1. Quantum Computing Lecture 19 Robert Mann

2. Nuclear Magnetic Resonance Quantum Computers Qubit representation: spin of an atomic nucleus Unitary evolution: using magnetic field pulses applied to spins in a strong magnetic field. Chemical bonds between atoms couple the spins State preparation: using a strong magnetic field to polarize the spins Readout: using magnetic-moment induced free induction decay signals

3. Nuclear Magnetic Resonance Q.C. Physical Apparatus (uniform to 1 part in 10 9 ) Liquid sample Regard as an ensemble of n-bit quantum computers Computer Typical Experiment 1.Wait a few minutes for the sample to come to thermal equilibrium 2.Send RF pulses to manipulate nuclear spins into desired state. 3. Switch off the amps and switch on the pre- amplifier to measure the free-induction decay

4. N.M.R.Q.C. Hamiltonian-I Main interaction: Schroedinger Equation solution a single qubit rotation about axis!

5. N.M.R.Q.C. Hamiltonian-II Spin-spin interactions: Dipolar interaction: Averages out to zero over a spherical volume Through-bond interaction: Valid for weak bond couplings

6. N.M.R.Q.C. Hamiltonian-III Thermal Equilibrium: For n spins, the density matrix is a mixture of the pure states Magnetization Readout: For k th spin Decoherence effects: Models decoherence effects due to inhomogeneities, thermalization, etc.

7. N.M.R.Q.C. Hamiltonian-IV Full n -spin Hamiltonian:

8. N.M.R.Q.C. Computation-I Refocussing: 2 -spin Hamiltonian Refocussing – spins gyrating at different frequencies come back to same point on the Bloch sphere Cnot: Sufficient for a CNot gate!

9. N.M.R.Q.C. Computation-II Preparation of pure states: Use Cnots to make a circuit P Consider a general unitary operator U Equivalent to a pure state!

10. N.M.R.Q.C. Computation-III Readout: Basic problem – NMR output is over an ensemble of molecules, and so we obtain only ensemble averages of results Example: Quantum factoring produces output, a random rational # From this we use a continued fraction expansion to get, then check result to get NMR output is an ensemble average So we can’t get Solution: Have each molecule do the continued fraction expansion Take output only from those molecules which verify problem Result will be the ensemble average

11. N.M.R.Q.C. Experiments-I State Tomography: A measurement of the density matrix Send in NMR pulses to measure Arbitrary #’s For a single qubit: Quantum Logic Gates: A 2-qubit proton-Carbon system in Chloroform Frequencies are 500 MHz and 125 MHz J-coupling frequency is J=215Hz Decoherence times are T 1 = 18s, T 2 = 7s (proton) T 1 = 25s, T 2 = 0.3s (Carbon) CNot gate has been experimentally demonstrated

12. N.M.R.Q.C. Experiments-II Quantum Algorithms: Again, using Chloroform Oracle: Hadamard: Circuit P: C H Cl Grover: Experimentally checked for all possible cases Measure each density matrix took 27 repetitions (3 for pure state prep, 9 for tomo- graphic reconstruction) Maximum # of iterations is 7 so far, taking 35 ms

13. N.M.R.Q.C. Drawbacks 1.Total Signal decreases exponentially as the number of qubits distilled into a pure state 2. Structure of the molecule constrains the computer architecture – determining what qubits interact with each other