Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms.

Fifty years of developments in nuclear magnetic resonance (NMR) have resulted in an unrivaled degree of control of the dynamics of coupled two-level quantum systems. This coherent control of nuclear spin dynamics has recently been taken to a new level, motivated by the interest in quantum information processing. NMR has been the workhorse for the experimental implementation of quantum protocols, allowing exquisite control of systems up to seven qubits in size. This article surveys and summarizes a broad variety of pulse control and tomographic techniques which have been developed for, and used in, NMR quantum computation. Many of these will be useful in other quantum systems now being considered for the implementation of quantum information processing tasks.

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NMR techniques for quantum control and computation

a ) p. 131 The discussion between eqs. (5.14) and (5.15) is incorrect (dA should be made as large as possible!). b ) p. 256 In the figure, the numbers 6) and 7) occur twice. c ) p. 292 At the end of section 12.5, it should be the space of isospectral 0Hermitian matrices. d ) p. 306 A ”Tr” is missing in eq. (13.43). e ) p. 327, Eq. (14.64b) is 〈Trρ〉B = N(14N+10) (5N+1)(N+3) should be 〈Trρ〉B = 8N+7 (N+2)(N+4)

https://chaos.if.uj.edu.pl/~karol/erratum10.pdf

Geometry of Quantum States

Stable quantum bits, capable both of storing quantum information for macroscopic time scales and of integration inside small portable devices, are an essential building block for an array of potential applications. We demonstrate high-fidelity control of a solid-state qubit, which preserves its polarization for several minutes and features coherence lifetimes exceeding 1 second at room temperature. The qubit consists of a single ^(13)C nuclear spin in the vicinity of a nitrogen-vacancy color center within an isotopically purified diamond crystal. The long qubit memory time was achieved via a technique involving dissipative decoupling of the single nuclear spin from its local environment. The versatility, robustness, and potential scalability of this system may allow for new applications in quantum information science.

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Room-Temperature Quantum Bit Memory Exceeding One Second

It is control that turns scientific knowledge into useful technology: in physics and engineering it provides a systematic way for driving a dynamical system from a given initial state into a desired target state with minimized expenditure of energy and resources As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantum-enhanced sensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry, magnetic resonance (spectroscopy as well as medical imaging), quantum information processing and quantum simulation In this communication, state-of-the-art quantum control techniques are reviewed and put into perspective by a consortium of experts in optimal control theory and applications to spectroscopy, imaging, as well as quantum dynamics of closed and open systems We address key challenges and sketch a roadmap for future developments

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Training Schrödinger’s cat: quantum optimal control

In this paper, we study the design of pulse sequences for nuclear magnetic resonance spectroscopy as a problem of time optimal control of the unitary propagator. Radio-frequency pulses are used in coherent spectroscopy to implement a unitary transfer between 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. Here, we give an analytical characterization of such time optimal pulse sequences applicable to coherence transfer experiments in multiple-spin systems. We have adopted a general mathematical formulation, and present many of our results in this setting, mindful of the fact that new structures in optimal pulse design are constantly arising. From a general control theory perspective, the problems we want to study have the following character. Suppose we are given a controllable right invariant system on a compact Lie group. What is the minimum time required to steer the system from some initial point to a specified final point? In nuclear magnetic resonance (NMR) spectroscopy and quantum computing, this translates to, what is the minimum time required to produce a unitary propagator? We also give an analytical characterization of maximum achievable transfer in a given time for the two-spin system.

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Time optimal control in spin systems

Cartan decomposition of SU(2n) and control of spin systems

The efficiency of dipole-dipole coupling driven coherence transfer experiments in solid-state NMR spectroscopy of powder samples is limited by dispersion of the orientation of the internuclear vectors relative to the external magnetic field. Here we introduce general design principles and resulting pulse sequences that approach full polarization transfer efficiency for all crystallite orientations in a powder in magic-angle-spinning experiments. The methods compensate for the defocusing of coherence due to orientation dependent dipolar coupling interactions and inhomogeneous radio-frequency fields. The compensation scheme is very simple to implement as a scaffold (comb) of compensating pulses in which the pulse sequence to be improved may be inserted. The degree of compensation can be adjusted and should be balanced as a compromise between efficiency and length of the overall pulse sequence. We show by numerical and experimental data that the presented compensation protocol significantly improves the efficiency of known dipolar recoupling solid-state NMR experiment.

https://arxiv.org/pdf/quant-ph/0601022.pdf

Composite Dipolar Recoupling: Anisotropy Compensated Coherence Transfer in Solid-State NMR

In a wide variety of settings, the measurement of nuclear magnetic resonance (NMR) effects has proven to be a remarkably effective for investigating unknown structures on both large and small scales. Over the years a large body of technique has been developed for improving the sensitivity and resolution of NMR measurements and many recent advances in biochemistry and medicine are dependent on the sophisticated signal processing techniques now used routinely. From a system theoretic perspective, problems in this area can be thought of as identification problems involving bilinear systems. Many ingenious techniques, such as the "two dimensional" Fourier transform procedure have been developed based on particular types of input patterns. Because of the low signal to noise ratios inherent in NMR, the optimization of such methods requires the use of stochastic models for the dynamics and measurement processes. We take a fresh look at problems in this area with the view of finding computational procedures that will determine the inputs which will optimize specific performance measures. In particular, we explore performance measures related to conditional entropy, and in this way develop a formalism for establishing the mathematical limits on what can be accomplished with better input design.

Navin Khaneja

Papers

Time optimal control in spin systems

Cartan decomposition of SU(2n) and control of spin systems

Composite Dipolar Recoupling: Anisotropy Compensated Coherence Transfer in Solid-State NMR

Optimal input design for NMR system identification

Relaxation Optimized Polarization-Coherence Transfer in Ising Spin Chains