PRL 112, 050503 (2014)
PHYSICAL REVIEW LETTERS
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Implementation of Dynamically Corrected Gates on a Single Electron Spin in Diamond
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Xing Rong,1,2 Jianpei Geng,1 Zixiang Wang,1 Qi Zhang,1 Chenyong Ju,1,2 Fazhan Shi,1,2 Chang-Kui Duan,1,2,* and Jiangfeng Du1,2,†
Hefei National Laboratory for Physics Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China 2 Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China (Received 22 July 2013; published 6 February 2014; corrected 10 February 2014) Precise control of an open quantum system is critical to quantum information processing but is challenging due to inevitable interactions between the quantum system and the environment. We demonstrated experimentally a type of dynamically corrected gates using only bounded-strength pulses on the nitrogen-vacancy centers in diamond. The infidelity of quantum gates caused by a nuclear-spin bath is reduced from being the second order to the sixth order of the noise-to-control-field ratio, which offers greater efficiency in reducing infidelity. The quantum gates have been protected to the limit essentially set by the spin-lattice relaxation time T 1 . Our work marks an important step towards fault-tolerant quantum computation in realistic systems. DOI: 10.1103/PhysRevLett.112.050503
PACS numbers: 03.67.Pp, 03.67.Ac, 42.50.Dv, 76.30.Mi
Quantum information processing can provide a dramatic speed-up over a classical computer for certain problems [1]. One of the most urgent demands in quantum computation is to realize noise-resistant universal quantum gates for qubits. Strategies including quantum error correction [2–4], a decoherence-free subspace [5,6], and dynamical decoupling (DD) [7] have been developed to accomplish this task. Compared with the other two strategies, the resource requirements for the DD are modest [8], and no extra qubits are required. DD uses stroboscopic qubit flips to average out the coupling to the environment. Protecting quantum states by DD from decoherence has been extensively experimentally demonstrated recently [9–12]. However, a more crucial task in quantum information processing is to protect quantum gates from decoherence. This is challenging due to the noncommutability between quantum gates and the dynamical decoupling pulses, which tend to destroy each other. Theoretical schemes for protecting quantum gates, termed dynamically corrected gates (DCGs), have been proposed [13–17]. However, it was only until very recently that a few experimental implementations of decoherence protected gates by using bang-bang (arbitrarily strong instantaneous control) pulses appeared [18–20]. The performance of these protected quantum gates is extended to the limit set by T 2. In this Letter, we adopted a type of DCG, namely, SUPCODE [17], to overcome the deterioration of quantum gates by the fluctuation of the static magnetic field in diamond. Since bounded-strength pulses are used in our experiments, the disadvantages of bang-bang pulses in other schemes, such as being extremely unrealistic for implementation and having poor spectral selectivity [21], can be avoided. Theoretical analysis and experimental 0031-9007=14=112(5)=050503(5)
study show that the quantum gate error is suppressed to the sixth order of the fluctuation-field-to-control-field ratio. The performance of quantum gates has been greatly promoted to the bound of any multiple dynamical decoupling pulse sequence set essentially by the spin-lattice relaxation time T 1 . The Hamiltonian describing a general single-qubit unitary operation on an electron-spin qubit in the rotating frame can be described as H C ¼ 2πω1 n · S, where S ¼ ðSx ; Sy ; Sz Þ is the spin vector operator of the qubit, n is a three-dimensional vector, and the strength ω1 is a real parameter. When the qubit is subjected to the noisy environment, the total Hamiltonian can be expressed as H ¼ H S þ HSB þ HB þ HC , where HS ¼ 2πΩ0 Sz is the system Hamiltonian and Ω0 is the off-resonance frequency. HSB stands for the qubit-environment coupling, and HB is the environmental Hamiltonian. Herein, we describe the environment as a nuclear-spin bath and P the coupling as a pure dephasing interaction HSB ¼ 2π k bk Sz I kz , where I kz is the spin operator of the kth nuclear spin and bk is the strength of the hyperfine interaction between the qubit and the kth nuclear spin. Figure 1(a) depicts an electron spin that interacts with the nuclear-spin bath. The hyperfine interaction with the nuclear spins results inPa random local magnetic field (Overhauser field) δ ¼ k bk I kz of typical strength of the order of magnitude of about 1 MHz in solids, which depends on the samples (see Ref. [22]). The thermal distribution of the Overhauser field causes rapid free induction decay (FID) of the electron-spin coherence. There are also dynamical fluctuations of the local Overhauser field driven by pairwise nuclear-spin flip flop. Because the dynamical fluctuations in δ are much slower
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FIG. 1 (color online). (a) A spin qubit interacting with the nuclear-spin bath. (b) Schematic of the optically detected magnetic resonance setup. The external magnetic field B0 was applied by a permanent magnet, while the microwave pulse was carried by a waveguide. (c) Energy-level diagram of the NV center with external magnetic field B0 (upper panel) and a continuous-wave spectrum of j0i↔j1i with B0 ¼ 513 G (lower panel). (d) FID of the electron spin. Experimental data (orange line) are fitted by y ¼ 0.5 − 0.5 exp½−ðt=T 2 Þ2 cosð2πωtÞ (blue line) with ω ¼ 0.5 MHz and T 2 ¼ 6.56 ð0.17Þ μs.
than the typical gate time, we take δ as a random timeindependent variable [17,23]. For simplicity, we consider the on-resonance case (Ω0 ¼ 0) and set the axis n to the x axis. Then, the Hamiltonian H can be written as H ¼ 2πðδSz þ ω1 Sx Þ. We define the fidelity of the gate Fg ¼ TrðAB−1 Þ=2, where A ¼ expð−i2πω1 Sx τÞ is the ideal gate operation and B ¼ expð−iHτÞ is the gate operation in the presence of dephasing noise, where τ is the gate time. The case of interest in implementing DCGs is δ ≪ ω1 . The infidelity of the gate Δ ¼ 1 − Fg for the plain case (rectangle pulse) is of the second order in δ=ω1 (see Table I). Since the nuclear spins flip, the qubit experiences a different effective δSz in every experiment once a new sequence is started. TABLE I. Infidelity of the SUPCODE π gate as a function of δ=ω1 . The words “plain” and “n piece” stand for a normal rectangle pulse and an n-piece SUPCODE pulse, respectively (see Ref. [22]). Sequence Plain Three piece Five piece Nine piece
Infidelity Δ 0.5ðδ=ω1 Þ2 þ Oðδ=ω1 Þ4 11:1ðδ=ω1 Þ4 þ Oðδ=ω1 Þ6 64:1ðδ=ω1 Þ6 þ Oðδ=ω1 Þ8 317 237ðδ=ω1 Þ8 þ Oðδ=ω1 Þ10
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When we average over many sequences, as required to build statistics, δSz will cause the decay of the Rabi oscillations even in the absence of the instability of control field HC [23]. Herein, we apply SUPCODE to protect the quantum gates from the decoherence effect aroused by the static fluctuation of the magnetic field. We denote the rotation of an angle θ around the axis in the equatorial plane with azimuth ϕ as Rϕ ðθÞ. The sequence of the five-piece SUPCODE π gate is f½τ1 − R0 ð2πω1 τ2 Þ − τ3 − R0 ð2πω1 τ2 Þ − τ1 × 2g, where τ1 ¼ 1.05τ0 , τ2 ¼ 0.625τ0 , and τ3 ¼ 1.71τ0 , with τ0 ¼ 1=ω1 . The details of other pulse sequences are included in the Supplemental Material [22]. The infidelity of the five-piece SUPCODE gate is calculated to be of the sixth order in δ=ω1 (see Ref. [22]). Table I summarized the infidelity of several SUPCODE π gates. Because of a large coefficient in the leading term, the nine-piece SUPCODE only provides a better performance when δ=ω1 ≲ 1% (see Ref. [22]), where the infidelity of the five-piece SUPCODE is below 10−4. Hence, we focus on the five-piece SUPCODE in this Letter. The setup and experimental scheme are schematically shown in Fig. 1(b). A B0 of 513 G was adopted so as to achieve effective polarization of the nitrogen nuclear spin in the nitrogen-vacancy (NV) center, which is confirmed by the continuous-wave spectrum shown in Fig. 1(c). Quantum states j0i and j1i are encoded as a qubit, which can be manipulated via microwave pulses. A near-surface NV in a typical IIa bulk diamond and 0.1% 13 C is used. The undesired couplings between the qubit and the surrounding 13 C nuclear spins lead to the dephasing effect. The resulting FID of the qubit is depicted in Fig. 1(d), where a dephasing time T 2 ¼ 6.56 ð0.17Þ μs is obtained. We first scrutinize the performance of SUPCODE under dephasing noises. We experimentally investigated the π rotation about the x axis (NOT gate) via a plain pulse and via three- and five-piece SUPCODEs. The state of the electron spin was first initialized to j0i by laser. After applying the NOT gate on the initial state, the state fidelity pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [1] of the resultant state ρf is Fs ¼ h1jρf j1i and the infidelity of state is δf ¼ 1 − Fs . For the case of the NOT gate on the initial state j0i, the infidelity of the states δf equals the infidelity of quantum gates Δ (see Ref. [22]), so the performance of the NOT gate can be characterized by state fidelity [11]. We are aiming to investigate the robustness of SUPCODE against the noise stemming from the quasistatic fluctuation of the magnetic field, which is simulated by detuning the frequency of the microwave from the on-resonance frequency. The value of δ ranges from 0 to 6 MHz, and the strength of the control field is set at 20 MHz, which is verified by Rabi nutation experiments. The three different pulse sequences for the π gate are given in Fig. 2(a), and their performances are plotted as functions of δ=ω1 in Fig. 2(b). Lines are theoretical predictions, while green triangles, black rectangles, and red circles are the
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FIG. 3 (color online). Universal control of a single spin by the five-piece SUPCODE pulse. The left panels of (a) and (b) show the Bloch-sphere diagrams of the spin vectors rotating around (a) x and (b) y, respectively. The right panels are the corresponding detected coherent oscillations of the qubit driven by the five-piece SUPCODE pulses. Light red (blue) lines are the theoretical predictions with X (Y) detection, and red (blue) rectangles are the corresponding experimental data. FIG. 2 (color online). (a) Pulse sequences corresponding to a plain sequence and to three- and five-piece SUPCODE sequences. The calculation of time durations τi (i ¼ 1, 2, 3, and 4) is included in the Supplemental Material [22] with τ0 ¼ 1=ω1 and ω1 ¼ 20 ¼ MHz. (b) The measured and theoretically predicted state infidelities of the destination states as functions of δ=ω1 . Experimental data fit the theoretical predictions (lines) well. Some deviations are caused by statistical fluctuations of photon numbers, with the uncertainty plotted as a dashed gray line.
experimental data for a plain pulse and three- and five-piece SUPCODE π pulses, respectively. Each data point is averaged by 7.2 × 107 times, and the uncertainty due to the statistical fluctuation of the photon number has been plotted as a dashed line. The experimental data fit the theoretical predictions (lines) well. Our results show that three-piece (five-piece) SUPCODE pulses reduce the error to the fourth (sixth) order in δ=ω1 , in agreement with theoretical predictions. Second, we show that universal single-qubit gates can be achieved by SUPCODE. Universal control of a single qubit requires the ability to realize precisely rotations around two different axes of the Bloch sphere. By successively applying the five-piece SUPCODE pulses on the electron spin, we can force the spin vector to rotate along the x (y) axis by setting the phase of the microwave pulse to 0 (π=2). The following readout procedure consists of a detecting π=2 microwave pulse and a laser pulse. The phase of the detecting microwave is set to 0 (π=2) so that the detection is sensitive to the Y (X) coherence of the quantum states. If the qubit is driven to rotate along the x (y) axis, the spin vector will rotate in the y-z (x-z) plane (see Fig. 3), and coherent oscillations are observable by setting the phase of the detecting microwave pulse to 0 (π=2), while there will be no oscillations with the phase of the detecting microwave pulse being set to π=2 (0). Figure 3(a) [Fig. 3(b)] plots the coherent oscillations driven
by the five-piece SUPCODE pulse around the x (y) axis. Red circles (blue rectangles) with error bars are experimental data under X (Y) detection, and light red (light blue) lines are from the theoretical predictions. Third, we show that the performance of the quantum gate can be protected towards the T 1 limit via SUPCODE. To minimize the influence of the instability of the control field HC [23], we adopted a microwave pulse of an amplitude corresponding to ω1 ¼ 1 MHz. The quantum oscillation driven by SUPCODE is plotted in Fig. 4(a). A decay time constant T DCG ¼ 690 ð40Þ μs is derived from the experimental data (black crosses), which is 2 orders of magnitude longer than the dephasing time T 2 . The Rabi oscillation (blue diamonds) driven by the normal rectangle pulse in Fig. 4(a) of ω1 ¼ 1 MHz shows a decay time T 2 0 ¼ 135 ð10Þ μs, which is also much shorter than T DCG . This verified that SUPCODE can largely suppress the dephasing effect during the gate time. It is noted that T DCG achieved here has reached the spin-lattice relaxation time in the rotating frame T 1ρ ¼ 660 ð80Þ μs, which is determined by a spin-locking experiment. T 1ρ is the upper bound achievable by any multiple dynamical decoupling pulse sequence for T 2 [24]. To confirm this, we applied Carr-Purcell-Meiboom-Gill (CPMG) sequences on the qubit to prolong the coherence time with ω1 ¼ 10 MHz. The scaling of the extended coherence times with the number of pulses N is depicted in Fig. 4(c) as blue rectangles. We found that the coherence time can be efficiently prolonged by CPMG sequences with the scaling T 2 ∝ N 0.33 until it approaches the saturation value for N > 100 at T 1ρ . The related decay times are summarized in Fig. 4(d). The coherence time without any dynamical decoupling control
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FIG. 4 (color online). (a) Comparison of the quantum oscillation driven by five-piece SUPCODE pulses with the one driven by plain pulses. Black crosses are the SUPCODE experimental data, and red lines are the envelopes of the function 0.5 þ 0.5 expð−T=T DCG Þ cosðωTÞ (light gray line), with T DCG ¼ 690 ð40Þ μs. The measured (blue diamonds) quantum oscillation by the plain pulses (Rabi oscillation) was fitted by 0.5 þ 0.5 expð−T=T 02 Þ cosðω0 TÞ (blue lines) with T 02 ¼ 135 ð10Þ μs. (b) The decay of the fidelity of five-piece SUPCODEs obtained via quantum process tomography. The black line is the theoretically predicted fidelity when only the T 1ρ process is taken into account. The shaded region is calculated with the uncertainty of the measured T 1ρ . Blue triangles are experimental fidelities by QPT, which shows a T 1ρ -limited decay. (c) Extended coherence times via CPMGs. N stands for the number of the π pulses. After applying more than 1000 pulses, the prolonged coherence times are limited by T 1ρ ¼ 660 ð80Þ μs. (d) Comparison of the decay times.
pulse is T 2 ¼ 6.56 ð0.17Þ μs. It can be prolonged to T 2 ¼ 123:2 ð8.8Þ μs by a single refocusing π pulse, which only enables robust quantum-state storage against the dephasing effect. The Rabi oscillation driven by a normal pulse, which can be used to realize the quantum gate, has a decay time T 20 ¼ 135 ð10Þ μs when ω1 ¼ 1 MHz. In comparison, the decay time of the quantum oscillation via SUPCODE is T DCG ¼ 690 ð40Þ μs, which equals T 1ρ ¼ 660 ð80Þ μs within the uncertainty. Finally, we examine the performance of the SUPCODEs quantitatively by quantum process tomography (QPT). Any quantum operation can be considered as mapping the input quantum state ρi to the output ρo ¼ εðρi Þ ¼ P † m;n χ mn Am ρi An with fAm g ¼ fI; σ x ; σ y ; σ z g, where σ x , σ y , and σ z are Pauli operators. The matrix χ completely and uniquely describes the process ε and can be experimentally reconstructed by QPT. The average fidelity of the quantum gate [25] is defined by 1 1 X Fðε; UÞ ¼ þ Tr½Uσ j U † εðσ j Þ; 2 12 j¼x;y;z
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where U is the ideal operation. We applied five-piece SUPCODE π=2 gates successively on the electron spins and performed QPT when the gates were repeated for M ¼ 0, 27, 54, 81, 108, and 135 times. The variation of gate fidelities is depicted as a function of the operation time in Fig. 4(b). Experimental data of the gate fidelity (blue triangles) were obtained by using QPT. We also plot the theoretical predicted variation of gate fidelities (black line and shaded region) with only T 1ρ relaxation being considered. The fact that the experimental data match the theoretical prediction indicates that the performance of the SUPCODE is limited by T 1ρ. The details of experimental QPT and theoretical calculations can be found in the Supplemental Material [22]. The fidelity of each five-piece SUPCODE π=2 gate is derived to be 0.9961 (2) (see Ref. [22]), close to the requirement of the fault-tolerant quantum computation [26], while the gate time (5.063 μs) is comparable to T 2 . As T 1ρ or the limit of T 2 can be prolonged by 2–3 orders of magnitude in proportion to T 1 (i.e., ∼0.5T 1 [27]) by lowering the temperature in a wide range [27–29], the DCGs demonstrated here can be considered as essentially T 1 limited. Further improvement of the performance of DCGs is feasible by prolonging T 1 via lowering temperature. To summarize, we applied SUPCODE to greatly suppress the error aroused by the static fluctuation of the magnetic field during quantum gates. The performance of the quantum gates, quantitatively measured by QPT, has been protected to the limit of T 1ρ . Our work is expected to have various applications in quantum information processing, high resolution spectroscopy [30,31], and various quantum metrologies [32], where high-fidelity quantum gates are required. Since the fluctuation of the static magnetic field exists in many important quantum qubit systems, such as trapped ions [33,34], superconducting qubits [35,36], and phosphorus doped in silicon [37–39], the DCGs implemented here may also be applied to these systems. Although our work focuses on single-qubit DCGs, theoretical schemes for two-qubit DCGs are available [16] for experimental implementation in the future. Our experimental implementation of high-fidelity and essentially T 1 -limited DCGs marks an important step towards realistic fault-tolerant quantum computation. We thank Yiqun Wang in the Suzhou Institute of Nano-Tech and Nano-Bionics for fabricating the coplanar waveguide and Ren-Bao Liu from the Chinese University of Hong Kong for helpful discussions. This work was supported by the National Key Basic Research Program of China (Grant No. 2013CB921800), the National Natural Science Foundation of China (Grants No. 11227901, No. 11275183, No. 11274299, No. 11104262, No. 91021005, and No. 10834005), the “Strategic Priority Research Program (B)” of the CAS (Grant No. XDB01030400), and the Fundamental Research Funds for the Central Universities.
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[email protected] [email protected] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2000). P. W. Shor, Phys. Rev. A 52, R2493 (1995). A. M. Steane, Phys. Rev. Lett. 77, 793 (1996). E. Knill and R. Laflamme, Phys. Rev. A 55, 900 (1997). L. M. Duan and G. C. Guo, Phys. Rev. Lett. 79, 1953 (1997). D. A. Lidar, I. L. Chuang, and K. B. Whaley, Phys. Rev. Lett. 81, 2594 (1998). L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. 82, 2417 (1999). H. K. Ng, D. A. Lidar, and J. Preskill, Phys. Rev. A 84, 012305 (2011). J. Du, X. Rong, N. Zhao, Y. Wang, J. Yang, and R. B. Liu, Nature (London) 461, 1265 (2009). Y. Wang, X. Rong, P. Feng, W. Xu, B. Chong, J.-H. Su, J. Gong, and J. Du, Phys. Rev. Lett. 106, 040501 (2011). G. de Lange, Z. H. Wang, D. Riste, V. V. Dobrovitski, and R. Hanson, Science 330, 60 (2010). C. A. Ryan, J. S. Hodges, and D. G. Cory, Phys. Rev. Lett. 105, 200402 (2010). K. Khodjasteh and L. Viola, Phys. Rev. Lett. 102, 080501 (2009). K. Khodjasteh, D. A. Lidar, and L. Viola, Phys. Rev. Lett. 104, 090501 (2010). J. R. West, D. A. Lidar, B. H. Fong, and M. F. Gyure, Phys. Rev. Lett. 105, 230503 (2010). J. P. Kestner, X. Wang, L. S. Bishop, E. Barnes, and S. D. Sarma, Phys. Rev. Lett. 110, 140502 (2013). X. Wang, L. S. Bishop, J. P. Kestner, E. Barnes, K. Sun, and S. Das Sarma, Nat. Commun. 3, 997 (2012). A. M. Souza, G. A. Alvarez, and D. Suter, Phys. Rev. A 86, 050301 (2012). T. van de Sar et al., Nature (London) 484, 82 (2012). G.-Q. Liu et al., Nat. Commun. 4, 2254 (2013). L. Viola and E. Knill, Phys. Rev. Lett. 90, 037901 (2003).
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[22] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.112.050503 for information about the instrumentation, details of the calculations, and experimental procedures. [23] H. De Raedt, B. Barbara, S. Miyashita, K. Michielsen, S. Bertaina, and S. Gambarelli, Phys. Rev. B 85, 014408 (2012). [24] B. Naydenov, F. Dolde, L. T. Hall, C. Shin, H. Fedder, L. C. L. Hollenberg, F. Jelezko, and J. Wrachtrup, Phys. Rev. B 83, 081201 (2011). [25] M. A. Nielsen, Phys. Lett. A 303, 249 (2002). [26] E. Knill, Nature (London) 434, 39 (2005). [27] N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, and R. L. Walsworth, Nat. Commun. 4, 1743 (2013). [28] A. Jarmola, V. M. Acosta, K. Jensen, S. Chemerisov, and D. Budker, Phys. Rev. Lett. 108, 197601 (2012). [29] M. Loretz, T. Rosskopf, and C. L. Degen, Phys. Rev. Lett. 110, 017602 (2013). [30] J. R. Maze et al., Nature (London) 455, 644 (2008). [31] G. Balasubramanian et al., Nature (London) 455, 648 (2008). [32] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 96, 010401 (2006). [33] M. J. Biercuk, H. Uys, A. P. VanDevender, N. Shiga, W. M. Itano, and J. J. Bollinger, Nature (London) 458, 996 (2009). [34] S. Kotler, N. Akerman, Y. Glickman, A. Keselman, and R. Ozeri, Nature (London) 473, 61 (2011). [35] J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. Fitch, D. G. Cory, Y. Nakamura, J.-S. Tsai, and W. D. Oliver, Nat. Phys. 7, 565 (2011). [36] M. H. Devoret and R. J. Schoelkopf, Science 339, 1169 (2013). [37] B. E. Kane, Nature (London) 393, 133 (1998). [38] J. J. Pla, K. Y. Tan, J. P. Dehollain, W. H. Lim, J. J. L. Morton, D. N. Jamieson, A. S. Dzurak, and A. Morello, Nature (London) 489, 541 (2012). [39] J. J. L. Morton, D. R. McCamey, M. A. Eriksson, and S. A. Lyon, Nature (London) 479, 345 (2011).
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Implementation of Dynamically Corrected Gates on a Single Electron Spin in Diamond
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Xing Rong,1,2 Jianpei Geng,1 Zixiang Wang,1 Qi Zhang,1 Chenyong Ju,1,2 Fazhan Shi,1,2 Chang-Kui Duan,1,2,* and Jiangfeng Du1,2,†
Hefei National Laboratory for Physics Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China 2 Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China (Received 22 July 2013; published 6 February 2014; corrected 10 February 2014) Precise control of an open quantum system is critical to quantum information processing but is challenging due to inevitable interactions between the quantum system and the environment. We demonstrated experimentally a type of dynamically corrected gates using only bounded-strength pulses on the nitrogen-vacancy centers in diamond. The infidelity of quantum gates caused by a nuclear-spin bath is reduced from being the second order to the sixth order of the noise-to-control-field ratio, which offers greater efficiency in reducing infidelity. The quantum gates have been protected to the limit essentially set by the spin-lattice relaxation time T 1 . Our work marks an important step towards fault-tolerant quantum computation in realistic systems. DOI: 10.1103/PhysRevLett.112.050503
PACS numbers: 03.67.Pp, 03.67.Ac, 42.50.Dv, 76.30.Mi
Quantum information processing can provide a dramatic speed-up over a classical computer for certain problems [1]. One of the most urgent demands in quantum computation is to realize noise-resistant universal quantum gates for qubits. Strategies including quantum error correction [2–4], a decoherence-free subspace [5,6], and dynamical decoupling (DD) [7] have been developed to accomplish this task. Compared with the other two strategies, the resource requirements for the DD are modest [8], and no extra qubits are required. DD uses stroboscopic qubit flips to average out the coupling to the environment. Protecting quantum states by DD from decoherence has been extensively experimentally demonstrated recently [9–12]. However, a more crucial task in quantum information processing is to protect quantum gates from decoherence. This is challenging due to the noncommutability between quantum gates and the dynamical decoupling pulses, which tend to destroy each other. Theoretical schemes for protecting quantum gates, termed dynamically corrected gates (DCGs), have been proposed [13–17]. However, it was only until very recently that a few experimental implementations of decoherence protected gates by using bang-bang (arbitrarily strong instantaneous control) pulses appeared [18–20]. The performance of these protected quantum gates is extended to the limit set by T 2. In this Letter, we adopted a type of DCG, namely, SUPCODE [17], to overcome the deterioration of quantum gates by the fluctuation of the static magnetic field in diamond. Since bounded-strength pulses are used in our experiments, the disadvantages of bang-bang pulses in other schemes, such as being extremely unrealistic for implementation and having poor spectral selectivity [21], can be avoided. Theoretical analysis and experimental 0031-9007=14=112(5)=050503(5)
study show that the quantum gate error is suppressed to the sixth order of the fluctuation-field-to-control-field ratio. The performance of quantum gates has been greatly promoted to the bound of any multiple dynamical decoupling pulse sequence set essentially by the spin-lattice relaxation time T 1 . The Hamiltonian describing a general single-qubit unitary operation on an electron-spin qubit in the rotating frame can be described as H C ¼ 2πω1 n · S, where S ¼ ðSx ; Sy ; Sz Þ is the spin vector operator of the qubit, n is a three-dimensional vector, and the strength ω1 is a real parameter. When the qubit is subjected to the noisy environment, the total Hamiltonian can be expressed as H ¼ H S þ HSB þ HB þ HC , where HS ¼ 2πΩ0 Sz is the system Hamiltonian and Ω0 is the off-resonance frequency. HSB stands for the qubit-environment coupling, and HB is the environmental Hamiltonian. Herein, we describe the environment as a nuclear-spin bath and P the coupling as a pure dephasing interaction HSB ¼ 2π k bk Sz I kz , where I kz is the spin operator of the kth nuclear spin and bk is the strength of the hyperfine interaction between the qubit and the kth nuclear spin. Figure 1(a) depicts an electron spin that interacts with the nuclear-spin bath. The hyperfine interaction with the nuclear spins results inPa random local magnetic field (Overhauser field) δ ¼ k bk I kz of typical strength of the order of magnitude of about 1 MHz in solids, which depends on the samples (see Ref. [22]). The thermal distribution of the Overhauser field causes rapid free induction decay (FID) of the electron-spin coherence. There are also dynamical fluctuations of the local Overhauser field driven by pairwise nuclear-spin flip flop. Because the dynamical fluctuations in δ are much slower
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FIG. 1 (color online). (a) A spin qubit interacting with the nuclear-spin bath. (b) Schematic of the optically detected magnetic resonance setup. The external magnetic field B0 was applied by a permanent magnet, while the microwave pulse was carried by a waveguide. (c) Energy-level diagram of the NV center with external magnetic field B0 (upper panel) and a continuous-wave spectrum of j0i↔j1i with B0 ¼ 513 G (lower panel). (d) FID of the electron spin. Experimental data (orange line) are fitted by y ¼ 0.5 − 0.5 exp½−ðt=T 2 Þ2 cosð2πωtÞ (blue line) with ω ¼ 0.5 MHz and T 2 ¼ 6.56 ð0.17Þ μs.
than the typical gate time, we take δ as a random timeindependent variable [17,23]. For simplicity, we consider the on-resonance case (Ω0 ¼ 0) and set the axis n to the x axis. Then, the Hamiltonian H can be written as H ¼ 2πðδSz þ ω1 Sx Þ. We define the fidelity of the gate Fg ¼ TrðAB−1 Þ=2, where A ¼ expð−i2πω1 Sx τÞ is the ideal gate operation and B ¼ expð−iHτÞ is the gate operation in the presence of dephasing noise, where τ is the gate time. The case of interest in implementing DCGs is δ ≪ ω1 . The infidelity of the gate Δ ¼ 1 − Fg for the plain case (rectangle pulse) is of the second order in δ=ω1 (see Table I). Since the nuclear spins flip, the qubit experiences a different effective δSz in every experiment once a new sequence is started. TABLE I. Infidelity of the SUPCODE π gate as a function of δ=ω1 . The words “plain” and “n piece” stand for a normal rectangle pulse and an n-piece SUPCODE pulse, respectively (see Ref. [22]). Sequence Plain Three piece Five piece Nine piece
Infidelity Δ 0.5ðδ=ω1 Þ2 þ Oðδ=ω1 Þ4 11:1ðδ=ω1 Þ4 þ Oðδ=ω1 Þ6 64:1ðδ=ω1 Þ6 þ Oðδ=ω1 Þ8 317 237ðδ=ω1 Þ8 þ Oðδ=ω1 Þ10
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When we average over many sequences, as required to build statistics, δSz will cause the decay of the Rabi oscillations even in the absence of the instability of control field HC [23]. Herein, we apply SUPCODE to protect the quantum gates from the decoherence effect aroused by the static fluctuation of the magnetic field. We denote the rotation of an angle θ around the axis in the equatorial plane with azimuth ϕ as Rϕ ðθÞ. The sequence of the five-piece SUPCODE π gate is f½τ1 − R0 ð2πω1 τ2 Þ − τ3 − R0 ð2πω1 τ2 Þ − τ1 × 2g, where τ1 ¼ 1.05τ0 , τ2 ¼ 0.625τ0 , and τ3 ¼ 1.71τ0 , with τ0 ¼ 1=ω1 . The details of other pulse sequences are included in the Supplemental Material [22]. The infidelity of the five-piece SUPCODE gate is calculated to be of the sixth order in δ=ω1 (see Ref. [22]). Table I summarized the infidelity of several SUPCODE π gates. Because of a large coefficient in the leading term, the nine-piece SUPCODE only provides a better performance when δ=ω1 ≲ 1% (see Ref. [22]), where the infidelity of the five-piece SUPCODE is below 10−4. Hence, we focus on the five-piece SUPCODE in this Letter. The setup and experimental scheme are schematically shown in Fig. 1(b). A B0 of 513 G was adopted so as to achieve effective polarization of the nitrogen nuclear spin in the nitrogen-vacancy (NV) center, which is confirmed by the continuous-wave spectrum shown in Fig. 1(c). Quantum states j0i and j1i are encoded as a qubit, which can be manipulated via microwave pulses. A near-surface NV in a typical IIa bulk diamond and 0.1% 13 C is used. The undesired couplings between the qubit and the surrounding 13 C nuclear spins lead to the dephasing effect. The resulting FID of the qubit is depicted in Fig. 1(d), where a dephasing time T 2 ¼ 6.56 ð0.17Þ μs is obtained. We first scrutinize the performance of SUPCODE under dephasing noises. We experimentally investigated the π rotation about the x axis (NOT gate) via a plain pulse and via three- and five-piece SUPCODEs. The state of the electron spin was first initialized to j0i by laser. After applying the NOT gate on the initial state, the state fidelity pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [1] of the resultant state ρf is Fs ¼ h1jρf j1i and the infidelity of state is δf ¼ 1 − Fs . For the case of the NOT gate on the initial state j0i, the infidelity of the states δf equals the infidelity of quantum gates Δ (see Ref. [22]), so the performance of the NOT gate can be characterized by state fidelity [11]. We are aiming to investigate the robustness of SUPCODE against the noise stemming from the quasistatic fluctuation of the magnetic field, which is simulated by detuning the frequency of the microwave from the on-resonance frequency. The value of δ ranges from 0 to 6 MHz, and the strength of the control field is set at 20 MHz, which is verified by Rabi nutation experiments. The three different pulse sequences for the π gate are given in Fig. 2(a), and their performances are plotted as functions of δ=ω1 in Fig. 2(b). Lines are theoretical predictions, while green triangles, black rectangles, and red circles are the
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FIG. 3 (color online). Universal control of a single spin by the five-piece SUPCODE pulse. The left panels of (a) and (b) show the Bloch-sphere diagrams of the spin vectors rotating around (a) x and (b) y, respectively. The right panels are the corresponding detected coherent oscillations of the qubit driven by the five-piece SUPCODE pulses. Light red (blue) lines are the theoretical predictions with X (Y) detection, and red (blue) rectangles are the corresponding experimental data. FIG. 2 (color online). (a) Pulse sequences corresponding to a plain sequence and to three- and five-piece SUPCODE sequences. The calculation of time durations τi (i ¼ 1, 2, 3, and 4) is included in the Supplemental Material [22] with τ0 ¼ 1=ω1 and ω1 ¼ 20 ¼ MHz. (b) The measured and theoretically predicted state infidelities of the destination states as functions of δ=ω1 . Experimental data fit the theoretical predictions (lines) well. Some deviations are caused by statistical fluctuations of photon numbers, with the uncertainty plotted as a dashed gray line.
experimental data for a plain pulse and three- and five-piece SUPCODE π pulses, respectively. Each data point is averaged by 7.2 × 107 times, and the uncertainty due to the statistical fluctuation of the photon number has been plotted as a dashed line. The experimental data fit the theoretical predictions (lines) well. Our results show that three-piece (five-piece) SUPCODE pulses reduce the error to the fourth (sixth) order in δ=ω1 , in agreement with theoretical predictions. Second, we show that universal single-qubit gates can be achieved by SUPCODE. Universal control of a single qubit requires the ability to realize precisely rotations around two different axes of the Bloch sphere. By successively applying the five-piece SUPCODE pulses on the electron spin, we can force the spin vector to rotate along the x (y) axis by setting the phase of the microwave pulse to 0 (π=2). The following readout procedure consists of a detecting π=2 microwave pulse and a laser pulse. The phase of the detecting microwave is set to 0 (π=2) so that the detection is sensitive to the Y (X) coherence of the quantum states. If the qubit is driven to rotate along the x (y) axis, the spin vector will rotate in the y-z (x-z) plane (see Fig. 3), and coherent oscillations are observable by setting the phase of the detecting microwave pulse to 0 (π=2), while there will be no oscillations with the phase of the detecting microwave pulse being set to π=2 (0). Figure 3(a) [Fig. 3(b)] plots the coherent oscillations driven
by the five-piece SUPCODE pulse around the x (y) axis. Red circles (blue rectangles) with error bars are experimental data under X (Y) detection, and light red (light blue) lines are from the theoretical predictions. Third, we show that the performance of the quantum gate can be protected towards the T 1 limit via SUPCODE. To minimize the influence of the instability of the control field HC [23], we adopted a microwave pulse of an amplitude corresponding to ω1 ¼ 1 MHz. The quantum oscillation driven by SUPCODE is plotted in Fig. 4(a). A decay time constant T DCG ¼ 690 ð40Þ μs is derived from the experimental data (black crosses), which is 2 orders of magnitude longer than the dephasing time T 2 . The Rabi oscillation (blue diamonds) driven by the normal rectangle pulse in Fig. 4(a) of ω1 ¼ 1 MHz shows a decay time T 2 0 ¼ 135 ð10Þ μs, which is also much shorter than T DCG . This verified that SUPCODE can largely suppress the dephasing effect during the gate time. It is noted that T DCG achieved here has reached the spin-lattice relaxation time in the rotating frame T 1ρ ¼ 660 ð80Þ μs, which is determined by a spin-locking experiment. T 1ρ is the upper bound achievable by any multiple dynamical decoupling pulse sequence for T 2 [24]. To confirm this, we applied Carr-Purcell-Meiboom-Gill (CPMG) sequences on the qubit to prolong the coherence time with ω1 ¼ 10 MHz. The scaling of the extended coherence times with the number of pulses N is depicted in Fig. 4(c) as blue rectangles. We found that the coherence time can be efficiently prolonged by CPMG sequences with the scaling T 2 ∝ N 0.33 until it approaches the saturation value for N > 100 at T 1ρ . The related decay times are summarized in Fig. 4(d). The coherence time without any dynamical decoupling control
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FIG. 4 (color online). (a) Comparison of the quantum oscillation driven by five-piece SUPCODE pulses with the one driven by plain pulses. Black crosses are the SUPCODE experimental data, and red lines are the envelopes of the function 0.5 þ 0.5 expð−T=T DCG Þ cosðωTÞ (light gray line), with T DCG ¼ 690 ð40Þ μs. The measured (blue diamonds) quantum oscillation by the plain pulses (Rabi oscillation) was fitted by 0.5 þ 0.5 expð−T=T 02 Þ cosðω0 TÞ (blue lines) with T 02 ¼ 135 ð10Þ μs. (b) The decay of the fidelity of five-piece SUPCODEs obtained via quantum process tomography. The black line is the theoretically predicted fidelity when only the T 1ρ process is taken into account. The shaded region is calculated with the uncertainty of the measured T 1ρ . Blue triangles are experimental fidelities by QPT, which shows a T 1ρ -limited decay. (c) Extended coherence times via CPMGs. N stands for the number of the π pulses. After applying more than 1000 pulses, the prolonged coherence times are limited by T 1ρ ¼ 660 ð80Þ μs. (d) Comparison of the decay times.
pulse is T 2 ¼ 6.56 ð0.17Þ μs. It can be prolonged to T 2 ¼ 123:2 ð8.8Þ μs by a single refocusing π pulse, which only enables robust quantum-state storage against the dephasing effect. The Rabi oscillation driven by a normal pulse, which can be used to realize the quantum gate, has a decay time T 20 ¼ 135 ð10Þ μs when ω1 ¼ 1 MHz. In comparison, the decay time of the quantum oscillation via SUPCODE is T DCG ¼ 690 ð40Þ μs, which equals T 1ρ ¼ 660 ð80Þ μs within the uncertainty. Finally, we examine the performance of the SUPCODEs quantitatively by quantum process tomography (QPT). Any quantum operation can be considered as mapping the input quantum state ρi to the output ρo ¼ εðρi Þ ¼ P † m;n χ mn Am ρi An with fAm g ¼ fI; σ x ; σ y ; σ z g, where σ x , σ y , and σ z are Pauli operators. The matrix χ completely and uniquely describes the process ε and can be experimentally reconstructed by QPT. The average fidelity of the quantum gate [25] is defined by 1 1 X Fðε; UÞ ¼ þ Tr½Uσ j U † εðσ j Þ; 2 12 j¼x;y;z
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where U is the ideal operation. We applied five-piece SUPCODE π=2 gates successively on the electron spins and performed QPT when the gates were repeated for M ¼ 0, 27, 54, 81, 108, and 135 times. The variation of gate fidelities is depicted as a function of the operation time in Fig. 4(b). Experimental data of the gate fidelity (blue triangles) were obtained by using QPT. We also plot the theoretical predicted variation of gate fidelities (black line and shaded region) with only T 1ρ relaxation being considered. The fact that the experimental data match the theoretical prediction indicates that the performance of the SUPCODE is limited by T 1ρ. The details of experimental QPT and theoretical calculations can be found in the Supplemental Material [22]. The fidelity of each five-piece SUPCODE π=2 gate is derived to be 0.9961 (2) (see Ref. [22]), close to the requirement of the fault-tolerant quantum computation [26], while the gate time (5.063 μs) is comparable to T 2 . As T 1ρ or the limit of T 2 can be prolonged by 2–3 orders of magnitude in proportion to T 1 (i.e., ∼0.5T 1 [27]) by lowering the temperature in a wide range [27–29], the DCGs demonstrated here can be considered as essentially T 1 limited. Further improvement of the performance of DCGs is feasible by prolonging T 1 via lowering temperature. To summarize, we applied SUPCODE to greatly suppress the error aroused by the static fluctuation of the magnetic field during quantum gates. The performance of the quantum gates, quantitatively measured by QPT, has been protected to the limit of T 1ρ . Our work is expected to have various applications in quantum information processing, high resolution spectroscopy [30,31], and various quantum metrologies [32], where high-fidelity quantum gates are required. Since the fluctuation of the static magnetic field exists in many important quantum qubit systems, such as trapped ions [33,34], superconducting qubits [35,36], and phosphorus doped in silicon [37–39], the DCGs implemented here may also be applied to these systems. Although our work focuses on single-qubit DCGs, theoretical schemes for two-qubit DCGs are available [16] for experimental implementation in the future. Our experimental implementation of high-fidelity and essentially T 1 -limited DCGs marks an important step towards realistic fault-tolerant quantum computation. We thank Yiqun Wang in the Suzhou Institute of Nano-Tech and Nano-Bionics for fabricating the coplanar waveguide and Ren-Bao Liu from the Chinese University of Hong Kong for helpful discussions. This work was supported by the National Key Basic Research Program of China (Grant No. 2013CB921800), the National Natural Science Foundation of China (Grants No. 11227901, No. 11275183, No. 11274299, No. 11104262, No. 91021005, and No. 10834005), the “Strategic Priority Research Program (B)” of the CAS (Grant No. XDB01030400), and the Fundamental Research Funds for the Central Universities.
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