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/courses/cs467 Lecture 15 (2005)

2. 2 Contents Grover’s quantum search algorithm Optimality of Grover’s algorithm

3. 3 Grover’s quantum search algorithm Optimality of Grover’s algorithm

4. 4 Quantum search problem Given: a black box computing f : {0,1} n  {0,1} Goal: determine if f is satisfiable (if  x  {0,1} n s.t. f(x) = 1 ) In positive instances, it makes sense to also find such a satisfying assignment x Classically, using probabilistic procedures, order 2 n queries are necessary to succeed—even with probability ¾ (say) x1x1 xnxn yy xnxn x1x1  y  f ( x 1,...,x n )  UfUf Grover’s quantum algorithm that makes only O(  2 n ) queries Query: [Grover ’96]

5. 5 Applications of quantum search The function f could be realized as a 3-CNF formula: Alternatively, the search could be for a certificate for any problem in NP 3-CNF-SAT FACTORING P NP PSPACE co-NP The resulting quantum algorithms appear to be quadratically more efficient than the best classical algorithms known

6. 6 Prelude to Grover’s algorithm: two reflections = a rotation  11 22 11 22 Consider two lines with intersection angle  : reflection 1 reflection 2 Net effect: rotation by angle 2 , regardless of starting vector

7. 7 Grover’s algorithm: description I x1x1 xnxn yy xnxn x1x1  y  [ x = 0...0 ]  U0U0 x1x1 xnxn yy xnxn x1x1  y  f ( x 1,...,x n )  UfUf U f  x  = (  1) f(x)  x  H H H H X X X X X X Hadamard Basic operations used: Implementation? U 0  x  = (  1) [ x = 0...0]  x 

8. 8 Grover’s algorithm: description II 1.construct state H  0...0  2.repeat k times: apply  H U 0 H U f to state 3. measure state, to get x  {0,1} n, and check if f (x) =1 00 00   UfUf HHU0U0 HUfUf HU0U0 H iteration 1 iteration 2... (The setting of k will be determined later)

9. 9 Grover’s algorithm: analysis I and N = 2 n and a = | A | and b = | B | Let A = { x  {0,1} n : f (x) = 1 } and B = { x  {0,1} n : f (x) = 0 } Letand BB AA H  0...0  Consider the space spanned by  A  and  B  Interesting case: a << N  goal is to get close to this state

10. 10 Grover’s algorithm: analysis II Algorithm: (  H U 0 H U f ) k H  0...0  BB AA H  0...0  Observation: U f is a reflection about  B  : U f  A  =   A  and U f  B  =  B  Question: what is  H U 0 H ? Partial proof:  H U 0 H H  0...0  =  H U 0  0...0  =  H (   0...0  ) = H  0...0  U 0 is a reflection about H  0...0 

11. 11 Grover’s algorithm: analysis III Algorithm: (  H U 0 H U f ) k H  0...0  BB AA H  0...0   22 22 22 22 Since  H U 0 H U f is a composition of two reflections, it is a rotation by 2 , where sin (  )=  a/N   a/N When a = 1, we want (2k+1)(1/  N)   / 2, so k  (  / 4)  N More generally, it suffices to set k  (  / 4)  N/a Question: what if a is not known in advance?

12. 12 Grover’s quantum search algorithm Optimality of Grover’s algorithm

13. 13 Optimality of Grover’s algorithm Proof (of a slightly simplified version): f U0U0 U1U1 U2U2 U3U3 UkUk fff and that a k -query algorithm is of the form where U 0, U 1, U 2,..., U k, are arbitrary unitary operations Theorem: any quantum search algorithm for f : {0,1} n  {0,1} must make  (  2 n ) queries to f (if f is used as a black-box) f |x|x (  1) f(x) | x  Assume queries are of the form | 0...0 

14. 14 Optimality of Grover’s algorithm Define f r : {0,1} n  {0,1} as f r ( x ) = 1 iff x = r frfr U0U0 U1U1 U2U2 U3U3 UkUk frfr frfr frfr |0|0 | ψ r,k  Consider versus I U0U0 U1U1 U2U2 U3U3 UkUk III |0|0 | ψ r, 0  We’ll show that, averaging over all r  {0,1} n, || | ψ r,k   | ψ r, 0  ||  2 k /  2 n

15. 15 Optimality of Grover’s algorithm Consider I U0U0 U1U1 U2U2 U3U3 UkUk Ifrfr frfr |0|0 | ψ r,i  k  ik  i i | ψ r,k   | ψ r, 0  = ( | ψ r,k   | ψ r,k  1  ) + ( | ψ r,k  1   | ψ r,k  2  ) +... + ( | ψ r,1   | ψ r,0  ) Note that || | ψ r,k   | ψ r, 0  ||  || | ψ r,k   | ψ r,k  1  || +... + || | ψ r,1   | ψ r,0  || which implies

16. 16 Optimality of Grover’s algorithm I U0U0 U1U1 U2U2 U3U3 UkUk Ifrfr frfr |0|0 | ψ r,i  query i query i +1 || | ψ r,i   | ψ r,i -1  || = | 2  i,r |, since query only negates | r  I U0U0 U1U1 U2U2 U3U3 UkUk IIfrfr |0|0 query i query i +1 | ψ r,i-1  Therefore, || | ψ r,k   | ψ r, 0  || 

17. 17 Optimality of Grover’s algorithm Therefore, for some r  {0,1} n, the number of queries k must be  (  2 n ), in order to distinguish f r from the all-zero function Now, averaging over all r  {0,1} n, (By Cauchy-Schwarz) This completes the proof

18. 18