13 CHCl 3 Ensemble NMR Implementation of the Deutsch- Jozsa - - PowerPoint PPT Presentation

13CHCl3 Ensemble NMR

Implementation of the Deutsch- Jozsa Algorithm

Dennis V. Perepelitsa with Brian J. Pepper 27 April 2007 8.14 Junior Lab MIT Department of Physics

Outline of Talk

• 1. Introduction
• 2. Quantum Computation
• 3. Initial State Preparation
• 4. Deutsch-Jozsa Algorithm
• 5. Experimental Setup
• 6. Calibration
• 7. Experimental Results
• 8. Error Analysis
• 9. Conclusion

... Introduction DVP 4/27/2007

Introduction

• 1985: Feynman proposes quantum computation,

characterizes quantum logic gates.

• 1985: Deutsch characterizes the universal quantum

computer.

• 1992: Deutsch and Jozsa present an algorithm that

requires less operations on a quantum computer.

• 1997: Gershenfeld and Chuang propose an

implementation using bulk-ensemble spin systems.

• 1997: Cory, et. al. propose using NMR spectroscopy.

... Quantum Computation DVP 4/27/2007

Quantum Computation

• Key Idea #1: Every pair of spin eigenstates is a qubit.
• Key Idea #3: Implement a function f with a unitary operator Uf.
• Key Idea #2: Operations are implemented with RF pulses.
• Important example: controlled-not. Implements f(x) = x.
• Ensemble NMR --> measure expectation values.

... Initial State Preparation DVP 4/27/2007

Initial State Preparation

The Boltzmann distribution of the energy eigenspectrum is Problem: We need the pure state |00> ! Solution: Time-average to obtain a pseudopure state.

• Take three trials, cyclically permuting the non-|00> elements.
• The average of the three is the action on the pure state |00> !
• Finally, fire a “read out” pulse and observe the FID signal.

... Deutsch-Jozsa Algorithm

Density matrix leads to spectrum

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Deutsch-Jozsa Algorithm (1992)

• Classically: Look at both sides to decide if a coin is fair. (2 evaluations)
• Quantum computation can do better!
• Enumerate (and implement) all possible functions:
• Apply pulse sequence to |00>
• If f(0) = f(1): We see |00>
• If f(0) != f(1): We see |10>

... Experimental Setup DVP 4/27/2007

Experimental Setup

(Heavily modified from 8.14 Quantum Information Processing lab guide )

... Calibration DVP 4/27/2007

Calibration

... Calibration (cont)

( xkcd.com – used under the Creative Commons License 2.5 )

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Calibration

DVP 4/27/2007 ... Calibration (cont)

Fourier transform into frequency domain.

Calibration

... Experimental Results (cont)

180-T-90 pulse sequence to obtain T1

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Vary pulse width and measure net magnetization to find pw-90

Experimental Results

... Experimental Results (cont) DVP 4/27/2007

Calibration from the thermal state proton carbon Theoretical Prediction: Uf1, Uf2 (6,0) (6,0) Uf3, Uf4 (-6,0) (0,6)

Experimental Results

... Experimental Results (cont)

Uf1 Uf2

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Experimental Results

... Error Analysis

Uf4 Uf3

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Error Analysis

• Integrate absolute peaks numerically.
• Random error calculated from 20 repeated “baseline” spectra ~ 1%
• Uncertainty from fluctuations in the background ~ .5%
• Calibrate from thermal state, propagate error.

But:

• Deviation from theoretical prediction typically ~10-20%.
• Our error does not cover this!

Significant systematics:

• Asymmetric lineshapes due to improper shimming.
• Imperfect pulse widths – deterioration over time.
• Inhomogeneous magnetic field.
• Inhomogeneous RF field.

... Conclusion DVP 4/27/2007

Conclusion

We have:

• Presented an implementation of Deutsch-Jozsa using ensemble NMR.
• Calibrated computing system.
• Successfully performed DJ on all four functions.
• Discussed sources of error.

Stay tuned:

• Brian Pepper on Grover's Algorithm.

Questions? DVP 4/27/2007