Realization of Quantum Hadamard Gate by Applying Optimal Control Fields to a Spin Qubit

Transcription

1 00 nd International Conference on Mechanical and Electronics Engineering (ICMEE 00) Realiation of Quantum Hadamard Gate b Appling Optimal Control Fields to a Spin Qubit Sh Mohammad Nejad Nanoptronics Research Center School of Electrical Engineering Iran Universit of Science & Technolog Tehran Iran shahramm@iustacir M Mehmandoost Nanoptronics Research Center School of Electrical Engineering Iran Universit of Science & Technolog Tehran Iran mehmandoost@eleciustacir Abstract In this paper we stud the manipulation of a two level quantum mechanical sstem which is the basic unit of representing data in quantum information science a qubit This research is motivated b the design of the quantum Hadamard gate for solid state spin qubits We consider the problem of driving the initial state of a qubit into a desired final state in a specified time while minimiing the energies of the control fields Optimal control theor is emploed to define optimum control fields which satisf both dnamical constraints and optimalit conditions Time evolution of the sstem is calculated within the Schrödinger equation and a gradient method is used for tailoring optimal control fields Inde terms solid state spin qubits; quantum Hadamard gate; optimal control theor; gradient methods I INTRODUCTION The new emerging field of quantum computation has attracted much attention in the past two decades The basic principle behind quantum computation is to utilie quantum mechanical sstems as building blocks of a computing machine a quantum computer With the help of quantum mechanical principles such as superposition and entanglement quantum computers can efficientl perform certain computational tasks much faster than their classical counterparts [] In general an two level quantum mechanical sstem can be used for the purpose of phsical realiation of quantum computers [] Several different schemes have been investigated to this aim eg nuclear magnetic resonance quantum computer [3] quantied states of superconducting devices [4] trapped ions [5] etc Although the knowledge acquired b these studies is so beneficial to our understanding of quantum computing sstems but the problem of scalabilit is one of their major obstacles Large networks of various etremel small semiconductor devices are integrated on wafers nowadas Therefore solid state quantum computation is the most promising candidate that ma overcome the challenging problem of scalabilit [6] One of the most significant proposals for the phsical realiation of quantum computers is based on the spin states of single electrons confined to semiconductor quantum dots [6] Quantum dots are small regions in semiconductors that can be filled with few electrons up to ero and confine their motions in all three dimensions Quantum dots formed in semiconductor heterostructure based on GaAs technolog have attracted much interest in recent ears [7] Different quantum logic gates can be realied b manipulating the spin states of the electrons confined to these quantum dots Single and two qubit /$600 C 00 IEEE operations can be achieved b appling pulsed local electromagnetic fields and electrical gating of the tunneling barrier between neighboring quantum dots respectivel [8] Solid state spin based devices are epected to dominate the field of quantum computation b eploiting the developed field of semiconductor devices fabrication technolog The aim of this paper is to realie quantum Hadamard gate b appling tailored control fields to the spin of a single electron confined to a semiconductor quantum dot Quantum Hadamard gate is one of the most useful quantum logic gates It maps a basis state to a superposition of basis states with equal weights This propert makes it useful in man quantum computing algorithms [9] We have to consider phsical restrictions in order to realie practical quantum logic gates Undesired interaction of solid state qubits with environment leads to decoherence Thus gate operation times should be much shorter than decoherence times [6] Therefore we need to perform transition of states in specified times We would also like to keep the control fields small in order not to disturb neighboring spins that at the same time are performing some other logic operations [0] Optimal control theor provides a powerful set of tools and concepts to deal with quantum control problems In the present work we appl optimal control theor to the problem of steering state of a spin qubit in a specified time while minimiing an energ-tpe cost functional Due to comple nonlinear dnamics of the sstem it is inevitable to use numerical methods to solve this problem The paper is organied as follows In Section II we will briefl review a solid state spin qubit Mathematical model of quantum Hadamard gate which describes the dnamics of the sstem will be derived in section III Optimal control of the sstem is formulated within Pontragin's principle in Section IV In Section V we present an iterative numerical procedure in the form of a gradient method to define optimal control fields Concluding remarks are given in section VI II SOLID STATE SPIN QUBITS Qubit is the basic unit of representing data in quantum information science Despite having two possible states like its classical analogue either or a qubit can be in a superposition of basis states: () where and are comple numbers restricted b the V-9 Volume

2 00 nd International Conference on Mechanical and Electronics Engineering (ICMEE 00) 0 θ φ Hamiltonian of the sstem is given b []: (7) with as the eternal control field which is defined b (8) Figure Bloch sphere representation of qubit states [] normaliation condition: () State of a qubit is a vector in a two dimensional vector space over comple numbers States and are computational basis states and form an orthonormal basis for this vector space Measuring the state of a qubit gives either the result with probabilit or the result with probabilit Considering the normaliation condition () the qubit state can be rewritten as: (3) Representing qubit state b a point on a sphere of unit radius called Bloch sphere would be useful Speciall in the case of spin qubits where the point on the sphere denotes spin's direction This scheme is shown in Fig The suggestion to use spin states of individual electrons confined to semiconductor quantum dots as qubits spin qubits was made for the first time b Daniel Loss and David P DiVinceno in their remarkable proposal [6] These quantum dots are formed in a two dimensional electron gas b surface electrical gating Electrostatic potentials confine electrons in small circles Electron's spin-up and spin-down correspond to and basis states respectivel Single qubit operations can be achieved b appling pulsed local electromagnetic fields to electrons in single quantum dots Two qubit operations can be accomplished b changing the height of the tunneling barrier between neighboring coupled quantum dots b a purel electrical gating [8] III MATHEMATICAL MODEL Assume a single electron confined to a quantum dot As mentioned before the spin-up and spin-down of the electron are denoted as and respectivel The underling Hilbert space is (4) The wave function of the electron has two components: which satisfies the Schrödinger equation (5) where and are the Bohr magneton and the effective g-factor of the semiconductor material respectivel and is the applied magnetic field The in (7) is the spin operator of the electron: (9) where and are usual Pauli matrices: (0) and and are the unit vectors in directions of and Unitar rotation gate is described beautifull in []: choosing the control field such that with and satisfing () () (3) as the direction of the applied field the Hamiltonian of the sstem becomes (4) Note that to suppress the unwanted effect of the Lorent force which produces an accumulated phase change [] a constraining condition on the direction of the magnetic field has been imposed ie The time evolution operator of the quantum sstem which is equal to corresponding quantum gate will be where ()/ ()() [ ()/]() () (5) (6) V-93 Volume (6)

3 Considering eigenvalues of (6) whichh are evolution of the sstem becomes: which is the unitar rotation gate Note that the boldface one in (7) is the identit matri Quantum Hadamard gate is a one input one output gate that causes a rotation followed b a reflection of the Bloch sphere representation of the qubit state An eample of Hadamard operation is presented in Fig 3 First the point representing the qubit state on the Bloch sphere is rotated b around the ais Then the resulting state will be reflected b the plane Matri operatorr of the Hadamard gate is given b It is obvious that the Hadamard gate cannot be implemented using (7) directl But we can decompose (8) into two canonical matrices which are feasible on (7) B choosing appropriate control fields Hadamard gate can be realied in two steps As shown in Fig control fields are chosen such that the first control field in direction of satisfies and second control field in direction of satisfies The corresponding matri operators become 00 nd International Conference on Mechanical and Electronics Engineering (ICMEE 00) & and the resulting matri is the Hadamard gate with an unwanted phase factor: time (7) (8) (9) (0) () Figure Directions of control fields applied to an electron spin confinedd to a quantum dot (gre circle) IV QUANTUM OPTIMAL CONTROL PROBLEM Let time t be in the interval for fied Dnamics of the sstem is described b the controlled Schrödinger equation: where both the wave function and the Hamiltonian are comple quantities In order to epress the problem in terms of real quantities onl we write [3]: and B placing (4) and (5) into (3) we obtain B defining ( t ) ( t ) dnamics of the sstem can be written as a sstem of nonautonomous linear ordinar differential equations [ 4]: and (3) (4) (5) (6) (7) () where and is a real valued 0 matri (8) 0 + Figure 3 Bloch sphere representationn of Hadamard gate operation [] First the initial state is rotated around the ais b The resulting state will be reflected b the plane in second step giving the final state V-94 Volume

4 00 nd International Conference on Mechanical and Electronics Engineering (ICMEE 00) The optimal control problem is to design a control field which steers our sstem from an initial state to a desired final state at specified final time Note that and correspond to and states respectivel This problem can be formulated in terms of a cost functional that also takes into account practical constraints such as energ of the control fields: (9) where the first term represents measure of distance of the final state from the desired one and the second integral term penalies the field fluenc is a weight function which forces the control field to approach ero near the end points of the time interval in accordance with realit where magnetic fields are of finite duration For this purpose we use the shape function [5] [6]: / ()/ (30) where the positive constants and are the penalt parameters and is a rise time First order optimalit conditions in the form of Pontragin's minimum principle offer an optimal solution of this problem These conditions are formulated using a Hamiltonian which in our problem has the following form: (3) where is called the co-state vector Pontragin's minimum principle states that necessar conditions to simultaneousl minimie and satisf (8) are as follows [7]: (3) (33) (34) The set of state and co-state equations (3) and (33) together with the algebraic equation (34) form a boundar value problem which is etremel difficult to solve analticall because of the apparent nonlinearit in the sstem of differential equations General procedure to solve the problem is as follows: choose an initial guess for the control field and solve state equations for b integration forward in time Using obtained integrate co-state equations backward in time Use (34) to update the control field and run this loop until the desired cost function value is achieved The numerical method to solve this problem is described in details in the following section V NUMERICAL ALGORITHM Numericall there eist two main classes of methods to solve optimal control problems direct and indirect [8] An indirect method forms the Hamiltonian of the sstem and solves the necessar conditions of optimalit in the form of a boundar value problem Here we will use a gradient method which b an iterative algorithm improves estimates of the control field histories so as to come closer to satisf optimalit and boundar conditions [9] A first order gradient algorithm for solving our problem is presented below [0]: () Choose an initial guess for the control field in the time interval Using the control field histor () integrate the state equations (3) from to with initial conditions () () 3 Calculate b substituting from step () () into Using this value of () () as the initial condition and obtained in the previous step integrate co-state equations (33) from to () () 4 If where is a preselected small positive constant terminate the procedure Otherwise () () () () set and go to step Here is the step sie Various numerical methods can be used to integrate the Schrödinger equation and the associated co-state sstem Here we used eplicit Runge-Kutta-Fehlberg method of order 4(5) to carr out the numerical integrations [] Optimum control fields and corresponding evolution of spin state are shown in Fig 4 Note that the mapping shown in Fig 4 which is the transformation of two superposition states into a final basis state is not the basic application of Hadamard gate Hadamard gate is mainl used for transforming a basis state into the superposition of basis states Two superposition states are chosen as the input in order to illustrate the two stage operation of the Hadamard gate The second stage would have been eliminated in our simulation if we had chosen a basis state as the input The value of the performance measure as a function of the number of iterations is shown in Fig 5 VI CONCLUDING REMARKS We addressed the realiation of the quantum Hadamard gate based on the spin of a single electron confined to a semiconductor quantum dot Hadamard gate cannot be implemented using in plane control fields directl thus we decomposed the Hadamard operator into two canonical forms which are feasible b appling in plane fields B deriving the Hamiltonians of the sstem we accessed to the dnamics of the quantum sstem The problem of finding tailored control fields which drive the quantum sstem in a specified time while minimiing the energ of the control fields is formulated within optimal control theor and Pontragin's minimum principle Using a gradient based iterative numerical procedure we solved this optimal control problem In this paper we studied the ideal case of an isolated solid state spin qubit subject to the eternal control fields While in practice there eist several tpes of interactions with environment which lead to decoherence Major sources of decoherence in solid state spin qubits are addressed in [] The effect of these undesired interactions on dnamics of the quantum sstem can be considered b adding uncontrolled Hamiltonians to the Schrödinger equation Optimal control theor can be used to design control fields in order to realie robust quantum gates even in the presence of an environment [3] V-95 Volume

5 00 nd International Conference on Mechanical and Electronics Engineering (ICMEE 00) (a) (b) Probabilit Densit Probabilit Densit Time Time Figure 4 Time evolution of the spin states during quantum Hadamard gate operation (a) Initial state turns into state b appling first optimal control field (b) Resulting spin states will be flipped b appling second optimal control field giving the final state Note that is the probabilit densit of the basis state and is the probabilit densit of the basis state (a) (b) Final Cost Value Final Cost Value Cost Functional Value Cost Functional Value Iteration Number Iteration Number Figure 5 The value of the cost functional as a function of the number of iterations First fift iterations for (a) the first and (b) the second control fields An etremel small value is chosen for the termination constant in order not to terminate the iterative procedure before fift iterations REFERENCES [] M A Nielsen and I L Chuang Quantum Computation and Quantum Information Cambridge Univ Press (000) [] D P DiVinceno The phsical implementation of quantum computation Fortschr Phs 48 pp (000) [3] N Gershenfeld and I L Chuang Bulk spin-resonance quantum computation Science 75 pp (997) [4] A Shnirman G Schön and Z Hermon Quantum manipulation of small Josephson junctions Phs Rev Lett 79 pp (997) [5] J I Cirac and P Zoller Quantum computation with cold trapped ions Phs Rev Lett 74 pp (995) [6] D Loss and D P DiVinceno Quantum computation with quantum dots Phs Rev A 57 pp 0-6 (997) [7] R Hanson L P Kouwenhoven J R Petta S Tarucha and L M K Vanderspen Spins in few-electron quantum dots Rev Mod Phs 79 pp 7-65 (007) [8] G Burkard D Loss and D P DiVinceno Coupled quantum dots as quantum gates Phs Rev B 59 pp (999) [9] D J Shepherd On the role of Hadamard gates in quantum circuits Quantum Inf Process 5 No 3 pp 6-77 (006) [0] D D'Alessandro and M Dahleh Optimal control of two-level quantum sstems IEEE Transactions on Automatic Control vol 46 pp (00) [] G Burkard H Engel D Loss Spintronics and quantum dots for quantum computing and quantum communication Fortschr Phs 48 pp (000) [] G Chen D A Church B-G Englert and M S Zubair Mathematical models of contemporar elementar quantum computing devices CRM Proceedings and Lecture Notes 33 pp 79-8 (003) [3] D D'Alessandro Introduction to Quantum Control and Dnamics CRC Press (007) [4] L Lara A numerical method for solving a sstem of nonautonomous linear ordinar differential equations Applied Mathematics and Computation vol 70 pp (005) [5] H Jirari F W J Hekking and O Buisson Optimal control of superconducting N-level quantum sstems Europhsics Letters vol (009) [6] H Jirari and W Pöt Optimal coherenct control of dissipating N-level sstems Phs Rev A (005) [7] L S Pontragin V G Boltanskii R V Gamkrelide and E F Mishchenko The Mathematical Theor of Optimal Processes Interscience Publishers New York (96) [8] J T Betts Practical Methods for Optimal Control Using Nonlinear Programming SIAM Philadelphia (00) [9] A E Brson and Y Ho Applied Optimal Control: Optimiation Estimation and Control Hemisphere New York (975) [0] D E Kirk Optimal Control Theor: An Introduction Dover Publications (004) [] E Hairer S Nørsett and G Wanner Solving Ordinar Differental Equations I: Nonstiff Problems Springer New York (008) [] L Chirolli and G Burkard Decoherence in solid state qubits Advances in Phsics Vol 3 pp 5-85 (008) [3] A Pechen D Prokhorenko and H Rabit Control landscapes for two-level open quantum sstems J Phs A (008) V-96 Volume