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arXiv:cond-mat/0304674 · Physical review. B, Condensed matter

Grover algorithm for large nuclear spins in semiconductors

Michael Niklaus Leuenberger, Daniel Loss

Published Oct 16, 2003• 47 citations
arXivHTMLPDF

Abstract

We show a possible way to implement the Grover algorithm in large nuclear spins $1/2<I<~9/2$ in semiconductors. The Grover sequence is performed by means of multiphoton transitions that distribute the spin amplitude between the nuclear spin states. They are distinguishable due to the quadrupolar splitting, which makes the nuclear spin levels nonequidistant. We introduce a generalized rotating frame for an effective Hamiltonian that governs the nonperturbative time evolution of the nuclear spin states for arbitrary spin lengths I. The larger the quadrupolar splitting, the better the agreement between our approximative method using the generalized rotating frame and exact numerical calculations.

DOI
10.1103/physrevb.68.165317

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Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences•Sep 3, 2012•15 citations
We show a possible way to implement the Grover algorithm in large nuclear spins 1/2<I< 9/21/2<I<~9/21/2<I< 9/2 in semiconductors. The Grover sequence is performed by means of multiphoton transitions that distribute the spin amplitude between the nuclear spin states. They are distinguishable due to the quadrupolar splitting, which makes the nuclear spin levels nonequidistant. We introduce a generalized rotating frame for an effective Hamiltonian that governs the nonperturbative time evolution of the nuclear spin states for arbitrary spin lengths I. The larger the quadrupolar splitting, the better the agreement between our approximative method using the generalized rotating frame and exact numerical calculations.
We propose an implementation for quantum information processing based on coherent manipulations of nuclear spins I=3/2 in GaAs semiconductors. We describe theoretically an NMR method which involves multiphoton transitions and which exploits the nonequidistance of nuclear spin levels due to quadrupolar splittings. Starting from known spin anisotropies we derive effective Hamiltonians in a generalized rotating frame, valid for arbitrary I, which allow us to describe the nonperturbative time evolution of spin states generated by magnetic rf fields. We identify an experimentally observable regime for multiphoton Rabi oscillations. In the nonlinear regime, we find Berry phase interference.
We discuss methods of quantum state tomography for solid-state systems with a large nuclear spin I=3/2I=3/2I=3/2 in nanometer-scale semiconductors devices based on a quantum well. Due to quadrupolar interactions, the Zeeman levels of these nuclear-spin devices become nonequidistant, forming a controllable four-level quantum system (known as quartit or ququart). The occupation of these levels can be selectively and coherently manipulated by multiphoton transitions using the techniques of nuclear magnetic resonance (NMR) [Yusa et al., Nature (London) 434, 1001 (2005)]. These methods are based on an unconventional approach to NMR, where the longitudinal magnetization Mz{M}_{z}Mz​ is directly measured. This is in contrast to the standard NMR experiments and tomographic methods, where the transverse magnetization Mxy{M}_{xy}Mxy​ is detected. The robustness against errors in the measured data is analyzed by using the condition number based on the spectral norm. We propose several methods with optimized sets of rotations yielding the highest robustness against errors, as described by the condition number equal to 1, assuming an ideal experimental detection. This robustness is only slightly deteriorated, as given by the condition number equal to 1.05, for a more realistic ``noisy'' Mz{M}_{z}Mz​ detection based on the standard cyclically ordered phase sequence (CYCLOPS) method.
A continuous, analog version of the Grover algorithm is realized using NMR. The system studied is 23Na{}^{23}\mathrm{Na}23Na in a liquid-crystal medium. The presence of quadrupolar coupling makes the spin I=3/2I=3/2I=3/2 nucleus a two-qubit system. Applying a specially designed pulse sequence, the time evolution of the spin-density operator is described in an interaction representation that has no external time-dependent radio-frequency fields. This approach is used to implement one instance of the continuous Grover search for the transform of a uniform state to a target state, and the implementation provides a clear physical interpretation of the algorithm. The experimental results are in good agreement with the theory.
Physical implementation of quantum-information processing by liquid-state nuclear magnetic resonance, using weakly coupled spin-12\frac{1}{2}21​ nuclei of a molecule, is well established. Nuclei with spin>1/2\mathrm{spin}>1/2spin>1/2 oriented in liquid-crystalline matrices is another possibility. Such systems have multiple qubits per nuclei and large quadrupolar couplings resulting in well separated lines in the spectrum. So far, creation of pseudopure states and logic gates has been demonstrated in such systems using transition selective radio-frequency pulses. In this paper we report two developments. First, we implement a quantum algorithm that needs coherent superposition of states. Second, we use evolution under quadrupolar coupling to implement multiqubit gates. We implement the Deutsch-Jozsa algorithm on a spin-32\frac{3}{2}23​ (2 qubit) system. The controlled-NOT operation needed to implement this algorithm has been implemented here by evolution under the quadrupolar Hamiltonian. To the best of our knowledge, this method has been implemented for the first time in quadrupolar systems. Since the quadrupolar coupling is several orders of magnitude greater than the coupling in weakly coupled spin-12\frac{1}{2}21​ nuclei, the gate time decreases, increasing the clock speed of the quantum computer.
Nuclear magnetic resonance is viewed as an important technique for the implementation of many quantum information algorithms and protocols. Although the most straightforward approach is to use the two-level system composed of spin 1 nuclei as qubits, quadrupolar nuclei, which possess a spin greater than 1 , are being used as an alternative. In this study, we show some unique features of quadrupolar systems for quantum information processing, with an emphasis on the ability to execute efficient quantum state tomography (QST) using only global rotations of the spin system, whose performance is shown in detail. By preparing suitable states and implementing logical operations by numerically optimized pulses together with the QST method, we follow the stepwise execution of Grover’s algorithm. We also review some work in the literature concerning the relaxation of pseudo-pure states in spin 3 systems as well as its modelling in both the Redfield and Kraus formalisms. These data are used to discuss differences in the behaviour of the quantum correlations observed for two-qubit systems implemented by spin 1 and quadrupolar spin 3 systems, also presented in the literature. The possibilities and advantages of using nuclear quadrupole resonance experiments for quantum information processing are also discussed.
Nuclear magnetic resonance is viewed as an important technique for the implementation of many quantum information algorithms and protocols. Although the most straightforward approach is to use the two-level system composed of spin 1/2 nuclei as qubits, quadrupolar nuclei, which possess a spin greater than 1/2, are being used as an alternative. In this study, we show some unique features of quadrupolar systems for quantum information processing, with an emphasis on the ability to execute efficient quantum state tomography (QST) using only global rotations of the spin system, whose performance is shown in detail. By preparing suitable states and implementing logical operations by numerically optimized pulses together with the QST method, we follow the stepwise execution of Grover's algorithm. We also review some work in the literature concerning the relaxation of pseudo-pure states in spin 3/2 systems as well as its modelling in both the Redfield and Kraus formalisms. These data are used to discuss differences in the behaviour of the quantum correlations observed for two-qubit systems implemented by spin 1/2 and quadrupolar spin 3/2 systems, also presented in the literature. The possibilities and advantages of using nuclear quadrupole resonance experiments for quantum information processing are also discussed.