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Acta Physica Sinica

Subspace quantum process tomography via nuclear magnetic resonance

Yao Xi-Wei, Zeng Bi-Rong, Qin Liu, Mu Xiao-Yang, Xingcheng Lin, Yang Chun, Pan Jian, Chen Zhong

Published Jan 1, 2010• 3 citations
Publisher

Abstract

Experimental investigation of subspace quantum process tomography in three-spin system was implemented via nuclear magnetic resonance. A quantum process was characterized by measuring a complete set of input states and corresponding outputs. The method using ancillary qubit remarkably reduces the number of the initial input states. And the pulse sequences used in this paper have fewer J-coupling evolutions. The experiment time was shortened and quantum decoherence of the system was weakened efficiently.

DOI
10.7498/aps.59.6837

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Experimental investigation of subspace quantum process tomography in three-spin system was implemented via nuclear magnetic resonance. A quantum process was characterized by measuring a complete set of input states and corresponding outputs. The method using ancillary qubit remarkably reduces the number of the initial input states. And the pulse sequences used in this paper have fewer J-coupling evolutions. The experiment time was shortened and quantum decoherence of the system was weakened efficiently.
With the development of quantum information and quantum computation science, quantum information processor has been widely used in different areas such as quantum simulation, quantum computation and quantum metrology and so on. To make quantum computer come true, we need to increase the number of controllable qubits of the system and improve the controllability to perform complex quantum manipulation. As a good experimental testbed for quantum information processing, nuclear magnetic resonance (NMR) spin system provides rich and sophisticated quantum control methods. In recent years a lot of multi-qubit experiments have been performed on the platform and a series of experimental technologies have been developed. In this paper, we firstly explain the difficulties of multi-qubit NMR experiments. Then by focusing on the experiment of 7-qubit labelled pseudo-pure state preparation and other relevant experiments, we review the technologies in multi-qubit experiments. Using the radio frequency selective method, the inhomogeneities of the radio frequency pulses are reduced and the spectral resolution is improved. After performing 1/2 spin selective sequence, we can regard the three methyl protons in the sample of crotonic acid as a single 1/2 spin nucleus and treat the whole molecule as a 7-qubit quantum information processor. We utilize Gauss pulses, Hermite pulses, composite pulses and gradient ascent pulse engineering (GRAPE) pulses to implement basic /2 and rotation operations. The GRAPE pulses are calculated by subspace GRAPE program to speed up the computation greatly. The errors of the basic pulses caused by chemical shift and J-coupling evolution can be estimated by the program of pulse compilation. It divides the errors of the pulses into a series of post-errors and pre-errors. A program of sequence compilation is used to eliminate the accumulated error of the whole pulse sequence, reduce the number of pulses and optimize the experimental duration. A variety of methods of quantum state tomography have been proposed to improve the efficiency of reading out information about quantum state. As an experimental example, we combine the above experimental technologies and perform the experiment of 7-qubit labelled pseudo-pure state preparation by using the method of cat state preparation. The sequence of cat state preparation consists of three steps:encoding procedure, phase cycling and decoding procedure. We use 14 experiments to realize the phase cycling and acquire the final 7-qubit labelled pseudo-pure state. The total duration of experimental sequence is about 132 ms. All the readout spectra have the similar shapes to the theoretical expectations. Finally we give an outlook for further research in this direction.
NMR quantum computing with qubit systems represented by nuclear spins (I=12) in small molecules in liquids has led to the most successful experimental quantum information processors so far. We use the quadrupolar spin-32 sodium nuclei of a NaNO3 single crystal as a virtual two-qubit system. The large quadrupolar coupling in comparison with the environmental interactions and the usage of strongly modulating pulses allow us to manipulate the system fast enough and at the same time keeping the decoherence reasonably slow. The experimental challenge is to characterize the "calculation" behavior of the quantum processor by process tomography which is here adapted to the quadrupolar spin system. The results of a selection of quantum gates and algorithms are presented as well as a detailed analysis of experimental results.
In the present work the first virtual 2 Qubit solid state NMR quantum processor, using a spin- 3/2 Na quadrupolar nucleus in a NaNO3 single crystal, is presented. Methods from liquid NMR quantum information processing, like strongly modulating pulses, are adopted to the quadrupolar spin system to implement the unitary operations and to generate pseudo pure states. Experimentally a set of quantum algorithms as well as the decoherence process is characterized by quantum process tomography which is developed for the quadrupolar spin system. A detailed analyses of the basic conditions of the experimental quantum process tomography procedure is presented, showing that even in such non-Markovian systems this method is valid.
We perform quantum state tomography (QST) in a three-qubit system consisting of strongly coupled nuclear spins, known in the NMR literature as ABXABXABX systems. We find that the number of experiments necessary to perform QST in such systems can be reduced with respect to those containing three qubits weakly coupled, which reduces the experimental effort required for the complete density-matrix reconstruction. To test the procedure we implement the full protocol for quantum teleportation. The tomographic results demonstrate that the density matrix can be reconstructed with fewer operations and high fidelity.
The standard procedure for quantum process tomography (QPT) requires a series of experiments. Each experiment involves initialization of the system to a particular basis state, applying the quantum process $\ensuremath{\varepsilon}$ on the system, and finally characterizing the output state by quantum state tomography (QST). The output states collected for a complete set of basis states enable us to calculate the $\ensuremath{\chi}$ matrix characterizing the process $\ensuremath{\varepsilon}$. The standard procedure for QST itself requires independent experiments, each involving measurement of a set of commuting observables. Thus QPT procedure demands a number of independent measurements and, moreover, this number increases rapidly with the size of the system. However, in ensemble systems, the total number of independent measurements can be greatly reduced with the availability of ancilla qubits. Ancilla-assisted process tomography (AAPT) has earlier been shown to require a single QST of system-ancilla space. Ancilla-assisted quantum state tomography (AAQST) has also been shown to perform QST in a single-scan measurement of an ensemble system. Here we combine AAPT with AAQST to realize a single-scan QPT (SSPT), a procedure to characterize a general quantum process in a single ensemble measurement. We demonstrate experimental SSPT by characterizing several single-qubit processes using a three-qubit NMR quantum register. Furthermore, using the SSPT procedure, we experimentally characterize the twirling process and compare the results with theory.
Quantum process tomography is a procedure by which the unknown dynamical evolution of an open quantum system can be fully experimentally characterized. We demonstrate explicitly how this procedure can be implemented with a nuclear magnetic resonance quantum computer. This allows us to measure the fidelity of a controlled-NOT logic gate and to experimentally investigate the error model for our computer. Based on the latter analysis, we test an important assumption underlying many models of quantum error correction, the independence of errors on different qubits.