Coupling-selective quantum optimal control in weak-coupling NV- 13 C system

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 13 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 13 C nuclear spins through weak-coupling with the NV center. In the experiments, the iSWAP † gate with selective optimal quantum control is implemented in a time-span of T ctrl = 170.25 µ s, which is comparable to the phase decoherence time T 2 = 203 µ s . 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.


Introduction
Currently, quantum information processing system is at its NISQ period, which is characterized with medium sized number of qubits, noisy gates, and short life time. Performing as many as possible quantum gates is extremely important in these devices. Typical NISQ devices include superconducting qubits and NV centers. One type of noise is the one caused by weak coupling with the environment. These noises are difficult to suppress using optimal control techniques. Since the decoupling operation will require the same length of time as the life time of the system. NV center is a typical system of that kind.
NV center is a defect consisting of a substitional nitrogen atom and adjacent vacancy in diamond. The spin state of NV center can be conveniently initialized and read-out with laser illumation [1] , and the triplet ground state can be coherently manipulated with resonant microwaves [2]. Nuclear spins around an NV center are also exploited as qubits [3]. The strong coupling nuclear spins, with distinguishable frequency splitting in optical detected magnetic resonance (ODMR) spectrum, can be manipulated with frequency-selective pulse. The NV center and nuclear spin system is a promising platform for quantum information processing [4,5]. Many quantum algorithms [6,7], quantum error corrections [8,9], and quantum simulations [10] have been realized with NV center system. The system is an excellent platform for detecting ultra-weak magnetic fields [11][12][13].
There are many factors affecting the control performance in NV center system. Firstly, the thermal distribution and state-dependent evolution of the 13 C nuclear spin bath are the major decoherence Full list of author information is available at the end of the article mechanisms of the NV electron spin in high purified diamond. The weak coupling spins, with coupling strength smaller than the inhomogeneous broadening, are selectively controlled with dynamic decoupling (DD) sequence filtering the unwanted spins. Secondly, the cross-talk between different nuclear spin states disturbs the control process in high-density ODMR spectrum. Thirdly, taking the real circumstance into consideration, the experimental factors, such as the microwave amplitude error and frequency detuning error, can also limit the control performance. To achieve high fidelity and robust control on these multi-qubit systems, many control methods [6,[14][15][16][17][18], suppression of phase noise in the of control hardware [19], topological dynamical decoupling [20], measurement-based feedback control [21], and holonomic gates [22], have been proposed and been tested in the experiments.
Quantum optimal control theory (OCT) has been extensively utilized to improve the control performance in multi-qubit systems. Gradient ascent pulse engineering (GRAPE) method [23] is widely used in magnetic resonance control. The cross-talk effect was suppressed with the OCT method [8,16]. The robust OCT method, which is a multi-objective optimization, can suppress the quasi-static errors, such as the thermal noise and the control amplitude error, and prolong greatly the NV coherence time over T * 2 [7,18]. Optimal control method can be combined with average Hamiltonian theory (AHT) to suppress the decoherence effect dynamically [24,25] .
However, in NV-13 C weak coupling system, the required OCT control time can be much longer than both T * 2 of the NV electron spin and the period of the nuclear bath evolution [26]. At this time scale, the quasi-static assumption about the coupling noise is completely valid. An optimal control method that decouples the evolving bath is essential for multi-qubit control in such NV system, especially for the weak coupling case.
To accomplish the multi-qubit optimal control in NV center system, in particularly the weak-coupling system, we develop an efficient coupling-selective OCT (COCT), which decouples the system-environment and takes the evolution of the 13 C bath into consideration. The optimization task of multi-qubit system in a long period can be divided into multiple tasks in smaller periods so as to further improve the effect. The method is experimentally tested in a NV-13 C weak-coupling system, and it has been shown that the lifetime has been prolonged significantly, and control rotation gate with five times of the phase decoherence time of NV qubit is completed with high fidelity.

The NV system
The ground state of a NV center is a spin triplet with m s = 0, ±1 . The m s = −1 and m s = 0 states of the NV center are chosen as the |0� and |1� states of the electron qubit. The system is subject to a static magnetic field, and the Hamiltonian in the rotating frame is where � x/y (t) is the Rabi frequency of the x/y component, and I x/y = σ x/y /2 is the corresponding spin operator.
The 13 C nuclear spins around the NV center interact through the electron spin with dipolar coupling. At the same time, the 13 C nuclear spins precess in the static magnetic field, and is the isotropic (anisotropic) coupling strength between the NV center and the ith nuclear spin, and ω i n is the Zeeman splitting of the ith nuclear spin. The weak 13 C-13 C interaction has been omitted in Eq. (2).
The 13 C nuclear spin can be divided qubit nuclear spin ( i ∈ S ) and the others ( j ∈ B ) as the environment. The total Hamiltonian is where ω i n ( ω j n ) is the Zeeman splitting of the ith ( jth ) nuclear spin.
The hyperfine interaction of the 13 C bath can be represented as an effective field B noise = j∈B (γ j x I j x + γ j z I j z ) applied on the electron spin. The fluctuation of this noise field results from both the thermal distribution and the dynamic evolution of the 13 C bath.

OCT in the weak-coupling NV system
For the weak-coupling system, the needed control time is comparable to T 2 time. In this time regime, bath evolution H B = j∈B ω j n I j z fluctuates, and the robust OCT method, considering only the quasi-static thermal noise, fails. Here, we give a coupling selective OCT that is suited for quantum control in the weak-coupling system. The essential idea is to combine average Hamiltonian theory (AHT) and OCT so as to suppress unwanted coupling and strengthen the wanted coupling. We use the propagator method and expand it using a time-dependent expansion. Then, we optimize the target so as to achieve our goal of the optimal control.
(1) Propagator of target system The system Hamiltonian is The function to optimize is fidelity of the evolution defined as is the evolution of the target system, U w is the objective quantum gate, and N D is the dimension of the target system.
(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 subsystem composed of the target system and one 13 C nuclear spin in the bath as where I 2 and I S are the identity operator in the nuclear spin and target system Hilbert space, respectively, and for the bath spin, the Ramsey frequency ω n approximated as ω 0 = γ n B.
(3) Decoupling optimization To suppress the unwanted evolution of the target system, we should decouple the system and make Q to the form I S ⊗ V , where V can be an operator in the bath space.
The decoupling optimization is performed from the lowest order of Q. The 1st order term disturbing the target system evolution is To make Q into I S ⊗ V form, we minimize the norm of Q x/z . The objective function is where α k is the penalty parameter, and N D is the dimension of the system subspace. The first part of the objective function is maximize to realize target evolution. The second part is minimized to decouple the bath. The objective function can be optimized with the gradient ascent method (see the Supplement).
Higher order decoupling method is given in the Supplement.
(4) Sub-sequence setting By dividing the whole pulse sequence into many sub-sequences and decoupling in each sub-sequence, the decoupling performance can be further improved.
We divide the whole pulse sequence into N equal widths sub-sequences, and set the objective function as where U i S is the propagator in the ith subsequence ( is the corresponding term of perturbed propagator, and α k is the penalty parameter. The first part of the objective function make the total evolution U N S · · · U 1 S approach target evolution. The optimization of second part decouples the bath in each subsequence.
After optimization, the system of the electron spin and the chosen 13 C spin together realize objective propagator U w . The unselected 13 C spins will rotate independently, Because the optimization process needs only the algebra structure of the coupling term, the decoupling optimization is valid for every nuclear spin in the bath.

Experiment
We demonstrate the viability of the new COCT method in a weak-coupling NV-13 C system , illustrated in Fig. 1(a), experimentally. The NV center is in a type IIa diamond, with a 511 G static magnetic field applied on the [1 1 1] axis. The m s = 0 and m s = +1 states constitute the electron qubit. The sample contains two distinguishable 13 C nuclear spins as shown in Fig. 1(b). These nuclear spins can be detected by applying a dynamic decoupling sequence. Varying the delay time τ between the π pulses, the two distinguishable coherence dips can be seen clearly, which reveal the existence of coupled nuclear spins. The fitted coupling strengths of these two nuclear spins are shown in Fig. 1(b), and they are much smaller than the measured inhomogeneous broadening ≈ 380 kHz.
Using the COCT method, we realized the NV-13 C control, such as the iSWAP † gates, the Control-R φ ±x gates. We illustrate the performance of NV-C 1 gates in the main text, and the results of NV-C 2 control are shown in the Supplement material. Utilizing these 2-qubit gates, we prepared an entanglement state on the two nuclear spins. We present them in the following.
(1) iSWAP † gate An iSWAP † gate swaps the quantum state of the electron spin and the nuclear spin, the matrix representation is The optimized NV-C 1 iSWAP † gate consists of 30 decoupled sub-sequences, each with the same shape, the (13) waveforms are shown in Fig. 2(a), (b). The total length of the pulse sequence is 30 × 5675 ns = 170.25 µ s, which is comparable to the phase decoherence time T 2 = 203 µs.
In order to test the selectivity of the searched pulse sequence, we simulated the pulse sequence on a system with various values of γ x and γ z parameters. The simulation results are illustrated in Fig. 2(d), (e). We simulated the two-qubit system with a weak coupling with the following Hamiltonian where H NV (t) changes as the optimized sequence. The fidelities between the simulated propagators and the iSWAP † gate (the propagator I 2 ⊗ e −i[(ω 0 +γ z /2)I z +γ x /2I x ]·T ) are represented as different colors in (a) ((b)).
The iSWAP † gate is a widely used gate for state preparation and readout measurement. The performance of this iSWAP † gate is tested in the nuclear Ramsey experiment ( Fig. 2(f )). By preparing the electron spin in 0 state and the nuclear spin in a superposition state, the nuclear spin processes at a frequency of ω 0 ≈ γ n B . At last, the nuclear state is swapped to the electron spin for the readout (Fig. 2(c)).
(2) Control-R φ ±x gate Another important two-qubit gate is the Control-R φ ±x gate, which can be described as The optimized pulse sequence for each C-R π/28 ±x gate lasts for 5670 ns. At the same time, the C-R π/2 ±x gate consisting of 14 C-R π/28 ±x gates is also optimized. The waveforms (14) Fig. 1 (a) The schematic of the 3 qubits system. The two detectable weak coupling 13 Cs are taken as qubits. And the rest nuclear spins are taken as an environment, which should be decoupled during the control process. We also demonstrated the selectivity of the C-R π/2 ±x gate using numerical simulations, in Fig. 3(a), (b). The drast selectivity of the isotropic coupling strength γ z guarantees the coherence keeping capability of the pulse sequence.
The coherence protection capability of the COCT is demonstrated in experiment shown in Fig. 3(c). Preparing the electron spin in superposition state 1 √ 2 (|0� + |1�) , and performing C-R π/28 ±x gates successively, we can observe the oscillation of the electron coherence ( Fig. 3(d)). The electron spin coherence is kept over 1.02(8) ms, which is 5 times longer than the phase coherence time T 2 ≈ 203 µs. Utilizing the iSWAP † gate for the state preparation and readout of the nuclear spin, we can observe the nuclear rotation with the C-R π/28 ±x gates application (Fig. 3(e)). The nuclear spin rotates along opposite directions under different electron spin states in the experiment (Fig. 3(f )).

Discussion
We realized optimal control in weak coupling system for the first time. In our optimization, the hyperfine interaction between the evolving bath and the central spin is decoupled. The performance of the new method is demonstrated in experiments. The advantages of the new The diagram to observe the nuclear rotation with C-R φ ±x gates applied. By preparing the electron spin in different states, the nuclear spin rotates along opposite axis with the application of the C-R φ ±x gates. The superposition state on C 1 is prepared with similar method in Fig.2(c). (f) The nuclear rotation under repetitively applying the C-R φ ±x gate, with different electron states. The red (blue) points are the result with electron spin in m s = 0 ( m s = +1 ) state method are being a high fidelity control and the robustness to quasi-static noise, of the optimal control is inherited by the new method.
The radio frequency field can also be applied to manipulate the nuclear spins directly. In this work, the flip-flop between 13 C spins has been ignored for the small dipolar interaction. With the applied radio frequency field, the decoupling of the 13 C-13 C interaction can also be introduced into the optimization.
The performance of the optimized pulse sequence can be further improved with the feedback control [18]. As the low temperature can extend the coherence time of the NV electron spin [27,28], the experiments in low temperature should be greatly improved.
The COCT method which presented here is universal, and it can also be extended to other weak-coupling quantum systems. Recently, quantum optimal control is applied in quantum sensing application, especially in NV system. The COCT method can also be applied in this area.