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arXiv:quant-ph/0005076 · Physical Review A

Spatially encoded pseudopure states for NMR quantum-information processing

Yehuda Sharf, Timothy F. Havel, David G. Cory

Published Oct 18, 2000• 23 citations

Abstract

Quantum-information processing by liquid-state NMR spectroscopy uses pseudopure states to mimic the evolution and observations on true pure states. A new method of preparing pseudopure states is described, which involves the selection of spatially labeled states of an ancilla spin with which the spin system of interest is correlated. This permits a general procedure to be given for the preparation of pseudopure states on any number of spins, subject to the limitations imposed by the loss of signal from the selected subensemble. The preparation of a single pseudopure state is demonstrated by carbon and proton NMR of ${}^{13}\mathrm{C}$-labeled alanine. With a judicious choice of magnetic-field gradients, the method further allows encoding of up to ${2}^{N}$ pseudopure states in independent spatial modes in an $N+1$ spin system. Fast encoding and decoding schemes are demonstrated for the preparation of four such spatially labeled pseudopure states.

DOI: 10.1103/physreva.62.052314

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Quantum-information processing by liquid-state NMR spectroscopy uses pseudopure states to mimic the evolution and observations on true pure states. A new method of preparing pseudopure states is described, which involves the selection of spatially labeled states of an ancilla spin with which the spin system of interest is correlated. This permits a general procedure to be given for the preparation of pseudopure states on any number of spins, subject to the limitations imposed by the loss of signal from the selected subensemble. The preparation of a single pseudopure state is demonstrated by carbon and proton NMR of

{}^{13}\mathrm{C}

-labeled alanine. With a judicious choice of magnetic-field gradients, the method further allows encoding of up to

{}^{13}\mathrm{C}

pseudopure states in independent spatial modes in an

N+1

spin system. Fast encoding and decoding schemes are demonstrated for the preparation of four such spatially labeled pseudopure states.

A method for creating pseudopure states in NMR is described that combines spatial averaging and logical labeling techniques. This method uses one labeling qubit and any desired pseudopure state of any number of qubits can be created in the subsystem manifolds of the labeling qubit. The method uses a nonselective $\ensuremath{\pi}/2$ radio frequency pulse on all the nonlabeling qubits to equalize the populations of all the states within each subsystem, followed by a single transition-selective $\ensuremath{\pi}$ pulse on one of the transitions of the labeling qubit. The transverse magnetization created during the process is destroyed by a magnetic field gradient pulse, yielding a subsystem pseudopure state. The method is demonstrated experimentally on a system of three qubits.

Quantum-information processing is carried out using dipolar coupled spins and high-resolution nuclear magnetic resonance (NMR). The systems chosen are the dipolar coupled methyl protons of

{\mathrm{CH}}_{3}\mathrm{CN}

partially oriented in a liquid crystalline matrix yielding a two-qubit system and dipolar coupled

{}^{13}\mathrm{C}

and methyl protons of

{\mathrm{CH}}_{3}\mathrm{CN}

also partially oriented in the liquid crystalline matrix, yielding a three-qubit system. The dipolar coupled protons of oriented

{\mathrm{CH}}_{3}

group are chemically and magnetically identical and their eigenstates can be divided into a set of quartet states (symmetric A) and a pair of doublet (E) states. We describe here a method for selectively retaining the magnetization of the symmetric states, yielding two and three qubit systems. We create pseudopure states using single-quantum-transition selective pulses and implement two- and three-qubit gates using one- and two-dimensional NMR.

The use of nuclear magnetic resonance (NMR) to carry out quantum information processing (QIP) often requires the preparation, transformation, and detection of pseudopure states. In our previous work, it was shown that the use of pairs of pseudopure states (POPS) as a basis for QIP is very convenient because of the simplicity in experimental execution. It is now further demonstrated that the product of the NMR spectra corresponding to two sets of POPS that share a common pseudopure state has the same peak frequencies as those of the common (single) pseudopure state. Examples of applying two different quantum logic gates to a 5-qubit system are given.

Quantum state preparation plays an equally important role with quantum operations and measurements in quantum information processing. The previous methods of preparing initial state for bulk quantum computation all have inevitable disadvantages, such as, requiring multiple experiments, causing loss of signals, or requiring molecules with restrictive structure. In this work, three kinds of quantum circuits are introduced to prepare the pseudo-pure states of (

n-1

) qubits in the Hilbert space of

n

coupled spins which merely need the assist of one ancilla spin and two experiments independent of

n

. Being without gradient fields effectively avoids the reduction of the signals. Our methods have no special requirements on the structure of the used molecules. To test these methods more comprehensively, we experimentally demonstrate the preparation of the labeled pseudo-pure states using heteronuclear 2-qubit and homonuclear 4-qubit nuclear magnetic resonance quantum information processor.

Nuclear magnetic resonance spectroscopy has demonstrated significant experimental progress toward the development of quantum computations. The developments so far have taken place mainly through the use of spin 12 nuclei. In this paper we describe the use of a spin 32 nucleus, oriented in a liquid crystal matrix for the creation of pseudopure states and the implementation of a complete set of two-qubit reversible logic gates using single-quantum transition-selective pulses, extending the range of practice of NMR toward quantum computation.