Generalization of Quantum Search Algorithm

1. Generalization of Quantum Search Algorithm DebasisSadhukhan M.Sc. Physics, IIT Bombay

2. Quantum Algorithm • Canonical Grover’s Quantum Search Algorithm • Generalized Quantum Search Operator: Selective Inversions of Two States • Generalized Quantum Search Operator: Selective Inversions of Three States Plan of Talk

3. An algorithm is a well defined procedure or a set of instructions to perform an information processing task. • Complexity Classes: P , NP • Quantum algorithms are those that uses quantum mechanical principles at the time of it’s execution. • Advantage of Quantum Algorithm: Superposition principle allows a quantum algorithm to exploit an exponentially large number of quantum components using only polynomial recourses. Hard to design ! Quantum Algorithms

4. Unsorted Database Search Problem A database is nothing but a collection of items. Number of items: N Classical Search: Average number of query to succeed: N/2 In the worst case, we need N-1 query to succeed. If there are M solutions, then we require O(N/M) trials to succeed. But using Grover’s Quantum Search Algorithm require only trial to succeed. Grover’s Algorithm

5. Practical demonstration for the superiority of quantum search over classical search

6. Start with • Apply Hadamard gate on each state to get uniform superposition of all states • Apply • Apply The Operations

7. Let’s start with the uniform superposition of all the states. i.e. the source state . • Our solution is the target state . G = , where, & Iteration Required (m): Geometric Visualization

8. Now, so, • So, no. of iteration required is Example

9. Generalized Quantum Search Operator: Selective Inversion of Two States Grover’s algorithm drives a quantum computer from a known initial state (source state) to an unknown final state (target state) by using selective phase inversions of these states. Grover's operator performs the selective inversions only on a unique source state and a unique target state. Here, in this project, we make our search operator to perform selective inversions on more than one source and target states. The generalized algorithm is simply a successive iteration of the new Generalized Grover's search operator , where and

10. The generalized search operator: Selective phase inversions of two source states and two target states. The operator , where, and Here, By the property of Walsh-Hadamard transform, Formulation

11. Let us take, We need to construct a four dimensional orthonormal basis to analyze the operation. New Basis: Where, and Analysis with specific Example

12. The Generalized search operator found in the new four dimensional orthonormal basis is, = = Where . Analysis is similar to the case of canonical Grover’s Algorithm. The Generalized Search Operator

13. Analysis for all to be positive or negative: We choose, Define, is perpendicular to So, does not induce any coupling between these states and . Then the search problem effectively reduces to the conventional Grover's search problem where we have one unique source states and multiple target sates and Summary: The results are mainly distributed in two main category: • The problem is reducible into two Grover's search problem in two different subspace, spanned by and. • The analysis of the problem is similar to a Grover's search problem which can be analyzed in conventional basis for all to be positive or negative.

14. Formulation: The new search operator is the selective phase inversions of three source states and three target states. The operator , where, and And Again we need to find out the new search operator in a 6 dimensional orthonormal basis. But, this problem is hard to analyze analytically, so we perform numerical calculations on some specific cases. Generalized Quantum Search Operator: Selective Inversion of Three States

15. We find orthogonal basis states , , using Gram-Schmidt orthogonalization. The initial states in the basis will be Analysis

16. Now, we construct the matrix in the new orthonormal basis and find the eigenvalues and eigenvectors. Result: So, for the three states also we can categorized our results in two different classes. Now, we have taken some specific examples of this two classes and analyze them numerically. Result

17. For the first example we have chosen . This case corresponds to the case of two different . For n=16, we got, Same for all the target states. Example - 1

18. For different n, we have, Plot: Formula: Example-1

19. For the second example we have chosen . This case corresponds to the case of three different . For n=16, Plot: For & , For , Example - 2

20. For different n, we got Formula: For For , Example – 2

21. The maximum amplitude of the target states after the optimal number of iteration is independent of the dimension of the search space. • Depending on the maximum amplitude, we need to iterate the generalized Grover’s operator suitable number of times to make the maximum amplitude nearly equal to 1. • For each case, the required number of iteration to get the target states with almost 100% probability is the same as the time taken by the canonical Grover's search operator. • So, the performance of our new generalized algorithm is the same as the conventional Grover’s search algorithm that performs selective inversions only on a unique source state and a unique target state. Summary

22. Thank You