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Journal of Tsinghua University(Science and Technology)

Measurement of the nuclear spin state in Kane's solid-state quantum computer

Gui‐Lu Long

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

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.

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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 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.
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.
PACS. 03.65.Ta – Foundations of quantum mechanics; measurement theory. PACS. 03.65.Wj – State reconstruction, quantum tomography. Abstract. – We propose a method for the tomographic reconstruction of qubit states for a general class of solid-state systems in which the Hamiltonians are represented by spin operators, e.g., with Heisenberg-, XXZ-, or XY-type exchange interactions. We analyze the implementation of the projective operator measurements, or spin measurements, on qubit states. All the qubit states for the spin Hamiltonians can be reconstructed by using experimental data. Quantum information processing requires the effective measurement of quantum states. However, a single quantum measurement can only obtain partial information of a quantum state. The reconstruction of a quantum state requires measuring a complete set of observables on an ensemble of identically prepared copies of the system. This method called quantum state tomography [1], is very important because any unknown state can be ascertained by tomographic measurements. Moreover, the full description of qubit states can increase the accuracy of quantum operations. Tomographic measurements have been experimentally implemented for, e.g., the nuclear spin state of an NMR system [2], the electromagnetic field and photon
Abstract Advances in quantum information processing can open a way for numerous applications of the processing in various fields of science and technology: communication, precision measurement, computing, nano-scale detectors, and sensors. Classical and various quantum correlations have been studied in real spin 1/2 systems. The nonlocality measures provide a novel classification scheme for bipartite states, highlighting that nonlocality is a quantum resource distinct from other types of quantum correlations. We first studied the temperature and field dependencies of nonlocality measure in the three fictitious spin 1/2 system, which represents a nuclear spin-7/2 placed in magnetic and inhomogeneous electric fields. The relationship between nonlocality and other quantum correlations (entanglement and geometric discord) was studied. The Hamiltonian and spin operators for a spin 7/2 are represented in the basis formed by the Kronecker products of the Pauli matrices. This transformation allows us to represent a spin 7/2 as a system of three coupled fictitious spins 1/2 and, from the quantum information point of view, as an equivalent system of three coupling qubits. Well-developed methods were used to calculate measures of quantum correlations. For example, we consider 59Co (spin 7/2) in the compounds [Co(NH3)5Cl]Cl2 and Ca3Co2O6. The interaction between the fictitious spins of 59Co nuclei depends on the magnitude and direction of the external magnetic field. Other potential cases of quantum correlations can be realized based on nuclear spins 7/2 in solid-state systems, for instance, such as 51V, 165Ho, or 123Sb.