Quantum Computing Vorlesung Quantum Computing SS 08 1

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Quantum Computing Vorlesung Quantum Computing SS ‘ 08 1

Quantum Computing Vorlesung Quantum Computing SS ‘ 08 1

Quantum Computing with NMR Nuclear magnetic resonance State preparation in an ensemble Quantum Fourier

Quantum Computing with NMR Nuclear magnetic resonance State preparation in an ensemble Quantum Fourier transform A B 12 A B 02 B 01 A finding prime factors –Shor’s algorithm solid state concepts Vorlesung Quantum Computing SS ‘ 08 2

NMR quantum computer Vorlesung Quantum Computing SS ‘ 08 3

NMR quantum computer Vorlesung Quantum Computing SS ‘ 08 3

the qubits in liquid NMR Jones in http: //arxiv. org/abs/quant-ph/0106067 magnetic moment of nucleus

the qubits in liquid NMR Jones in http: //arxiv. org/abs/quant-ph/0106067 magnetic moment of nucleus much smaller than of electron (1/1000) measuring magnetic moment of a single nucleus not possible for reasonable S/N 1018 spins qubit: spin 1/2 nucleus Vorlesung Quantum Computing SS ‘ 08 4

spins in a magnetic field Eint = -mz. B 0 = - g. NIz.

spins in a magnetic field Eint = -mz. B 0 = - g. NIz. B 0 = -g. Nm. IħB 0 Dm. I = 1 energy m. I = -1/2 B 0 m. I = 1/2 magnetic field DE = hn = - g. NħB 0 ~ 300 MHz (B 0 = 7 T, 1 H) population difference ~ 5∙ 10 -5 Vorlesung Quantum Computing SS ‘ 08 5

spin dynamics d. M = g M(t) x B dt 0 B= 0 Bz

spin dynamics d. M = g M(t) x B dt 0 B= 0 Bz d. Mx (My(t)Bzz=- M Mxzcos( (t)Byw ) Lt) + Mysin(w. Lt) =g. M y dt d. My = g-g(M - Mxy(t)B cos(zw ) Lt) - Mxsin(w. Lt) Mzx(t)Bxz = dt d. Mz =0 g (Mx(t)By - My(t)Bx) dt Vorlesung Quantum Computing SS ‘ 08 6

spin-lattice relaxation T 1 spin system is in excited state |1 relaxation to ground

spin-lattice relaxation T 1 spin system is in excited state |1 relaxation to ground state due to spin-phonon interaction read-out within T 1 nuclei: T 1 ~ hours – days electrons: T 1 ~ ms |0 d. Mz M - M 0 = g (Mx(t)By - My(t)Bx) - z dt T 1 Vorlesung Quantum Computing SS ‘ 08 7

spin-spin relaxation T 2 magnetization in x, y-plane (superposition) |1 |1 superposition decays because

spin-spin relaxation T 2 magnetization in x, y-plane (superposition) |1 |1 superposition decays because of dephasing T 1 relaxation to ground state d. Mx Mx = g (My(t)Bz - Mz(t)By) dt T 2 |0 |0 d. My M = g (Mz(t)Bx - Mx(t)Bz) - y dt T 2 Vorlesung Quantum Computing SS ‘ 08 8

spin manipulation Bloch equations d. Mx M = g (My(t)Bz - Mz(t)By) - x

spin manipulation Bloch equations d. Mx M = g (My(t)Bz - Mz(t)By) - x dt T 2 d. My M = g (Mz(t)Bx - Mx(t)Bz) - y dt T 2 d. Mz Mz - M 0 = g (Mx(t)By - My(t)Bx) dt T 1 magnetic field rotating in x, y-plane B 1 cos wt B = B 1 sin wt B 0 Vorlesung Quantum Computing SS ‘ 08 B 1<<B 0 9

spin flipping in lab frame http: //www. wsi. tu-muenchen. de/E 25/members/Hans. Huebl/animations. htm Vorlesung

spin flipping in lab frame http: //www. wsi. tu-muenchen. de/E 25/members/Hans. Huebl/animations. htm Vorlesung Quantum Computing SS ‘ 08 10

NMR technique Lieven Vandersypen, Ph. D thesis: http: //arxiv. org/abs/quant-ph/0205193 z y x B

NMR technique Lieven Vandersypen, Ph. D thesis: http: //arxiv. org/abs/quant-ph/0205193 z y x B 0 ~ 7 -10 T cos wt B 1 cos wt Brf = 2 = B 1 sin wt + B 1 -sin wt 0 0 Vorlesung Quantum Computing SS ‘ 08 11

pulsed magnetic resonance precessing spin changes flux in coils inducing a voltage signal damped

pulsed magnetic resonance precessing spin changes flux in coils inducing a voltage signal damped with 1/T 2* Fast Fourier Transform (FFT) Hanning window + zero filling on resonance off resonance Lorentz shaped resonance with HWHM = 1/T 2* Vorlesung Quantum Computing SS ‘ 08 12

FID spectrum Vorlesung Quantum Computing SS ‘ 08 13

FID spectrum Vorlesung Quantum Computing SS ‘ 08 13

selective excitation Pulse shapes Lieven Vandersypen, Ph. D thesis: http: //arxiv. org/abs/quant-ph/0205193 Vorlesung Quantum

selective excitation Pulse shapes Lieven Vandersypen, Ph. D thesis: http: //arxiv. org/abs/quant-ph/0205193 Vorlesung Quantum Computing SS ‘ 08 14

rotating frame cos wt sin wt 0 x r y = - sin wt

rotating frame cos wt sin wt 0 x r y = - sin wt cos wt 0 z 0 0 1 x y z z applied RF generates a circularly polarized RF field, which is static in the rotating frame r Brf = r cos wt sin wt 0 - sin wt cos wt 0 0 0 1 Brf = B 1 wt x wt y yr cos wt B 1 sin wt + B 1 -sin wt 0 0 cos 2 wt 1 0 + B 1 -sin 2 wt 0 0 Vorlesung Quantum Computing SS ‘ 08 xr constant counter-rotating at twice RF 15

chemical shift Cory et al. : Fortschr. Phys. 48 (2000) 9 -11, 875 The

chemical shift Cory et al. : Fortschr. Phys. 48 (2000) 9 -11, 875 The 13 C protons feel a different effective magnetic field depending on the chemical environment local electron currents shield the field the Zeeman splitting changes and thus the resonance frequency Eint = -ħB 0 S i Vorlesung Quantum Computing SS ‘ 08 g. N(i)m. I(i) (1 -si) 16

coupling between nuclear spins Cory et al. : Fortschr. Phys. 48 (2000) 9 -11,

coupling between nuclear spins Cory et al. : Fortschr. Phys. 48 (2000) 9 -11, 875 Eint = -ħB 0 S g. N(i)m. I(i) (1 -si) + ħSJij m. I(i)m. I(j) i i≠j Ecoup = ħ Jij m. I(i) m. I(j) Vorlesung Quantum Computing SS ‘ 08 17

state preparation calculation |Y 0 H U read-out H-1 time Y|A|Y time system cannot

state preparation calculation |Y 0 H U read-out H-1 time Y|A|Y time system cannot be cooled to pure ground state a mixed ensemble is described by the density matrix r = |y y| Vorlesung Quantum Computing SS ‘ 08 18

density matrix a |y = a|00 + b|01 + g|10 + d|11 = b

density matrix a |y = a|00 + b|01 + g|10 + d|11 = b g d a * b* g* d*) = (a r = |y y| = b g d aa* ab* ag* ad* ba* bb* bg* bd* ga* gb* gg* gd* da* db* dg* dd* pure state: only one state in diagonal occupied with P=1 → Tr(r 2) =1 mixed state: states |yi occupied with Pi → Tr(r 2) <1 Vorlesung Quantum Computing SS ‘ 08 19

states in an ensemble energy m. I = -1/2 magnetic field m. I =

states in an ensemble energy m. I = -1/2 magnetic field m. I = 1/2 level occupation follows Boltzmann statistics p~e Vorlesung Quantum Computing SS ‘ 08 –E/k. BT =e -mz. B 0/k. BT = for | e-b for | eb 20

pseudo pure states r = p- | | + p+ | | = with

pseudo pure states r = p- | | + p+ | | = with e -mz. B 0/k. BT p- 0 1 = eb + e-b 0 p+ eb 0 0 e-b m z. B 0 ≈1 k. BT 1 r = 2 n 1 0 0 1 1 + 2 n b 0 0 -b reduced density matrix can be written r = 2 -n (1 + r. D) access population scales with 2 -n (n: number of qubits) Vorlesung Quantum Computing SS ‘ 08 21

qubit representation 1 r = 2 n r|0 r|1 1 0 0 1 1

qubit representation 1 r = 2 n r|0 r|1 1 0 0 1 1 + 2 n b 0 0 -b 1 0 = |0 0| = = 1 2 0 0 = |1 1| = = 1 2 0 1 identity is omitted Vorlesung Quantum Computing SS ‘ 08 1 0 Iz = 1 2 1 0 0 -1 + 1 2 0 1 1 0 1 0 - 1 2 0 1 0 -1 = 1 1 + Iz 2 = 1 1 - Iz 2 r|0 = |0 0| = Iz r|1 = |1 1| = -Iz 22

time development preparation calculation |Y 0 H read-out H-1 U time Y|A|Y time Liouville

time development preparation calculation |Y 0 H read-out H-1 U time Y|A|Y time Liouville – von Neumann equation ^ r] iħ r = [H, t r(t) = ^t -i. H e ħ Vorlesung Quantum Computing SS ‘ 08 r(t=0) ^ i. Ht e ħ ^ r(t=0) U ^ †(t) = U(t) 23

time development 1018 copies of the same nuclear spin 1 req= 2 1 0

time development 1018 copies of the same nuclear spin 1 req= 2 1 0 1 +2 0 1 b 0 0 -b b=- m z. B 0 ħ w L = 2 k. BT 0 B= 0 B 0 w. L ^ 1 + k T Iz B rotate spin to x, y plane by applying p RF pulse 2 1 r(0+)= 2 r(t) = ^t -i. H e ħ w. L ^ 1+ k T Ix B ^ i. Ht rr(0+) e ħ eq Vorlesung Quantum Computing SS ‘ 08 ^ = w ^I H L z w. L ^ 1 = 2 1 + k T Ix cos w. Lt + ^Iy sin w. Lt B 24

refocusing 1 r(0+)= 2 2 r(t) -1 w. L ^ 1+ k T Ix

refocusing 1 r(0+)= 2 2 r(t) -1 w. L ^ 1+ k T Ix B ^t -i. H e ħ ^ Ix ^ i. Ht e ħ if w. L ≠ wr, e. g. , due to inhomogeneous B 0, the spin picks up a phase ^ ^ = Ix cos Dw. Lt + Iy sin Dw. Lt applying second RF pulse px inverts y-component: 2 r(t+) -1 ^Ix cos Dw. Lt - ^Iy sin Dw. Lt w. L Vorlesung Quantum Computing SS ‘ 08 25

2 qubits Cory et al. : Physica D 120 (1998), 82 2, 3 -dibromo-thiophene

2 qubits Cory et al. : Physica D 120 (1998), 82 2, 3 -dibromo-thiophene a Jab b Vorlesung Quantum Computing SS ‘ 08 26

Simple CNOT spin a b | | CNOT operation |10 |11 |00 | |

Simple CNOT spin a b | | CNOT operation |10 |11 |00 | | energy |11 |10 |01 |00 | p | | | spin levels individually addressable p pulse inverts spin population Vorlesung Quantum Computing SS ‘ 08 27

coupling between nuclear spins Cory et al. : Fortschr. Phys. 48 (2000) 9 -11,

coupling between nuclear spins Cory et al. : Fortschr. Phys. 48 (2000) 9 -11, 875 ^ ^a ^b H = (wa. Iz + wb. Iz + wc. I^zc) ^ a^ b ^ a^ c chemical shift ^ b^ c + 2 p(Jab. Iz Iz +Jac. Iz Iz +Jbc. Iz Iz ) Vorlesung Quantum Computing SS ‘ 08 qubit coupling (always on) 28

CNOT with Alanine Cory et al. : Fortschr. Phys. 48 (2000) 9 -11, 875

CNOT with Alanine Cory et al. : Fortschr. Phys. 48 (2000) 9 -11, 875 | a i | b i Jab Y: -90 | c i UNO = e UCNOT Y: 90 NO Y: 180 operation i p I - i J t. I a. I c - i p I z z ħ y ħ ħ y e e i p. I b i p. I a - i 2 x ħ 2 z ħ e eħ =e e Vorlesung Quantum Computing SS ‘ 08 X: -90 | b o -Y: 180 | c o |1 e i J t. I a. I c z z ħ p Iz a Ix b e -i p 4 i p I - i J t I a. I b - i p I z z ħ 2 y ħ e | a o Z: -90 e |0 29