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Acta Physica Polonica A

Quantum Gates Based on Polarization Transfer NMR Spectroscopy

Ali Güleç, Semīha Bahçelī

Published Jun 1, 2005• 1 citation
Publisher

Abstract

We first introduce an alternative way to study the insensitive nuclei enhanced by polarization and distortionless enhancement by polarization transfer NMR experiments with the product operator descriptions for weakly coupled IS (I = 1/2, S = 1/2) spin systems as a quantum controlled not gate and a not gate, respectively.In this framework we apply the density matrix operators obtained from the analytical descriptions of the mentioned pulse sequences by using product operator formalism step by step on the NMR spin product states represented in Dirac's notation.

DOI
10.12693/aphyspola.107.983

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We first introduce an alternative way to study the insensitive nuclei enhanced by polarization and distortionless enhancement by polarization transfer NMR experiments with the product operator descriptions for weakly coupled IS (I = 1/2, S = 1/2) spin systems as a quantum controlled not gate and a not gate, respectively.In this framework we apply the density matrix operators obtained from the analytical descriptions of the mentioned pulse sequences by using product operator formalism step by step on the NMR spin product states represented in Dirac's notation.
This paper describes how to derive nuclear magnetic resonance (NMR) pulse sequences suitable as ensemble quantum computing logical gates by known systematic tools for design of pulse sequences in NMR spectroscopy. This is in contrast to the current trend of developing NMR pulse sequences for ensemble quantum computing independent of established NMR spectroscopic technology, although many of the pertinent spin manipulations are already known in NMR spectroscopy. The SWAP operation is discussed in detail and one version is only (2J)−1long, which compares favorably with 3(2J)−1 in a recent implementation (Linden N, Barjat H, Kupce E, Freeman R. Chem Phys Lett 1999; 307:198–204). Alternatively, the number of π pulses can be reduced in that sequence. © 2000 John Wiley & Sons, Inc. Concepts Magn Reson 12: 389–395, 2000
This paper describes how to derive nuclear magnetic resonance (NMR) pulse sequences suitable as ensemble quantum computing logical gates by known systematic tools for design of pulse sequences in NMR spectroscopy. This is in contrast to the current trend of developing NMR pulse sequences for ensemble quantum computing independent of established NMR spectroscopic technology, although many of the pertinent spin manipulations are already known in NMR spectroscopy. The SWAP operation is discussed in detail and one version is only (2J)−1long, which compares favorably with 3(2J)−1 in a recent implementation (Linden N, Barjat H, Kupce E, Freeman R. Chem Phys Lett 1999; 307:198–204). Alternatively, the number of π pulses can be reduced in that sequence. © 2000 John Wiley & Sons, Inc. Concepts Magn Reson 12: 389–395, 2000
In NMR quantum computing, spin states of spin-1/2 nuclei are called qubits. Quantum logic gates are represented by unitary matrices. As a universal gate, controlled-NOT (CNOT) is a two-qubit gate. For the IS (I = 1/2 and S = 1/2) spin system, two-qubit CNOT gate is represented by a 4 × 4 matrix. SWAP logic gate, which exchanges two quantum states, is constructed by CNOT gates. In this study, first, four-qubit CNOT gates are constructed for the IS (I = 3/2, S = 3/2) spin system. Then, by using these CNOT gates, a four-qubit SWAP logic gate is found. As an application and verification, an obtained SWAP logic gate is applied to the matrix representation of product operators for the IS (I = 3/2, S = 3/2) spin system. SWAP logic gate can also be presented by an NMR pulse sequence. By using the product operator theory, the pulse sequence of the SWAP logic gate is applied to product operators of the IS (I = 3/2, S = 3/2) spin system. The expected exchange results are obtained for both the matrix representation and the pulse sequence of SWAP logic gate.
The density matrix is a very convenient way to do calculations for a large number of spin systems and is developed from the fundamental ideas of quantum mechanics. The idea of pure and mixed states is introduced and used to develop an expression for the single-spin density matrix as it applies to NMR spectroscopy. Multiple-spin density matrices are produced by using the outer product of single-spin matrices and their commutation properties are examined. A very useful form of the scalar coupling density operator is developed and used, along with the chemical shift operator, to calculate the spectrum of a weakly coupled two-spin system. The density operator of spin-1 is explored briefly. Density matrix calculations in perturbed systems are investigated in preparation for relaxation calculations. Strong coupling is discussed and compared to the weak coupling approximation, using the full scalar coupling Hamiltonian and matrix diagonalization techniques.
Logic gates such as the controlled-NOT (c-NOT) and Toffoli gates play a key role in quantum information processing (QIP) and quantum computing. A natural extension of such gates would necessarily operate on one quantum bit (qubit) conditional on the state of the remaining qubits in the system. We show that such selective gates, termed $(\mathrm{controlled}{)}^{n}\ensuremath{-}\mathrm{NOT}$ gates, or ${\mathrm{c}}^{n}\ensuremath{-}\mathrm{NOT}$ gates, are convenient in nuclear magnetic resonance (NMR) implementations of QIP and are straightforward to implement. NMR pulse sequences for these gates can be built using classical methods as well as insights from geometric algebra. These methods yield equivalent NMR pulse sequences for the generation of ${\mathrm{c}}^{n}\ensuremath{-}\mathrm{NOT}$ gates for any number of control spins. In this work, a catalog of ${\mathrm{c}}^{n}\ensuremath{-}\mathrm{NOT}$ gates for systems of as many as 16 spins is provided along with an experimental implementation of a ${\mathrm{c}}^{3}\ensuremath{-}\mathrm{N}\mathrm{O}\mathrm{T}$ gate on a four spin system, 13C{}^{13}\mathrm{C}13C alanine.
The density matrix is a very convenient way to do calculations for a large number of spin systems and is developed from the fundamental ideas of quantum mechanics. The idea of pure and mixed states is introduced and used to develop an expression for the single-spin density matrix as it applies to NMR spectroscopy. Multiple-spin density matrices are produced by using the outer product of single-spin matrices and their commutation properties are examined. A very useful form of the scalar coupling density operator is developed and used, along with the chemical shift operator, to calculate the spectrum of a weakly coupled two-spin system. The density operator of spin-1 is explored. Density matrix calculations in perturbed systems are investigated in preparation for relaxation calculations. Strong coupling is discussed and compared to the weak coupling approximation, using the full scalar coupling Hamiltonian and matrix diagonalisation techniques.