Квантовые цепи и кубиты

Transcription

1 Квантовые цепи и кубиты Твердотельные наноструктуры и устройства для квантовых вычислений Лекция А.В. Устинов Karlsruhe Insttute of Technology, Germany Russan Quantum Center, Russa

2 Alexey Ustnov Sold-state qubts

3 Trapped ons (Wneland) Alexey Ustnov Sold-state qubts 3

4 Rydberg atoms n a cavty (Haroche) Alexey Ustnov Sold-state qubts 4

5 Quantum crcuts and qubts: Plan for 6 lectures Nov. 7 Nov. Dec. 5. Quantum nformaton processng, mcroscopc realzatons of qubts. Condensed matter qubts, quantum dots, damond qubts 3. Superconductors, Josephson devces, macroscopc quantum coherence 4. Phase qubts, flux qubts 5. Charge qubts, quantronum and transmon 6. Fluxonum and phase slp qubt. Crcut quantum electrodynamcs Alexey Ustnov Sold-state qubts 5

6 Outlne of lecture a Quantum nformaton processng what s a qubt? basc dea about quantum computng spn ½ partcle and Bloch sphere qubt manpulaton; quantum gates decoherence Alexey Ustnov Sold-state qubts 6

7 Classcal bts and quantum bts A classcal computer operates wth bts A quantum computer operates wth quantum bts called qubts energy coordnate two-level system possble states: and possble states: Ψ α + β Alexey Ustnov Sold-state qubts 7

8 Qubt descrpton as a spn ½ partcle Quantum states Ψ Wavefuncton α + β α β The Bloch sphere two bass states α and β are complex numbers In prncple, any two-level system whch acts quantum mechancally can be used as a qubt. Alexey Ustnov Sold-state qubts 8

9 Alexey Ustnov Sold-state qubts 9 A sngle spn n a magnetc feld Ψ Ψ Ψ E H dt d ( ) z N z B E B B µ ± + + Ψ β α β α ) ( ) ( ) ( z z y y x x N z N y x N y x N z N B B B B B B B B B H σ σ σ µ µ µ µ µ ; ; z y x σ σ σ x y Β magnetc feld See, for example, The Feynman Lectures on Physcs, Vol. III Ch., z

10 The Bloch sphere model for a qubt µ Ν z θ H ( Bx, By, Bz ) t / magnetc feld Ψ( t) e Ψ() Β Tme evoluton of a qubt Any two-level system can be mapped to a sngle spn n an external magnetc feld φ y Β x Classcal analogy: Spn µ Ν precesses about the magnetc feld vector Β wth frequency ω µ N B µ Ν Alexey Ustnov Sold-state qubts

11 The Bloch sphere model for a qubt µ Ν z θ magnetc feld Β Ψ here α + β θ α cos e θ β sn e φ / + φ / x φ y Standard qubt operatons: rotaton by θ θ cos R( θ ) θ sn θ sn θ cos e.g., so-called π/ and π pulses Alexey Ustnov Sold-state qubts

12 Alexey Ustnov Sold-state qubts Qubt operatons (gates) smply move the ponter on the Bloch sphere ntal state fnal state NOT NOT NOT Example : Sngle-qubt gates ()

13 Alexey Ustnov Sold-state qubts NOT NOT NOT ntal state fnal state Example : square root of the ΝΟΤ gate NOT NOT NOT You can check that Sngle-qubt gates ()

14 Sngle-qubt gates (3) Example 3: Hadamard transformaton ntal state fnal state H H + ( ) H ( ) Alexey Ustnov Sold-state qubts 4

15 Alexey Ustnov Sold-state qubts 5 General two bt state 3 α α α α Ψ Ψ n j j j α n j α j Two-qubt bass states General n-bt state, where n qubts can represent a superposton of n states, e.g, 5 qubts ~ 5 states Mult-qubt states 7 qubts more states than atoms n the unverse (~ 8 )

16 Product states Product states Some two-qubt states can be obtaned as a product of sngle-qubt states, e.q. Ψ ( ) ( + ) ( + ) Entangled states Entangled state can NOT be obtaned as a product of sngle-qubt states Examples: Ψ ( + ) Bell state measurement wth probablty ½ result wth probablty ½ Ψ ( ) EPR (Ensten Podolsky Rosen) par Alexey Ustnov Sold-state qubts 6

17 Schrödnger cat Entangled states cannot be factored nto a product of ndvdual qubt states An example of entangled state s the 'Schrödnger-cat' states of N quantum bts: Ψ 'lvecat' + cat N N 'deadcat' Alexey Ustnov Sold-state qubts 7

18 Entangled states: artst s vew Interacton of a two-level atom and a feld atom feld nteracton Ψ atom + feld atom feld entangled state Alexey Ustnov Sold-state qubts 8

19 Two-qubt gate: Controlled NOT A B Controlled NOT gate: Control bt Target bt A B CNOT The CNOT gate flps the second qubt (the target qubt) f and only f the frst qubt (the control qubt) s. A( α, φ, θ) Unversal -qubt gate: e e α cosθ ( α + φ ) cosθ e e ( α α φ ) cosθ cosθ For quantum computaton t s requred to have at least one two-qubt gate + sngle-qubt gates Alexey Ustnov Sold-state qubts 9

20 Quantum bts (Qubts) Quantum system wth two states Wavefuncton Ψ α + β α, β complex numbers E Ψ Ψ α α +, β β probabltes to measure > or > normalzaton snce global phase s not observable, rewrte: Ψ a ϕ + e b a, b real numbers, ϕ phase A potental well Qubts can be n a superposton of equal superposton of true and false: and a qubt has a contnuum of possble states Ψ ( + ) Alexey Ustnov Sold-state qubts

21 Decoherence A qubt can be dsturbed n two dfferent ways: Characterstc tmes. Exchange energy wth the envronment by stmulated emsson or absorpton. T energy relaxaton (mxng) tme t θ changes. Lost of qubt phase memory T φ dephasng tme φ changes T φ s mportant for manpulaton: The qubt must stay coherent T s mportant for measurement: The states must not be mxed before we can retreve the result Alexey Ustnov Sold-state qubts

22 Sources of decoherence Actve sources absorpton heat nose Passve sources emsson couplng to excted states external degrees of freedom photons, phonons, quaspartcles How can we reduce the decoherence? coolng solatng from envronment (sheldng, flterng) talorng the envronment Alexey Ustnov Sold-state qubts

23 Outlne of lecture b Mcroscopc realzatons of qubts > > DVncenzo's crtera molecular spns and lqud state NMR trapped atoms trapped ons photons Alexey Ustnov Sold-state qubts 3

24 Requrements for quantum computer hardware: DVncenzo s crtera Identfable qubts and the ablty to scale them up n number The ablty to prepare the ground state of whole system Low decoherence - less than -4 per elementary quantum gate operaton Realzaton of quantum gates to control of the system Hamltonan The ablty to perform quantum measurement of the qubts to obtan the result of the computaton D. P. DVncenzo, Topcs n Quantum Computers, n Mesoscopc Electron Transport, (ed. L. Kowenhoven, G. Schön, and L. Sohn), NATO ASI Seres E, (Kluwer Ac. Publ., Dordrecht, 997); cond-mat/966. Alexey Ustnov Sold-state qubts 4

25 Choce of qubt characterstcs Degree of freedom charge, spn, polarzaton, magnetc flux, etc. Two-level system solaton from other levels Representaton of one qubt ensemble or sngle system Manpulaton photons, phonons, voltage pulses, etc. Read out Counter, sgnal amplfer, etc. Operaton tme ps... ms Decoherence tme ns... s Couplng between qubts Alexey Ustnov Sold-state qubts 5

26 Possble "hardware" for a quantum computer Mcroscopc systems ensemble of spns (NMR) ons n electromagnetc traps neutral atoms photons n cavtes Macroscopc or mesoscopc systems spns n sold-state nanodevces electrons on superflud helum magnetc nanopartcles charge, phase or flux n superconductors Alexey Ustnov Sold-state qubts 6

27 Advantages and dsadvantages of mcroscopc qubts Advantages Dsadvantages models are well establshed well-defned dentcal qubts low decoherence Integratng of many qubts n a large crcut s a very dffcult task Alexey Ustnov Sold-state qubts 7

28 Quantum computaton usng NMR Chloroform CHCl 3 qubts H Cl 3 C Cl Cl Deutsch algorthm demonstrated. Alexey Ustnov Sold-state qubts 8

29 Quantum computaton usng NMR Degree of freedom nuclear spn Representaton of qubt ensemble of 3 spns Manpulaton RF-pulses -5 MHz Read out RF-detector H Operaton tme -3 s Decoherence tme up to 4 s Couplng spn-spn (not tunable) Cl 3 C Cl Cl Alexey Ustnov Sold-state qubts 9

30 Fundamentals of NMR In a magnetc feld spn energy levels are splt by an energy Ε ( Bz ) E ± N Bz B µ µ Ν z θ magnetc feld Β An RF feld wll excte transtons between the two spn levels when hν E For nuclear spns at Β Tesla ν MHz φ y For electron spns at Β Tesla ν GHz x Quantum gates are realzed by keepng the RF feld on durng the approprate tme nterval Alexey Ustnov Sold-state qubts 3

31 Rab oscllaton Apply a resonant ( ω Rab oscllaton occurs. Rab s formula: ( E E ) P( ) ) feld to a quantum two-level system The state s nverted (so called π-pulse), or NOT gate large drve ampltude An equal superposton s created (π/-pulse) or Haddamard gate Ψ ( + ) small drve ampltude Alexey Ustnov Sold-state qubts 3

32 Spn Echo a refocusng technque correctng dephasng all spns start wth the same phase spns evolve wth dfferent veloctes -> dephasng the 8 pulse nverts the stuaton: fastest spns are set behnd slower ones all spns meet agan after the tme t Spn Echo Alexey Ustnov Sold-state qubts 3

33 Spn Echo a refocusng technque correctng dephasng was frst observed by Erwn Hahn E. L. Hahn, Physcal Revew 8, 58 (95) Spn Echo Alexey Ustnov Sold-state qubts 33

34 General system schematc for an NMR quantum computer H Hardware for a table-top NMR quantum computer: Magnet Transmtter Recever Pulse controller Cl 3 C Cl Cl superconductng magnet NMR setup Y. Magure, et al. IBM Systems J. 39, 83 () Alexey Ustnov Sold-state qubts 34

35 An example of the quantum computer molecule IBM 7-qubt quantum computer A perfluorobutadenyl ron complex wth the nner two carbons 3 C-labelled. L. M. K. Vandersypen, et al. Nature 44, 883 () Alexey Ustnov Sold-state qubts 35

36 NMR realzaton of Shor s quantum algorthm: factorng of N5 7 qubt quantum processor L. M. K. Vandersypen, et al. Nature 44, 883 () Alexey Ustnov Sold-state qubts 36