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Quantum Fourier transform (QFT) is a quantum analogue of the classical discrete Fourier transform. It is a fundamental quantum gate in quantum algorithms which has an exponential advantage over the classical computation and has been excessively studied. Normally, an n-qubit quantum Fourier transform could be resolved into the tensor product of n single-qubit operations, and each operation could be implemented by a Hadamard gate and a controlled phase gate. Then the complexity of an n-qubit QFT is of order O(n2). To reduce the complexity of quantum operations, optimal control (OC) method has recently been used successfully to find the minimum time for implementing a quantum operation. Up to now, two types of quantum optimal control methods have been presented, i.e. analytical and numerical methods. The analytical approach is to change the problem of efficient synthesis of unitary transformations into the geometrical one of finding the shortest paths. Numerical optimal control procedures are based on the gradient methods (GRAPE, Gradient Ascent Pulse Engineering) and Krotov methods. Notable application mainly focus on nuclear magnetic resonance fields, including imaging, liquid-state NMR, solid-state NMR, and NMR quantum computation. One obvious advantage of optimal control NMR quantum computation is that the OC unitary evolution transformation pulse sequences are normally shorter than the conventional corresponding ones. Here we use the optimal control method to find the minimum duration for implementing QFT quantum gate. A linear spin chain with nearest-neighbor Ising interaction is used to find the optimization. And the optimized pulse sequence is experimentally demonstrated on an NMR quantum information processor. By using optimal control method with numerical calculation, a three-qubit QFT in an indirect-linear-coupling chain system is optimized. The duration of the OC QFT is obviously shorter than that of conventional approaches. The OC pulse sequence has been experimentally implemented on a liquid-state NMR spectrometer. To verify the optimally controlled pulse sequence for the three-qubit QFT, different initial states are assumed. The accuracy of the OC pulse sequence could be demonstrated by the consistency of theoretical simulation spectra with the experimental results. The good consistency between the simulation and the experimental spectra demonstrates that the OC QFT is of high fidelity.
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[1] Nielsen M A, Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)
[2] Long G L 2010 Physics 39 0
[3] Fu X Q, Bao W S, Li F D, Zhang Y C 2014 Chin. Phys. B 23 020306
[4] Weinstein Y S, Pravia M A, Fortunato E M, Lloyd S, Cory D G 2001 Phys. Rev. Lett. 86 1889
[5] Shor P 1994 Algorithms for quantum computation: discrete logarithms and factoring. Proc. 35th Ann. Symp. on Found. Of Comp. Sci. (IEEE Comp. Soc. Press) pp124-134
[6] Ekert A, Jozsa R 1996 Rev. of Mod. Phy 68 733
[7] D'Ariano G M, Macchiavello C, Sacchi M F 1998 Phys. Lett. A 248 103
[8] Cooley J W, Tukey J W 1965 Math Comput. 19 297
[9] Pang C Y, Hu B Q 2008 Chin. Phys. B 17 3220
[10] Fang X M, Zhu X W, Feng M, MaoX A, Du D 2000 Chin. Sci. Bull. 45 1071
[11] Yaakov S, Weinstein W, Lloyd S, Cory D G 2001 Phys. Rev. Lett. 86 1889
[12] Yu L B, Xue Z Y 2010 Chin. Phys. Lett. 27 070305
[13] Ren G, Du J M, Yu H J 2014 Chin. Phys. B 23 024207
[14] Zheng S B 2007 Common. Theor. Phys. 47 1049
[15] Huang D Z, Chen Z G, Guo Y 2009 Common. Theor. Phys. 51 221
[16] Beth T, Verfahren der schnellen Fourier-Transformation. Teubner, Stuttgart, 1984
[17] Khaneja N, Li J S, Kehlet C, Luy B, Glaser S J 2004 Proc. Natl. Acad. Sci. USA 101 14742
[18] Khaneja N, Heitmann B, Spörl A, Yuan H, Schulte-Herbrüggen T, Glaser S J 2007 Phys. Rev. A 75 012322
[19] Carlini A, Koike T 2013 J. Phys. A: Math. Theor. 46 045307
[20] Khaneja N, Reiss T, Kehlet C, Schulte-Herbruggen T, Glaser S J 2005 J . Magn. Reson. 172 296
[21] Maximov I, Tosner Z, Nielsen N C 2008 J. Chem. Phys. 128 184505
[22] Tosner Z, Vosegaard T, Kehlet C T, Khaneja N, Glaser S J, Nielsen N C 2009 J. Magn. Reson. 197 120
[23] Li Z K, Yung M H, Chen H W, Lu D W, Whitfield J D, Peng X H, Aspuru-Guzik A, Du J F 2011 Sci. Rep. 1 88
[24] Lu D W, Xu N Y, Xu R X, Chen HW, Gong J B, Peng X H, Du J F 2011 Phys. Rev. Lett. 107 020501
[25] Feng G R, Xu G F, Long G L 2013 Phys. Rev. Lett. 110 190501
[26] Feng G R, Lu Y, Hao L, Zhang F H, Long G L 2013 Sci. Rep. 3 2232
[27] Wei D X, Spörl A, Chang Y, Khaneja N, Yang X D, Glaser S J 2014 Chem. Phys. Lett. 612 143
[28] Schulte-Herbrüggen T, Spörl A, Khaneja N, Glaser S J 2005 Phys. Rev. A 72 042331
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[1] Nielsen M A, Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)
[2] Long G L 2010 Physics 39 0
[3] Fu X Q, Bao W S, Li F D, Zhang Y C 2014 Chin. Phys. B 23 020306
[4] Weinstein Y S, Pravia M A, Fortunato E M, Lloyd S, Cory D G 2001 Phys. Rev. Lett. 86 1889
[5] Shor P 1994 Algorithms for quantum computation: discrete logarithms and factoring. Proc. 35th Ann. Symp. on Found. Of Comp. Sci. (IEEE Comp. Soc. Press) pp124-134
[6] Ekert A, Jozsa R 1996 Rev. of Mod. Phy 68 733
[7] D'Ariano G M, Macchiavello C, Sacchi M F 1998 Phys. Lett. A 248 103
[8] Cooley J W, Tukey J W 1965 Math Comput. 19 297
[9] Pang C Y, Hu B Q 2008 Chin. Phys. B 17 3220
[10] Fang X M, Zhu X W, Feng M, MaoX A, Du D 2000 Chin. Sci. Bull. 45 1071
[11] Yaakov S, Weinstein W, Lloyd S, Cory D G 2001 Phys. Rev. Lett. 86 1889
[12] Yu L B, Xue Z Y 2010 Chin. Phys. Lett. 27 070305
[13] Ren G, Du J M, Yu H J 2014 Chin. Phys. B 23 024207
[14] Zheng S B 2007 Common. Theor. Phys. 47 1049
[15] Huang D Z, Chen Z G, Guo Y 2009 Common. Theor. Phys. 51 221
[16] Beth T, Verfahren der schnellen Fourier-Transformation. Teubner, Stuttgart, 1984
[17] Khaneja N, Li J S, Kehlet C, Luy B, Glaser S J 2004 Proc. Natl. Acad. Sci. USA 101 14742
[18] Khaneja N, Heitmann B, Spörl A, Yuan H, Schulte-Herbrüggen T, Glaser S J 2007 Phys. Rev. A 75 012322
[19] Carlini A, Koike T 2013 J. Phys. A: Math. Theor. 46 045307
[20] Khaneja N, Reiss T, Kehlet C, Schulte-Herbruggen T, Glaser S J 2005 J . Magn. Reson. 172 296
[21] Maximov I, Tosner Z, Nielsen N C 2008 J. Chem. Phys. 128 184505
[22] Tosner Z, Vosegaard T, Kehlet C T, Khaneja N, Glaser S J, Nielsen N C 2009 J. Magn. Reson. 197 120
[23] Li Z K, Yung M H, Chen H W, Lu D W, Whitfield J D, Peng X H, Aspuru-Guzik A, Du J F 2011 Sci. Rep. 1 88
[24] Lu D W, Xu N Y, Xu R X, Chen HW, Gong J B, Peng X H, Du J F 2011 Phys. Rev. Lett. 107 020501
[25] Feng G R, Xu G F, Long G L 2013 Phys. Rev. Lett. 110 190501
[26] Feng G R, Lu Y, Hao L, Zhang F H, Long G L 2013 Sci. Rep. 3 2232
[27] Wei D X, Spörl A, Chang Y, Khaneja N, Yang X D, Glaser S J 2014 Chem. Phys. Lett. 612 143
[28] Schulte-Herbrüggen T, Spörl A, Khaneja N, Glaser S J 2005 Phys. Rev. A 72 042331
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