lecturenotes12 - CSE 599d - Quantum Computing Grover’s...

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Unformatted text preview: CSE 599d - Quantum Computing Grover’s Algorithm Dave Bacon Department of Computer Science & Engineering, University of Washington After Peter Shor demonstrated that quantum computers could efficiently factor, great interest arose in finding other problems for which quantum algorithms could outperform the best known classical algorithms. One of the early offshoots of this work was an algorithm invented by Lov Grover in 1996. Here we will describe Grover’s algorithm and show that it is, in a query complexity manner, the optimal quantum algorithm. I. GROVER’S ALGORITHM Suppose that we have a function f ( x ) from { , 1 } n to { , 1 } which is zero on all inputs except for a single (marked) item x : f ( x ) = δ x,x . By querying this function you wish to find the marked item x . This is like finding a needle in a haystack. Certainly if you have no information about the particular x , then finding this marked item is very difficult. In the worst case it will take 2 n- 1 queries to find x for a deterministic algorithm. Well what if we turn this haystack into a quantum haystack? This is the question that Grover asked. In the quantum version of Grover’s problem we have access to a function which computes f ( x ) in our standard reversible manner: U f = X x,y ∈{ , 1 } n | x ih x | ⊗ | y ⊕ f ( x ) ih y | . (1) Using our standard trick of phase kickback, we can feed |-i = 1 √ 2 ( | i - | 1 i ) into the y register of this unitary. If we do this, then the effect on the first register is still unitary and is given by V f = X x ∈{ , 1 } n (- 1) f ( x ) | x ih x | = X x ∈{ , 1 } n (- 1) δ x,x | x ih x | (2) We will therefore assume that we have query access to V f . Consider the two vectors | x i and | ψ i = 1 √ 2 n ∑ x ∈{ , 1 } n | x i . These vectors are not orthogonal, h x | ψ i = 1 √ 2 n , however they are linearly independent. Thus we can for a two dimensional basis for the subspace of vectors which are linear superpositions of these two basis elements. One such choice of basis consists of | x i and | ψ i = 1 √ 2 n- 1 X x ∈{ , 1 } n ,x 6 = x | x i (3) We note that | ψ i = r 2 n- 1 2 n | ψ i + 1 √ 2 n | x i . (4) Now notice that V f preserve the subspace | x i , | ψ i : V f | x i =-| x i V f | ψ i = | ψ i . (5) Now consider the unitary W = 2 | ψ ih ψ | - I . This is unitary? Yes because (2 | ψ ih ψ | - I )(2 | ψ ih ψ | - I ) † = 4 | ψ ih ψ | - 2 | ψ ih ψ | - 2 | ψ ih ψ | + I = I . W also preserves the subspace spanned by | x i and | ψ i : W | x i = (2 | ψ ih ψ | - I ) | x i = 2 √ 2 n | ψ i - | x i = 2 p (2 n- 1) 2 n | ψ i + 2 2 n- 1 | x i W | ψ i = (2 | ψ ih ψ | - I ) | ψ i = 2 r 2 n- 1 2 n | ψ i - | ψ i = 2(2 n- 1) 2 n- 1 | ψ i + 2 √ 2 n- 1 2 n | x i =- 2 2 n- 1 | ψ i + 2 √ 2 n- 1 2 n | x i (6) 2 We can rewrite this as a rotation about an angle θ : W | x i =- cos θ | x i + sin θ | ψ i W | ψ i = sin θ | x i + cos θ | ψ i (7) where sin θ = 2 √ 2 n- 1 2 n (8) If we combine W...
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This note was uploaded on 11/06/2011 for the course CSE 599 taught by Professor Staff during the Fall '08 term at University of Washington.

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lecturenotes12 - CSE 599d - Quantum Computing Grover’s...

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