1. The Integration Algorithm A quantum computer could integrate a function in less computational time then a classical computer... The integral of a one dimensional function, f(x), is the area between the f(x) and the x-axis. y = f(x) x y

2. Integration via Summation y=f(x) y x The integral, I, can be approximated by a sum, S. Taking more equally spaced points in the summation, leads to a better the approximation of the integral. y=f(x) y x

3. Summation y x We first evaluate the sum where M is the number of points used in the approximation. This sums the height of all the boxes. Multiplying this by the width of each box gives the area under the boxes. Defining, we see that S is equal to the average value of f(a).

4. Quantum Averaging The average of a function can be found on a quantum computer in the following way... Initial state of quantum computer 1 work qubitlog 2 (M) function qubits - these qubits store the number for which we will evaluate the function, f(a).

5. The Hadamard Transform The Hadamard transform, H, takes a qubit from a ‘classical’ 0 or 1 state, to a superposition of 0 and 1. Hence, Hadamards on all function qubits in the initial state of our quantum computer will give an equal superposition of all possible states, a, allowing us to evaluate f(a) for all input states.

6. Quantum Averaging We now conditionally rotate the work qubit by an amount f(a) depending on the state of the function qubits. This puts our quantum computer into the state... If we now perform another set of Hadamards on the function qubits the state will have an amplitude of from which we can get S.

7. Quantum Averaging via NMR Measurement of a quantum system in a superposition state is probabilistic. Therefore, we can only extract the amplitude of a particular state by repeated experiments and measurements of the system. The more experiments the closer we can estimate the amplitude. An NMR quantum information processor allows us to read out the entire state of our system exactly - allowing us to bypass methods necessary to amplify the amplitude.

8. Integration Gate Sequence H H H H H H H H work bit function bits evaluate f(a) Extract amplitude of state Sequence of conditional rotations - rotate work bit by some angle if the function bit is 1.

9. Integrating Sinusoidal Functions H H H H H H H H work bit function bits Extract amplitude of state a is stored as a binary number. Thus the sequence to evaluate f(a) is a series of conditional gates that rotate the work bit by an amount. To integrate a sinusoidal function between 0 and 1 would require each state, a, to conditionally rotate the work bit by, where

10. Integration of Actual integration yields: The integration algorithm taking the four data points shown above yields: 0 1 1

11. work bit function bits Integrating H H H H conditional rotations Extract amplitude of state 0 1 1

12. Integration Algorithm for Pseudo pure state Hadamard on function bits Conditional rotation from least significant function bit Conditional rotation from most significant function bit Hadamard on function bits Bits 1 and 3 are function bits. Amplitude of state =.433

13. Integration of Actual integration yields: The integration algorithm taking the four data points shown above yields: 0 1 1

14. 0 1 1 Integrating H H H H work bit function bits Extract amplitude of state Controlled-NOT gate

15. Initial state Integration Algorithm Using CNOT Hadamard on function bits CNOT31 Hadamard on function bits Amplitude of state =.5

16. Quantum Information Processing using NMR B 0000 1111 Nuclear Spins as qubits High field magnet RF Wave sample test tube Spectrometer ADC for data acquisition RF synthesizer and amplifier Gradient control wave guides I S J IS 2-3 Dibromothiophene 9.6 T RF wave

17. Internal Hamiltonian The evolution of a spin system is generated by Hamiltonians –Internal Hamiltonian: H int =  I I z +  S S z +2   J IS I z S z spin-spin coupling interaction with B field I S J IS 2-3 Dibromothiophene 9.6 T

18. External Hamiltonian –Experimentally Controlled Hamiltonian: –Total Hamiltonian: H ext (t) =  RFx (t)·(I x +S x )+  RFy (t)·(I y +S y ) H total (t) controlled via H ext (t) I S J IS 2-3 Dibromothiophene 9.6 T RF wave spins couple to RF field H total (t) = H int + H ext (t)

19. The Alanine Spin System C1C1 C2C2 C3C3 J 12 = 54.1 J 13 = -1.3 J 23 = 35.0

20. Radio Frequency Pulses RF pulses are designed to implement a single unitary operator on any number of spins. A computer program designed for the specific spin system is used to search for such a pulse based on the parameters: duration of pulse, power, phase, and frequency offset. time RF nutation rate (radians) This pulse implements a Hadamard gate on the second and third spins.

21.  Start with an initial state and some extra spins Single bit errors become correlated errors   Encode  No Error  Flip Bit 1  Flip Bit 2  Flip Bit 3 Decode Measure the extra bits to collapse to one error and learn what error occurred. Then correct it. Never need to know the original state! Quantum Error Correction

22. Decoherence Free Subspace Information Noise strength (Hz) Encoded Un-Encoded EngineeredNoise    EncodeDecode

23. Noise Strength (Hz) Encoded, Y, Z Noise No Encoding, Y Noise Information Weak Noise Noiseless Subsystem Experiment Strong Noise Limit Z-X Noise 0.24 Un-Encoded 0.70 NS-Encoded No Noise 0.70 Z-X Noise Z-Y Noise Info

24. Tomography Not all elements of the density matrix are observable on an NMR spectra. To observe the other elements of the density matrix requires repeating the experiment 7 times with readout pulses appended to the pulse program. This is done without changing any other parameters of the pulse program.

25. Creation of a Pseudo-Pure State Pseudo-pure state thermal state72 o spin 2 rotation and gradient Control2 90 o y on 1 & 3 gradientFake ‘swap’ 1 &2 Add some identity

26. NMR Simulation Pseudo-pure state Hadamard on function bits Conditional rotation from least significant function bit Conditional rotation from most significant function bit Hadamard on function bits Simulator correlation -.92

27. NMR CNOT Simulation Pseudo-pure state Hadamard on function bits CNOT31 Simulator correlation -.99

28. NMR Experiment Pseudo-pure state projection =.98 Hadamard on function bits CNOT31 correlation =.97 correlation =.92correlation =.91

29. Integration Results The element gives the result of the integration. element Amplitude =.497

30. Conclusions Concrete mapping between integration algorithm and NMR QIP implementation. Sufficient control with current NMR quantum information processors to execute integration in small Hilbert spaces. NMR QIP version of algorithm does not require amplitude amplification. General approach for integrating sinusoidal functions.