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/lp/springer-journals/coupling-selective-quantum-optimal-control-in-weak-coupling-nv-13-CQxE3UAxrx
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- Springer Journals
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- Copyright © The Author(s) 2023
- ISSN
- 2309-4710
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- 2309-4710
- DOI
- 10.1007/s43673-022-00072-1
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Abstract
Quantum systems are under various unwanted interactions due to their coupling with the environment. Efficient control of quantum system is essential for quantum information processing. Weak-coupling interactions are ubiq- uitous, and it is very difficult to suppress them using optimal control method, because the control operation is at a time scale of the coherent life time of the system. Nitrogen-vacancy (NV ) center of diamond is a promising platform for quantum information processing. The C nuclear spins in the bath are weakly coupled to the NV, rendering the manipulation extremely difficulty. Here, we report a coupling selective optimal control method that selectively suppresses unwanted weak coupling interactions and at the same time greatly prolongs the life time of the wanted quantum system. We applied our theory to a 3 qubit system consisting of one NV electron spin and two C nuclear spins through weak-coupling with the NV center. In the experiments, the iSWAP gate with selective optimal quan- tum control is implemented in a time-span of T = 170.25 μ s, which is comparable to the phase decoherence time ctrl T = 203 μs . The two-qubit controlled rotation gate is also completed in a strikingly 1020(80) μ s, which is five times of the phase decoherence time. These results could find important applications in the NISQ era. Keywords: Optimal control theory, NV center, NISQ, Quantum control NV center is a defect consisting of a substitional nitro- 1 Introduction gen atom and adjacent vacancy in diamond. The spin Currently, quantum information processing system is at state of NV center can be conveniently initialized and its NISQ period, which is characterized with medium read-out with laser illumation [1] , and the triplet ground sized number of qubits, noisy gates, and short life state can be coherently manipulated with resonant time. Performing as many as possible quantum gates microwaves [2]. Nuclear spins around an NV center are is extremely important in these devices. Typical NISQ also exploited as qubits [3]. The strong coupling nuclear devices include superconducting qubits and NV cent- spins, with distinguishable frequency splitting in optical ers. One type of noise is the one caused by weak coupling detected magnetic resonance (ODMR) spectrum, can with the environment. These noises are difficult to sup - be manipulated with frequency-selective pulse. The NV press using optimal control techniques. Since the decou- center and nuclear spin system is a promising platform pling operation will require the same length of time as for quantum information processing [4, 5]. Many quan- the life time of the system. NV center is a typical system tum algorithms [6, 7], quantum error corrections [8, 9], of that kind. and quantum simulations [10] have been realized with NV center system. The system is an excellent platform for detecting ultra-weak magnetic fields [11–13]. There are many factors affecting the control per- *Correspondence: xypan@aphy.iphy.ac.cn; gllong@mail.tsinghua.edu.cn formance in NV center system. Firstly, the ther- Beijing Academy of Quantum Information Sciences, 100193 Beijing, China mal distribution and state-dependent evolution of Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, 100190 Beijing, China the C nuclear spin bath are the major decoherence Full list of author information is available at the end of the article © The Author(s) 2022. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Zhang et al. AAPPS Bulletin (2023) 33:2 Page 2 of 8 mechanisms of the NV electron spin in high purified 2 The NV system diamond. The weak coupling spins, with coupling The ground state of a NV center is a spin triplet with strength smaller than the inhomogeneous broadening, m = 0,±1 . The m =−1 and m = 0 states of the NV s s s are selectively controlled with dynamic decoupling center are chosen as the |0� and |1� states of the electron (DD) sequence filtering the unwanted spins. Secondly, qubit. The system is subject to a static magnetic field, and the cross-talk between different nuclear spin states the Hamiltonian in the rotating frame is disturbs the control process in high-density ODMR H (t) NV spectrum. Thirdly, taking the real circumstance into = � (t)I + � (t)I (1) x x y y 2π consideration, the experimental factors, such as the microwave amplitude error and frequency detun- where � (t) is the Rabi frequency of the x/y component, x/y ing error, can also limit the control performance. and I = σ /2 is the corresponding spin operator. x/y x/y To achieve high fidelity and robust control on these 13 The C nuclear spins around the NV center interact multi-qubit systems, many control methods [6, 14–18], through the electron spin with dipolar coupling. At the suppression of phase noise in the of control hardware 13 same time, the C nuclear spins precess in the static [19], topological dynamical decoupling [20], meas- magnetic field, and urement-based feedback control [21], and holonomic H + H gates [22], have been proposed and been tested in the NV-C C i i i i i i =|0��0|⊗ (γ I + γ I ) + ω I , x x z z n z experiments. 2π i i Quantum optimal control theory (OCT) has been (2) extensively utilized to improve the control performance i i where γ (γ ) is the isotropic (anisotropic) coupling z x in multi-qubit systems. Gradient ascent pulse engineer- strength between the NV center and the ith nuclear spin, ing (GRAPE) method [23] is widely used in magnetic and ω is the Zeeman splitting of the ith nuclear spin. The resonance control. The cross-talk effect was suppressed 13 13 weak C- C interaction has been omitted in Eq.(2). with the OCT method [8, 16]. The robust OCT method, The C nuclear spin can be divided qubit nuclear spin which is a multi-objective optimization, can suppress ( i ∈ S ) and the others ( j ∈ B ) as the environment. The the quasi-static errors, such as the thermal noise and total Hamiltonian is the control amplitude error, and prolong greatly the NV coherence time over T [7, 18]. Optimal control method H(t) H (t) NV 2 i i i i i i = + |0��0|⊗ (γ I + γ I ) + ω I z z x x n z can be combined with average Hamiltonian theory 2π 2π i∈S i∈S (AHT) to suppress the decoherence effect dynamically j j j j j j +|0��0|⊗ (γ I + γ I ) + ω I [24, 25] . z z x x n z j∈B j∈B However, in NV- C weak coupling system, the (3) required OCT control time can be much longer than ∗ i both T of the NV electron spin and the period of the where ω ( ω ) is the Zeeman splitting of the ith ( jth ) 2 n nuclear bath evolution [26]. At this time scale, the nuclear spin. quasi-static assumption about the coupling noise is The hyperfine interaction of the C bath can be rep- j j j j completely valid. An optimal control method that resented as an effective field B = (γ I + γ I ) noise x x z z j∈B decouples the evolving bath is essential for multi-qubit applied on the electron spin. The fluctuation of this noise control in such NV system, especially for the weak cou- field results from both the thermal distribution and the pling case. dynamic evolution of the C bath. To accomplish the multi-qubit optimal control in NV center system, in particularly the weak-coupling sys- 2.1 OCT in the weak‑coupling NV system tem, we develop an efficient coupling-selective OCT For the weak-coupling system, the needed control time (COCT), which decouples the system-environment and is comparable to T time. In this time regime, bath evo- j j takes the evolution of the C bath into consideration. lution H = ω I fluctuates, and the robust OCT B z j∈B The optimization task of multi-qubit system in a long method, considering only the quasi-static thermal noise, period can be divided into multiple tasks in smaller fails. Here, we give a coupling selective OCT that is suited periods so as to further improve the effect. The method for quantum control in the weak-coupling system. The is experimentally tested in a NV- C weak-coupling sys- essential idea is to combine average Hamiltonian theory tem, and it has been shown that the lifetime has been (AHT) and OCT so as to suppress unwanted coupling prolonged significantly, and control rotation gate with and strengthen the wanted coupling. We use the propa- five times of the phase decoherence time of NV qubit is gator method and expand it using a time-dependent completed with high fidelity. Zhang et al. AAPPS Bulletin (2023) 33:2 Page 3 of 8 expansion. Then, we optimize the target so as to achieve The decoupling optimization is performed from the low - our goal of the optimal control. est order of Q. The 1st order term disturbing the target system evolution is (1) Propagator of target system The system Hamiltonian is Q = dt U(t ) · I ⊗ I · U (t ). (10) x/z 1 1 z x/z 1 i i i i i i H (t) = H (t) +|0��0|⊗ (γ I + γ I ) + ω I S NV z z x x n z To make Q into I ⊗ V form, we minimize the norm of i∈S i∈S Q . The objective function is (4) x/z The function to optimize is fidelity of the evolution defined as † |Tr(U · U )| want † − α · Tr(Q · Q ) k k (11) 2 k † 2 �U · U (T )� w S k∈{x,z} F = S (5) where α is the penalty parameter, and N is the dimen- k D −i H (t )dt S 1 1 U (t) = T · e sion of the system subspace. The first part of the objective where is the evolution of the function is maximize to realize target evolution. The sec - target system, U is the objective quantum gate, and N w D ond part is minimized to decouple the bath. The objec - is the dimension of the target system. tive function can be optimized with the gradient ascent method (see the Supplement). (2) System‑bath coupling The hyperfine interaction between the electron spin and the bath spins disturbs the propagator of the target system. We represent the subsys- Higher order decoupling method is given in the tem composed of the target system and one C nuclear Supplement. spin in the bath as (4) Sub‑sequence setting By dividing the whole pulse H (t) = H (t) ⊗ I + |0��0|⊗ (γ I + γ I ) + I ⊗ ω I , SB S 2 z z x x S n z sequence into many sub-sequences and decoupling in (6) each sub-sequence, the decoupling performance can be where I and I are the identity operator in the nuclear spin 2 S further improved. and target system Hilbert space, respectively, and for the bath spin, the Ramsey frequency ω approximated as ω = γ B. n 0 n We divide the whole pulse sequence into N equal widths sub-sequences, and set the objective function as Taking the decoupled Hamiltonian as the reference † N 1 2 |Tr(U · U ··· U )| want S S i † i H (t) = H (t) ⊗ I + I ⊗ ω I , S 2 S 0 z (7) ref − α ·�Q · Q � k k i=1 k∈{x,z} and (12) H (t) = H (t) + H (t), SB ref c (8) where U is the propagator in the ith sub- sequence ( [(i − 1)T /N , iT /N ] ) and the propagator Q in the toggling frame can be expanded iT /N i † Q = dt U(t ) · I ⊗ I · U (t ) is the cor- 1 1 z 1 k (i−1)T /N as responding term of perturbed propagator, and α is the penalty parameter. The first part of the objective func - −i [H (t )+H (t )]dt ref 1 c 1 1 Q =Te N 1 tion make the total evolution U ··· U approach target S S (9) evolution. The optimization of second part decouples the =I ⊗ I − i U (t ) H U (t )dt + ··· , S 2 1 c 1 1 0 bath in each subsequence. where the referenced propagator After optimization, the system of the electron spin and U (t) = T · exp −i H (s)ds , and the coupling term ref 13 the chosen C spin together realize objective propagator H =|0��0|⊗ (γ I + γ I ) = (I/2 + I ) ⊗ (γ I + γ I ). c z z x x z z z x x U . The unselected C spins will rotate independently, j j j −i[(ω +γ /2)I +γ /2I ]·T 0 z z x x U = e . (3) Decoupling optimization To suppress the unwanted evolution of the target system, we should decouple the Because the optimization process needs only the algebra system and make Q to the form I ⊗ V , where V can be an structure of the coupling term, the decoupling optimiza- operator in the bath space. tion is valid for every nuclear spin in the bath. Zhang et al. AAPPS Bulletin (2023) 33:2 Page 4 of 8 3 Experiment waveforms are shown in Fig. 2(a), (b). The total length of We demonstrate the viability of the new COCT method the pulse sequence is 30 × 5675 ns = 170.25 µ s, which is in a weak-coupling NV- C system , illustrated in comparable to the phase decoherence time T = 203 µ s. Fig. 1(a), experimentally. The NV center is in a type IIa diamond, with a 511 G static magnetic field applied on In order to test the selectivity of the searched pulse the [1 1 1] axis. The m = 0 and m =+1 states consti- sequence, we simulated the pulse sequence on a system s s tute the electron qubit. The sample contains two distin - with various values of γ and γ parameters. The simula - x z guishable C nuclear spins as shown in Fig. 1(b). These tion results are illustrated in Fig. 2(d), (e). We simulated nuclear spins can be detected by applying a dynamic the two-qubit system with a weak coupling with the fol- decoupling sequence. Varying the delay time τ between lowing Hamiltonian the π pulses, the two distinguishable coherence dips can H(t, γ , γ ) = H (t) + S ⊗ (γ I + γ I ) + ω I , x z NV z z z x x 0 z be seen clearly, which reveal the existence of coupled (14) nuclear spins. The fitted coupling strengths of these two where H (t) changes as the optimized sequence. The NV nuclear spins are shown in Fig. 1(b), and they are much fidelities between the simulated propagators and the smaller than the measured inhomogeneous broadening † −i[(ω +γ /2)I +γ /2I ]·T z z x x iSWAP gate (the propagator I ⊗ e ) ≈ 380 kHz. 13 are represented as different colors in (a) ((b)). Using the COCT method, we realized the NV- C con- † The iSWAP gate is a widely used gate for state prepa- trol, such as the iSWAP gates, the Control-R gates. ±x ration and readout measurement. The performance of We illustrate the performance of NV-C gates in the main this iSWAP gate is tested in the nuclear Ramsey experi- text, and the results of NV-C control are shown in the ment (Fig. 2(f )). By preparing the electron spin in 0 state Supplement material. Utilizing these 2-qubit gates, we and the nuclear spin in a superposition state, the nuclear prepared an entanglement state on the two nuclear spins. spin processes at a frequency of ω ≈ γ B . At last, the 0 n We present them in the following. nuclear state is swapped to the electron spin for the read- † † out (Fig. 2(c)). (1) iSWAP gate An iSWAP gate swaps the quantum state of the electron spin and the nuclear spin, the matrix (2) Control‑ R gate Another important two-qubit gate ±x representation is is the Control-R gate, which can be described as ±x 10 0 0 −iI φ 00 − i 0 e 0 iSWAP = C-R = (13) (15) ±x +iI φ 0 − i 00 0 e 00 0 1 π/28 The optimized pulse sequence for each C- R gate lasts ±x π/2 The optimized NV- C iSWAP gate consists of 30 decou- for 5670 ns. At the same time, the C-R gate consist- ±x π/28 pled sub-sequences, each with the same shape, the ing of 14 C-R gates is also optimized. The waveforms ±x Fig. 1 (a) The schematic of the 3 qubits system. The two detectable weak coupling Cs are taken as qubits. And the rest nuclear spins are taken as an environment, which should be decoupled during the control process. (b) The coherence of the NV electron spin as a function of τ in DD sequence. The green (red) line is the simulated curve with the 1st (2nd) nuclear parameters Zhang et al. AAPPS Bulletin (2023) 33:2 Page 5 of 8 Fig. 2 COCT pulse sequence and experimental demonstration of the iSWAP gate. (a) A sub-sequence of the COCT pulse sequence for the iSWAP gate. (b) The same as (a) for . (c) Circuit for nuclear Ramsey experiment using iSWAP gate. Initially, electron spin in |0� and nuclear spin is in (|0�+|1�) =|X+� . After delay τ , nuclear spin evolves into cosθ|0� + sinθ|1� . iSWAP gate transfer the nuclear spin state into the electron spin, and then is measured. (d) Selectivity test of the COCT sequence. The fidelity of the iSWAP gate. It reaches its maximum at the experimental value of γ = 65 kHz, γ = − 35.6 kHz (with crosspoint). (e) For system without the weak-coupling, the same COCT pulse sequence does not suppress the x z unwanted evolution, the fidelity remains high most of the time. (f ) Ramsey precession experiment. ISWAP successfully swap the state of nuclear spin to the electron spin. The Ramsey oscillation is clearly demonstrated in the experiment The coherence protection capability of the COCT is of the pulse sequences are shown in Fig. 2(c), (d) of demonstrated in experiment shown in Fig. 3(c). Prepar- Supplement. ing the electron spin in superposition state (|0�+|1�) , π/28 π/2 and performing C-R gates successively, we can We also demonstrated the selectivity of the C-R gate ±x ±x observe the oscillation of the electron coherence using numerical simulations, in Fig. 3(a), (b). The drast (Fig. 3(d)). The electron spin coherence is kept over selectivity of the isotropic coupling strength γ guar- 1.02(8) ms, which is 5 times longer than the phase coher- antees the coherence keeping capability of the pulse ence time T ≈ 203 µs . sequence. Zhang et al. AAPPS Bulletin (2023) 33:2 Page 6 of 8 π π Fig. 3 Numerical simulation and experimental demonstration of the C-R gate. (a), (b) The simulation to demonstrate the selectivity of the C-R ±x ±x gate. The isotropic (anisotropic) coupling strength increases along x (y) axis. The fidelity between the simulated propagator and the objective propagator (the independent propagator) is represented as different colors in (a) (b) . The white cross is the objective coupling strength of the NV-C C-R gate. The selected range of isotropic coupling γ is about 10 kHz. (c) The diagram to test the capability of coherence keeping of the C-R ±x ±x gate. We prepare the electron spin in superposition state and apply C-R gates repeatedly. (d) The coherence of the NV electron spin as a function ±x of the gate number. The fitted coherence time is about 1.02 ms. (e) The diagram to observe the nuclear rotation with C- R gates applied. By ±x preparing the electron spin in different states, the nuclear spin rotates along opposite axis with the application of the C- R gates. The superposition ±x state on C is prepared with similar method in Fig.2(c). (f ) The nuclear rotation under repetitively applying the C-R gate, with different electron ±x states. The red (blue) points are the result with electron spin in m = 0 ( m = +1 ) state s s 4 Discussion Utilizing the iSWAP gate for the state preparation We realized optimal control in weak coupling system for and readout of the nuclear spin, we can observe the π/28 the first time. In our optimization, the hyperfine inter - nuclear rotation with the C-R gates application ±x action between the evolving bath and the central spin is (Fig. 3(e)). The nuclear spin rotates along opposite decoupled. The performance of the new method is dem - directions under different electron spin states in the onstrated in experiments. The advantages of the new experiment (Fig. 3(f )). Zhang et al. AAPPS Bulletin (2023) 33:2 Page 7 of 8 Author details method are being a high fidelity control and the robust - Beijing Academy of Quantum Information Sciences, 100193 Beijing, China. ness to quasi-static noise, of the optimal control is inher- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, ited by the new method. Chinese Academy of Sciences, 100190 Beijing, China. State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua The radio frequency field can also be applied to manip - University, 100084 Beijing, China. Frontier Science Center for Quantum Infor- ulate the nuclear spins directly. In this work, the flip-flop mation, 100084 Beijing, China. Beijing Academy of Quantum Information, between C spins has been ignored for the small dipolar 100085 Beijing, China. interaction. With the applied radio frequency field, the Received: 4 September 2022 Accepted: 7 December 2022 13 13 decoupling of the C- C interaction can also be intro- duced into the optimization. The performance of the optimized pulse sequence can be further improved with the feedback control [18]. As References the low temperature can extend the coherence time of 1. A. Gruber, A. Dräbenstedt, C. Tietz, L. Fleury, J. Wrachtrup, C. Von Borc- zyskowski, Scanning confocal optical microscopy and magnetic reso- the NV electron spin [27, 28], the experiments in low nance on single defect centers. Science. 276(5321), 2012–2014 (1997) temperature should be greatly improved. 2. G. Fuchs, V. Dobrovitski, D. Toyli, F. Heremans, D. Awschalom, Gigahertz The COCT method which presented here is universal, dynamics of a strongly driven single quantum spin. Science. 326(5959), 1520–1522 (2009) and it can also be extended to other weak-coupling quan- 3. F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, J. 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Journal
AAPPS Bulletin – Springer Journals
Published: Jan 5, 2023
Keywords: Optimal control theory; NV center; NISQ; Quantum control