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Grover. Part 2

Grover. Part 2. Anuj Dawar. Components of Grover Loop. The Oracle -- O The Hadamard Transforms -- H The Zero State Phase Shift -- Z. Inputs oracle. This is action of quantum oracle. We need to initialize in a superposed state. This is a typical way how oracle operates.

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Grover. Part 2

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  1. Grover. Part 2 Anuj Dawar

  2. Components of Grover Loop • The Oracle -- O • The Hadamard Transforms -- H • The Zero State Phase Shift -- Z

  3. Inputs oracle This is action of quantum oracle We need to initialize in a superposed state

  4. This is a typical way how oracle operates This is a typical way how oracle operation is described Encodes input combination with changed sign in a superposition of all

  5. Role of Oracle • We want to encode input combination with changed sign in a superposition of all states. • This is done by Oracle together with Hadamards. • We need a circuit to distinguish somehow globally good and bad states.

  6. Vector of Hadamards

  7. All information of oracle is in the phase but how to read it? Flips the data phase This is value of oracle bit This is just an example of a single minterm, but can be any function

  8. Flips the oracle bit when all bits are zero Rewriting matrix Z to Dirac notation, you can change phase globally This is state of all zeros

  9. Here you have all components of Grover’s loop This is a global view of Grover. Repeatitions of G In each G

  10. Generality • Observe that a problem is described only by Oracle. • So by changing the Oracle you can have your own quantum algorithm. • You can still improve the Grover loop for particular special cases

  11. Grover iterate has two tasks: (1) invert the solution states and (2) invert all states about the mean Here we explain in detail what happens inside G. This can be generalized to G-like circuits proof

  12. Here we prove that |> <  |used inside HZH calculates the mean a

  13. From previous slide ( ) ( ) What does it mean invert all states about the mean? This proof is easy and it only uses formalisms that we already know.

  14. Positive or negative amplitudes in other explanations Amplitudes of bits after Hadamard For every bit All possible states

  15. Amplitudes of bits after one stage of G This value based on previous slide

  16. This slides explains mechanism of Grover-like algorithms

  17. Additional Exercise You can verify it also in simulation

  18. Here we calculate analytically when to stop The equations taken from the previous slides “Grover Iterate” For marked state For unmarked state

  19. recursion We want to find how many times to iterate We found k from these equations

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