**Matthew Hayward**

- Contents
- Motivation for Study of Quantum Computing

- The Quantum Computer

- Performing Computations
- Grover's Algorithm
- An Illustration of Grover's Algorithm
- Outline of Proof of Correctness of Grover's Algorithm
- Operator to Create Equal Superposition of States
- Operator to Rotate Phase
- Inversion About Average Operator
- Proof that Operations are Unitary
- Proof that Algorithm Increases Amplitude of Desired State
- A Special Case

- Open Questions
- How Many Iterations are Required
- Searching for More Than One Item
- Optimality of Grover's Algorithm
- Implications on P = NP

- Conclusion
- Bibliography