Grover. Part 2

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Grover. Part 2
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Components of Grover Loop

• The Oracle -- O
• The Hadamard Transforms -- H
• The Zero State Phase Shift -- Z

O is an Oracle
H is Hadamards
H is Hadamards
Z is Zero State Phase Shift
Grover Iterate
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Inputs oracle
This is action of quantum oracle
We need to initialize in a superposed state
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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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Notation Reminder
a
Control with value a1
a
a
Control with value a0
Control with value a0
equivalent
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Zero State Phase Shift Circuit
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
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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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2 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
With accuracy to phase
1 0 0 0
0 -1 0 0
0 0 -1 0
0 0 0 -1
-1 0 0 0
0 -1 0 0
0 0 -1 0
0 0 0 -1


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Here you have all components of Grovers loop
This is a global view of Grover. Repeatitions of G
In each G
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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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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
proof
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Will be explained in next slide
Explanation of the first part of Grover iterate
formula
Here we prove that ?gt lt ? used inside HZH
calculates the mean
a
Vector of mean values
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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.
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Positive or negative amplitudes in other
explanations
Amplitudes of bits after Hadamard
For every bit
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 the basic mechanism of the
Grover-like algorithms
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Additional Exercise
You can verify it also in simulation
This is a lot calculations, requires matrix
multiplication
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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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recursion
We want to find how many times to iterate
We found k from these equations
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But you can do better if you have knowledge, for
instance the upper bound of chromatic number in
graph coloring
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Grover search example.

• Here is an example of Grover search for n 3
  qubits, where N 2n 8.
• We omit reference to qubit n1, which is in state
  1 /v2 (0gt-1gti) and does not change.
• The dimension of the unitary operators for this
  example is thus 2n 8 also.)

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oracle

• (Remember that numbering starts with 0 and ends
  with 7, so that the -1 here is in the slot for
  5gt.)
• This matrix reverses the sign on state 5gt, and
  leaves the other states unchanged.

• Suppose the unknown number is agt 5gt.
• The matrix or black box oracle Ufa is

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• The Walsh matrix W8 is

Now we use normalization
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The matrix -Uf0 is
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This matrix changes the sign on all states except
0gt. Finally, we have the repeated step RsRa in
the Grover algorithm
oracle
hadamards
shift
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After second rotation we get
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Summary and our work
When you know anything about the problem
(symmetry, observation, bounds, function within
some classification class) you can design a
better Grover like algorithm but for your data
only. This is enough in real life like CAD or
Image Processing, since data are always specific,
not the worst case data as in Mathematic proofs
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Problem for students

• Build the Grover algorithm for ternary quantum
  logic.
• First you need to generalize Hadamard transform
  to Chrestenson transform.
• Next you need to have some kind of ternary
  reversible gates to build oracle.
• The same gates will be used for Zero State Phase
  Shift circuit.