Control of open quantum systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Nuclear Engineering, February 2005.Includes bibliographical references (p. 93-102).This thesis describes the development, investigation and experimental implementation via liquid state nuclear magnetic resonance techniques of new methods for controlling open quantum systems. First, methods that improve coherent control through the use of both strong control fields and detailed knowledge of the subsystem's Hamiltonian are demonstrated. With the aid of numerical search methods, pulsed irradiation schemes are obtained that perform accurate, arbitrary, selective gates on multi-qubit systems. For systems of 3 and 4 qubits, simulations show that the control sequences faithfully implement unitary operations with gate fidelities on the order of 0.999 while experimentally determined correlations of 0.99 were obtained. The technique is then extended to account for the incoherent errors arising from the slow variation of control parameters. It is demonstrated in this study that such errors can be greatly counteracted directly from the design of the time-dependent control fields if some knowledge about the incoherence source is available. The results obtained show a substantial decrease of the non-unitary features normally caused by incoherent noise. The methods are applicable to a variety of experimental studies in quantum information processing.(cont.) To test the control techniques, we carried out two benchmark experiments, namely an entanglement transfer and an entanglement swapping experiment performed on a 4-qubit system. The second experiment, while more complex, yields significantly better results, thereby showing the improvement made by the further development of the control techniques. To optimally protect a quantum system against various decoherent errors, it is essential to design methods to acquire knowledge about them. It is in this context that we then develop a robust method for quantum process tomography for measuring relaxation superoperators and Lindblad operators, which is experimentally tested. Finally, we explore both theoretically and experimentally the concatenation of a quantum error correction code with a decoherence-free subspace scheme. Using the two techniques, a 4-qubit quantum system is efficiently protected against a noise containing partial symmetry. To date, this is the first experimental demonstration of such a concatenation scheme.by Nicolas Boulant.Ph.D

Experimental Implementation of a Concatenated Quantum Error-Correcting Code

Concatenated coding provides a general strategy to achieve the desired level of noise protection in quantum information storage and transmission. We report the implementation of a concatenated quantum error-correcting code able to correct against phase errors with a strong correlated component. The experiment was performed using liquid-state nuclear magnetic resonance techniques on a four spin subsystem of labeled crotonic acid. Our results show that concatenation between active and passive quantum error-correcting codes offers a practical tool to handle realistic noise contributed by both independent and correlated errors.Comment: 4 pages, 2 encapsulated eps figures. REVTeX4 styl

Quantum Process Tomography of the Quantum Fourier Transform

The results of quantum process tomography on a three-qubit nuclear magnetic resonance quantum information processor are presented, and shown to be consistent with a detailed model of the system-plus-apparatus used for the experiments. The quantum operation studied was the quantum Fourier transform, which is important in several quantum algorithms and poses a rigorous test for the precision of our recently-developed strongly modulating control fields. The results were analyzed in an attempt to decompose the implementation errors into coherent (overall systematic), incoherent (microscopically deterministic), and decoherent (microscopically random) components. This analysis yielded a superoperator consisting of a unitary part that was strongly correlated with the theoretically expected unitary superoperator of the quantum Fourier transform, an overall attenuation consistent with decoherence, and a residual portion that was not completely positive - although complete positivity is required for any quantum operation. By comparison with the results of computer simulations, the lack of complete positivity was shown to be largely a consequence of the incoherent errors during the quantum process tomography procedure. These simulations further showed that coherent, incoherent, and decoherent errors can often be identified by their distinctive effects on the spectrum of the overall superoperator. The gate fidelity of the experimentally determined superoperator was 0.64, while the correlation coefficient between experimentally determined superoperator and the simulated superoperator was 0.79; most of the discrepancies with the simulations could be explained by the cummulative effect of small errors in the single qubit gates.Comment: 26 pages, 17 figures, four tables; in press, Journal of Chemical Physic

Robust Control of Quantum Information

Errors in the control of quantum systems may be classified as unitary, decoherent and incoherent. Unitary errors are systematic, and result in a density matrix that differs from the desired one by a unitary operation. Decoherent errors correspond to general completely positive superoperators, and can only be corrected using methods such as quantum error correction. Incoherent errors can also be described, on average, by completely positive superoperators, but can nevertheless be corrected by the application of a locally unitary operation that ``refocuses'' them. They are due to reproducible spatial or temporal variations in the system's Hamiltonian, so that information on the variations is encoded in the system's spatiotemporal state and can be used to correct them. In this paper liquid-state nuclear magnetic resonance (NMR) is used to demonstrate that such refocusing effects can be built directly into the control fields, where the incoherence arises from spatial inhomogeneities in the quantizing static magnetic field as well as the radio-frequency control fields themselves. Using perturbation theory, it is further shown that the eigenvalue spectrum of the completely positive superoperator exhibits a characteristic spread that contains information on the Hamiltonians' underlying distribution.Comment: 14 pages, 6 figure

Design of Strongly Modulating Pulses to Implement Precise Effective Hamiltonians for Quantum Information Processing

We describe a method for improving coherent control through the use of detailed knowledge of the system's Hamiltonian. Precise unitary transformations were obtained by strongly modulating the system's dynamics to average out unwanted evolution. With the aid of numerical search methods, pulsed irradiation schemes are obtained that perform accurate, arbitrary, selective gates on multi-qubit systems. Compared to low power selective pulses, which cannot average out all unwanted evolution, these pulses are substantially shorter in time, thereby reducing the effects of relaxation. Liquid-state NMR techniques on homonuclear spin systems are used to demonstrate the accuracy of these gates both in simulation and experiment. Simulations of the coherent evolution of a 3-qubit system show that the control sequences faithfully implement the unitary operations, typically yielding gate fidelities on the order of 0.999 and, for some sequences, up to 0.9997. The experimentally determined density matrices resulting from the application of different control sequences on a 3-spin system have overlaps of up to 0.99 with the expected states, confirming the quality of the experimental implementation.Comment: RevTeX3, 11 pages including 2 tables and 5 figures; Journal of Chemical Physics, in pres

Electromagnetic and RF pulse design simulation based optimization of an eight-channel loop array for 11.7T brain imaging

Purpose: Optimization of transmit array performance is crucial in ultra-high-field MRI scanners such as 11.7T because of the increased RF losses and RF nonuniformity. This work presents a new workflow to investigate and minimize RF coil losses, and to choose the optimum coil configuration for imaging. Methods: An 8-channel transceiver loop-array was simulated to analyze its loss mechanism at 499.415 MHz. A folded-end RF shield was developed to limit radiation loss and improve the B+ 1 efficiency. The coil element length, and the shield diameter and length were further optimized using electromagnetic (EM) simulations. The generated EM fields were used to perform RF pulse design (RFPD) simulations under realistic constraints. The chosen coil design was constructed to demonstrate performance equivalence in bench and scanner measurements. Results: The use of conventional RF shields at 11.7T resulted in significantly high radiation losses of 18.4%. Folding the ends of the RF shield combined with optimizing the shield diameter and length increased the absorbed power in biological tissue and reduced the radiation loss to 2.4%. The peak B+ 1 of the optimal array was 42% more than the reference array. Phantom measurements validated the numerical simulations with a close match of within 4% of the predicted B+ 1 . Conclusion: A workflow that combines EM and RFPD simulations to numerically optimize transmit arrays was developed. Results have been validated using phantom measurements. Our findings demonstrate the need for optimizing the RF shield in conjunction with array element design to achieve efficient excitation at 11.7T

High tip angle approximation based on a modified Bloch-Riccati equation

When designing a radio-frequency pulse to produce a desired dependence of magnetization on frequency or position, the small flip angle approximation is often used as a first step, and a Fourier relation between pulse and transverse magnetization is then invoked. However, common intuition often leads to linear scaling of the resulting pulse so as to produce a larger flip angle than the approximation warrants\u2014with surprisingly good results. Starting from a modified version of the Bloch\u2013Riccati equation, a differential equation in the flip angle itself, rather than in magnetization, is derived. As this equation has a substantial linear component that is an instance of Fourier's equation, the intuitive approach is seen to be justified. Examples of the accuracy of this higher tip angle approximation are given for both constant- and variable-phase pulsesPeer reviewed: YesNRC publication: Ye

Two-spoke placement optimization under explicit specific absorption rate and power constraints in parallel transmission at ultra-high field

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