computers and Quantum Communications (QCQS ’98), edited by C. P. Williams, Lecture Notes in Computer Science Vol. 1509 (Springer-Verlag, Berlin, 1999).
39.
Y. S.
Weinstein
,
M. A.
Pravia
,
E. M.
Fortunato
,
S.
Lloyd
, and
D. G.
Cory
, “
Implementation of the quantum Fourier transform
,”
Phys. Rev. Lett.
86
,
1889
1891
(
2001
).
40.
N. D. Mermin, “Notes for physicists of the theory of quantum computation,” 1999, available from http://www.lassp.cornell.edu/lassp_data/NMermin.html.
41.
P. W.
Shor
, “
Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
,”
SIAM J. Comput.
26
,
1484
1509
(
1997
).
42.
R.
Schack
, “
Using a quantum computer to investigate quantum chaos
,”
Phys. Rev. A
57
,
1634
1635
(
1998
).
43.
Y. S. Weinstein, S. Lloyd, J. V. Emerson, and D. G. Cory, “Experimental implementation of the quantum baker’s map,” Phys. Rev. Lett. (submitted).
44.
M. D.
Price
,
T. F.
Havel
, and
D. G.
Cory
, “
Multiqubit logic gates in NMR quantum computing
,”
New J. Phys.
2
,
10
(
2000
), or http://njp.org.
45.
R. P.
Feynman
, “
Simulating physics with computers
,”
Int. J. Theor. Phys.
21
,
467
488
(
1982
).
46.
S.
Lloyd
, “
Universal quantum simulator
,”
Science
273
,
1073
1078
(
1996
).
47.
S. S.
Somaroo
,
C.-H.
Tseng
,
T. F.
Havel
,
R.
Laflamme
, and
D. G.
Cory
, “
Quantum simulations on a quantum computer
,”
Phys. Rev. Lett.
82
,
5381
5384
(
1999
).
48.
A. M.
Steane
, “
Introduction to quantum error correction
,”
Philos. Trans. R. Soc. London, Ser. A
456
,
1739
1756
(
1998
).
49.
M. D.
Price
,
S. S.
Somaroo
,
C.-H.
Tseng
,
J. C.
Gore
,
A. F.
Fahmy
,
T. F.
Havel
, and
D. G.
Cory
, “
Construction and implementation of NMR quantum logic gates for two-spin systems
,”
J. Magn. Reson.
140
,
371
378
(
1999
).
50.
Y.
Sharf
,
D. G.
Cory
,
T. F.
Havel
,
S. S.
Somaroo
,
E.
Knill
,
R.
Laflamme
, and
W. H.
Zurek
, “
A study of quantum error correction by geometric algebra and liquid-state NMR spectroscopy
,”
Mol. Phys.
98
,
1347
1363
(
2000
).
51.
Y.
Sharf
,
T. F.
Havel
, and
D. G.
Cory
, “
Quantum codes for controlling coherent evolution
,”
J. Chem. Phys.
113
,
10878
10885
(
2000
).
52.
C.-H.
Tseng
,
S.
Somaroo
,
Y.
Sharf
,
E.
Knill
,
R.
Laflamme
,
T. F.
Havel
, and
D. G.
Cory
, “
Quantum simulation with natural decoherence
,”
Phys. Rev. A
62
,
032309
(
2000
).
53.
C. H.
Tseng
,
S.
Somaroo
,
Y.
Sharf
,
E.
Knill
,
R.
Laflamme
,
T. F.
Havel
, and
D. G.
Cory
, “
Quantum simulation of a three-body interaction Hamiltonian on an NMR quantum computer
,”
Phys. Rev. A
61
,
012302
(
2000
).
54.
G.
Teklemariam
,
E. M.
Fortunato
,
M. A.
Pravia
,
T. F.
Havel
, and
D. G.
Cory
, “
NMR analog of the quantum disentanglement eraser
,”
Phys. Rev. Lett.
86
,
5845
5849
(
2001
).
This content is only available via PDF.
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.