Jacob%20D.%20Biamonte - PowerPoint PPT Presentation

Title: Jacob%20D.%20Biamonte


1
Grovers Search
Jake Biamonte
Should be one in four search

• Jacob D. Biamonte
• biamonte_at_ieee.org
• Portland Quantum Logic Group

2
Karnaugh Maps for quantum operations
H
X
a0 b0
X
H
3
Karnaugh Maps for quantum operations
H
X
a b
X
H
folding
1 1
1 0
1 0 0 0
1 1 1 1
1 1 1 1
0 0
1 1 1 1
1 1 0 0
1 1 1 1
0 0
1 1 1 1
10 1 0
10 1 0
1 1 1 1
1 1 0 0
01 0 1
01 0 1
1 1 1 -1

• 0
• 0 1

10 -10
10 -10
01 0-1
01 0-1
Hadamard
wire
4
Karnaugh Maps for quantum operations
H
X
a b
X
H
folding
1 1
1 0
1 0 0 0
0 0 0 1
1 1 1 1
0 0
0 0
0 0
1 1 1 1
1 1 0 0
0 0 1 0
0 0
1 0
1 1 1 1
0 1
00 1 0
00 1 0
1 0 0 0
0 0 1 0
00 0 1
00 0 1
0 1 1 0

• 0
• 0 1

01 0 0

0 0 1 0
0 0 0 1
10 00
10 0 0
10 0 0

01 00
01 0 0
00 0 1
00 1 0
NOT
wire
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Karnaugh Maps for quantum operations
H
X
a b
X
H
mirroring
folding
1 1
1 0
1 0 0 0
0 0 0 1
0 0 1 0
0 0
0 0
0 0
1 1 1 1
1 1 0 0
0 0 1 0
0 0
1 0
0 0
1 1 1 1

0 1
1 0
10 0 0
0 0 0 1
0 0 1 0
01 0 0

00 0 1
00 1 0
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Why is this important?

• This presentation shows clearly how to perform a
  so called 1 in 4 search
• We start out with the basics

1 in 4 search
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Pick your needle and I will find you a haystack
The point of this slide is to show examples of 4
different oracles. Grovers search can tell
between these oracles in a single iteration,
classically we would need 3 iterations.
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Properties of the oracle
Let f 0,12 ? 0,1 have the property that
there is exactly one x ? 0,12 for which f (x)
1
Goal find x ? 0,12 for which f (x) 1
Classically 3 queries are necessary
Quantumly ?
Only after 3 tests can we determine with
certainty that the oracles is 1 for only a single
input value x
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A 1-4 search can chose between 4 oracles in one
iteration
Black box for 1-4 search
Start by creating phases in superposition of all
inputs to f
Input state to query (?00? ?01? ?10?
?11?)(?0? ?1?)
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Time
?0?
X
f
H
H
X
M
H
?0?
X
X
M
H
H
H
H
H
?1?
M
H
H
This slide illustrates how the state of the
system is changed as it propagates through the
quantum network implementation of Grovers Search
algorithm.
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Time
?0?
X
f
H
H
X
M
H
?0?
X
X
M
H
H
H
H
H
?1?
M
H
H
ab c 0 1
ab c 0 1
ab c 0 1
ab c 0 1
ab c 0 1
ab c 0 1
0.3 0,3
00 01 11 10
0.3 0,3
00 01 11 10
00 01 11 10
00 01 11 10
00 01 11 10
00 01 11 10
0.3 0,3
- 0.3 0,3
0 0
0 0
0.3 0,3
0.3 0,3
0.3 0,3
0.3 0,3
- 0.5 0,5
- 0.5 0,5
0.3 0,3
- 0.3 0,3
- 0.3 0,3
0.3 - 0,3
0 0
0.5 - 0.5
0.3 0,3
0.3 0,3
0.3 0,3
0.3 0,3
0.5 0,5
0 0
12
Time
?0?
X
f
H
H
X
M
H
?0?
X
X
M
H
H
H
H
H
?1?
M
H
H
Hadamard of affine function
Ibverters flip between 00 and 11
Inverter flips second bit when first is 1
Hadamard addis in 00 and 11
ab c 0 1
ab c 0 1
ab c 0 1
Ibverters flip between 00 and 11
ab c 0 1
ab c 0 1
ab c 0 1
ab c 0 1
00 01 11 10
00 01 11 10
00 01 11 10
-0.3 0.3
- 0.3 0.3
00 01 11 10
00 01 11 10
00 01 11 10
00 01 11 10
0.3 0,3
- 0.3 0,3
0 0
0 0
0.3 -0.3
0.3 - 0.3
0.3 0,3
0.3 0,3
- 0.5 0,5
- 0.5 0,5
-1
-0.3 0.3
- 0.3 0.3
- 0.3 0,3
0.3 - 0,3
0 0
0.5 - 0.5
0.3 -0.3
0.3 - 0.3
0.3 0,3
0.3 0,3
0.5 0,5
0 0
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Time
?0?
X
f
H
H
X
M
H
?0?
X
X
M
H
H
H
H
H
?1?
M
H
H
??00? ?00? ?01? ?10? ?11?
??01? ?00? ?01? ?10? ?11?
??10? ?00? ?01? ?10? ?11?
??11? ?00? ?01? ?10? ?11?
After Hadamard the solution is known in Hilbert
space by having value -1. But it is hidded from us
The state corresponding to the input to the
oracle that has a output result of 1 is tagged
with a negative 1.
14
QuIDDPro Script --- Density states

• grover4.qpbiamonte_at_ieee.orgstate
  cb("001")cb("001")'state hadamard(3)stateha
  damard(3)'echo("define a needle in a
  haystack") Oracle 1 xi (1,1)
  ------    ------   
  --()--oracle
  cu_gate(sigma_x(1), "c1c2x3", 3)state
  oraclestateoracle'echo("apply CPS")state
  kron(hadamard(2), identity(1))statekron(hadamard
  (2), identity(1))'state kron(sigma_x(2),
  identity(1))statekron(sigma_x(2),
  identity(1))'state cu_gate(hadamard(1), "x2",
  3)statecu_gate(hadamard(1), "x2", 3)'state
  cu_gate(sigma_x(1), "c1x2", 3)statecu_gate(sigma
  _x(1), "c1x2", 3)'state cu_gate(hadamard(1),
  "x2", 3)statecu_gate(hadamard(1), "x2",
  3)'state kron(sigma_x(2), identity(1))statekr
  on(sigma_x(2), identity(1))'state
  hadamard(3)statehadamard(3

15
QuIDDPro Script --- State Vector

• state cb("001")state hadamard(3)stateecho
  ("define a needle in a haystack")
  Oracle 1 xi (1,1) ------   
  ------    --()--oracle
  cu_gate(sigma_x(1), "c1c2x3", 3)state
  oraclestateecho("apply CPS")state
  kron(hadamard(2), identity(1))statestate
  kron(sigma_x(2), identity(1))statestate
  cu_gate(hadamard(1), "x2", 3)statestate
  cu_gate(sigma_x(1), "c1x2", 3)statestate
  cu_gate(hadamard(1), "x2", 3)statestate
  kron(sigma_x(2), identity(1))statestate
  hadamard(3)state

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We found Grover!