Source: http://aoot.osa.org/oe/abstract.cfm?uri=oe-27-6-8267
Timestamp: 2019-04-19 02:20:57+00:00

Document:
High-fidelity qubit initialization is of significance for efficient error correction in fault tolerant quantum algorithms. Combining two best worlds, speed and robustness, to achieve high-fidelity state preparation and manipulation is challenging in quantum systems, where qubits are closely spaced in frequency. Motivated by the concept of shortcut to adiabaticity, we theoretically propose the shortcut pulses via inverse engineering and further optimize the pulses with respect to systematic errors in frequency detuning and Rabi frequency. Such protocol, relevant to frequency selectivity, is applied to rare-earth ions qubit system, where the excitation of frequency-neighboring qubits should be prevented as well. Furthermore, comparison with adiabatic complex hyperbolic secant pulses shows that these dedicated initialization pulses can reduce the time that ions spend in the excited state by a factor of 6, which is important in coherence time limited systems to approach an error rate manageable by quantum error correction. The approach may also be applicable to superconducting qubits, and any other systems where qubits are addressed in frequency.
A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: Towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324 (2012).
J. Bylander, Microtechnology and Nanoscience, Chalmers University of Technology, Kemivägen 9, Gothenburg, Sweden, SE-41296 (personal communication, 2018).
N. Didier, E. A. Sete, M. P. da Silva, and C. Rigetti, “Analytical modeling of parametrically modulated transmon qubits,” Phys. Rev. A 97, 022330 (2018).
N. Ohlsson, R. K. Mohan, and S. Kröll, “Quantum computer hardware based on rare-earth-ion-doped inorganic crystals,” Opt. Commun. 201(1–3), 71–77 (2002).
L. Rippe, B. Julsgaard, A. Walther, Y. Ying, and S. Kröll, “Experimental quantum-state tomography of a solid-state qubit,” Phys. Rev. A 77, 022307 (2008).
K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003 (1998).
N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergmann, “Laser-induced population transfer by adiabatic passage techniques,” Rnnu. Rev. Phus. Chem. 52, 763 (2001).
P. Král, I. Thanopulos, and M. Shapiro, “Colloquium: Coherently controlled adiabatic passage,” Rev. Mod. Phys. 79(1), 53 (2007).
M. H. Levitt, “Composite pulses,” Progress in NMR Spectroscopy 18, 61–122 (1986).
B. T. Torosov, S. Guérin, and N. V. Vitanov, “High-Fidelity Adiabatic Passage by Composite Sequences of Chirped Pulses,” Phys. Rev. Lett. 106, 233001 (2011).
V. S. Mallinovsky and D. J. Tannor, “Simple and robust extension of the stimulated Raman adiabatic passage technique to N-level systems,” Phys. Rev. A 56, 4929 (1997).
G. S. Vasilev, A. Kuhn, and N. V. Vitanov, “Optimum pulse shapes for stimulated Raman adiabatic passage,” Phys. Rev. A 80, 013417 (2009).
A. Walther, B. Julsgaard, L. Rippe, Y. Ying, S. Kröll, R. Fisher, and S. Glaser, “Extracting high fidelity quantum computer hardware from random systems,” Phys. Scr. T137, 014009 (2009).
K. Kobzar, S. Ehni, T. E. Skinner, S. J. Glaser, and B. Luy, “Exploring the limits of broadband 90° and 180° universal rotation pulses,” J. Magn. Reson. 225, 142–160 (2012).
T. Nöbauer, A. Angerer, B. Bartels, M. Trupke, S. Rotter, J. Schmiedmayer, Florian Mintert, and J. Majer, “Smooth optimal quantum control for robust solid-state spin magnetometry,” Phys. Rev. Lett. 115, 190801 (2015).
L. Van-Damme, D. Schraft, Genko T. Genov, D. Sugny, T. Halfmann, and S. Guérin, “Robust NOT gate by single-shot-shaped pulses: Demonstration of the efficiency of the pulses in rephasing atomic coherences,” Phys. Rev. A 96, 022309 (2017).
X. Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Guéry-Odelin, and J. G. Muga, “Fast Optimal Frictionless Atom Cooling in Harmonic Traps: Shortcut to Adiabaticity,” Phys. Rev. Lett. 104, 063002 (2010).
X. Chen and J. G. Muga, “Engineering of fast population transfer in three-level systems,”, Phys. Rev. A 86, 033405 (2012).
X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riescenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011).
M V Berry, “Transitionless quantum driving,” J. Phys. A: Math. Theor. 42(36), 365303 (2009).
X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to adiabatic passage in two and three level atoms,” Phys. Rev. Lett. 105, 123003 (2010).
S. Masuda and K. Nakamura, “Fast-forward of adiabatic dynamics in quantum mechanics,” Proc. R. Soc. A 466(2116), 1135–1154 (2010).
A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. 14, 093040 (2012).
D. Daems, A. Ruschhaupt, D. Sugny, and S. Guerin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett. 111, 050404 (2013).
Y. Du, Z. Liang, Y. Chao, X. Yue, Q. Lv, W. Huang, X. Chen, H. Yan, and S. Zhu, “Experimental realization of stimulated Raman shortcut-to-adiabatic passage with cold atoms,” Nat. Commun. 7, 12479 (2016).
Y. C. Li and X. Chen, “Shortcut to adiabatic population transfer in quantum three-level systems: effective two-level problems and feasible counter-diabatic driving,” Phys. Rev A 94, 063411 (2016).
A. Baksic, Hugo Ribeiro, and A. A. Clerk, “Speeding up adiabatic quantum state transfer by using dressed states,” Phys. Rev. Lett. 116, 230503 (2016).
B. B. Zhou, A. Baksic, Hugo Ribeiro, C. G. Yale, F. J. Heremans, Paul C. Jerger, A. Auer, G. Burkard, A. A. Clerk, and D. Awschalom, “Accelerated quantum controal sign superadiabatic dynamics in a solid-state lambda system,” Nature Phys. 13, 330 (2017).
H. L. Mortensen, J. Jakob, W. H. Sørensen, K. Mølmer, and J. F. Sherson, “Fast state transfer in a Λ-system: a shortcut-to-adiabaticity approach to robust and resource optimized control,” New J. Phys. 20, 025009 (2018).
Y. Ban, L. Jiang, Y. Li, L. Wang, and X. Chen, “Fast creation and transfer of coherence in triple quantum dots by using shortcuts to adiabaticity,” Opt. Express 26(24), 31137 (2018).
“Quantum Technologies foster a new initiative in Europe,” (2018). https://qt.eu/news/quantum-technologies-launch-press-release/ .
M. Zhong, M. P. Hedges, R. L. Ahlefeldt, J. G. Bartholomew, S. E. Beavan, S. M. Witting, J. J. Longdell, and M. J. Sellars, “Optically addressable nuclear spins in a solid with a six-hour coherence time,” Nature 517, 177–180 (2015).
R. W. Equall, Y. Sun, R. L. Cone, and R. M. Macfarlane, “Ultraslow optical dephasing in Eu3+:Y2SiO5,” Phys. Rev. Lett. 72, 2179 (1994).
A. Walther, L. Rippe, Y. Yan, J. Karlsson, D. Serrano, A. N. Nilsson, S. Bengtsson, and S. Kröll, “High-fidelity readout scheme for rare-earth solid-state quantum computing,” Phys. Rev. A. 92, 022319 (2015).
I. Roos and K. Mølmer, “Quantum computing with an inhomogeneously broadened ensemble of ions: Suppression of errors from detuning variations by specially adapted pulses and coherent population trapping,” Phys. Rev. A 69, 022321 (2004).
K. Paul and A. K. Sarma, “Shortcut to adiabatic passage in a waveguide coupler with a complex-hyperbolic-secant scheme,” Phys. Rev. A 91, 053406 (2015).
S. Tseng and X. Chen, “Engineering of fast mode conversion in multimode waveguides,” Opt. Lett. 37(24), 5118–5120 (2012).
H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458 (1969).
Y. Z. Lai, J. Q. Liang, H. J. W. Müller-Kirsten, and J. G. Zhou, “Time-dependent quantum systems and the invariant Hermitian operator,” Phys. Rev. A 53, 3691 (1996).
R. W. Equall, R. L. Cone, and R. M. Macfarlane, “Homogeneous broadening and hyperfine structure of optical transitions in Pr3+:Y2SiO5,” Phys. Rev. B 52, 3963 (1995).
N. V. Vitanov, K.-A. Suominen, and B.W. Shore, “Creation of coherent atomic superpositions by fractional stimulated Raman adiabatic passage,” J. Phys. B 32, 4535–4546 (1999).
U. Güngördü, Y. Wan, M. A. Fashihi, and M. Nakahara, “Dynamical invariants for quantum control of four-level systems,” Phys. Rev. A 86, 062312 (2012).
N. V. Vitanov, K.-A. Suominen, and B.W. Shore, “Hamiltonian design to prepare arbitrary states of four-level systems,” Phys. Rev. A 97, 013830 (2018).
Y. C. Li, X. Chen, J G Muga, and E Ya Sherman, “ubit gates with simultaneous transport in double quantum dots,” New J. Phys. 20, 113029 (2018).
Fig. 1 Schematic energy levels of a three-level lambda system. The qubit is represented by two long-lived ground state levels |0〉 and |1〉, where |1〉 is initially populated. Qubit levels are coupled through optical transitions |0〉 − |e〉 and |1〉 − |e〉, which possibly exhibits an inhomogeneous broadening. Ωp and Ωs denote the respective Rabi frequencies. φ is a time-independent phase factor of Ωs.
Fig. 2 (a) The relevant energy levels of Pr ions in an Y2SiO5 crystal. The qubit is represented by two ground state levels |0〉 and |1〉, where |1〉 is initially populated. Qubit levels are coupled through optical transitions |0〉 − |e〉 and |1〉 − |e〉, both of which have an inhomogeneous FWHM linewidth of 170 kHz. (b) A schematic diagram of the absorption spectrum of a qubit in a zero absorption spectral window. Peak 1–3 represent absorption from |0〉 to each level of the excited states, and peak 4–5 from |1〉 to the two lower levels in excited state, respectively. The distance from peak 2 or peak 5 to the edges of the pit is 3.9 MHz.
Fig. 3 (a) Time dependence of Rabi frequencies, where the solid-red (dashed-blue) line denotes Ωp (Ωs). (b) Time evolution of the population on level |1〉 (Solid-red line), |0〉 (dashed-green line) and |e〉 (dash-dotted-blue line). a2 = −1.10, a6 = 0.06 and a8 = 0.02, which are optimized to achieve high robustness against frequency detuning and minimize the off-resonant excitations. | ψ tg 〉 = 1 2 ( | 1 〉 + i | 0 〉 ).
Fig. 4 Dependence of fidelity of achieving | ψ tg 〉 = 1 2 ( | 1 〉 + i | 0 〉 ) on frequency detuning. solid-red line: fidelity achieved with the shortcut pulses developed in this work with optimized parameters (a2 = −1.10, a6 = 0.06 and a8 = 0.02). Solid-blue line: the fidelity achieved by the complex hyperbolic secant pulses used previously. The insert is a magnification of the center frequency range.
Fig. 5 Dependence of fidelity of achieving | ψ tg 〉 = 1 2 ( | 1 〉 + i | 0 〉 ) on the fluctuations in Rabi frequencies for shortcut pulses (a) and complex hyperbolic secant pulses (b), where η describes the relative change in Ωp,s. Blue-dashed and green-dashed curves show the dependences under conditions of no detuning, and solid-red and solid-purple for the cases where detuning is 170 kHz.
Fig. 6 Population of final state |ψ(tf)〉 in level |1〉 (solid-red), |e〉 (dash-dotted-blue) and |0〉 (dashed-green), and as a function of detuning frequencies with optimized an values. The off-resonant excitation at 3.5 MHz is less than 2.0%, which can be further reduced by sacrificing the width of the robust region in the center.
Fig. 7 (a) Dependence of the fidelity (F) for achieving the target state i |e〉 on frequency detuning Δ. (b) State evolutions (solid-red line) on a Bloch sphere where Δ = 170 kHz. The dashed lines are the three perpendicular great circles in u–v, u–w and v–w plane.
Fig. 8 Optimization of an values. Fidelity (a–c) and off-resonant excitations in |0〉 state (d–f) as function of frequency detuning while scanning pulse parameters. (a) and (d) scan a2, a6 = a8 = 0; (b) and (e) Scan a6, a2 = −1.10, and a8 = 0; (c) and (f) Scan a8, a2 = −1.10, a6 = 0.06.
Fig. 9 Fidelity (a) and off-resonant excitations (b) on |0〉 state as a function of frequency detuning in the three scanning steps. Dash-dotted-blue lines: a2 = −1.1, a6 = a8 = 0. Dashed-green lines: a2 = −1.1, a6 = 0.06, and a8 = 0. Solid-red lines: a2 = −1.1, a6 = 0.06, and a8 = 0.02.

References: V. 
 V. 

V. 
 V. 
 V. 
 V.