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arXiv:quant-ph/0012032 · Physical Review A

Realization of quantum process tomography in NMR

Andrew M. Childs, Isaac L. Chuang, Debbie Leung

Published Jun 13, 2001• 205 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.64.012314

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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.
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Developments of quantum techniques have drastically improved the control ability of current quantum processors. Characterizing quantum simulations performed on these processors has become a vital topic in quantum information processing. Traditional characterizing techniques such as quantum process and state tomography fully characterize quantum simulations, but always with the exponentially increased resource cost, unfortunately. It may place their applications only in small-scale quantum systems. Meanwhile, many potential characterizing methods have been put forward in the past decades. In this work we experimentally test and compare different methods for characterizing quantum simulations of Ising Hamiltonians on the nuclear magnetic resonance (NMR) platform, including the recent quantum Hamiltonian tomography with polynomial complexity to estimate the generator of quantum simulations and nonrepetitive direct fidelity estimation with the Monte Carlo method to estimate the fidelity of dynamical states. We also experimentally estimate the parameters of Hamiltonian via full quantum process tomography and quantum state tomography for the comparison. These methods have merits and demerits in accuracy, the required resource, and prior knowledge. Our experiments test the performance of different methods while providing a beneficial reference for characterizing quantum simulations in the future.
Quantum information processing has been the subject of countless discoveries since the early 1990's. It is believed to be the way of the future for computation: using quantum systems permits one to perform computation exponentially faster than on a regular classical computer. Unfortunately, quantum systems that not isolated do not behave well. They tend to lose their quantum nature due to the presence of the environment. If key information is known about the noise present in the system, methods such as quantum error correction have been developed in order to reduce the errors introduced by the environment during a given quantum computation. In order to harness the quantum world and implement the theoretical ideas of quantum information processing and quantum error correction, it is imperative to understand and quantify the noise present in the quantum processor and benchmark the quality of the control over the qubits. Usual techniques to estimate the noise or the control are based on quantum process tomography (QPT), which, unfortunately, demands an exponential amount of resources. \n \nThis thesis presents work towards the characterization of noisy processes in an efficient manner. The protocols are developed from a purely abstract setting with no system-dependent variables. To circumvent the exponential nature of quantum process tomography, three different efficient protocols are proposed and experimentally verified. The first protocol uses the idea of quantum error correction to extract relevant parameters about a given noise model, namely the correlation between the dephasing of two qubits. Following that is a protocol using randomization and symmetrization to extract the probability that a given number of qubits are simultaneously corrupted in a quantum memory, regardless of the specifics of the error and which qubits are affected. Finally, a last protocol, still using randomization ideas, is developed to estimate the average fidelity per computational gates for single and multi qubit systems. \n \nEven though liquid state NMR is argued to be unsuitable for scalable quantum information processing, it remains the best test-bed system to experimentally implement, verify and develop protocols aimed at increasing the control over general quantum information processors. For this reason, all the protocols described in this thesis have been implemented in liquid state NMR, which then led to further development of control and analysis techniques.
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.
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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.