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Grover. Part 2

Grover. Part 2. 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

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Page 1: Grover. Part 2. 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

Grover. Part 2

Page 2: Grover. Part 2. 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

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

Page 3: Grover. Part 2. 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

Inputs oracle

We need to initialize in a superposed state

This is action of quantum oracle

Page 4: Grover. Part 2. 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

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

Page 5: Grover. Part 2. 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

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.

Page 6: Grover. Part 2. 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

Vector of Hadamards

Page 7: Grover. Part 2. 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

Notation Reminder

a Control with value a=1

a Control with value a=0

a

equivalent

Page 8: Grover. Part 2. 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

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

Zero State Phase Shift Circuit

Page 9: Grover. Part 2. 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

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

Page 10: Grover. Part 2. 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

2 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

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

+ =

With accuracy to phase

Page 11: Grover. Part 2. 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

In each G

This is a global view of Grover. Repeatitions of G

Here you have all components of Grover’s loop

Page 12: Grover. Part 2. 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

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

Page 13: Grover. Part 2. 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

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

Page 14: Grover. Part 2. 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

Here we prove that |> < | used inside HZH calculates the mean

a Vector of mean values

Will be explained in next slide

Explanation of the first part of Grover iterate formula

Page 15: Grover. Part 2. 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

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?

Page 16: Grover. Part 2. 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

For every bit

Amplitudes of bits after Hadamard

Positive or negative amplitudes in other explanations

All possible states

Page 17: Grover. Part 2. 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

Amplitudes of bits after one stage of G

This value based on previous slide

Page 18: Grover. Part 2. 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

This slides explains the basic mechanism of the Grover-like algorithms

Page 19: Grover. Part 2. 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

You can verify it also in simulation

Additional Additional ExerciseExercise

This is a lot calculations, requires matrix multiplication

Page 20: Grover. Part 2. 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

Here we calculate analytically when to stop

The equations taken from the previous slides “Grover Iterate”

For marked state

For unmarked state

Page 21: Grover. Part 2. 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

We found k from these equations

recursionWe want to find how many times to iterate

Page 22: Grover. Part 2. 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
Page 23: Grover. Part 2. 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

But you can do better if you have knowledge, for instance the upper bound of chromatic number in graph coloring

Page 24: Grover. Part 2. 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

Grover search example.

• Here is an example of Grover search for n = 3 qubits, where N = 2n =8. – We omit reference to qubit n+1, which is in

state 1 /√2 (|0>−|1>i) and does not change.

• The dimension of the unitary operators for this example is thus 2n = 8 also.)

Page 25: Grover. Part 2. 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

• (Remember that numbering starts with 0 and ends with 7, so that the -1 here is in the slot for |5>.)

• This matrix reverses the sign on state |5>, and leaves the other states unchanged.

•Suppose the unknown number is |a> = |5>. •The matrix or black box oracle Ufa is

oracle

Page 26: Grover. Part 2. 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

•The Walsh matrix W8 is

Now we use normalization

Page 27: Grover. Part 2. 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

The matrix −Uf0 is

Page 28: Grover. Part 2. 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

This matrix changes the sign on all states except |0>.

Finally, we have the repeated step RsRa in the

Grover algorithm:

oracle

shift

hadamards

Page 29: Grover. Part 2. 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
Page 30: Grover. Part 2. 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
Page 31: Grover. Part 2. 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

After second rotation we get

Page 32: Grover. Part 2. 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

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

Page 33: Grover. Part 2. 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

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.