1. 1 Introduction to Quantum Information Processing CS 467 / CS 667 Phys 467 / Phys 767 C&O 481 / C&O 681 Richard Cleve DC 3524 cleve@cs.uwaterloo.ca Course web site at: http://www.cs.uwaterloo.ca/~cleve Lecture 6 (2005)

2. 2 Contents Phase estimation problem Algorithm for the phase estimation problem Order-finding via phase estimation

3. 3 Phase estimation problem Algorithm for the phase estimation problem Order-finding via phase estimation

4. 4 Generalized controlled- U gates U aa bb aa U abU ab a1a1 U : amam a1a1 : amam b1b1 : bnbn U a 1  a m  b  Example:  1101  0101    1101  U 1101  0101 

5. 5 Phase estimation problem U is a unitary operation on n qubits    is an eigenvector of U, with eigenvalue e 2  i  ( 0 ≤  < 1 ) Output:  ( m -bit approximation) Input: black-box for U n qubits m qubits and a copy of   

6. 6 Phase estimation problem Algorithm for the phase estimation problem Order-finding via phase estimation

7. 7 Algorithm for phase estimation (I) U  H H H 00 00 00 Starts off as:  (  0  +  1  ) (  0  +  1  ) … (  0  +  1  )    = (  000  +  001  +  010  +  011  + … +  111  )    = (  0  +  1  +  2  +  3  + … +  2 m  1  )     00 … 0     (0 + e2i1 + (e2i)22 + (e2i)33 + … + (e2i)2m12m1)  (0 + e2i1 + (e2i)22 + (e2i)33 + … + (e2i)2m12m1)   a  b    a  U a  b  

8. 8 Quantum Fourier transform where  = e 2  i / N (for n qubits, N = 2 n )

9. 9 Computing the QFT (I) H 4 84 8164 321684 H H H H H m Quantum circuit for F 32 : Gates: For F 2 n costs O(n 2 ) gates reverse order

10. 10 Computing the QFT (II) One way on seeing why this circuit works is to first note that F 2 n  a 1 a 2 …a n  = (  0  + e 2  i  (0. a n )  1  )…(  0  + e 2  i  (0. a 2 …a n )  1  ) (  0  + e 2  i  (0. a 1 a 2 …a n )  1  ) It can then be checked that the circuit produces these states (with qubits in reverse order) for all computational basis states  a 1 a 2 …a n  Exercise: (a) prove the above equation from the definition of the QFT; (b) confirm that the circuit produces these states

11. 11 Algorithm for phase estimation (II) Note: this is exactly the state generated by our preliminary algorithm for phase estimation when  = 0. a 1 a 2 …a m Therefore, if  = 0. a 1 a 2 …a m then applying F † to this state yields  a 1 a 2 …a m  (from which  can be deduced exactly) What  if is not of this nice form? Example:  = ⅓ = 0.0101010101010101…

12. 12 Phase estimation problem Algorithm for the phase estimation problem Order-finding via phase estimation

13. 13 Order-finding algorithm (I) Let M be an m -bit integer Def: Z M * = {x  {1,2,…, M  1} : gcd( x,M ) = 1 } Z M * is a group under multiplication modulo M Example: Z 21 * = { 1,2,4,5,8,10,11,13,16,17,19,20 } For a  Z M *, consider all powers of a : a 0, a 1, a 2, a 3, … Ex: For a = 10, the sequence is: 1, 10, 16, 13, 4, 19, 1, 10, 16, … Def: ord M ( a ) is the minimum r > 0 such that a r = 1 ( mod M ) Ex: ord 21 (10) = 6 Order-finding problem: given a and M, find ord M ( a )

14. 14 Order-finding algorithm (II) Define: U (an operation on m qubits) as: U  y  =  a y mod M  Define: Then

15. 15 Order-finding algorithm (III) U n qubits 2n qubits corresponds to the mapping:  x  y    x  a x y mod M  Moreover, this mapping can be implemented with roughly O( n 2 ) gates The phase estimation algorithm yields a 2 n - bit estimate of 1/ r From this, a good estimate of r can be calculated … Problem: how do we construct state  1  to begin with?

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