Download Introduction to Quantum Computation

Introduction to Quantum
Computation
Neil Shenvi
Department of Chemistry
Yale University
Talk Outline
Quantum
Random
Walks
Background
What is Quantum Computation?
Quantum Algorithms
Noise in Grover’s
Decoherence and Noise
Algorithm
Implementations
O
Applications
Decoherence in Spin
Systems
Background: Classical Computation
Input
Computation
Output
2 + 2
C:\Hello.exe
4
Hello World!
What is the essence of computation?
Classical Computation Theory
Church-Turing Thesis: Computation is anything that can be done by a
Turing machine. This definition coincides with our intuitive ideas of
computation: addition, multiplication, binary logic, etc…
What is a Turing machine?
…0100101101010010110…
Input
Finite State Automaton
(control module)
Infinite
tape
Computation
…0000001011111111100…
Read/Write
head
…0100101101010010110…
…1110010110100111101…
Output
Classical Computation Theory
What kind of systems can perform
universal computation?
Desktop computers
Cellular automata
Billiard balls
DNA
These can all be shown to be
equivalent to each other and to
a Turing machine!
The Big Question: What next?
Talk Outline
Background
What is Quantum Computation?
Quantum Algorithms
Decoherence and Noise
Implementations
Applications
What Is Quantum Computation?
Conventional computers, no matter how exotic, all obey the laws of
classical physics.
On the other hand, a quantum computer obeys the laws of quantum physics.
The Bit
The basic component of a classical computer is the bit, a single
binary variable of value 0 or 1.
1
0
0
1
At any given time, the value
of a bit is either ‘0’ or ‘1’.
The state of a classical computer is described by some
long bit string of 0s and 1s.
0001010110110101000100110101110110...
The Qubit
A quantum bit, or qubit, is a two-state system which
obeys the laws of quantum mechanics.
Valid qubit states:
Spin-½ particle
=|1
=|0
| = |0
| = |1
| = (|0- ei/4 |1)/2
| = (2|0- 3ei5/6 |1)/13
The state of a qubit | can be thought of as a vector in
a two-dimensional Hilbert Space, H2, spanned by the
Basis vectors |0 and |1.
Computation with Qubits
How does the use of qubits affect computation?
Classical Computation
Data unit: qubit
Data unit: bit
= ‘1’
= ‘0’
=|1
=|0
Valid states:
Valid states:
x = ‘0’ or ‘1’
| = c1|0 + c2|1
x=1
x=0
Quantum Computation
| = |0
0
0
1
1
| = |1
| = (|0 + |1)/√2
Computation with Qubits
How does the use of qubits affect computation?
Classical Computation
Quantum Computation
Operations: unitary
Valid operations:
Operations: logical
Valid operations:
in
NOT =
0
1
1
0
1-bit
σX =
0 1
σy =
0
i
-i
0
1-qubit
out
1 0
in
in
0
1
0
0
0
1
2-bit
out
2-qubit
CNOT =
1 0
0 -1
Hd =1
√2
1 0 0 0
0 1
AND =
σz =
0 1 0 0
0 0 0 1
0 0 1 0
1 1
1 -1
Computation with Qubits
How does the use of qubits affect computation?
Classical Computation
Quantum Computation
Measurement: deterministic
Measurement: stochastic
State
x = ‘0’
x = ‘1’
Result of measurement
‘0’
‘1’
State
| = |0
| = |1
| = |0- |1
2
Result of measurement
‘0’
‘1’
‘0’
‘1’
50%
50%
More than one qubit
Single qubit
Two qubits
|00,|01,|10,|11
|0,|1
Hilbert
space
Arbitrary
state
Operator
H2 =
1 0
0 ,1
c
| = c1|0 + c2|1 = 1
c2
u11 u12 c1
U| =
u21 u22 c2
1
0
0
0
H2 2 = H2H2 =
| =
,
0
1
0
0
c1|00 + c2|01 +
=
c3|10 + c4|11
U| =
u11
u21
u31
u41
u12
u22
u32
u42
u13
u23
u33
u43
u14
u24
u34
u44
,
c1
c2
c3
c4
c1
c2
c3
c4
0
0
1
0
,
0
0
0
1
Quantum Circuit Model
Example Circuit
Two-qubit
operation
One-qubit
operation
|0
σx
|1
Measurement
CNOT
|1
‘1’
|0
|0
|1
‘1’
1
0
0
0
0
0
1
0
0
0
0
1
0
0
0
1
σx  I =
0
0
1
0
01
00
00
10
0
1
0
0
CNOT =
1
0
0
0
0
1
0
0
0
0
0
1
0
0
1
0
Quantum Circuit Model
Example Circuit
50%
50%
|0
+ |1
______
√2
|0
σx
|0
+ |1
______
√2
|0
1/√2
0
1/√2
0
1/√2
0
1/√2
0
CNOT
?
?
‘0’
1/√2
0
0
1/√2
1
0
0
0
Separable state:
can be written as
tensor product
Entangled state:
cannot be written
as tensor product
| = |  |
| ≠ |  |
‘0’
or
or
‘1’
‘1’
0
0
0
1
Some Interesting Consequences
Reversibility
Since quantum mechanics is reversible (dynamics are unitary),
quantum computation is reversible.
|00000000
|
|00000000
Quantum Superordinacy
All classical quantum computations can be performed by a quantum
computer.
U
No cloning theorem
It is impossible to exactly copy an unknown quantum state
|
|0
|
|
Talk Outline
Background
What is Quantum Computation?
Quantum Algorithms
Decoherence and Noise
Implementations
Applications
Quantum Algorithms: What can
quantum computers do?
Grover’s search algorithm
Quantum random walk search algorithm
Shor’s Factoring Algorithm
Grover’s Search Algorithm
Imagine we are looking for the solution to a problem with
N possible solutions. We have a black box (or ``oracle”) that
can check whether a given answer is correct.
Question: I’m thinking of a number between 1 and 100. What is it?
78
Oracle
No
3
Oracle
Yes
Grover’s Search Algorithm
Classical computer
Quantum computer
1
Oracle
No
2
Oracle
No
Superposition over all N possible inputs.
3
Oracle
Yes
Using Grover’s algorithm, a quantum computer can
find the answer in N queries!
...
The best a classical computer
can do on average is N/2 queries.
1+2+3+...
Oracle
No+No+Yes+No+...
Grover’s Search Algorithm
Pros:
Can be used on any unstructured search problem, even
NP-complete problems.
Cons:
Only a quadratic speed-up over classical search.
O(N) iterations
Hd
…
Hd
O
Hd σz Hd
Hd
Hd
…
Hd
…
|0
O
Hd σz Hd
Hd
Hd
…
Hd
Hd
…
|0
|0
Hd
Hd
…
…
…
The circuit is not complicated, but it doesn’t provide an immediately
intuitive picture of how the algorithm works. Are there any more
intuitive models for quantum search?
Quantum Random Walk Search
Algorithm
Idea: extend classical random walk formalism to quantum mechanics
Classical random walk:
A
pt 1  A  pt
Aij  Pr( j  i )
Quantum random walk:
C
| t 1  U | t 
U  S C
Moves walkers
based on coin
Flips coin
pt 1
pt
| t 
S
| t 1
Quantum Random Walk Search
Algorithm
To obtain a search algorithm, we use our “black box” to apply a different
type of coin operator, C1, at the marked node
C1
C0
C0=
1
2
1 -1 -1 -1
-1 1 -1 -1
-1 -1 1 -1
-1 -1 -1 1
C1=
-1 0 0 0
0 -1 0 0
0 0 -1 0
0 0 0 -1
Quantum Random Walk Search
Algorithm
Pros:
As general as Grover’s search algorithm.
Cons:
Same complexity as Grover’s search algorithm.
Slightly more complicated in implementation
Slightly more memory used
Interesting Feature: Search algorithm flows naturally
out of random walk formalism. Motivation for new QRWbased algorithms?
Shor’s Factoring Algorithm
Makes use of quantum Fourier Transform, which is exponentially
faster than classical FFT.
Find the factors of: 57
Find the factors of:
16238476016501762387610762691722612171239872103974621876187
12073623846129873982634897121861102379691863198276319276121
3 x 19
whimper
All known algorithms for factoring an n-bit number on a
classical computer take time proportional to O(n!).
But Shor’s algorithm for factoring on a quantum computer
takes time proportional to O(n2 log n).
Shor’s Factoring Algorithm
The details of Shor’s factoring algorithm are more complicated than
Grover’s search algorithm, but the results are clear:
with a classical computer
# bits
factoring in 2006
factoring in 2024
factoring in 2042
1024
105 years
38 years
3 days
2048
5x1015 years
1012 years
3x108 years
4096
3x1029 years
7x1025 years
2x1022 years
with potential quantum computer
(e.g., clock speed 100 MHz)
# bits
# qubits
# gates
factoring time
1024
5124
3x109
4.5 min
2048
10244
2X1011
36 min
4096
20484
X1012
4.8 hours
R. J. Hughes, LA-UR-97-4986
Talk Outline
Background
What is Quantum Computation?
Quantum Algorithms
Decoherence and Noise
Implementations
Applications
Decoherence and Noise
What happens to a qubit when it interacts with an environment?
Environment
Quantum computer
H  H0 V
V
H 0  B 1, z
V   A j 1   j
j
σ1
σ2
σ3
…
Quantum information is lost through decoherence.
σN
Types of Decoherence
T1 processes: longitudinal relaxation, energy is lost to the environment
V
T2 processes: transverse relaxation, system becomes entangled with
the environment
+
V
+
What are the effects of decoherence?
Effects of Environment on Quantum
Memory
T1 – timescale of
longitudinal relaxation
T2 – timescale of
transverse relaxation
Fidelity of stored information decays with time.
Ideal
oracle
O
Noisy
oracle
O
Grover’s algorithm success rate
Effects of Environment on Quantum
Algorithms
n = # of qubits
Errors accumulate, lowering success rate of algorithm
Suppressing Decoherence
1. Remove or reduce V, i.e. build a better computer
System isolated from environment
2. Increase B, i.e. increase level splitting
|1
E
When E >> V, decoherence
is small
E
|0
B
3. Use decoherence free subspace (DFS)
4. Use pulse sequence to remove decoherence
Talk Outline
Background
What is Quantum Computation?
Quantum Algorithms
Decoherence and Noise
Implementations
Applications
Some Proposed Implementations for QC
NMR
Ion trap
B
Kane
Proposal
Optical Lattice
The Loss-Divincenzo Proposal
D. Loss and D.P. DiVincenzo, Phys. Rev. A 57, 120 (1998);
G. Burkhard, H.A. Engel, and D. Loss, Fortschr. der Physik 48, 965 (2000).
Solid State Electron Spin Qubit
Electron wavefunction
Si28 (no spin)
Phosphorus
impurity
Dipolar coupling
Si29
(spin ½)
Hyperfine coupling
Silicon lattice
External Magnetic
Field, B
System Hamiltonian
Hyperfine coupling
Dipolar coupling
Electron
spin
N nuclear
spins
H   S BS z   I BI jz   Aj S  I j 
j
~1011 Hz / T
~107 Hz / T
j
~105 Hz
b
jk
I j  Ik
( j ,k )
~102 Hz
Hyperfine-Induced Longitudinal
Decay
 Bc 
1
S z (t ) 
 8 
2
B
Critical field for electron
spin relaxation:
Bc 

A
j j
 S  I
For B > Bc, T1 is infinite
2
Hyperfine-Induced Transverse Decay
Free evolution
Spin echo pulse sequence
Spin echo pulse sequence removes nearly all dephasing!
Talk Outline
Background
What is Quantum Computation?
Quantum Algorithms
Decoherence and Noise
Implementations
Applications
Applications
Factoring – RSA encryption
Quantum simulation
Spin-off technology – spintronics, quantum
cryptography
Spin-off theory – complexity theory, DMRG
theory, N-representability theory
Acknowledgements
Dr. Julia Kempe, Dr. Ken Brown, Sabrina
Leslie, Dr. Rogerio de Sousa
Dr. K. Birgitta Whaley
Dr. Christina Shenvi
Dr. John Tully and the Tully Group