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Bulletin of the Lebedev Physics Institute

Simulation of NMR spectra of 31P in the spinless 28Si matrix and problems of the design of a solid-state quantum computer

Aleksander I. Golovashkin, S. N. Dobryakov, Vladimir V. Privezentsev, Andrei M Tskhovrebov

Published Jan 1, 2007• 0 citations
Publisher

Abstract

The problem of numerical simulation of NRM spectra of the donor impurity in silicon is solved in the context of the fact that the system of nuclear spins of 31 P is planned to be used to organize working registers of a solid-state quantum computer. The proposed simulation method is shown to be applicable in the case of a quantitative increase of the NMR register and its topological complication. A scheme of readout of the results of “quantum counting” at a level allowing determination (with the help of SQUID) of the state of an individual spin is proposed.

DOI
10.3103/s1068335607010010

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The problem of numerical simulation of NRM spectra of the donor impurity in silicon is solved in the context of the fact that the system of nuclear spins of 31 P is planned to be used to organize working registers of a solid-state quantum computer. The proposed simulation method is shown to be applicable in the case of a quantitative increase of the NMR register and its topological complication. A scheme of readout of the results of “quantum counting” at a level allowing determination (with the help of SQUID) of the state of an individual spin is proposed.
As an ensemble scheme of solid-state NMR quantum computers the extension of Kane's many-qubits silicon scheme based on the array of 31 P donor atoms are spaced lengthwise of the strip gates is considered. The possible planar topology of such ensemble quantum computer is suggested. The estimation of the output NMR signal was performed and it was shown that for the number N>=10^5 of ensemble elements involving L~10^3 qubits each, the standard NMR methods are usable. As main mechanisms of decoherence for low temperature (<0.1K), the adiabatic processes of random modulation of qubit resonance frequency determined by secular part of nuclear spin hyperfine interaction with electron magnetic moment of basic atom and dipole-dipole interaction with nuclear moments of neighboring impurity atoms was considered, It was made estimations of allowed concentrations of magnetic impurities and of spin temperature whereby the required decoherence suppression is obtained. Semiclassical decoherence model of two qubit entangled states is also presented. As another variant of the solid-state ensemble quantum computer, the gateless architecture of cellular-automaton with antiferromagnetically ordered electron spins is also discussed here.
It is discussed the decoherence problems in ensemble large-scale solid state NMR quantum computer based on the array of 31P donor atoms having nuclear spin I=1/2. It is considered here, as main mechanisms of decoherence for low temperature (<0.1K), the adiabatic processes of random modulation of qubit resonance frequency determined by secular part of nuclear spin interaction with electron spin of the basic atoms, with impurity paramagnetic atoms and also with nuclear spins of impurity and of spin temperature whereby the required decoherence suppression is obtained. It is discussed the random phase error suppression in the ensemble quantum register basic states.
A numerical method is introduced to determine the nuclear magnetic resonance frequency of a donor (31P)(^{31}\mathrm{P})(31P) doped inside a silicon substrate under the influence of an applied electric field. This phosphorus donor has been suggested for operation as a qubit for the realization of a solid-state scalable quantum computer. The operation of the qubit is achieved by a combination of the rotation of the phosphorus nuclear spin through a globally applied magnetic field and the selection of the phosphorus nucleus through a locally applied electric field. To realize the selection function, it is required to know the relationship between the applied electric field and the change of the nuclear magnetic resonance frequency of phosphorus. In this study, based on the wave functions obtained by the effective-mass theory, we introduce an empirical correction factor to the wave functions at the donor nucleus. Using the corrected wave functions, we formulate a first-order perturbation theory for the perturbed system under the influence of an electric field. In order to calculate the potential distributions inside the silicon and the silicon dioxide layers due to the applied electric field, we use the multilayered Green's functions and solve an integral equation by the moment method. This enables us to consider more realistic, arbitrary shape, and three-dimensional qubit structures. With the calculation of the potential distributions, we have investigated the effects of the thicknesses of silicon and silicon dioxide layers, the relative position of the donor, and the applied electric field on the nuclear magnetic resonance frequency of the donor.
Problems in realization of silicon-based solid-state NMR quantum computer with ensemble addressing to qubits are considered. It is presented the extension of Kane's scheme to ensemble approach version with strip gates. For the initialization of nuclear quantum states it is proposed to use the solid-state effect in ENDOR technique whereby the nuclear spins can be practically fully polarized or, that is the same, indirect cooled to the spin temperature less than ~ 1 mK. It is suggested the possible planar silicon topology of such ensemble quantum computer and shown that the measurement with standard NMR methods signal of L ~ 10^3 qubit system may be achieved for a number of ensemble components N >= 10^5. As another variant of ensemble silicon quantum computer the gateless architecture of cellular-automaton is also considered. The decoherence of quantum states in the ensemble quantum computers and ways of its suppression is also discussed.
The procedure for simulating the nuclear magnetic resonance spectrum linked to the spin system of a molecule for a certain nucleus entails diagonalizing the associated Hamiltonian matrix. As the dimensions of said matrix grow exponentially with respect to the spin system's atom count, the calculation of the eigenvalues and eigenvectors marks the performance of the overall process. The aim of this paper is to provide a formalization of the complete procedure of the simulation of a spin system's NMR spectrum while also explaining how to diagonalize the Hamiltonian matrix with a quantum computer, thus enhancing the overall process's performance. Two well-known quantum algorithms for calculating the eigenvalues of a matrix are analyzed and put to the test in this context: quantum phase estimation and the variational quantum eigensolver. Additionally, we present simulated results for the later approach while also addressing the hypothetical noise found in a physical quantum computer.