Books (NMR): Spin dynamics: basics of nuclear magneac resonance, M. H. LeviF, Wiley, 200. The principles of nuclear magneasm, A. Abragam, Oxford, 96. Principles of magneac resonance, C. P. Slichter, Springer, 990. Reviews (): NMR techniques for quantum control and computa>on, LMK Vandersypen and I. Chuang, Reviews of Modern Physics (2004) vol. 76 (4) pp. 037-069. Quantum informa>on processing using nuclear and electron magne>c resonance: review and prospects, et al., 2007. arxiv: 070.447. NMR Based Quantum Informa>on Processing: Achievements and Prospects, D. G. Cory et al., 2000. arxiv:quant- ph/000404.
Spin- /2 nucleus NMR freq (at 0 T) abundance H 426 MHz 99.9% 3 C 07.% 5 N 43 0.4% 9 F 40 00% 28 Si 85 4.7% 3 P 75 00% Nuclei with an odd (even) number of nucleons possess half- integer (integer) spin Nuclei with an even number of protons and neutrons are spinless
H NMR Hamiltonian: 2 coupled spins in liquid- state NMR (e.g. 3 C- labelled Chloroform) ( =) H 0 = ( 2 ω σ z + ω 2 σ 2z + πj 2 σ z σ ) 2z 3 C Chloroform molecule, CHCl 3 Larmor frequency (rad/s): External magneac field (T): Coupling strength (Hz): J kl ω k = γ k B B Chemical shielding: For the same isotope, γ j = γ ( 0 δ ) j for nucleus at the j th chemical site. Ability to address different nuclei as qubits
Larmor precession H 0 = ω σ z 2 U(t) = e iω tσ z 2 = Take ψ = + 2 U(t) ψ = e iω t 2 + e +iω t 2 2 + e +iωt =e iω t 2 2 + i = +
H Eigenstates of the Hamiltonian H 0 = ( 2 ω σ z + ω 2 σ 2z + πj 2 σ z σ ) 2z : H state 2: 3 C eigenvalue ( 2 ω + ω + πj 2 2) 2 ω + ω 2 πj 2 ( 2 ω ω πj 2 2) ( ) ( 2 ω ω + πj 2 2) ω = 2π 426MHz ω 2 = 2π 07MHz J 2 200Hz B =0T ( 2 ω + ω 2) ( 2 ω + ω 2 ) ( 2 ω ω 2) ( 2 ω ω 2 ) + π 2 J 2 π 2 J 2 π 2 J 2 + π 2 J 2 3 C Chloroform molecule, CHCl 3
NMR transifons (spectrum) ( 2 ω + ω 2) + π 2 J 2 2 2 ( 2 ω + ω 2) ( 2 ω ω 2) π 2 J 2 π 2 J 2 2πJ 2πJ ( 2 ω ω 2) + π 2 J 2 ω ω 2 frequency These are magneac dipole transiaons, and are thus driven by an oscillaang (RF) magneac field Need transverse spin operators σ x,σ y ( )
RF Hamiltonian: H rf (t) = Ω(t) 2 θ(t) = ω rf t + φ [ σ x cos(θ(t)) + σ y sin(θ(t)) ] RF amplitude RF frequency B x cos(ω rf t) Re B + e iω rf t ( )
RotaFng reference frame transformafon * R(t) = e iω rf tσ z 2 e iω 2rf tσ 2z 2 ψ R = R(t) ψ lab id ψ lab dt = H lab ψ lab id ψ R dt = H R ψ R H lab = H 0 + H rf (t) * H R = 2 ω ω rf *This is for the case of two isotopes doubly- rotaang frame [( )σ z + ( ω 2 ω 2rf )σ 2z + πj 2 σ z σ 2z ] + [ 2 Ω (t) ( σ x cos(φ ) + σ y sin(φ ))] + [ 2 Ω (t) σ cos(φ 2 ( ) + σ sin(φ ) 2x 2 2y 2 )]
RotaFng frame = We had Larmor precession in Lab frame: U(t) ψ = e iω t 2 + e +iωt 2 But in the rotaang frame (on resonance): + i R(t) U(t) + U(t) ψ R = U(t)R(t) ψ = + 2 =
Resonant pulses as Bloch sphere rotafons Consider a one- spin system on resonance: ω = ω rf H R = [( 2 ω ω rf )σ z + Ω (t)( σ x cos(φ ) + σ y sin(φ ))] In NMR, we can control both RF phase ( φ) and amplitude ( Ω) in Ame, so we can perform arbitrary single spin rotaaons U(t) = e iω t ( σ x cos(φ )+σ y sin(φ ) ) / 2 R y (π /2) φ = π /2 Ωt = π /2 R x (π /2) φ = π Ωt = π /2 - Y X ( 0 + ) 2 Y - Y X Y ( 0 + i ) 2
Example of two- qubit control: the CNOT gate H(t) = ( 2 ω σ z + ω 2 σ 2z + πjσ z σ 2z ) + H rf (t) Zeeman coupling RF pulses (control)
q q 2 2 3 4 5 R z (90) R y (90) R y ( 90) R x (90) U = e iπ 2 I 2 σ y 2 t = π /2J Ame U 2 = e i π 2 σ z σ 2z / 2 U 3 = e iπ 2 σ z 2 I 2 U 4 = e +iπ 2 I σ 2y 2 U CNOT = U 5 U 4 U 3 U 2 U = 0 0 I 2 + σ 2x U 5 = e iπ 2 I σ 2x 2
The thermal state A general mixed state: ρ = k a k ψ k ψ k Thermal equilibrium (Boltzmann distribuaon): ρ th = e H 0 kt Tr e H 0 ( kt ) In the high temperature limit ( ) H 0 << kt ρ th 2 I H 0 I kt 2 4kT ω σ z + ω 2 σ 2z ( ) (two- spin case) ω kt ~ 0 5 at room temperature in NMR
The thermal state ρ th 2 I H 0 I kt 2 4kT ω σ z + ω 2 σ 2z ( ) The idenaty term cannot be changed by pulses or observed So we define a deviaaon density matrix: ρ dev = H 0 kt ω σ z + ω 2 σ 2z (the J coupling term is ~0-6 Ames smaller than Zeeman energy) σ z indicates a populaaon difference between spin- up and spin- down states
Pseudopure states To iniaalize an NMR quantum info. processor, we want a pure state or at least one that behaves like a pure state: ρ pp = I 2 α ψ ψ Example: making a H 3 C pseudopure state 5 0 0 0 0 3 0 0 ρ dev 4σ z + σ 2z = 0 0 3 0 0 0 0 5
Example: making a H 3 C pseudopure state Apply three permutaaon circuits and sum results: ρ dev ρ dev 5 0 0 0 0 3 0 0 0 0 3 0 0 0 0 5 5 0 0 0 0 5 0 0 0 0 3 0 0 0 0 3 5 0 0 0 0 5 0 0 = 20 00 00 5 I 0 0 5 0 0 0 0 5 ρ dev 5 0 0 0 0 3 0 0 0 0 5 0 0 0 0 3 ʹ ρ dev 00 00
Measurement in NMR Bulk magneazaaon of j th isotope: B // ˆ z M z j = µ j ( n j n ) j 90 degree ( readout ) pulse rotates magneazaaon to x- y plane: Z M z Ensemble of idenacal, non- interacang processor molecules - Y X M x Y
Measurement in NMR In the lab frame, magneazaaon precesses in the x- y plane, at the Larmor frequency Z - Y X M x Y the oscillaang magneac flux induces an emf in the solenoid coil via Faraday s law.
Measurement in NMR This corresponds to an expectaaon value measurement, with observables: M x = Tr ρ σ kx M y = Tr ρ σ ky k k We call these observables transverse magneazaaon
Measurement in NMR Example: - qubit tomography of an arbitrary state ρ dev Expt. ρ dev M x, M y P x,p y Expt. 2 ρ dev R y (π /2) M x P z Unfortunately, the # of expts grows exponenaally with # qubits for any implementaaon!
Measurement in NMR : FT spectrum V 0 M x (0) NMR transi>ons Voltage T 2 V 0 /e Fourier transform Δω ~ T 2 Time amp M x (0) Frequency free- inducaon signal ( ) M x (t) = Tr e ih 0 t ρ(0)e +ih 0 t σ x
RelaxaFon and decoherence General Kraus operator sum: ρ A k ρa k + k k A k + A k = I Z RelaxaAon Dephasing Spin operators σ x,σ y σ z X Y Example: pure dephasing A = e iφσ z ρ dφ e iφσ z ρe+iφσ z f (φ,t,t 2 ) Lorentzian with Ame- dependent FWHM ( ) a be t /T 2 b * e t /T 2 Dephasing on the Bloch sphere a
Pure dephasing Or in another form: Z Λ(ρ) = ( p) ψ ψ + pσ z ψ ψ σ z With Kraus operators: A 0 = pi Y A = pσ z p = e t /T 2 2 Dephasing on the Bloch sphere
RelaxaFon A 0 = p 0 0 η A = p 0 η 0 0 A 2 = η 0 p 0 A 3 = p 0 0 η 0 see e.g. Nielsen & Chuang ( Λ(ρ) = a a 0)e t /T + a 0 be t / 2T b * e t / 2T ( a 0 a)e t /T + ( a ) 0 For pure relaxaaon, T 2 = 2T In liq. state NMR, typically T 2 ~ 0.5-5 s T ~ 5-20 s
NMR Spectrometer
The Deutsch- Jozsa algorithm on two qubits H 0 R y 90 R z 90 3 C 0 R 80 90 x R y R y 90 τ = π 2J ( U 0,U 0 ) R x ±90
constant: f 00 balanced: f 0 balanced: f 0 constant: f Jones and Mosca, 998
State- of- the- art 2- qubit molecule: histadine Pseudopure state demonstrated (Negrevergne et al, PRL 96, 7050 (2006))
Benchmarking control ~0.005 average gate error for 3- qubit control; ~0-4 for single qubit C. A. Ryan, M. Laforest and R. Laflamme, Randomized benchmarking of single and mulaqubit control in liquid- state NMR quantum informaaon processing, New J. Phys.. 03034 (2009)
Control in solid- state NMR 4 itera>ons of heat- bath algorithmic cooling C. A. Ryan,, O. Moussa, and R. Laflamme, PRL 00, 4050 (2008)
Books (NMR): Spin dynamics: basics of nuclear magneac resonance, M. H. LeviF, Wiley, 200. The principles of nuclear magneasm, A. Abragam, Oxford, 96. Principles of magneac resonance, C. P. Slichter, Springer, 990. Reviews (): NMR techniques for quantum control and computa>on, LMK Vandersypen and I. Chuang, Reviews of Modern Physics (2004) vol. 76 (4) pp. 037-069. Quantum informa>on processing using nuclear and electron magne>c resonance: review and prospects, et al., 2007. arxiv: 070.447. NMR Based Quantum Informa>on Processing: Achievements and Prospects, D. G. Cory et al., 2000. arxiv:quant- ph/000404.