1. Quantum Search Algorithms for Multiple Solution Problems EECS 598 Class Presentation Manoj Rajagopalan

2. Outline 1.Recap of Grover’s algorithm for the unique solution case 2.Grover’s algorithm for multiple solutions: multiplicity known 3.Quantum search algorithm for multiple solutions: multiplicity unknown 4.Quantum counting to determine multiplicity

3. References 1.Quantum Computing and Quantum Information textbook 2.“A fast quantum mechanical algorithm for database search”, LK Grover, 1996 3.“Tight bounds on quantum searching”, M Boyer, G Brassard, P Hoyer, 1996 4.“Quantum counting”, G Brassard, P Hoyer, A Tapp, 1998

4. 1.n = # qubits in the system 2.N = # of possible values of n qubits = 2 n 3.M = multiplicity of solution 4.k = probability amplitude of system in solution state 5.l = probability amplitude of system in non-solution state 6.A = set of indices that denote solutions (good states) 7.B = set of indices denoting bad states 8.  = rotation angle corresponding to Grover operator Notation

5. Given F:{0,1} n  {0,1}, find i 0  F(i 0 )=1 and  i  i 0 F(i)=0 1.Set up initial state |0   n 2.Apply the Hadamard transform H  n |0   |  = Let i 0 be the solution:|  = k |i 0  + 3.Grover operator made of 4 steps Apply the oracle Apply H  n Conditional phase shift: Apply H  n Grover’s Algorithm for Unique Solution Case

6. Unique Solution Case Recap (…contd) 4.Apply the Grover operator. After j iterations, Need bound on the number of iterations

7. Unique Solution Case Recap (…contd) Let sin 2  = 0 <   For k m = 1, (2m+1)  =  /2 => For large N,   sin  =  m 

8. Multiple Solutions: Multiplicity known Given F:{0,1} n  {0,1}, find all i  {0,1} n  F(i)=1 M = number of solutions > 1 Define ‘good states’ A = {i | F(i) = 1}|A| = M ‘bad states’ B = {j | F(j) = 0 }|B| = N - M Suffices to tackle good and bad states as groups k = probability amplitude of each solution (element of set A) l = probability amplitude of each element of set B Mk 2 + (N-M)l 2 = 1

9. Multiple Solutions: Multiplicity known Grover’s algorithm for the multiple solution case Structurally the same as that in the case of unique solution 1.Set up initial state |0   n 2.Apply the Hadamard transform 3.Apply Grover operator repeatedly Apply the oracle Apply H  n Conditional phase shift Apply H  n Differs in the oracle implementation: Oracle lends a relative phase shift of –1 to all solutions

10. Multiple Solutions: Multiplicity known Define After j iterations:

11. Multiple Solutions: Multiplicity known Let m = upper bound on number of iterations We want l m = 0 cos ((2m+1)  ) = =>  | cos(2m+1)  |  | sin  |  Probability of failure after exactly m iterations (N-M) l m 2 = cos 2 ((2m+1)  )  sin 2  = Negligible for M << N

12. Multiple Solutions: Multiplicity known For M << N,   sin  Knowing M, we can predetermine the upper bound on the number of iterations, m. Unique solution problem is a special case of this for M=1.

13. Multiple Solutions: unknown Multiplicity Number of iterations required to obtain a solution with significant confidence depends on the solution’s multiplicity. If M is not known, then there is no way of telling how many iterations will suffice. Take m = to be on the safe side? (max # iterations) No! Probability of success minuscule when M = 4a 2 where a is a small integer.

14. Multiple Solutions: unknown Multiplicity Modified procedure for unknown M: 1.Initialize m = 1 and = 8/7 (actually 1 < < 4/3) 2.Choose integer j such that 0  j  m 3.Apply j iterations of Grover’s algorithm 4.Measure and let outcome be i 5.If F(i) == 1 then solution found: exit program 6.Else m = min( m, ): goto step #2 Theorem: This algorithm finds a solution in O( )

15. Multiple Solutions: unknown Multiplicity For M > 3N/4 constant expected time by classical sampling For 0 < M  3N/4, runtime = O( ) For M << N, runtime < 6 times runtime_if_M_were_known Knowing the number of solutions helps in reducing runtime. This motivates quantum counting

16. Quantum Counting Aim: To determine the number of solutions M to an N item unstructured search problem Classical computing consults the oracle  (N) times to determine M Quantum computing can combine Grover’s algorithm and phase estimation to determine M much faster! Why count? Fast estimation of M => rapid solution detection Is there a solution at all? NP-Complete problems

17. Quantum Counting Recall: The computational bases can be partitioned into two subsets, the ‘good states’ set A containing all the solutions, and Letting we get in the basis.

18. Quantum Counting Eigenvalues of G are e i2  and e i(2  -2  ) The value of  can be determined by phase estimation From , the value of M can be calculated PHASE ESTIMATION Given a unitary operator U and one of its eigenvectors, the phase  of its corresponding eigenvalue e i2  is determined

19. Quantum Counting Complexity of phase estimation algorithms Probability of success Error in M Absolute, max Runtime, P Evaluations of F P  4