Shanping YU,Na LI,Peng WEI,Zairong XI

Key Laboratory of Systems and Control,Institute of Systems Science,Academy of Mathematics and Systems Science,CAS,Beijing 100190,China

Time-optimal control for hybrid systems based on the nitrogen-vacancy center

Shanping YU,Na LI,Peng WEI?,Zairong XI

Key Laboratory of Systems and Control,Institute of Systems Science,Academy of Mathematics and Systems Science,CAS,Beijing 100190,China

Fast and high fidelity quantum control is the key technology of quantum computing.The hybrid system composed of the nitrogen-vacancy center and nearby Carbon-13 nuclear spin is expected to solve this problem.The nitrogen-vacancy center electron spin enables fast operations for its strong coupling to the control field,whereas the nuclear spins preserve the coherence for their weak coupling to the environment.In this paper,we describe a strategy to achieve time-optimal control of the Carbon-13 nuclear spin qubit by alternating controlling the nitrogen-vacancy center electron spin as an actuator.We transform the qubit gate operation into a switched system.By using the maximum principle,we study the minimum time control of the switched system and obtain the time-optimal control of the qubit gate operation.We show that theXgate andYgate operations are within 10μs while the fidelity reaches 0.995.

Time-optimal control,nitrogen-vacancy center,qubit gate operations,switched system

1 Introduction

Nowadays,systems consisting ofelectron and nuclear spins have a very promising progress in quantum technologies[1-10].Nuclear spins associated with a lot of important defect electron spins in solids offer us natural systems,such as fullerene qubits[1],phosphorus donors in Si[2],and most recently nitrogen-vacancy(NV)centers in diamond[3-6].The defect electron spin enables fast operations for its strong coupling to the external controller,whereas the nuclear spins preserve the isolation and coherence for their weak coupling to external fields.Therefore,the system can achieve fast and high fidelity control[11-19].Nitrogen-vacancy center in diamond has become a powerful candidate system for quantum computers,due to its optical and quantum properties.Several control schemes have been proposed.By applying the π pulse to the electron spin,the nuclear spin can reach arbitrary target state and the switched control is universal[3].By alternating rotations for equal times,the decoherence-protected gate operation is obtained[4,5].However these schemes are not time-optimal.The indirect control of the nuclear spin is identified to be faster than the direct control of the nuclear spin[6].Therefore,to improve the speed of quantum gate operation is a very important problem to be studied.

In this paper,we focus on a special implementation based on the electron spin of single NV center in diamond as the actuator and nearby Carbon-13(13C)nuclear spin as the qubit.The quantum gate operation of the13Cnuclear spin is reached by a sequence of alternating rotations,which is caused by applying π pulse sequence to the NV center.Assuming that the π pulses are instantaneous,the π pulses can be modeled as switching signals.The quantum gate operation can thus be converted into a switched system.We study the solution properties of the switched system,which is used to simplify necessary conditions of the time-optimal control.By using the maximum principle,the time-optimal control of switched systems is obtained.Since the NV center is either in the|0〉or|1〉state,it is unnecessary to consider the problem of decoherence.

This paper is organized as follows.In Section 2,the quantum system is described and the quantum gate operation is modeled as a switched system.In Section 3,the solution to the switched system and the properties of the solution are presented.In Section 4,the necessary conditions of time-optimal control of the quantum gate operations are obtained by using the maximum principle.In Section 5,simulation results of four gate operations are given.In Section 6,conclusions of the main results are shown.

2 The system model

The quantum system consists of a single electron spin of NV center as a actuator and a13Cnuclear spin as the qubit by their hyperfine interaction[6].In a suitable rotating frame,the Hamiltonian of the system can be written as

|0〉and|1〉are the NV centerelection spin state,?Ix,?Iyand ?Izare the13Cnuclear spin operator.When the NV center is in the state|0〉,13Cnuclear spin is rotating around axes →n0with angular velocity ω0.When the NV center is in the state|1〉,13Cnuclear spin is rotating around axe →n1with angular velocity ω1.The axes and angular velocity in the two states are respectively defined as

Thus the nuclear spin evolves by rotating around two distinct axes,depending on the electronic spin state.Assume that the initial state of the NV center is|0〉,the time evolution operator?Uof the13Cnuclear spin is

Concatenating rotations about the two axes is enough to achieve full control lability of the nuclear spin[3].Therefore,we can indirectly control13Cnuclear spin rotations by flipping the NV center electron spin state with π pulse[6].

We use SU(2)group to characterize the rotation.ElementsU∈SU(2)are matrices with complex entries satisfying det(U)=1[20].A possible parametrization ofUis

and thezis satisfy the relation

This is the quaternion representation ofU,and is related to the group of rotations SO(3)[20].There exists θ ∈ [0,4π]and →n∈S2(0,1)(the unit sphere centered at the origin)such that

The quantum Hamiltonian of a system takes the form

then the time evolution operator?U(t,t0)belongs to the group SU(2)and respect a rotation.Inserting(7)into the Schr¨odinger equation for the time evolution operator

we can obtain the following differential equations

andz=[z1z2z3z4]T.Differential equations(12)with condition(8)can characterize arbitrary rotation,which will be used in the next subsection.

Assume that the pulse is instantaneous.By putting the pulse as a switching signal,the system can be translated into a switched system

3 Properties of pulse sequences

The solution to the differential equation(12)with the initial statez0=[1 0 0 0]Tis

By computing,we can obtain

Therefore,the relationship(8)can be satisfied in the process of system evolution.

In the next section,we will use another functionp(t),which satisfies the same differentialequation asz(t)with a different initial value

Then we have

By computing,we can obtain

After several pulse sequences,the system can keep the above relationship.

4 Optimal time control

Consider the autonomous linear switched systems(14).The switched signal σ(t):[0,tf] → ζ ={0,1}is a piecewise constant function.

The control problem is to find an optimal switched signal σ＊(t)for a given target statezdto make it satisfy

and to minimizetf.The optimalobjection functionJ(z,σ)is

Put the system(14)into a bigger system,

u(t)=[u0(t)u1(t)]can be regarded as a piecewise continuous input control.The trajectory set of the embedded systems(25)contain the trajectories ofthe switched system(14).If the functionu(t)in the embedded systems(25)satisfy constraints

we can obtain the trajectories of the switched system(14).

The time-optimal control problem is described as follows:Given a target statezd,find the optimal controlu(t)＊and the corresponding optimal trajectoryz＊(t)of the embedded system(25)with limited conditions(25),such thatz(t)=z＊(t)satisfies the terminal condition(23),and minimize the optimal objection functionJ(z,u).The Hamilton function is

According to the maximum principle,we can obtain the following results:

1)z＊(t)satisfies

with the initial and terminal conditions

2)p(t)satisfies

with the terminal condition

whereg(z(tf))=zd-z(tf)is the terminal condition,and μ is undetermined.

3)The controlu＊satisfies

and we can obtain

where Θ={u(t)|uqi(t)=1,uqj(t)=0,qi?q j,when σ(t)=qi}is the admissible control function set.

4)The terminal timet＊fsatisfies

and we can obtain

Inserting(37)into(36),we can obtain

Assume the optimal controlu＊in a certain period of time[tj,tj+1)is

According to(34),we can obtain

Similarly,when the optimal controlu＊in a certain period of time[tk,tk+1]is

we can obtain

then the optimal controlu＊satisfies

According to(40)and(42),the switching time occurs when

In summary,the necessary conditions for minimum time control is

Choosing appropriatep(0)satisfying condition(38)and switch whenP(t)equals to zero,we can obtain the optimal solution to the time-optimal control within a certain error tolerance.

5 Simulation

and switch atP(t)=0.When?＜0.005,the optimaltimetfis 4.67μs and the switched number is 8,as shown in Fig.1.

X-π gate:zd=[0 0 0 1]T.Choose

and switch atP(t)=0.When ?＜0.005,the optimal timetfis 8.39μs and the switched number is 14,as shown in Fig.2.

Fig.1gate.The above four graphs represent the time evolution of z1,z2,z3 and z4.The below graph represents the time evolution of error ?.By applying 8 times of π pulse,the error of Y-π gate is 0.0049 at 4.67μs.

Fig.2 X-π gate.The above four graphs represent the time evolution of z1,z2,z3 and z4.The below graph represents the time evolution of error ?.By applying 14 times of π pulse,the error of Y-π gate is 0.0045 at 8.39μs.

and switch atP(t)=0.When?＜0.005,the optimaltimetfis 4.70μs and the switched number is 8,as shown in Fig.3.

Y-π gate:zd=[0 0 1 0]T.Choose and switch atP(t)=0.When ?＜0.005,the optimal timetfis 9.29μs and the switched number is 16,as shown in Fig.4.

Fig.3gate.The above four graphs represent the time evolution of z1,z2,z3 and z4.The below graph represents the time evolution of error ?.By applying 8 times of π pulse,the error of Y-π gate is 0.0049 at 4.70μs.

Fig.4 Y-π gate.The above four graphs represent the time evolution of z1,z2,z3 and z4.The below graph represents the time evolution of error ?.By applying 16 times of π pulse,the error of Y-π gate is below 0.0049 at 9.29μs.

6 Conclusions

Based on the hybrid system composed of a nitrogenvacancy center and single Carbon-13 nuclear spin,the time-optimal control of qubit gate operations has been researched.By controlling the NVcenter,the13Cnuclear spin can be indirectly controlled.The quantum gate operation is transformed into a switched system.By using the maximum principle,the minimum time control of switched system isrealized and the time-optimalcontrol of the qubit gate operation is obtained.The NV center is on the population state,it is unnecessary to consider the decoherence of the NV center.

[1]J.J.L.Morton,A.M.Tyryshkin,A.Ardavan,et al.Bang-bang control of fullerene qubits using ultrafast phase gates.Nature Physics,2006,2(1):40-43.

[2]J.J.L.Morton,A.M.Tyryshkin,R.M.Brown,et al.Solid-state quantum memory using the 31P nuclear spin.Nature,2008,455(7216):1085-1088.

[3]J.S.Hodges,J.C.Yang,C.Ramanathan,et al.Universal control of nuclear spins via anisotropic hyperfine interactions.Physical Review A,2007,78(1):124-124.

[4]T.van der Sar,Z.H.Wang,M.S.Blok,et al.Decoherenceprotected quantum gates for a hybrid solid-state spin register.Nature,2012,484(7392):82-86.

[5]T.H.Taminiau,J.Cramer,T.V.D.Sar,etal.Universalcontroland error correction in multi-qubit spin registers in diamond.Nature Nanotechnology,2013,9(3):171-176.

[6]C.D.Aiello,P.Cappellaro.Time-optimal control by a quantum actuator.Physical Review A,2015,91(4):DOI 10.1103/Phys-RevA.91.042340.

[7]A.Reiserer,N.Kalb,M.S.Blok,et al.Robust quantumnetwork memory using decoherence-protected subspaces of nuclear spins.Physical ReviewX,2016,6(2):DOI 10.1103/PhysRevX.6.021040.

[8]P.Cappellaro,L.Jiang,J.S.Hodges,et al.Coherence and control of quantum registers based on electronic spin in a nuclear spin bath.Physical Review Letters,2009,102(21):DOI 10.1103/PhysRevLett.102.210502.

[9]S.Saito,X.Zhu,R.Ams¨uss,et al.Towards realizing a quantum memory for a superconducting qubit:storage and retrieval of quantum states.Physical Review Letters,2013,111(10):3019-3023.

[10]X.Y.L¨u,Z.L.Xiang,W.Cui,et al.Quantum memory using a hybrid circuit with flux qubits and NV centers.Physical Review A,2013,88(1):10.1103/PhysRevA.88.012329.

[11]F.Jelezko,J.Wrachtrup.Single defect centres in diamond:A review.Physica Status Solidi A-Applications and Materials Science,2006,203(203):3207-3225.

[12]C.Kurtsiefer,S.Mayer,P.Zarda,et al.Stable solid-state source of single photons.Physical Review Letters,2000,85(2):307-310.[13]R.Hanson,F.M.Mendoza,R.J.Epstein,et al.Polarization and readout of coupled single spins in diamond.Physical Review Letters,2006,97(8):DOI 10.1103/PhysRevLett.97.087601.

[14]G.Balasubramanian1,P.Neumann,D.Twitchen,et al.Ultralong spin coherence time in isotopically engineered diamond.Nature Materials,2009,8(5):383-387.

[15]A.Gruber,A.Dr¨abenstedt,C.Tietz,et al.Scanning confocal optical microscopy and magnetic resonance on single defect centers.Science,2012,276(5321):2012-2014.

[16]F.Jelezko,T.Gaebel,I.Popa,et al.Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate.Physical Review Letters,2004,93(13):DOI 10.1103/PhysRevLett.93.130501.

[17]G.D.Fuchs,V.V.Dobrovitski,D.M.Toyli,et al.Gigahertz dynamics ofa strongly driven single quantum spin.Science,2009,326(5959):1520-1522.

[18]L.Childress,M.V.Gurudev Dutt,J.M.Taylor,et al.Coherent dynamics ofcoupled electron and nuclearspin qubits in diamond.Science,2006,314(5797):281-285.

[19]J.R.Weber,W.F.Koehl,J.B.Varley,et al.Quantum computing with defects.MRS Bulletin,2013,299(10):64-71.

[20]A.Garon,S.J.Laser,D.Sugny.Time-optimal control of SU(2)quantum operations.Physical Review A,2013,88(4):3544-3549.

26 April 2017;revised 22 June 2017;accepted 23 June 2017

DOI 10.1007/s11768-017-7057-2

?Corresponding author.

E-mail:weipeng215@mails.ucas.ac.cn.Tel.:+15201051296;fax+86-10-82541972.

This work was supported by the National Natural Science Foundation of China(Nos.61227902,61573343)and the National Center for Mathematics and Interdisciplinary Sciences,CAS.

?2017 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag Berlin Heidelberg

Shanping YUwas born in Weihai,Shandong,in 1989.She received her B.Sc.degree in Mathematics from Central South University,Changsha,China,in 2012.She is currently pursuing the Ph.D degree in the Key Laboratory of Systems and Control,Chinese Academy of Sciences.Her research interests include quantum control theory and nitrogen-vacancy center.Email:yushanping12@mails.ucas.ac.cn.

Na LIwas born in Xiangyang,Hubei,in 1992.She received her B.Sc.degree in Mathematics from Beihang University,Beijing,China,in 2014.She is currently pursuing the Ph.D.degree in the Key Laboratory of Systems and Control,Chinese Academy of Sciences.Her research interests include nonlinear control,optimization and multiagent system for mobile targets tracking.Email:lina314@mails.ucas.ac.cn.

Peng WEIwas born in Xinyang,Henan,in 1992.He received his B.Sc.degree in Applied Mathematics from Henan University,Kaifeng,China,in 2015.He is currently pursuing the M.Sc.degree in Academy of Mathematics and Systems Science,Chinese Academy ofSciences,Beijing,China.His research interests include quantum control theory and graphene.E-mail:weipeng215@mails.ucas.ac.cn.

Zairong XIwas born in Hunan,China,in 1969.He received the M.Sc.degree in Mathematics from Zhengzhou University,Zhengzhou,China,in 1997,and the Ph.D.degree in Control Theory from the Chinese Academy of Sciences,Beijing,China,in 2000.He was a Post-doctoral Fellow in Tsinghua University,Beijing.In 2002,he joined the Laboratory of Systems and Control,Institute of Systems Science,Academy of Mathematics and Systems Science,Chinese Academy ofSciences.Currently,he is Associate Professor.His research interests include nonlinear systems analysis and synthesis,power systems control,mechanical systems,stability analysis,robust control,and quantum control.E-mail:zrxi@iss.ac.cn.