Sub-Riemannian geometry and time optimal control of three spin systems: Quantum gates and coherence transfer

By Navin Khaneja, Steffen J. Glaser, Roger W. Brockett. 2001

210 citations recorded by the source corpus.

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).

Subfields: Artificial Intelligence, Spectroscopy

View Paper: https://www.semanticscholar.org/paper/8e98f00dc7abda3fee6737980c0c0202399366e3

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Citations in the corpus

Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe (2022, background): “As a result, arbitrarily strong fields are not sufficient to speed up the system dynamics, and it is impossible to prepare a desired state or carry out a desired quantum gate in arbitrarily short time [326, 327].”
Controllability of the discrete-spectrum Schrödinger equation driven by an external field (2008, background): “(For optimal control results for finite dimensional quantum systems see, for instance, [15, 16, 17, 26].”
Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms. (2005, background): “It is qualitatively similar to the analytically derived sequence, which consists only of rf pulses irradiated at spin I2 [39].”
Local controllability of 1D linear and nonlinear Schr (2010, unknown): “We also refer to the following works about the controllability of finite dimensional quantum systems [2, 21, 22, 23, 24, 25, 26], by Agrachev, Boscain, Chambrion, Charlot, Gauthier, Guérin, Jauslin and Mason, [41] by Khaneja, Glaser and Brockett, [54] by Ramakrishna, Salapaka, Dahleh, Rabitz, [61] b”
NMR techniques for quantum control and computation (2004, background): “Such optimization problems have been extensively studied in mathematics s Brockett, 1981 d and have been solved explicitly for elementary quantum gates on two coupled spins s Khaneja et al. , 2001 d and a three-spin chain with nearest-neighbor couplings s Khaneja et al. , 2002 d .”
Training Schrödinger’s cat: quantum optimal control (2015, methodology): “Mathematical tools that were developed recently [4, 34, 44, 188] allowed to solve problems of increasing difficulty including fundamental control problems for closed [42, 43, 91, 92, 135, 209] and open quantum systems [35, 205, 238, 291, 400, 461].”
Time-optimal Control of Spin Systems (2006, background): “In Section 2.5 we are then specializing to the situation where the Lie algebra g has the additional property of being semisimple and belonging to a symmetric Lie algebra pair (g, k) (cf. Definition 1.2.5).”