Quantum Computing with NMR

Transcription

1 Quantum Computing with NMR Sabine Keiber, Martin Krauÿ June 3, 2009 Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

2 1 A Short Introduction to NMR 2 Di Vincenzo's Requirements 3 Experimental setup 4 Theoretical Background 5 Quantum gates 6 Experimental results 7 Strengths and Weaknesses 8 Recent research Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

3 History of NMR A Short Introduction to NMR History of NMR nuclear magnetic resonance discovered by Rabi in 1938 Nobel prize 1952 for Purcell and Bloch "for their development of new methods for nuclear magnetic precision measurements and discoveries in connection therewith" various applications chemistry: examination of molecule structures medicine: magnetic resonance imaging solid state physics: stucture of dierent materials Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

4 Principles of NMR A Short Introduction to NMR Principles of NMR charged particle with spin has magnetic moment µ external magnetic eld B 0 = B 0 e z depending on the spin orientation energy levels split up with V = µ z B 0 Zeeman Eect spin precesses around magnetic eld with lamor frequency ω L = γb 0 resonant electromagnetic pulses can excite transition between spin states averaging over spins resulting magnetization Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

5 Di Vincenzo's Requirements Di Vincenzo's Requirements 1 Well dened qubits 2 Initializiation to a pure state 3 Universal set of quantum gates 4 Long decoherence times 5 Qubit specic measurement Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

6 Qubits Di Vincenzo's Requirements Qubits two level system equals two states of a particle with spin s = 1 2 logical 0 represented by m s = +1/2 >, logical 1 represented by m s = 1/2 > nuclear spin of atoms in molecules freely oating liquid state NMR ordered in a lattice solid state NMR suitable nuclei: 1 H, 13 C, 19 F Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

7 Initialization Di Vincenzo's Requirements Initialization relaxation to the ground state by cooling in principle possible (optical pumping...) problem: the cooled liquid crystallizes structure of the system changes alternative at room temperature: pseudo pure states pseudo pure states = mixed states that behave in respect to the unitary transformations like pure states Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

8 Gates I Di Vincenzo's Requirements Gates qubits are selectively adressable because of dierent resonance frequencies ω L = γb heteronuclear: dierent gyromagnetic ration homonuclear: chemical shift complete set of unitary transformations available (RF pulses, spin-spin coupling) Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

9 Gates II Di Vincenzo's Requirements Gates Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

10 Decoherence Time Di Vincenzo's Requirements Decoherence Time reason for decoherence: spins interact with environment typical decoherence time 1 s typical gate times several ms number of feasible gates in order of 100 Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

11 Measurement Di Vincenzo's Requirements Measurement weak measurement instead of projective measurement no collaps of the wavefunction continuous measurement possible averaging over to spins "ensemble quantum computing" classical description: measurement of the induction voltage (free induction decay = FID) dierent qubits have dierent precession frequency Fourier transformation separates qubits in frequency space Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

12 Experimental setup Experimental setup superconducting coil homogeneous B 0 eld in z direction (10 to 15 T) RF coils exitation of nuclear spins in x-y plain and measurement of induced magnetisation Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

13 Hamiltonian Theoretical Background Hamiltonian of a Spin System Molecule with nuclear spins Interaction with external magnetic eld Indirect spin coupling through electrons Dipolar interaction between molecules averaged to zero because of molecular motion H = 1 2 i ω i σ (i) z + π 2 i j J ij σ (i) z σ (j) z Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

14 Theoretical Background Pseudo Pure States Thermal Equilibrium Density Matrix macroscopic spin ensemble ( spins!) mixed state use pseudo pure states density matrix at thermal equilibrium describes initial state: for one spin (H = 1 2 ω 0σ z ): ρ = ρ = 1 Z e βh ( 1 exp( 1 exp( 1 2 β ω 0) + exp( 1 2 β ω ) 0) 0 2 β ω 0) 0 exp( 1 2 β ω 0) Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

15 Approximation Theoretical Background Pseudo Pure States β typically 10 5, approximation of density matrix: Generalization for n qubits: ρ is diagonal signal 1 2 n ρ = β ω 0σ z ρ = 1 (1 βh) 2n Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

16 Temporal Averaging I Theoretical Background Pseudo Pure States Assume a mixed 2 qubit system: Initial state a ρ 1 = a b c d = 0 b c d where a + b + c + d = 1 ( Tr{ρ 1 } = 1) Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

17 Temporal Averaging II Theoretical Background Pseudo Pure States combination of CNOT operations similar initial states: a a ρ 2 = 0 c d 0, ρ 3 = 0 d b b c time evolution in three seperate experiments: ρ i (t) = U(t)ρ i U (t) Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

18 Theoretical Background Temporal Averaging III Pseudo Pure States time evolution is a linear operator: ρ(t) = ρ 1 (t) + ρ 2 (t) + ρ 3 (t) = U(t)(ρ 1 + ρ 2 + ρ 3 )U (t) pseudo pure state can be obtained: U(t) (1 a)1 + (4a 1) U (t) } {{ } Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

19 Theoretical Background Temporal Averaging IV Pseudo Pure States Measurement of σ (k) z (linearity of Trace): Tr{ρ i (t)σ (k) z } = Tr{ρ(t)σ (k) z } Tr{1σ (k) z } = 0 i Tr{U(t) ((1 a)1 + (4a 1) ) U (t)σ (k) z } = Tr{U(t) U (t)σ (k) z } same result as for initial pure state Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

20 RF Field Theoretical Background Rotating frame RF eld of the coil as superposition of counter-rotating elds: cos(ωt) cos(ωt) cos(ωt) B RF = 2B 1 0 = B 1 sin(ωt) + B 1 sin(ωt) switch to a rotating frame: cos(ωt) sin(ωt) 0 x = sin(ωt) cos(ωt) 0 x Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

21 Rotating frame Theoretical Background Rotating frame B RF in rotating frame: 1 cos(2ωt) B RF = B B 1 sin(2ωt) 0 0 only the static term has an eect quantum computing experiments use rotating frame Transformation of the State ψ = U 1 ψ with U(t) = e i ω 2 tσ z Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

22 Hamiltonian Theoretical Background Rotating frame transformation of the Hamiltonian: H = U 1 HU + i U 1 U = 2 ω Lσ z + 2 ω 1σ x with ω L = ω L ω and ω 1 = γb 1 eective eld: ω = (ω 1, 0, ω L ) Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

23 Evolution Theoretical Background Evolution spins precess around the static eld in the rotating frame Rabi opping: apply resonant RF eld (eective eld in xy plane) spins precess around eective eld spin moves to xy plane and then to -z possible eective Field in x or y direction by shift of RF phase used to implement quantum gates. Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

24 Spin Coupling Theoretical Background Refocusing spins of the qubits are coupled causes unintended time evolution has to be removed to gain controll can be achieved by refocusing method Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

25 Time Evolution Theoretical Background Refocusing example: two qubit interaction. Interaction Hamiltonian H I = ασ (1) z σ (2) z where α determines the strength of the coupling. Time evolution U I (t) = e iαtσ(1) z σ (2) z Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

26 Refocusing Theoretical Background Refocusing RF-Pulse that rotates the rst spin around the x-axis (rotating frame): π pulse R x1 (π) = e i π 2 σ(1) x = iσ (1) x series of RF pulses suppress the evolution after a time t: ( t ) ( t ) U I R x1 (π)u I R x1 (π) = can be considered a single-qubit NO gate used to decouple certain spins Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

27 Theoretical Background Refocusing Runners Analogy I Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

28 Runners Analogy II Theoretical Background Refocusing Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

29 z-rotation Gates z-rotation only rotations around x- and y-axis by RF pulse no z-axis rotation implementation by a series of x,y rotations: φ rotation around z-axis ( ) e i φ 2 σ e i φ 2 0 = = 0 e i φ e i π 4 σ φ x e i 2 σ π y ei 4 σ x 2 Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

30 NOT and Hadamard Gate Gates Single-qubit gates NOT gate implemented using a π-rotation: NOT gate ( ) ( ) e i π 2 σ x 0 i = = e i π i (The global phase can be neglected since it has no observable eect.) Hadamard gate implemented by a combination of rotations: Hadamard gate H = 1 ( ) 1 1 = e i π 2 σ π z ei 4 σ y Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

31 Gates CNOT Gate CNOT with Resonance Frequency CNOT gate inverts the second qubit if the rst one is in state 1 possible implementation: the resonace frequencies for the second qubit depends on the state of the rst one apply an RF pulse with the correct resonance frequenzy for the 10 and 11 state this ips the second spin to 11 resp. 10 it does not change the 00 and 01 state spectrum of second qubit Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

32 CNOT with RF Pulses I Gates CNOT Gate alternative implementation without using selective RF pulses non-selective rotations and precission (spin-coupling e iαtσ(1) z σ (2) z ) Start with a 00 state apply e i π 4 σ(2) y (ips the spin to the x-axis): ψ(0) = 1 2 ( 0 ( )) precission of the spin (rotating frame!): 0 (1) U I (t) = e iαtσ(1) z σ (2) z = e iαt(+1)σ(2) z ψ(t) = 1 2 ( 0 ( 0 e iαt + 1 e iαt ) ) Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

33 CNOT with RF Pulses II Gates CNOT Gate t = π 4α spin has moved to y-axis: ψ(0) = 1 ( 0 ((1 i) 0 + (1 + i) 1 )) 2 nal rotation e i π 4 σ(2) x spin returns to its initial state 00. Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

34 CNOT with RF Pulses III Gates CNOT Gate initial state 10 precission leads to an opposite orientated rotation: 1 (1) U I (t) = e iαtσ(1) z σ (2) z = e iαt( 1)σ(2) z ψ(t) = 1 2 ( 0 ( 0 e iαt + 1 e iαt ) ) nal state in this case: 11 Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

35 CNOT with RF Pulses IV Gates CNOT Gate CNOT operation on the Bloch sphere Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

36 CNOT with RF Pulses V Gates CNOT Gate eect of this operation: CNOT gate: in the same way: 00 00, , we have single-qubit gates and the CNOT gate. We can construct a full set of quantum gates. Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

37 Molecules used for QC Experimental results Molecules Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

38 Experimental results Algorithms Quantum Fourier-Transformation principle of quantum Fourier transform: j > 1 N 1 N k=0 x je 2πijk/N k > implemented by Weinstein et al goal: nd periodicity of state Ψ >= 1 2 ( 000 > > > >) uses gates: three Hadamard gates, three controlled phase shift gates accuracy of implementation: 62 to 80 % Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

39 Shore Algorithm I Experimental results Algorithms group at IBM Almaden Research Center implemented Shor's factoring algorithm using a custom designed molecule with ve 19 F and two 13 C spins number of resonance lines per spin: 2 6 = 64 chemical shift 1 khz sequence: factorization of 15 sequence lasted almost 1 s (order of decoherence time) group needed about 300 pulses Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

40 Shore Algorithm II Experimental results Algorithms Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

41 Errors Experimental results Errors coherent errors: real unitary transformations dier from ideal operators incoherent errors: inhomogenities across the sample dierent molecules evolve dierently (non unitary for ensemble) decoherent errors: coupling between qubit and environment (non unitary for single qubits) Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

42 Strengths and Weaknesses Strengths of liquid state NMR Strengths long experimental experience RF excitations tunable in a wide range easy experimental setup easy implementation of gates Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

43 Strengths and Weaknesses Weaknesses of liquid state NMR Weaknesses exponential signal loss in pseudo-pure states with number of qubits N number of resonance lines can increase with N2 N only molecules with strong chemical shift suitable when chemical shift is small bandwidth of the RF pulses must be small duration of the pulses is long decoherence Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

44 Recent research Solid-based NMR in diamand I possible advantages of solid state NMR higher polarization longer decoherence time stronger couplings between spins faster gate times Childress, Dutt et al. (Science 2006/2007): coupled electron and nuclear spin electronic spin triplet of nitrogen-vacancy (NV) centers (m s = ±1, 0) in diamond couples coherently to the nuclear spin of nearby 13 C nuclei Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

45 Recent research Solid-based NMR in diamand II NV center can be manipulated with optical and microwave excitation addressing and manipulating the nucleus via the electronic spin is possible m s = 0 no inuence on nuclear spins, free lamor precession m s = 1 hyperne interaction splitting between states 1, > and 1, >, inhibited lamor precession quantum registers: chemical shift results from dierent interactions of the nuclear spins with the NV center Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46

46 References G. Benenti, G. Casuti, G. Strini, Principles of Quantum Computation and Information, World Scientic (2007) M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2000) J. Stolze, D. Suter, Quantum Computing, Wiley VCH (2008) D. G. Cory, et al., NMR Based Quantum Information Processing: Achievements and Prospects Fortschr. Phys. 48 (2000) L. Childress, et al., Coherent Dynamics of Coupled Electron and Nuclear Spin Qubits in Diamond, Science 314 (2006) M. V. Gurudev Dutt, et al., Dynamics of Coupled Electron and Nuclear Spin Qubits in Diamond, Science 316 (2007) Sabine Keiber, Martin Krauÿ Quantum Computing with NMR June 3, / 46