grover. part 2 anuj dawar. components of grover loop the oracle -- o the hadamard transforms -- h...
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Grover. Part 2
Anuj Dawar
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Components of Grover Loop
• The Oracle -- O
• The Hadamard Transforms -- H
• The Zero State Phase Shift -- Z
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Inputs oracle
We need to initialize in a superposed state
This is action of quantum oracle
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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
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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.
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Vector of Hadamards
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This is value of oracle bit
Flips the data phase
All information of oracle is in the phase but how to read it?
This is just an example of a single minterm, but can be any function
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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
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In each G
This is a global view of Grover. Repeatitions of G
Here you have all components of Grover’s loop
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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
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proof
Here we explain in detail what
happens inside G. This can be
generalized to G-like circuits
Grover iterate has two tasks: (1) invert the solution states and (2) invert all states about the mean
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Here we prove that |> < | used inside HZH calculates the mean
a
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This proof is easy and it only uses formalisms that we already know.
(( ))
(( ))
From previous slide
What does it mean invert all states about the mean?
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For every bit
Amplitudes of bits after Hadamard
Positive or negative amplitudes in other explanations
All possible states
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Amplitudes of bits after one stage of G
This value based on previous slide
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This slides explains mechanism of Grover-like algorithms
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You can verify it also in simulation
Additional Additional ExerciseExercise
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Here we calculate analytically when to stop
The equations taken from the previous slides “Grover Iterate”
For marked state
For unmarked state
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We found k from these equations
recursionWe want to find how many times to iterate
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