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Grover’s Quantum Search Algorithm: Optimality and Applications

Learn about Grover’s quantum search algorithm, its optimality, and applications in quantum information processing. Understand how this algorithm provides quadratically more efficiency than classical algorithms. Explore the analysis and proof of the algorithm’s optimality.

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Grover’s Quantum Search Algorithm: Optimality and Applications

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  1. Introduction to Quantum Information ProcessingCS 467 / CS 667Phys 467 / Phys 767C&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. Contents • Grover’s quantum search algorithm • Optimality of Grover’s algorithm

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

  4. x1 x1 Uf xn xn y y  f(x1,...,xn) 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 2n queries are necessary to succeed—even with probability ¾ (say) Grover’s quantum algorithm that makes only O(2n) queries Query: [Grover ’96]

  5. PSPACE NP co-NP 3-CNF-SAT FACTORING P 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 The resulting quantum algorithms appear to be quadratically more efficient than the best classical algorithms known

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

  7. H X X X H X X X H x1 x1 x1 x1 H U0 Uf xn xn xn xn y y y  f(x1,...,xn) y  [x=0...0] Grover’s algorithm: description I Basic operations used: Uf x = (1) f(x)x Implementation? U0x = (1)[x=0...0]x Hadamard

  8. iteration 2 ... iteration 1 0 U0 Uf Uf U0 H H H H H 0  Grover’s algorithm: description II • construct state H0...0 • repeatktimes: • apply HU0HUfto state • 3. measure state, to get x{0,1}n, and check if f(x)=1 (The setting of k will be determined later)

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

  10. A B Grover’s algorithm: analysis II Algorithm: (HU0HUf)kH0...0 H0...0 Observation: Uf is a reflection about B: UfA =AandUfB =B Question: what is HU0H ? U0 is a reflection about H0...0 Partial proof: HU0HH0...0=HU00...0 =H(0...0) =H0...0

  11. A Since HU0HUf is a composition of two reflections, it is a rotation by 2, wheresin()=a/N  a/N More generally, it suffices to set k  (/4)N/a B Grover’s algorithm: analysis III Algorithm: (HU0HUf)kH0...0 2 2 2 H0...0 2  When a = 1, we want (2k+1)(1/N) /2 , so k  (/4)N Question: what if a is not known in advance?

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

  13. U0 f U1 f U2 f f U3 f Uk |x (1)f(x)|x Assume queries are of the form Optimality of Grover’s algorithm Theorem: any quantum search algorithm for f: {0,1}n {0,1}must make (2n) queries to f (if f is used as a black-box) Proof (of a slightly simplified version): and that a k-query algorithm is of the form |0...0 where U0, U1, U2, ..., Uk, are arbitrary unitary operations

  14. U0 U0 fr I U1 U1 I fr U2 U2 fr I U3 U3 fr I Uk Uk |0 |0 Optimality of Grover’s algorithm Define fr: {0,1}n {0,1} as fr (x) = 1 iff x = r Consider |ψr,k versus |ψr,0 We’ll show that, averaging over all r {0,1}n, |||ψr,k  |ψr,0|| 2k/2n

  15. |ψr,i U0 I U1 I U2 fr U3 fr Uk i ki |0 Note that |ψr,k  |ψr,0 =(|ψr,k  |ψr,k1)+(|ψr,k1  |ψr,k2)+ ... +(|ψr,1  |ψr,0) which implies |||ψr,k  |ψr,0||  |||ψr,k  |ψr,k1|| + ... + |||ψr,1  |ψr,0|| Optimality of Grover’s algorithm Consider

  16. query i query i+1 |ψr,i query i query i+1 U0 U0 I I U1 U1 I I U2 U2 I fr U3 U3 fr fr Uk Uk |ψr,i-1 |0 |0 Therefore, |||ψr,k  |ψr,0||  Optimality of Grover’s algorithm |||ψr,i  |ψr,i-1|| =|2i,r|,since query only negates|r

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

  18. THE END

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