This document summarizes the quantum algorithm known as Grover's algorithm for searching an unordered database. It explains that classically, searching an unsorted database of n items would require O(n) queries on average to find the single item satisfying some predicate f(). Grover's algorithm uses amplitude amplification in a quantum system to find this item using only O(√n) queries, providing a quadratic speedup over classical algorithms. It describes how the algorithm iteratively rotates the system state toward the target item using reflections about the average and target states. After O(√n) iterations, the target item can be measured with high probability.

1. Searching using Quantum Rules

2. System State is
captured by a
Probability
Our Two Worlds
Distribution
Σ vi = 1 , 0<=vi<=1
T is stochastic (non-neg, col sums 1)
Classical
Measurement
Quantum
|v|2 = v†v = Σ |vi|2 = 1
System State is T is Unitary T†T = I
captured by an
Amplitude
Distribution

3. What does this mean for
computation?

4. Simple Search
• Given n items, A[1..n]
• Exactly one satisfies f()=1
– all others have f()=0
• Find the unique object for which f()=1
• How many f() calls are needed?

5. Classical Algorithm
x  Random item in 1..n
Repeat
–
– If f(x)=1
• Stay at x
– Else
• Choose another (never previously picked) value for x,
each possible value equi-probably
K times

6. Classical Algorithm: Analysis
The state for
which f()=1
K needs to be np
before this prob
exceeds p

7. Grover’s Quantum Algorithm
x  Random item in 1..n
Repeat
– If f(x)=1
• then stay at x with amplitude -1
• else stay at x with amplitude 1
– Choose
– this x with amplitude a
– another value for x, each possible value with equal
amplitude, say b
K times

8. Unitariness Constraints
Diagonals A, non-
diagonals B

9. Unitariness Constraints
Diagonal Matrix, so
clearly Unitary

10. Algorithm Analysis
k

11. State Progression
For n=4, one
iteration suffices!
With certainity!!

12. State Progression in General
The state for
which f()=1
What progress have
we made?

13. Progress per Iteration Dot Product

14. Geometry
Destination
State Vector
[0 0……1]

15. Wrapping Up