Vorlesung Quantum Computing SS 08 1 Quantum Computing

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

Vorlesung Quantum Computing SS 08 3 NMR quantum computer

Vorlesung Quantum Computing SS 08 4 the qubits in liquid NMR Jones in magnetic moment of nucleus much smaller than of electron (1/1000) for reasonable S/N spins measuring magnetic moment of a single nucleus not possible qubit: spin 1/2 nucleus

Vorlesung Quantum Computing SS 08 5 spins in a magnetic field m I = -1/2 m I = 1/2 B0B0 energy magnetic field E = h = - N ħB 0 ~ 300 MHz (B 0 = 7 T, 1 H) E int = - z B 0 = - N I z B 0 = - N m I ħB 0 m I = 1 population difference ~

Vorlesung Quantum Computing SS 08 6 spin dynamics dM x dt = (M y (t)B z M z (t)B y ) dM y dt = (M z (t)B x M x (t)B z ) dM z dt = (M x (t)B y M y (t)B x ) = M y (t)B z = - M x (t)B z = dM dt = M(t) x B = M y cos( L t) - M x sin( L t) = M x cos( L t) + M y sin( L t) B = 0 0 BzBz

Vorlesung Quantum Computing SS 08 7 spin-lattice relaxation T 1 nuclei: T 1 ~ hours – days electrons: T 1 ~ ms spin system is in excited state relaxation to ground state due to spin-phonon interaction read-out within T 1 dM z dt = (M x (t)B y M y (t)B x ) M z M 0 T1T1

Vorlesung Quantum Computing SS 08 8 spin-spin relaxation T 2 magnetization in x,y-plane (superposition) superposition decays because of dephasing T 1 relaxation to ground state dM x dt = (M y (t)B z M z (t)B y ) dM y dt = (M z (t)B x M x (t)B z ) MxMx T2T2 MyMy T2T2

Vorlesung Quantum Computing SS 08 9 spin manipulation Bloch equations dM z dt = (M x (t)B y M y (t)B x ) M z M 0 T1T1 dM x dt = (M y (t)B z M z (t)B y ) MxMx T2T2 dM y dt = (M z (t)B x M x (t)B z ) MyMy T2T2 B = B 1 cos t B 1 sin t B0B0 magnetic field rotating in x,y-plane B 1 <<B 0

Vorlesung Quantum Computing SS spin flipping in lab frame

Vorlesung Quantum Computing SS NMR technique x y z B 0 ~ 7-10 T Lieven Vandersypen, PhD thesis: B rf = 2 = + B 1 cos t 0 0 cos t sin t 0 B1B1 cos t -sin t 0 B1B1

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

Vorlesung Quantum Computing SS FID spectrum

Vorlesung Quantum Computing SS selective excitation Pulse shapes Lieven Vandersypen, PhD thesis:

Vorlesung Quantum Computing SS rotating frame x y z x y z cos t sin t - sin t = r z y x xrxr yryr t t cos t sin t - sin t cos t sin t 0 B1B1 cos t -sin t 0 B1B1 + B rf = r cos 2 t 0 B rf = r B1B1 -sin 2 t B1B1 + constant counter-rotating at twice RF applied RF generates a circularly polarized RF field, which is static in the rotating frame

Vorlesung Quantum Computing SS chemical shift 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 E int = -ħB 0 N (i) m I (i) (1- i ) i Cory et al.: Fortschr. Phys. 48 (2000) 9-11, 875

Vorlesung Quantum Computing SS coupling between nuclear spins Cory et al.: Fortschr. Phys. 48 (2000) 9-11, 875 E coup = ħ J ij m I (i) m I (j) E int = -ħB 0 N (i)m I (i) (1- i ) i + ħ J ij m I (i)m I (j) ijij

Vorlesung Quantum Computing SS state preparation HH -1 calculation U preparation read-out |A| time a mixed ensemble is described by the density matrix = system cannot be cooled to pure ground state

Vorlesung Quantum Computing SS density matrix = = ( ) = pure state: only one state in diagonal occupied with P=1 mixed state: states i occupied with P i Tr(

Vorlesung Quantum Computing SS states in an ensemble level occupation follows Boltzmann statistics m I = -1/2 m I = 1/2 energy magnetic field p ~ e =e = –E/k B T - z B 0 /k B T ebeb e -b for

Vorlesung Quantum Computing SS pseudo pure states = p p = = p p 0 0 ebeb e -b 0 0 e b + e -b 1 - z B 0 /k B T with e 1 z B 0 kBTkBT = n2n 1 b -b0 0 2n2n 1 density matrix can be written = 2 -n (1 + ) access population scales with 2 -n (n: number of qubits) reduced density matrix

Vorlesung Quantum Computing SS qubit representation = n2n 1 b -b0 0 2n2n I z = 2 1 I z I z identity is omitted I z

Vorlesung Quantum Computing SS time development HH -1 calculation U preparation read-out |A| time Liouville – von Neumann equation H, ^ iħiħ t (t) = (t=0) = U (t) (t=0) U (t) ħ - i H t ^ e ħ i H t ^ e ^^

Vorlesung Quantum Computing SS time development eq = b -b copies of the same nuclear spin z B 0 kBTkBT b = = ħ L 2k B T B = 0 0 B0B0 rotate spin to x,y plane by applying RF pulse 2 (t) = ħ - i H t ^ e ħ i H t ^ e eq = L kBTkBT 1 IzIz ^ (0+) = L kBTkBT 1 IxIx ^ IyIy ^ 2 1 L kBTkBT 1 IxIx ^ cos L t +sin L t + = H = L I z ^^ (0+) eq

Vorlesung Quantum Computing SS refocusing (0+) = L kBTkBT 1 IxIx ^ 2 (t) -1 ħ - i H t ^ e ħ i H t ^ e IxIx ^ = IxIx ^ IyIy ^ cos L t + sin L t if L r, e.g., due to inhomogeneous B 0, the spin picks up a phase applying second RF pulse x inverts y-component: 2 (t+) -1 IyIy ^ IxIx ^ cos L t sin L t L

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

Vorlesung Quantum Computing SS energy Simple CNOT CNOT operation spin levels individually addressable spin b a pulse inverts spin population

Vorlesung Quantum Computing SS coupling between nuclear spins Cory et al.: Fortschr. Phys. 48 (2000) 9-11, 875 chemical shift qubit coupling (always on) ^ H = ( a I z a + b I z b + c I z c ) ^ ^ ^ + 2 (J ab I z a I z b +J ac I z a I z c +J bc I z b I z c ) ^^^^^^

Vorlesung Quantum Computing SS CNOT with Alanine Z:-90 a i b i c i Y:-90Y: 90X:-90Y:180-Y:180 a o b o c o J ab NO operation Cory et al.: Fortschr. Phys. 48 (2000) 9-11, 875 U NO = i ħ tI z a I z c J e i ħ J e i ħ I y e i ħ I y e -i 4 e i ħ IxbIxb 2 e i ħ IzaIza 2 e i ħ I z a I x b e i ħ I y 2 e i ħ IyIy 2 e i ħ J t I z a I z b e U CNOT =