NMR Quantum Computation

Transcription

1 NMR Quantum Computation C/CS/Phs 191: Quantum Information Science and Technolog 11/13/2003 Thaddeus Ladd Department of Applied Phsics Stanford Universit

2 Solution NMR Quantum Computation 4. Measurement is facilitated b huge ensemble of independent copies 1. A small number of stable qubits provided b distinct spin 1/2 nuclei B RF 3. Dipolar interactions controllable b RF 5. Decoherence times are long since nuclei interact onl weakl with environment 2. Perfect initialiation is not possible. Instead, pseudopure states must be used N. A. Gershenfeld and I. Chuang, Science 275, 350 (1997) D. G. Cor, A. F. Fahm, and T. F. Havel, Proc. Natl. Acad. Sci. USA 94, 1634 (1997)

3 Solution NMR QC: Wh Bother? From the start (1997), researchers knew solution NMR would not surpass the 10-qubit level. Wh bother going through the trouble of making solution NMR quantum computers then? Because classical computers are too slow to factor 15 and decide whether 3-bit functions are constant or balanced. These problems will soon to be solved b coffee cup NMR quantum computers in ever home! Right answer: NMR is eas, due to the 50+ ear histor and etensive commercial engineering of the technique. If we cannot do quantum computing with this plug and pla, high-q sstem, then our chances with atoms, semiconductor electrons, etc. are slim! NMR gives us a head start on solving the problems in quantum control that we will face with other architectures.

4 Outline Wh bother with solution NMR quantum computing? Basic phsics of NMR and ingredients for QC Nuclear magnetic moments in static and RF magnetic fields Nuclear-nuclear couplings Weak ensemble measurement Eperimental Overview Initialiation Phsical polariation Algorithmic cooling Pseudo-pure states Scaling Quantum logic RF implementations of the universal gate set Refocussing Surve of results The future of NMR quantum computing

5 Definitions and Units Subscripts indicate nuclei from different atoms of the molecule. (All Hamiltonians in units of angular frequenc)

6 The Hamiltonian Bigger smaller Independent of spin Term Motion Zeeman HperFine RadioFrequenc Dipolar Sie kt» 6 TH ω 0» 100 MH A» 1 MH ω 1» 10 kh D» 1 kh Role Bath for thermal relaation. Affects decoherence. Allows polariation. Establishes resonance. SNR. Chemical shielding and J-coupling. The knob. Allows spin rotations. Decoherence source.

7 Applied Magnetic Fields and the Rotating Frame RF frequenc ω is chosen to be different from resonant frequenc ω 0 =γb 0 b δω. Rotating frame rotates at ω New frame rotates at ω in spin subspace. Fastest, uninteresting motion (rotation about -ais at ω) removed. Terms oscillating at 2ω are neglected. Spin precession at ω 0 Spin precession at δω RF oscillating at ω RF counterrotating at 2ω Stationar RF Change δω, ω 1, and φ to achieve arbitrar rotation angle and ais!

8 Single Spin Rotations with the Densit Matri Consider eample of a spin ½ in pure state Suppose we appl on-resonant RF (δω=0) with phase φ=0 for time τ. M The last step is the result of some simple matri algebra; it is worth deriving in our spare time. (Result is independent of representation.) End result is standard rotation of a vector! Thus we write: Of particular importance are π-pulses, corresponding to Ωτ=π, which invert M, and π/2 pulses, corresponding to Ωτ=π/2, which are Hadamard gates.

9 The Hamiltonian Bigger smaller Independent of spin Term Motion Zeeman HperFine RadioFrequenc Dipolar Sie kt» 6 TH ω 0» 100 MH A» 1 MH ω 1» 10 kh D» 1 kh Role Bath for thermal relaation. Affects decoherence. Allows polariation. Establishes resonance. SNR. Chemical shielding and J-coupling. The knob. Allows spin rotations. Decoherence source.

10 The Chemical Shift Although the hperfine coupling constants can be large (~MH), this term ends up small because the sums over spin (S k ) and orbital (L k ) angular momentum of electrons vanish in molecular orbital ground states. The chemical shift arises as a first order perturbation of the ground state wave function due to the magnetic field Tin electron currents oppose magnetic fields Magnetic field seen b nucleus is reduced b a few parts per million, depending on the molecular orbital state Varies from nucleus to nucleus according to smmetr of molecule Gives quantitative information about molecular structure; Allows qubits of same nuclear species to be distinguished. j Molecular orbital nucleus j

11 The J-Coupling The J-coupling arises as a second order perturbation to the ground state wavefunction as a result of the effective field due to the nuclei. Magnetic field of one nucleus distorts electron orbital Tin electron currents change magnetic field at second nucleus The matri J gives quantitative information about molecular structure. Nonlinear term used for quantum logic. j j B eff / j / I > 0 -j -j B eff / j / I < 0 Components of effective magnetic field due to J- coupling not in -direction time-average to ero due to rapid Larmor rotatation of nuclei in large applied magnetic field.

12 The Hamiltonian Bigger smaller Independent of spin Term Motion Zeeman HperFine RadioFrequenc Dipolar Sie kt» 6 TH ω 0» 100 MH A» 1 MH ω 1» 10 kh D» 1 kh Role Bath for thermal relaation. Affects decoherence. Allows polariation. Establishes resonance. SNR. Chemical shielding and J-coupling. The knob. Allows spin rotations. Decoherence source.

13 The Dipolar Coupling: The Advantage of Liquids Couples nuclear spins. Coupling strength is ~1 kh. Includes relative distance r between nuclear spins,. The advantage of liquids is that, due to the thermal motion of the molecules in the liquid, the dipolar coupling averages to ero in first order. Residual dipolar coupling is leading cause of Decoherence, T 2 : Random magnetic fields due to nuclear dipoles on other molecules appl random -rotations Initiall parallel spins go out of phase due to random rotations Nuclear magnetiation from different molecules destructivel interferes Thermal relaation, T 1 : Sometimes, moving molecules interact for timescale of order 1/ω 0 (Larmor period) Such a random interaction is like a random, resonant RF applied field Tips spins with energ compensated b motional bath

14 The Hamiltonian in the Rotating Frame Bigger smaller We have complete control over single spin rotations. Spin couplings and relaation are alwas present. Select spin with RF frequenc Select angle with RF amplitude Select ais with RF phase

15 Free Induction Deca (FID) Consider a two spin molecule. (Neglect T 1 ). Suppose spins begin in ground state (parallel to field) Set δω 1 =0 b choosing RF frequenc to match chemical shifted Larmor frequenc of first nucleus. Appl π/2 pulse about -ais. R (π/2) Remove RF. J-coupling causes precession about - ais in rotating frame, at frequenc Jh I 2 i =J/2. Due to dephasing from the random dipolar coupling, the different spins of the ensemble start to interfere, and the magnetiation shrinks.

16 Free Induction Deca in Laborator Frame That was the rotating frame. In the laborator frame, Rotating magnetiation in coil of wire causes an EMF due to Farada s law of induction. V FFT t 1/πT 2 ω/2π + J 12 /4π f

17 Weak Ensemble Measurement Wait a minute!!! What happened to quantum mechanics?!? Wh doesn t measurement affect the spins? Wh can we measure all the components of the spin at once? Wh can we treat the spins as classical precessing magnetiation?!? 1. The measurement is ver weak The interaction of one nucleus with a macroscopic coil of wire is etremel small The inductive emf from one nucleus gives ver little information about that nucleus. This is good, because it means that the measurement circuit (the coil) does not cause much decoherence. Decoherence due to measurement circuitr is a big problem in man quantum computing architectures, but not solution NMR. However, it requires a ver, ver large ensemble of nuclei in order to get enough signal to measure anthing 2. The ensemble averages of quantum observables behave classicall under macroscopic forces as the ensemble gets ver large. Known as Ehrenfest s Theorem. Same theorem clarifies that large ensembles ma act quantum under microscopic forces (e.g. the J-coupling)

18 Eperimental Overview Ke eperimental aspect is RF engineering Commercial equipment is readil available Ke issues: Homogeneit of magnetic field Homogeneit of RF field Efficienc of probe

19 Eperimental Overview

20 Sample Molecules A Cat-State Benchmark on Seven Bit QC E. Knill et al., quant-ph/ Eperimental realiation of an order-finding algorithm with an NMR QC L. Vanderspen et al., PRL 85, 5452 (2000) 7-qubit molecule used to factor 15 b Shor s algorithm: I. Chuang et al., Nature 414, 883 (2001)

21 Real Timescales 7-qubit molecule used to factor 15 b Shor s algorithm: I. Chuang et al., Nature 414, 883 (2001)

22 Outline Wh bother with solution NMR quantum computing? Basic phsics of NMR and ingredients for QC Nuclear magnetic moments in static and RF magnetic fields Nuclear-nuclear couplings Weak ensemble measurement Eperimental Overview Initialiation Phsical polariation Algorithmic cooling Pseudo-pure states Scaling Quantum logic RF implementations of the universal gate set Refocussing Surve of results The future of NMR quantum computing

23 Phsical Polariation In thermal equilibrium, densit matri is For single spin, effectivel equivalent to spin in pure state with M scaled down b factor p 0. In this approimation, all the spins that are up and down cancel out and have no effect on the ecess pointing up. For multiple spins, equilibrium state is not like pure state.

24 The FID Revisited + For a pure state, we saw one peak. For the mied state, we see 2 n-1 peaks. FFT ω/2π -J 12 /4π ω/2π + J 12 /4π f

25 Is it practical to increase p 0? Mabe lowering the temperature, raising the field? Largest reasonable laborator static field: 12 T. Highest γ nucleus: protons at 43 MH/T. What temperature is needed for p 0 =50%? Onl helium is liquid at this low temperature! Cross relaation with cold paramagnetic electrons? Those same electrons lead to short T 1. Eotic methods to polarie nuclei: Chemical reaction with para-hdrogen Cross relaation with hperpolaried Xenon These have been done; the are not sufficient to bring nuclei close to ground state Efficient methods of dnamic nuclear polariation work in solids, though...

26 Algorithmic Cooling Entrop in n spins with polariation p 0? An algorithm cannot decrease the entrop, but it can redistribute it. Strateg: put all the entrop in n-k spins, and keep the remaining, ero-entrop k spins. NOT practical for solution NMR Onl useful if p 0 is increased substantiall

27 Pseudo-Pure Pure States: Temporal Averaging Eample: 2 equivalent spins U U U Average U

28 Pseudo-Pure Pure States: Logical Labeling Eample: 3 equivalent spins Pure state conditioned on first spin up (in 0 state)!

29 Scaling Issues with Pseudo-pure States ρ t ) = ρ ( t ) ρ ( t ) ρ ( ) ( t1 1+np of 2 n possibilities are initialied Best possible initialied signal scales as np/2 n-1, for small p, p where p = tanh(γ~b/2kt) 1 np of 2 n possibilities are antiinitialied U(t 1,t 2 ) ρ t ) = ρ ( t ) ρ ( t ) ρ ( ) ( t2 States ma alwas be written as separable, but with no classical path between them

30 Outline Wh bother with solution NMR quantum computing? Basic phsics of NMR and ingredients for QC Nuclear magnetic moments in static and RF magnetic fields Nuclear-nuclear couplings Weak ensemble measurement Eperimental Overview Initialiation Phsical polariation Algorithmic cooling Pseudo-pure states Scaling Quantum logic RF implementations of the universal gate set Refocussing Surve of results The future of NMR quantum computing

31 Universal Gate Set with RF Pulses We have alread seen that arbitrar single qubit rotations are effected b appropriate choice of RF phase for and direction RF offset for direction. Note: Offset with no RF amplitude = frame change. Universal logic is present if controlled-not gate is available The J-coupling (or ZZ interaction) is sufficient. Up to single qubit phases:

32 Controlled Controlled-NOT gate with ZZ interaction NOT gate with ZZ interaction R (π/2) R (π/2) R (π/2) R (π/2) R (π/2) R (π/2) R (π/2) R (π/2)

33 Refocussing X π π π The problem: the J coupling is alwas on. What if we want spins to be uncoupled? We notice: R (π) R (π)

34 Hadamard Decoupling ω 1 ω 2 ω All Uncoupled For decoupling man qubits, need to use π-pulses to alternate relative signs between all qubits Orthogonal sign matrices of sie n 2 eist for most n (Hadamard Conjecture, 1893) Selective recoupling can be done b putting signs in phase ω 1 ω 2 ω Coupled

35 Outline Wh bother with solution NMR quantum computing? Basic phsics of NMR and ingredients for QC Nuclear magnetic moments in static and RF magnetic fields Nuclear-nuclear couplings Weak ensemble measurement Eperimental Overview Initialiation Phsical polariation Algorithmic cooling Pseudo-pure states Scaling Quantum logic RF implementations of the universal gate set Refocussing Surve of results The future of NMR quantum computing

36 Algorithms Performed b Solution NMR 2-qubit Deutsch-Josa (1998) (Chuang, Jones) 2-qubit Grover (1998) (Chuang, Jones) 3-qubit Quantum Error Correction (1998) (Cor/Knill/Laflamme) 3-qubit Quantum Simulation (1999) (Cor) 3-qubit Deutsch-Josa (2000) (Knill/Laflamme) 3-qubit Grover (2000) (Chuang) 3-qubit Quantum Fourier Transform (2000) (Cor) 5-qubit Deutsch-Josa (2000) (Glaser) 5-qubit Order-Finding (2000) (Chuang) 5-qubit Quantum Error Correction (2001) (Knill/Laflamme) 7-qubit Shor s Algorithm (2001) (Chuang)

37 7-Qubit Shor s Algorithm: Details Pulse sequence optimied to correct for pulse errors. Gate sequence compiled to minimie number of gates. π pulse π/2 pulse Frame change (-rotation) I. Chuang et al., Nature 414, 883 (2001)

38 Results Thermal FID Pseudo-pure preparation (Temporal Averaging) Epected Measured a=11 Simulated with T 1, T 2 Epected Measured a=7 Simulated with T 1, T 2 I. Chuang et al., Nature 414, 883 (2001)

39 The Future of NMR QC No one is seriousl tring to scale solution NMR to more qubits Few qubit eperiments are still useful for testing new ideas in quantum information theor Adiabatic algorithms Geometric phases for pure states Algorithmic cooling Quantum Random Walks For future improvements of NMR QC, however, we must move towards solid-state implementations Several bulk ensemble approaches are being pursued: Dilute molecular crstals. Similar to solution, ecept solid-state effect cooling is possible. Difficult to grow good crstals of comple molecules. Semiconducting or insulating crstal lattice (e.g. silicon). Ensemble. Allows optical pumping for polariation. Uses magnetic field gradient to distinguish nuclei, adding scalabilit. Difficult to measure nuclear spins. Finall, approaches using single spins as qubits have seen the most attention... (net lecture!)