臺灣博碩士論文加值系統

要實現量子計算(quantum computation)，我們需要一組高保真(high-fidelity)而且堅固(robust)的量子邏輯閘(quantum gate)，來對抗量子位元(qubit)系統中的噪音(noise)並容許系統參數(system parameter)的不準確性(uncertainty)。堅固控制方法(robust control method)可以提供控制脈波(control pulse)來操控並實現高保真而且堅固的量子邏輯閘。可是大部分的堅固控制方法都假設在量子位元系統中的噪音強度並不隨時間而改變，然而這個假設並不總是對的。因此我們提供一套有系統的堅固控制方法，可以有效地處理隨機(stochastic)並且可隨時間改變(time-varying)的噪音。我們的方法可以同時處理多個不同的噪音來源(multiple sources of noise)，可以運用到不同的量子位元系統與不同的噪音模型，並提供連續性(smooth)的控制脈波來操作高保真而且堅固的量子邏輯閘以實現容錯量子計算(fault-tolerant quantum computation)。

接著，我們將此堅固控制方法運用到一個實際的量子位元系統：半導體量子點電子自旋(quantum-dot electron spin)量子位元。最近，澳洲實驗團隊將此量子位元系統建構在純化的同位素矽(isotopically purified silicon)半導體上來改善來自量子位元環境的噪音，並實現二位元(two-qubit)量子邏輯閘。然而，操控二位元量子邏輯閘會伴隨著電的噪音(electrical noise)，而這個噪音使得二位元量子邏輯閘誤差(gate error)無法達到實現容錯量子計算的門檻(threshold)。我們的堅固控制方法可提供最佳化控制脈波(optimal control pulse)，來操控可以抵抗電的噪音之二位元量子邏輯閘，使得邏輯閘誤差遠低於此門檻，並且堅固於來自於系統參數的不準確性。此外我們的最佳化控制脈波也考慮到實驗上對於控制脈波的限制，像是最強脈波強度(maximum pulse strength)限制，還有由波形產生器(waveform generator)的有限頻寬(finite-bandwidth)所造成的濾波效應(filtering effect)。更進一步，我們在同一個控制架構下提供實驗上可實現的最佳化控制脈波，來操控高保真而且堅固的二位元量子邏輯閘與單一位元量子邏輯閘(single-qubit quantum gate)，為實現大尺度(large-scale)容錯量子計算提供重要的一步。

To realize practical quantum computation, a set of high-fidelity universal quantum gates robust against noise and uncertainty in the qubit system is prerequisite. Constructing control pulses to operate quantum gates which meet this requirement is an important and timely issue. In most robust control methods, noise is assumed to be quasi-static, i.e., is time-independent within the gate operation time but can vary between different gates. But this quasi-static-noise assumption is not always valid. Here we develop a systematic method to find pulses for quantum gate operations robust against both low- and high-frequency (comparable to the qubit transition frequency) stochastic time-varying noise. Our approach, taking into account the noise properties of quantum computing systems, can output single smooth pulses in the presence of multisources of noise. Furthermore, our method can be applied to different system models and noise models, and will make essential steps toward constructing high-fidelity and robust quantum gates for fault-tolerant quantum computation (FTQC). We also discuss and compare the gate operation performance by our method with that by the filter-transfer-function method.

Then we apply our robust control method for a realistic system of electron spin qubits in semiconductor (silicon) quantum dots, a promising solid-state system compatible with existing manufacturing technologies for practical quantum computation. A two-qubit controlled-NOT (CNOT) gate, realized by a controlled-phase (C-phase) gate together with some single-qubit gates, has been experimentally implemented recently for quantum-dot electron spin qubits in isotopically purified silicon. But the infidelity of the two-qubit C-phase gate is, primarily due to the electrical control noise, still higher than the required error threshold for FTQC. Here we apply our robust control method to construct high-fidelity CNOT gates with single smooth control pulses robust against the electrical noise and the system parameter uncertainty. The experimental constraints on the maximum pulse strength due to the power limitation of the on-chip electron spin resonance (ESR) line and the filtering effects on the pulses due to the finite bandwidth of waveform generators are also accounted for. The robust and high-fidelity single-qubit gates, together with the two-qubit CNOT gates, can be performed within the same control framework in our scheme, paving the way for large-scale FTQC.

Acknowledgements I
Chinese abstract II
Abstract IV
1. Introduction 1
2. Robust quantum gates for stochastic time-varying noise 5
2.1. Ensemble average infidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2. Optimization method and noise suppression . . . . . . . . . . . . . . . . . . 8
2.3. Demonstration of our optimal control method . . . . . . . . . . . . . . . . . 10
2.3.1. Comparison with the quasi-static-noise method . . . . . . . . . . . . 11
2.3.1.1. Single-qubit gates . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1.2. Two-qubit gates . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2. Comparison with the filter-transfer-function method . . . . . . . . . 19
2.4. Generalization to open quantum system . . . . . . . . . . . . . . . . . . . . 21
3. Applications to quantum-dot electron spin qubits in isotopically purified silicon 28
3.1. Quantum-dot electron spin qubits . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2. Ideal system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3. Realistic system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4. Demonstration of our control scheme . . . . . . . . . . . . . . . . . . . . . 44
3.4.1. CNOT gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.2. Single-qubit gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4. Conclusion 56
A. : Derivation of Eqs. (2.12)-(2.14) 58
B. : Estimation of higher-order contributions 61
Bibliography 62

[1] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: Towards practical large-scale quantum computation,” Phys. Rev. A, vol. 86, p. 032324, Sep 2012.
[2] K. Khodjasteh and L. Viola, “Dynamically error-corrected gates for universal quantum computation,” Phys. Rev. Lett., vol. 102, p. 080501, Feb 2009.
[3] K. Khodjasteh and L. Viola, “Dynamical quantum error correction of unitary operations with bounded controls,” Phys. Rev. A, vol. 80, p. 032314, Sep 2009.
[4] K. Khodjasteh, D. A. Lidar, and L. Viola, “Arbitrarily accurate dynamical control in open quantum systems,” Phys. Rev. Lett., vol. 104, p. 090501, Mar 2010.
[5] K. Khodjasteh, H. Bluhm, and L. Viola, “Automated synthesis of dynamically corrected quantum gates,” Phys. Rev. A, vol. 86, p. 042329, Oct 2012.
[6] J. R. West, D. A. Lidar, B. H. Fong, and M. F. Gyure, “High fidelity quantum gates via dynamical decoupling,” Phys. Rev. Lett., vol. 105, p. 230503, Dec 2010.
[7] A. M. Souza, G. A. Álvarez, and D. Suter, “Experimental protection of quantum gates against decoherence and control errors,” Phys. Rev. A, vol. 86, p. 050301, Nov 2012.
[8] F. F. Fanchini, J. E. M. Hornos, and R. d. J. Napolitano, “Continuously decoupling a hadamard quantum gate from independent classes of errors,” Phys. Rev. A, vol. 76, p. 032319, Sep 2007.
[9] F. F. Fanchini, R. d. J. Napolitano, B. Çakmak, and A. O. Caldeira, “Protecting the sqrt-swap operation from general and residual errors by continuous dynamical decoupling,” Phys. Rev. A, vol. 91, p. 042325, Apr 2015.
[10] A. Z. Chaudhry and J. Gong, “Decoherence control: Universal protection of two-qubit states and two-qubit gates using continuous driving fields,” Phys. Rev. A, vol. 85, p. 012315, Jan 2012.
[11] X. Xu, Z. Wang, C. Duan, P. Huang, P. Wang, Y. Wang, N. Xu, X. Kong, F. Shi, X. Rong, and J. Du, “Coherence-protected quantum gate by continuous dynamical decoupling in diamond,” Phys. Rev. Lett., vol. 109, p. 070502, Aug 2012.
[12] J. Clausen, G. Bensky, and G. Kurizki, “Bath-optimized minimal-energy protection of quantum operations from decoherence,” Phys. Rev. Lett., vol. 104, p. 040401, Jan 2010.
[13] J. Clausen, G. Bensky, and G. Kurizki, “Task-optimized control of open quantum systems,” Phys. Rev. A, vol. 85, p. 052105, May 2012.
[14] M. Wenin and W. Pötz, “Minimization of environment-induced decoherence in quantum subsystems and application to solid-state-based quantum gates,” Phys. Rev. B, vol. 78, p. 165118, Oct 2008.
[15] P. Rebentrost, I. Serban, T. Schulte-Herbrüggen, and F. K. Wilhelm, “Optimal control of a qubit coupled to a non-markovian environment,” Phys. Rev. Lett., vol. 102, p. 090401, Mar 2009.
[16] B. Hwang and H.-S. Goan, “Optimal control for non-markovian open quantum systems,” Phys. Rev. A, vol. 85, p. 032321, Mar 2012.
[17] J.-S. Tai, K.-T. Lin, and H.-S. Goan, “Optimal control of quantum gates in an exactly solvable non-markovian open quantum bit system,” Phys. Rev. A, vol. 89, p. 062310, Jun 2014.
[18] S.-Y. Huang and H.-S. Goan, “Optimal control for fast and high-fidelity quantum gates in coupled superconducting flux qubits,” Phys. Rev. A, vol. 90, p. 012318, Jul 2014.
[19] Y. Chou, S.-Y. Huang, and H.-S. Goan, “Optimal control of fast and high-fidelity quantum gates with electron and nuclear spins of a nitrogen-vacancy center in diamond,” Phys. Rev. A, vol. 91, p. 052315, May 2015.
[20] R. Tycko, “Broadband population inversion,” Phys. Rev. Lett., vol. 51, pp. 775-777, Aug 1983.
[21] M. H. Levitt, “Composite pulses,” Progress in Nuclear Magnetic Resonance Spectroscopy, vol. 18, no. 2, pp. 61-122, 1986.
[22] S. Wimperis, “Broadband, narrowband, and passband composite pulses for use in advanced NMR experiments,” Journal of Magnetic Resonance, Series A, vol. 109, no. 2, pp. 221-231, 1994.
[23] H. K. Cummins, G. Llewellyn, and J. A. Jones, “Tackling systematic errors in quantum logic gates with composite rotations,” Phys. Rev. A, vol. 67, p. 042308, Apr 2003.
[24] K. R. Brown, A. W. Harrow, and I. L. Chuang, “Arbitrarily accurate composite pulse sequences,” Phys. Rev. A, vol. 70, p. 052318, Nov 2004.
[25] L. M. K. Vandersypen and I. L. Chuang, “NMR techniques for quantum control and computation,” Rev. Mod. Phys., vol. 76, pp. 1037-1069, Jan 2005.
[26] T. Ichikawa, M. Bando, Y. Kondo, and M. Nakahara, “Designing robust unitary gates: Application to concatenated composite pulses,” Phys. Rev. A, vol. 84, p. 062311, Dec 2011.
[27] M. Bando, T. Ichikawa, Y. Kondo, and M. Nakahara, “Concatenated composite pulses compensating simultaneous systematic errors,” Journal of the Physical Society of Japan, vol. 82, no. 1, p. 014004, 2013.
[28] T. Ichikawa, U. Güngördü, M. Bando, Y. Kondo, and M. Nakahara, “Minimal and robust composite two-qubit gates with Ising-type interaction,” Phys. Rev. A, vol. 87, p. 022323, Feb 2013.
[29] J. T. Merrill and K. R. Brown, “Progress in compensating pulse sequences for quantum computation,” in Quantum Information and Computation for Chemistry, pp. 241-294, John Wiley & Sons, Inc., 2014.
[30] X. Wang, L. S. Bishop, J. P. Kestner, E. Barnes, K. Sun, and S. Das Sarma, “Composite pulses for robust universal control of singlet-triplet qubits,” Nat. Commun., vol. 3, p. 997, Aug. 2012.
[31] J. P. Kestner, X. Wang, L. S. Bishop, E. Barnes, and S. Das Sarma, “Noise-resistant control for a spin qubit array,” Phys. Rev. Lett., vol. 110, p. 140502, Apr 2013.
[32] X. Wang, L. S. Bishop, E. Barnes, J. P. Kestner, and S. Das Sarma, “Robust quantum gates for singlet-triplet spin qubits using composite pulses,” Phys. Rev. A, vol. 89, p. 022310, Feb 2014.
[33] X. Wang, F. A. Calderon-Vargas, M. S. Rana, J. P. Kestner, E. Barnes, and S. Das Sarma, “Noise-compensating pulses for electrostatically controlled silicon spin qubits,” Phys. Rev. B, vol. 90, p. 155306, Oct 2014.
[34] X. Rong, J. Geng, Z. Wang, Q. Zhang, C. Ju, F. Shi, C.-K. Duan, and J. Du, “Implementation of dynamically corrected gates on a single electron spin in diamond,” Phys. Rev. Lett., vol. 112, p. 050503, Feb 2014.
[35] X.-C. Yang and X.Wang, “Noise filtering of composite pulses for singlet-triplet qubits,” Scientific Reports, vol. 6, p. 28996, July 2016.
[36] C. Chen, D. Dong, R. Long, I. R. Petersen, and H. A. Rabitz, “Sampling-based learning control of inhomogeneous quantum ensembles,” Phys. Rev. A, vol. 89, p. 023402, Feb 2014.
[37] D. Dong, C. Chen, B. Qi, I. R. Petersen, and F. Nori, “Robust manipulation of superconducting qubits in the presence of fluctuations,” Scientific Reports, vol. 5, p. 7873, Jan. 2015.
[38] D. Dong, C. Wu, C. Chen, B. Qi, I. R. Petersen, and F. Nori, “Learning robust pulses for generating universal quantum gates,” Scientific Reports, vol. 6, p. 36090, Oct. 2016.
[39] J.-S. Li and N. Khaneja, “Control of inhomogeneous quantum ensembles,” Phys. Rev. A, vol. 73, p. 030302, Mar 2006.
[40] J. H. Lee, E. Montano, I. H. Deutsch, and P. S. Jessen, “Robust site-resolvable quantum gates in an optical lattice via inhomogeneous control,” Nat. Commun., vol. 4, p. 2027, June 2013.
[41] E. Barnes, X. Wang, and S. Das Sarma, “Robust quantum control using smooth pulses and topological winding,” Scientific Reports, vol. 5, p. 12685, Aug. 2015.
[42] D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett., vol. 111, p. 050404, Jul 2013.
[43] R. L. Kosut, M. D. Grace, and C. Brif, “Robust control of quantum gates via sequential convex programming,” Phys. Rev. A, vol. 88, p. 052326, Nov 2013.
[44] B. E. Anderson, H. Sosa-Martinez, C. A. Riofrío, I. H. Deutsch, and P. S. Jessen, “Accurate and robust unitary transformations of a high-dimensional quantum system,” Phys. Rev. Lett., vol. 114, p. 240401, Jun 2015.
[45] G. Gordon, G. Kurizki, and D. A. Lidar, “Optimal dynamical decoherence control of a qubit,” Phys. Rev. Lett., vol. 101, p. 010403, Jul 2008.
[46] A. Ruschhaupt, X. Chen, D. Alonso, and J. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New Journal of Physics, vol. 14, no. 9, p. 093040, 2012.
[47] X.-J. Lu, X. Chen, A. Ruschhaupt, D. Alonso, S. Guérin, and J. G. Muga, “Fast and robust population transfer in two-level quantum systems with dephasing noise and/or systematic frequency errors,” Phys. Rev. A, vol. 88, p. 033406, Sep 2013.
[48] T. J. Green, J. Sastrawan, H. Uys, and M. J. Biercuk, “Arbitrary quantum control of qubits in the presence of universal noise,” New Journal of Physics, vol. 15, no. 9, p. 095004, 2013.
[49] A. Soare, H. Ball, D. Hayes, J. Sastrawan, M. C. Jarratt, J. J. McLoughlin, X. Zhen, T. J. Green, and M. J. Biercuk, “Experimental noise filtering by quantum control,” Nat. Phys., vol. 10, pp. 825-829, Nov. 2014.
[50] H. Ball and M. J. Biercuk, “Walsh-synthesized noise lters for quantum logic,” EPJ Quantum Technology, vol. 2, p. 11, 2015.
[51] G. A. Paz-Silva and L. Viola, “General transfer-function approach to noise filtering in open-loop quantum control,” Phys. Rev. Lett., vol. 113, p. 250501, Dec 2014.
[52] W. D. Oliver, “Quantum control: Engineering a revolution,” Nat. Phys., vol. 10, pp. 794-795, Nov. 2014.
[53] C. Kabytayev, T. J. Green, K. Khodjasteh, M. J. Biercuk, L. Viola, and K. R. Brown, “Robustness of composite pulses to time-dependent control noise,” Phys. Rev. A, vol. 90, p. 012316, Jul 2014.
[54] D. Loss and D. P. DiVincenzo, “Quantum computation with quantum dots,” Phys. Rev. A, vol. 57, pp. 120 126, Jan 1998.
[55] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science, vol. 309, p. 2180, Sept. 2005.
[56] F. H. L. Koppens, C. Buizert, K. J. Tielrooij, I. T. Vink, K. C. Nowack, T. Meunier, L. P. Kouwenhoven, and L. M. K. Vandersypen, “Driven coherent oscillations of a single electron spin in a quantum dot,” Nature, vol. 442, pp. 766-771, Aug. 2006.
[57] F. H. L. Koppens, K. C. Nowack, and L. M. K. Vandersypen, “Spin echo of a single electron spin in a quantum dot,” Phys. Rev. Lett., vol. 100, p. 236802, Jun 2008.
[58] D. J. Reilly, J. M. Taylor, J. R. Petta, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Suppressing spin qubit dephasing by nuclear state preparation,” Science, vol. 321, p. 817, Aug. 2008.
[59] S. Foletti, H. Bluhm, D. Mahalu, V. Umansky, and A. Yacoby, “Universal quantum control of two-electron spin quantum bits using dynamic nuclear polarization,” Nat. Phys., vol. 5, pp. 903-908, Dec. 2009.
[60] H. Bluhm, S. Foletti, D. Mahalu, V. Umansky, and A. Yacoby, “Enhancing the coherence of a spin qubit by operating it as a feedback loop that controls its nuclear spin bath,” Phys. Rev. Lett., vol. 105, p. 216803, Nov 2010.
[61] C. Barthel, J. Medford, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Interlaced dynamical decoupling and coherent operation of a singlet-triplet qubit,” Phys. Rev. Lett., vol. 105, p. 266808, Dec 2010.
[62] H. Bluhm, S. Foletti, I. Neder, M. Rudner, D. Mahalu, V. Umansky, and A. Yacoby, “Dephasing time of GaAs electron-spin qubits coupled to a nuclear bath exceeding 200μs,” Nat. Phys., vol. 7, pp. 109-113, Feb. 2011.
[63] K. C. Nowack, M. Shafiei, M. Laforest, G. E. D. K. Prawiroatmodjo, L. R. Schreiber, C. Reichl, W. Wegscheider, and L. M. K. Vandersypen, “Single-shot correlations and two-qubit gate of solid-state spins,” Science, vol. 333, p. 1269, Sept. 2011.
[64] R. Brunner, Y.-S. Shin, T. Obata, M. Pioro-Ladrière, T. Kubo, K. Yoshida, T. Taniyama, Y. Tokura, and S. Tarucha, “Two-qubit gate of combined single-spin rotation and interdot spin exchange in a double quantum dot,” Phys. Rev. Lett., vol. 107, p. 146801, Sep 2011.
[65] M. D. Shulman, O. E. Dial, S. P. Harvey, H. Bluhm, V. Umansky, and A. Yacoby, “Demonstration of entanglement of electrostatically coupled singlet-triplet qubits,” Science, vol. 336, p. 202, Apr. 2012.
[66] J. Medford, J. Beil, J. M. Taylor, S. D. Bartlett, A. C. Doherty, E. I. Rashba, D. P. DiVincenzo, H. Lu, A. C. Gossard, and C. M. Marcus, “Self-consistent measurement and state tomography of an exchange-only spin qubit,” Nat. Nanotech., vol. 8, pp. 654-659, Sept. 2013.
[67] J. M. Nichol, L. A. Orona, S. P. Harvey, S. Fallahi, G. C. Gardner, M. J. Manfra, and A. Yacoby, “High-fidelity entangling gate for double-quantum-dot spin qubits,” npj Quantum Information, vol. 3, no. 1, p. 3, 2017.
[68] E. A. Chekhovich, M. N. Makhonin, A. I. Tartakovskii, A. Yacoby, H. Bluhm, K. C. Nowack, and L. M. K. Vandersypen, “Nuclear spin effects in semiconductor quantum dots,” Nat. Mater., vol. 12, pp. 494-504, June 2013.
[69] B. M. Maune, M. G. Borselli, B. Huang, T. D. Ladd, P. W. Deelman, K. S. Holabird, A. A. Kiselev, I. Alvarado-Rodriguez, R. S. Ross, A. E. Schmitz, M. Sokolich, C. A. Watson, M. F. Gyure, and A. T. Hunter, “Coherent singlet-triplet oscillations in a silicon-based double quantum dot,” Nature, vol. 481, pp. 344-347, Jan. 2012.
[70] E. Kawakami, P. Scarlino, D. R. Ward, F. R. Braakman, D. E. Savage, M. G. Lagally, M. Friesen, S. N. Coppersmith, M. A. Eriksson, and L. M. K. Vandersypen, “Electrical control of a long-lived spin qubit in a Si/SiGe quantum dot,” Nat. Nanotech., vol. 9, pp. 666-670, Sept. 2014.
[71] M. Veldhorst, J. C. C. Hwang, C. H. Yang, A. W. Leenstra, B. de Ronde, J. P. Dehollain, J. T. Muhonen, F. E. Hudson, K. M. Itoh, A. Morello, and A. S. Dzurak, “An addressable quantum dot qubit with fault-tolerant control-fidelity,” Nat. Nanotech., vol. 9, pp. 981-985, Dec. 2014.
[72] M. Veldhorst, C. H. Yang, J. C. C. Hwang, W. Huang, J. P. Dehollain, J. T. Muhonen, S. Simmons, A. Laucht, F. E. Hudson, K. M. Itoh, A. Morello, and A. S. Dzurak, “A two-qubit logic gate in silicon,” Nature, vol. 526, pp. 410-414, Oct. 2015.
[73] K. Eng, T. D. Ladd, A. Smith, M. G. Borselli, A. A. Kiselev, B. H. Fong, K. S. Holabird, T. M. Hazard, B. Huang, P. W. Deelman, I. Milosavljevic, A. E. Schmitz, R. S. Ross, M. F. Gyure, and A. T. Hunter, “Isotopically enhanced triple-quantum-dot qubit,” Science Advances, vol. 1, no. 4, 2015.
[74] K. Takeda, J. Kamioka, T. Otsuka, J. Yoneda, T. Nakajima, M. R. Delbecq, S. Amaha, G. Allison, T. Kodera, S. Oda, and S. Tarucha, “A fault-tolerant addressable spin qubit in a natural silicon quantum dot,” Science Advances, vol. 2, no. 8, 2016.
[75] F. J. Dyson, “The radiation theories of Tomonaga, Schwinger, and Feynman,” Phys. Rev., vol. 75, pp. 486-502, Feb. 1949.
[76] J. A. Nelder and R. Mead, “A simplex method for function minimization,” The computer journal, vol. 7, no. 4, pp. 308-313, 1965.
[77] S. Finch and Y. Yt, Ornstein-Uhlenbeck process. Citeseer, 2004.
[78] A. V. Kuhlmann, J. Houel, A. Ludwig, L. Greuter, D. Reuter, A. D.Wieck, M. Poggio, and R. J. Warburton, “Charge noise and spin noise in a semiconductor quantum device,” Nat. Phys., vol. 9, pp. 570-575, Sept. 2013.
[79] Y. Li, N. Sinitsyn, D. L. Smith, D. Reuter, A. D. Wieck, D. R. Yakovlev, M. Bayer, and S. A. Crooker, “Intrinsic spin fluctuations reveal the dynamical response function of holes coupled to nuclear spin baths in (In, Ga)As quantum dots,” Phys. Rev. Lett., vol. 108, p. 186603, May 2012.
[80] R. A. Żak, B. Röthlisberger, S. Chesi, and D. Loss, “Quantum computing with electron spins in quantum dots,” arXiv preprint arXiv: 0906.4045, 2009.
[81] W. H. Lim, F. A. Zwanenburg, H. Huebl, M. Möttönen, K. W. Chan, A. Morello, and A. S. Dzurak, “Observation of the single-electron regime in a highly tunable silicon quantum dot,” Appl. Phys. Lett., vol. 95, p. 242102, Dec. 2009.
[82] T. Meunier, V. E. Calado, and L. M. K. Vandersypen, “Efficient controlled-phase gate for single-spin qubits in quantum dots,” Phys. Rev. B, vol. 83, p. 121403, Mar 2011.
[83] L. S. Theis, F. Motzoi, and F. K. Wilhelm, “Simultaneous gates in frequency-crowded multilevel systems using fast, robust, analytic control shapes,” Phys. Rev. A, vol. 93, p. 012324, Jan 2016.
[84] F. Motzoi, J. M. Gambetta, S. T. Merkel, and F. K. Wilhelm, “Optimal control methods for rapidly time-varying Hamiltonians,” Phys. Rev. A, vol. 84, p. 022307, Aug 2011.
[85] D. J. Egger and F. K. Wilhelm, “Adaptive hybrid optimal quantum control for imprecisely characterized systems,” Phys. Rev. Lett., vol. 112, p. 240503, Jun 2014.
[86] J. Kelly, R. Barends, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, I.-C. Hoi, E. Jeffrey, A. Megrant, J. Mutus, C. Neill, P. J. J. O''Malley, C. Quintana, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, “Optimal quantum control using randomized benchmarking,” Phys. Rev. Lett., vol. 112, p. 240504, Jun 2014.
[87] C. Ferrie and O. Moussa, “Robust and efficient in situ quantum control,” Phys. Rev. A, vol. 91, p. 052306, May 2015.
[88] E. Magesan, J. M. Gambetta, and J. Emerson, “Scalable and robust randomized benchmarking of quantum processes,” Phys. Rev. Lett., vol. 106, p. 180504, May 2011.
[89] A. D. Córcoles, J. M. Gambetta, J. M. Chow, J. A. Smolin, M. Ware, J. Strand, B. L. T. Plourde, and M. Steffen, “Process verification of two-qubit quantum gates by randomized benchmarking,” Phys. Rev. A, vol. 87, p. 030301, Mar 2013.
[90] S. Machnes, D. J. Tannor, F. K. Wilhelm, and E. Assémat, “Gradient optimization of analytic controls: the route to high accuracy quantum optimal control,” arXiv preprint arXiv: 1507.04261, 2015.
[91] Z. Chen, J. Kelly, C. Quintana, R. Barends, B. Campbell, Y. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Lucero, E. Jeffrey, A. Megrant, J. Mutus, M. Neeley, C. Neill, P. J. J. O''Malley, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, T. C. White, A. N. Korotkov, and J. M. Martinis, “Measuring and suppressing quantum state leakage in a superconducting qubit,” Phys. Rev. Lett., vol. 116, p. 020501, Jan 2016.
[92] J. Scheuer, X. Kong, R. S. Said, J. Chen, A. Kurz, L. Marseglia, J. Du, P. R. Hemmer, S. Montangero, T. Calarco, B. Naydenov, and F. Jelezko, “Precise qubit control beyond the rotating wave approximation,” New Journal of Physics, vol. 16, no. 9, p. 093022, 2014.
[93] F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, J. Biamonte, and J. Wrachtrup, “High- delity spin entanglement using optimal control,” Nat. Commun., vol. 5, p. 3371, Feb. 2014.
[94] X. Rong, J. Geng, F. Shi, Y. Liu, K. Xu, W. Ma, F. Kong, Z. Jiang, Y. Wu, and J. Du, “Experimental fault-tolerant universal quantum gates with solid-state spins under ambient conditions,” vol. 6, p. 8748, Nov. 2015.