Quantum Computation Sandy Irani Department of Computer Science University of California Irvine Simulating Physics with Computers Richard Feynman Keynote Talk 1 st Conference on ID: 358220
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Slide1
An Introduction toQuantum Computation
Sandy IraniDepartment of Computer ScienceUniversity of California, IrvineSlide2
“Simulating Physics with Computers”Richard Feynman –
Keynote Talk, 1st Conference on Physics and Computation, MIT, 1981
Slide3
“Simulating Physics with Computers”Richard Feynman –
Keynote Talk, 1st Conference on Physics and Computation, MIT, 1981
Is it possible to build computers that use the laws of quantum mechanics to compute?Slide4
Quantum Superposition
√2
1
+
√
2
1Slide5
Quantum Superposition
√2
1
-
√
2
1
-Slide6
Quantum SuperpositionSlide7
Implementations of a “Qubit”
Energy level of an atomSpin orientation of an electronPolarization of a photon.
NMR, Ion traps,…Slide8
Information: 1 Bit Example(Schrodinger’s Cat)
Classical Information:A bit is in state 0 or state 1
Classical Information with Uncertainty
Bit is 0 with probability p
0
Bit is 1 with probability p
1
State (p
0
, p
1
)
Quantum Information
State is a
superposition
over states 0 and 1
State is (
a
0
,
a
1
) where
a
0
,
a
1
are
complex
.
X=0
OR
X=1Slide9
Information: 1 Bit Example(Schrodinger’s Cat)
Classical Information:A bit is in state 0 or state 1
Classical Information with Uncertainty
Bit is 0 with probability p
0
Bit is 1 with probability p
1
State
(p
0
, p
1
)
Quantum Information
State is a
superposition
over states 0 and 1
State is (
a
0
,
a
1) where
a0
,
a
1
are
complex
.
X=1
with
prob
p
1
X=0
OR
X=0
with
prob
p
0
X=1Slide10
Information: 1 Bit Example(Schrodinger’s Cat)
Classical Information:A bit is in state 0 or state 1
Classical Information with Uncertainty
State
(p
0
, p
1
)
Quantum Information
State is partly 0 and partly 1
State is
(
a
0
,
a
1
)
where
a
0
, a
1 are complex
.
X=1
with
prob
p
1
X=0
OR
X=0
with
prob
p
0
a
1
+ a
0
X=1Slide11
Information: n Bit Example
Classical Information:State of n bits specified by a string x in {0,1}n
Classical Information with Uncertainty
State described by probability distribution over 2
n
possibilities
(p
0
, p
1
, …., p
2
n
-1
)
Quantum Information
State is a superposition over 2
n
possibilities
(
a
0
,
a1,…,
a2n
-1
), where a is complex
X = 011Slide12
Information: n Bit Example
Classical Information:State of n bits specified by a string x in {0,1}n
Classical Information with Uncertainty
State described by probability distribution over 2
n
possibilities
(p
0
, p
1
, …., p
2
n
-1
)
Quantum Information
State is a superposition over 2
n
possibilities
(
a
0
,
a1,…,
a2n
-1
), where a is complex
p
000
p
001
p
010
p
011
p
100
p
101
p
110
p
111Slide13
Information: n Bit Example
Classical Information:State of n bits specified by a string x in {0,1}n
Classical Information with Uncertainty
State described by probability distribution over 2
n
possibilities
(p
0
, p
1
, …., p
2
n
-1
)
Quantum Information
State is a superposition over 2
n
possibilities
(
a
0
,
a1,…,
a2n
-1
), where a is complex
a
000
+ a
001
+ a
010
+ a
011
+ a
100
+ a
101
+ a
110
+ a
111Slide14
A quantum kilobyte of data
(8192 qubits)Encodes 28192
complex numbers
2
8192
~ 10
2466
(Number of atoms in the universe ~ 10
82)Slide15
State of n
qubits (a0,…, a2
n
-1
)
stores 2
n
complex numbers:
Rich in information
How to use it?
How to access it?Slide16
Quantum Measurement
State of n qubits (a0,…, a
2
n
-1
)
If all n
qubits
are examined:
Outcome is
010
with probability
|
a
010
|
2
.
a
000
+ a
001
+ a
010
+ a
011
+ a
100
+ a
101
+ a
110
+ a
111Slide17
Quantum Measurement
State of n qubits (a0,…, a
2
n
-1
)
If all n
qubits
are examined:
Outcome is
010
with probability
|
a
010
|
2
.
The measurement causes the state of the system to change:
The state “collapses” to
010Slide18
Ingredients in ComputationStore information about a problem to be solved
Manipulate the information to solve the problemRead out an answer Slide19
Computer Circuits
AND
OR
AND
AND
OR
OR
AND
NOT
0/1
0/1
0/1
0/1
0/1
0/1
0/1
0/1
0/1
0/1
0/1
Input
OutputSlide20
Quantum Circuits
U
1
U
2
U
3
U
4
U
5
U
6
U
7
U
8
M
Time
Input
Read the output
by measurementSlide21
Interference
[Image from www.thehum.info, due to Dr. Glen MacPherson]Slide22
[Figure from gwoptics.org]Slide23
Quantum Gates: An Example
Operation H:
H
Input:
Output:
Input:
Output:Slide24
Qubit measurement
Measure:
With probability
Outcome =
With probability
Outcome =Slide25
Qubit measurement
Measure:
With probability
Outcome =
With probability
Outcome =Slide26
Quantum Interference: An Example|0
+ |1 |0 - |1 Slide27
Quantum Interference: An Example|0
+ |1 |0 - |1
Apply
Gate
HSlide28
Quantum Interference: An Example|0
+ |1 |0 - |1
(
|0
+
|1 )
Apply
Gate
HSlide29
Quantum Interference: An Example|0
+ |1 |0 - |1
(
|0
+
|1 )
(
|0
-
|1
)
+
Apply
Gate
HSlide30
Quantum Interference: An Example|0
+ |1 |0 - |1
(
|0
+
|1 )
(
|0
-
|1
)
+
Result:
|0
Apply
Gate
HSlide31
Quantum Interference: An Example|0
+ |1 |0 - |1
(
|0
+
|1 )
(
|0
-
|1
)
+
Result:
|0
Apply
Gate
H
Apply
Gate
HSlide32
Quantum Interference: An Example|0
+ |1 |0 - |1
(
|0
+
|1 )
(
|0
-
|1
)
+
Result:
|0
Apply
Gate
H
Apply
Gate
H
(
|0
+
|1
)Slide33
Quantum Interference: An Example|0
+ |1 |0 - |1
(
|0
+
|1 )
(
|0
-
|1
)
+
Result:
|0
Apply
Gate
H
Apply
Gate
H
(
|0
+
|1
)
(
|0
-
|1
)
-Slide34
Quantum Interference: An Example|0
+ |1 |0 - |1
(
|0
+
|1 )
(
|0
-
|1
)
+
Result:
|0
Apply
Gate
H
Apply
Gate
H
(
|0
+
|1
)
(
|0
-
|1
)
-
Result:
|1
Slide35
Quantum Circuits
U
1
U
2
U
3
U
4
U
5
U
6
U
7
U
8
M
Quantum Algorithms
Manipulate data so that negative interference causes wrong answers to have small amplitude and right answers to have high amplitude, so that when we measure output, we are likely to get the right answer.Slide36
FactoringGiven a positive integer, find its prime factorization.24 = 2 x 2 x 2 x 3
Input
OutputSlide37
FactoringRSA-210 = 245246644900278211976517663573088018467026787678332759743414451715061600830038587216952208399332071549103626827191679864079776723243005600592035631246561218465817904100131859299619933817012149335034875870551067Slide38
Can Quantum Computers Be Built?Key challenge: prevent
decoherence (interaction with the environment).Can factor N=15 on a quantum computerLarger problems will require
quantum error correcting codes
.Slide39
Computer Science <-> Quantum Mechanics
ClassicalSimulationOfQuantum Systems
Algorithm
Design
For
Quantum
Circuits
Physical
Implementation
Of
Quantum
Computers
Quantum
Cryptography
Quantum
Information
Theory
Quantum
Complexity
TheorySlide40
Quantum Computers for Simulation in Physics Slide41
My ResearchDesign efficient algorithms on a quantum (or classical) computer that will
provably compute properties of a quantum system.For what kinds of systems is this possible?
Or: give mathematical evidence that there is no efficient way to solve this problem.