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Journal of Physics Condensed Matter

Dynamics of nuclear spin measurement in a mesoscopic solid-state quantum computer

G. P. Berman, David K Campbell, Gary D. Doolen, Kirill E. Nagaev

Published Mar 15, 2000• 7 citations
Publisher

Abstract

We study numerically the process of nuclear spin measurement in a solid-state quantum computer of the type proposed by Kane, by calculating the quantum dynamics of two coupled nuclear spins on 31 P donors implanted in 28 Si. We estimate the time of the `quantum swap operation' - the minimum measurement time required for the reliable transfer of quantum information from the nuclear spin subsystem to the electronic subsystem. Our calculations show that for realistic values of the parameters this time is of the order of swap ~5 × 10 -5 s. We also calculate the probability of error for typical values of the external noise.

DOI
10.1088/0953-8984/12/13/306

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We study numerically the process of nuclear spin measurement in a solid-state quantum computer of the type proposed by Kane, by calculating the quantum dynamics of two coupled nuclear spins on 31 P donors implanted in 28 Si. We estimate the time of the `quantum swap operation' - the minimum measurement time required for the reliable transfer of quantum information from the nuclear spin subsystem to the electronic subsystem. Our calculations show that for realistic values of the parameters this time is of the order of swap ~5 × 10 -5 s. We also calculate the probability of error for typical values of the external noise.
We study numerically the process of nuclear spin measurement in a solid-state quantum computer of the type proposed by Kane by modeling the quantum dynamics of two coupled nuclear spins on 31^{31}31P donors implanted in silicon. We estimate the minimum measurement time necessary for the reliable transfer of quantum information from the nuclear spin subsystem to the electronic subsystem. We also calculate the probability of error for typical values of the external noise.
Deutsch formulated in 1985 a new computer based on quantummechanics. In quantum computers, the ones and zeros of classical digital computers are replaced by the quantum state of a two-level system which is named qubit. Computation, i.e., a sequence of unitary transformations, simultaneously affects each element of the superposition, generating a massive parallel data processing, albeit within one piece of quantum hardware. Practical quantum computer requires preventing decoherence (uncontrolled interaction of a quantum system with its surrounding environment) in order to maintain the superposition. In 1998, Kane proposed the scheme of implementing a quantum computer on an array of nuclear spins located on donors in silicon. Because nuclear spins are extremely well isolated from environment, operations on nuclear-spin qubits could have low error rates. Quantum logic gate is performed by transitions between the different energy levels of the electron-nuclear spin states, which is induced by a radio-frequency magnetic field B applied at a frequency resonant with the energy-level difference. In this study, we numerically simulated the quantum logic gate of Kane’s quantum computer, by neglecting the complications generated from the band structure in Si. For the calculation of the time evolution of the quantum logic gate, we compute the transition probability between the energy levels of electron-nuclear spin states under the influence of a time-dependent perturbation.
The measurement of the nuclear spin state in the Kane's solid-state quantum computer was investigated for a simple two-qubit system. To simplify the calculations, the Hamiltonian matrix was divided into five invariant subspaces based on the projection of the total spins along the external magnetic field. The calculations analyzed the relations between the eigenvalues and eigenstates of the Hamiltonian matrix and the interactions between the two electrons. The results show that measurement of the nuclear spin state by exchanging the nuclear and election spin wave functions is restricted. Only four of the sixteen possible states in a two-qubit system can be measured using this method.
We examine a 31P{}^{31}\mathrm{P}31P donor electron spin in a Si crystal to be used for the purpose of quantum computation. The interaction with an uncontrolled system of 29Si{}^{29}\mathrm{Si}29Si nuclear spins influences the electron-spin dynamics appreciably. The hyperfine field at the 29Si{}^{29}\mathrm{Si}29Si nuclei positions is noncollinear with the external magnetic field. Quantum operations with the electron wave function, i.e., using magnetic-field pulses or electrical gates, change the orientation of hyperfine field and disturb the nuclear-spin system. This disturbance produces a deviation of the electron spin qubit from an ideal state, at a short-time scale in comparison with the nuclear-spin diffusion time. For H${}_{\mathrm{ext}}\ensuremath{\approx}9$ T, the estimated error rate is comparable to the threshold value required by the quantum error correction algorithms. The rate is lower at higher external magnetic fields.
We propose a scheme for quantum information processing based on donor electron spins in semiconductors, with an architecture complementary to the original Kane proposal. We show that a na\"{\i}ve implementation of electron spin qubits provides only modest improvement over the Kane scheme, however through the introduction of global gate control we are able to take full advantage of the fast electron evolution timescales. We estimate that the latent clock speed is 100--1000 times that of the nuclear spin quantum computer with the ratio T2∕Tops{T}_{2}∕{T}_{\mathrm{ops}}T2​∕Tops​ approaching the 106{10}^{6}106 level.