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Physical Review A

Ensemble quantum-information processing by NMR: Spatially averaged logical labeling technique for creating pseudopure states

T. S. Mahesh, Anil Kumar

Published Jun 6, 2001• 25 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.64.012307

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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 \ensuremathπ/2\ensuremath{\pi}/2\ensuremathπ/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 \ensuremathπ\ensuremath{\pi}\ensuremathπ 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 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 13C{}^{13}\mathrm{C}13C-labeled alanine. With a judicious choice of magnetic-field gradients, the method further allows encoding of up to 2N{2}^{N}2N pseudopure states in independent spatial modes in an N+1N+1N+1 spin system. Fast encoding and decoding schemes are demonstrated for the preparation of four such spatially labeled pseudopure states.
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 CH3CN{\mathrm{CH}}_{3}\mathrm{CN}CH3​CN partially oriented in a liquid crystalline matrix yielding a two-qubit system and dipolar coupled 13C{}^{13}\mathrm{C}13C and methyl protons of 13CH3CN{}^{13}{\mathrm{CH}}_{3}\mathrm{CN}13CH3​CN also partially oriented in the liquid crystalline matrix, yielding a three-qubit system. The dipolar coupled protons of oriented CH3{\mathrm{CH}}_{3}CH3​ 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.
We report the first implementation of NMR quantum information processing in a static single-crystal solid, illustrating pseudopure state preparation and gate manipulations on a three-qubit system, U–13C2, N15-glycine (H215N13CH213COOH). We consider issues important to single-crystal NMR implementations of quantum information processing and introduce several innovations specific to such implementations. In particular, concurrent cross-polarization of both the C13 and N15 spins from proton magnetization enhances and equalizes populations on these nuclei in a single step. This multiple cross-polarization step simplifies preparation of pseudopure spin states relative to previously described multiple-pulse sequences. Also, proton decoupling, which prolongs coherence during spin-evolution and detection periods, is switched off to dephase unwanted coherences during preparation of pseudopure spin states. Such “relaxation averaging” (or T2 averaging) is an alternative to spatial averaging, which requires special hardware and large gradient strengths for small crystals.
The premise of nuclear magnetic resonance (NMR) quantum computing is that pseudopure quantum states, for which the populations of all energy levels but one are equal, can be used instead of pure quantum states. However, for systems with a large number of quantum bits (qubits), it would require elaborate pulse sequences with many steps to prepare these pseudopure states. To overcome this problem, a method using pairs of pseudopure states (POPS) instead of individual pseudopure states as the basis for NMR quantum computing is introduced here. The POPS approach has a significant advantage over the existing methods that, regardless of the number of qubits, only a single step is needed to prepare each pair of pseudopure states. A detailed analysis of a 16-level spin system is given, and the results of applying a c2{c}^{2}c2-NOT quantum logic gate (a controlled-NOT gate conditional upon the state of two other qubits) to all the POPS in the system are presented.
A critical step in experimental quantum information processing (QIP) is to implement control of quantum systems protected against decoherence via informational encodings, such as quantum error-correcting codes, noiseless subsystems, and decoherence-free subspaces. These encodings lead to the promise of fault-tolerant QIP, but they come at the expense of resource overheads. Part of the challenge in studying control over multiple logical qubits is that QIP testbeds have not had sufficient resources to analyze encodings beyond the simplest ones. The most relevant resources are the number of available qubits and the cost to initialize and control them. Here we demonstrate an encoding of logical information that permits control over multiple logical qubits without full initialization, an issue that is particularly challenging in liquid-state NMR. The method of subsystem pseudopure states will allow the study of decoherence control schemes on up to six logical qubits using liquid-state NMR implementations.