Quantum process tomography of the quantum Fourier transform.

The results of quantum process tomography on a three-qubit nuclear magnetic resonance quantum information processor are presented and shown to be consistent with a detailed model of the system-plus-apparatus used for the experiments. The quantum operation studied was the quantum Fourier transform, which is important in several quantum algorithms and poses a rigorous test for the precision of our recently developed strongly modulating control fields. The results were analyzed in an attempt to decompose the implementation errors into coherent (overall systematic), incoherent (microscopically deterministic), and decoherent (microscopically random) components. This analysis yielded a superoperator consisting of a unitary part that was strongly correlated with the theoretically expected unitary superoperator of the quantum Fourier transform, an overall attenuation consistent with decoherence, and a residual portion that was not completely positive-although complete positivity is required for any quantum operation. By comparison with the results of computer simulations, the lack of complete positivity was shown to be largely a consequence of the incoherent errors which occurred over the full quantum process tomography procedure. These simulations further showed that coherent, incoherent, and decoherent errors can often be identified by their distinctive effects on the spectrum of the overall superoperator. The gate fidelity of the experimentally determined superoperator was 0.64, while the correlation coefficient between experimentally determined superoperator and the simulated superoperator was 0.79; most of the discrepancies with the simulations could be explained by the cumulative effect of small errors in the single qubit gates.

[1]  Christof Zalka.  Simulating quantum systems on a quantum computer , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[2]  David G. Cory,et al.  Experimental demonstration of an entanglement swapping operation and improved control in NMR quantum-information processing , 2003 .

[3]  E M Fortunato,et al.  NMR analog of the quantum disentanglement eraser. , 2001, Physical review letters.

[4]  Measuring Quantum Optical Hamiltonians , 1998, quant-ph/9805032.

[5]  Lorenza Viola,et al.  Hadamard products of product operators and the design of gradient-diffusion experiments for simulating decoherence by NMR spectroscopy , 2000, quant-ph/0009010.

[6]  Isaac L. Chuang,et al.  Prescription for experimental determination of the dynamics of a quantum black box , 1997 .

[7]  Timothy F. Havel,et al.  NMR Based Quantum Information Processing: Achievements and Prospects , 2000, quant-ph/0004104.

[8]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[9]  Timothy F. Havel.  The Real Density Matrix , 2002, Quantum Inf. Process..

[10]  Vladimir Buzek,et al.  Dynamics of open quantum systems initially entangled with environment: Beyond the Kraus representation , 2001, quant-ph/0108136.

[11]  Timothy F. Havel,et al.  Design of strongly modulating pulses to implement precise effective Hamiltonians for quantum information processing , 2002, quant-ph/0202065.

[12]  Timothy F. Havel,et al.  Multiqubit logic gates in NMR quantum computing , 2000 .

[13]  Timothy F. Havel,et al.  Expressing the operations of quantum computing in multiparticle geometric algebra , 1998 .

[14]  Schumacher,et al.  Sending entanglement through noisy quantum channels. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[15]  Timothy F. Havel,et al.  Robust method for estimating the Lindblad operators of a dissipative quantum process from measurements of the density operator at multiple time points , 2003 .

[16]  Timothy F. Havel,et al.  Robust control of quantum information , 2003, quant-ph/0307062.

[17]  Don Coppersmith,et al.  The Data Encryption Standard (DES) and its strength against attacks , 1994, IBM J. Res. Dev..

[18]  R. Jozsa.  Quantum algorithms and the Fourier transform , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[19]  Peter Hawkes,et al.  Advances in Imaging and Electron Physics , 2002 .

[20]  D. Leung,et al.  Choi’s proof as a recipe for quantum process tomography , 2003 .

[21]  V. Buzek,et al.  Erratum: Dynamics of open quantum systems initially entangled with environment: Beyond the Kraus representation [Phys. Rev. A 64, 062106 (2001)] , 2003 .

[22]  P. Zoller,et al.  Complete Characterization of a Quantum Process: The Two-Bit Quantum Gate , 1996, quant-ph/9611013.

[23]  R. Schack.  Using a quantum computer to investigate quantum chaos , 1997, quant-ph/9705016.

[24]  G. Bodenhausen,et al.  Principles of nuclear magnetic resonance in one and two dimensions , 1987 .

[25]  E M Fortunato,et al.  Implementation of the quantum Fourier transform. , 2001, Physical review letters.

[26]  Timothy F. Havel.  Robust procedures for converting among Lindblad, Kraus and matrix representations of quantum dynamical semigroups , 2002, quant-ph/0201127.

[27]  Quantum computers in phase space , 2002, quant-ph/0204149.

[28]  Debbie W. Leung,et al.  Realization of quantum process tomography in NMR , 2000, quant-ph/0012032.

[29]  Seth Lloyd,et al.  Experimental implementation of the quantum baker's map. , 2002, Physical review letters.