Radio-frequency pulses are used in nuclear-magnetic-resonance spectroscopy to produce unitary transfer of states. Pulse sequences that accomplish a desired transfer should be as short as possible in order to minimize the effects of relaxation, and to optimize the sensitivity of the experiments. Many coherence-transfer experiments in NMR, involving a network of coupled spins, use temporary spin decoupling to produce desired effective Hamiltonians. In this paper, we demonstrate that significant time can be saved in producing an effective Hamiltonian if spin decoupling is avoided. We provide time-optimal pulse sequences for producing an important class of effective Hamiltonians in three-spin networks. These effective Hamiltonians are useful for coherence-transfer experiments in three-spin systems and implementation of indirect swap and ${\ensuremath{\Lambda}}_{2}(U)$ gates in the context of NMR quantum computing. It is shown that computing these time-optimal pulses can be reduced to geometric problems that involve computing sub-Riemannian geodesics. Using these geometric ideas, explicit expressions for the minimum time required for producing these effective Hamiltonians, transfer of coherence, and implementation of indirect swap gates, in a three-spin network are derived (Theorems 1 and 2). It is demonstrated that geometric control techniques provide a systematic way of finding time-optimal pulse sequences for transferring coherence and synthesizing unitary transformations in quantum networks, with considerable time savings (e.g., 42.3% for constructing indirect swap gates).
[1]
Mark W. Spong,et al.
Proceedings of the IEEE Conference on Decision and Control
,
1995
.
[2]
Theodore Euclid Djaferis,et al.
System theory : modeling, analysis and control
,
1999
.
[3]
E. Polak,et al.
System Theory
,
1963
.
[4]
Roger W. Brockett.
Explicitly solvable control problems with nonholonomic constraints
,
1999,
Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).
[5]
Navin Khaneja,et al.
On the Stochastic Control of Quantum Ensembles
,
2000
.
[6]
W. Warren.
Advances in magnetic and optical resonance
,
1990
.
[7]
M. L. Chambers.
The Mathematical Theory of Optimal Processes
,
1965
.
[8]
L. S. Pontryagin,et al.
Mathematical Theory of Optimal Processes
,
1962
.
[9]
Barenco,et al.
Elementary gates for quantum computation.
,
1995,
Physical review. A, Atomic, molecular, and optical physics.
[10]
G. Bodenhausen,et al.
Principles of nuclear magnetic resonance in one and two dimensions
,
1987
.
[11]
R. Brockett.
Control Theory and Singular Riemannian Geometry
,
1982
.
[12]
Physical Review
,
1965,
Nature.
[13]
Peter Hilton,et al.
New Directions in Applied Mathematics
,
1982
.