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Introduction to Quantum Computation
Neil Shenvi
Department of Chemistry
Yale University
Talk Outline
BackgroundWhat is Quantum Computation?Quantum AlgorithmsDecoherence and NoiseImplementationsApplications
Quantum Random Walks
O
Noise in Grover’sAlgorithm
Decoherence in Spin Systems
Background: Classical Computation
C:\Hello.exe Hello World!
Input Computation Output
What is the essence of computation?
2 + 2 4
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…
Infinite tape
Read/Write head
Finite State Automaton (control module)
…0000001011111111100…
Computation
…1110010110100111101… Output
…0100101101010010110… Input
Classical Computation Theory
What kind of systems can perform universal computation?
Desktop computers Billiard balls DNA
Cellular automata
These can all be shown to be equivalent to each other and to a Turing machine!
The Big Question: What next?
Talk Outline
BackgroundWhat is Quantum Computation?Quantum AlgorithmsDecoherence and NoiseImplementationsApplications
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
The state of a classical computer is described by some long bit string of 0s and 1s.
0001010110110101000100110101110110...
At any given time, the valueof a bit is either ‘0’ or ‘1’.
The Qubit
A quantum bit, or qubit, is a two-state system which obeys the laws of quantum mechanics.
=|1 =|0
Valid qubit states:
| = |0 | = |1| = (|0- ei/4 |1)/2 | = (2|0- 3ei5/6 |1)/13
Spin-½ particle
The state of a qubit | can be thought of as a vector in a two-dimensional Hilbert Space, H2, spanned by theBasis vectors |0 and |1.
Computation with Qubits
How does the use of qubits affect computation?
Classical Computation
Data unit: bit
x = 0 x = 1
0
1
0
1
Valid states:x = ‘0’ or ‘1’ | = c1|0 + c2|1
Quantum Computation
Data unit: qubit
Valid states:
| = |0 | = |1 | = (|0 + |1)/√2
=|1 =|0= ‘1’ = ‘0’
Computation with Qubits
0 1
1 0
How does the use of qubits affect computation?
Classical Computation
Operations: logicalValid operations:
AND =
0 i
-i 0
1 0
0 -1
1 1
1 -1
0 1
0
1
0 0
0 1
NOT =0 1
1 0
in
out
out
in
in
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
1-bit
2-bit
Quantum Computation
Operations: unitary
Valid operations:
σX =
σy =
σz =
Hd =
CNOT =
√21
1-qubit
2-qubit
Computation with Qubits
How does the use of qubits affect computation?
Classical Computation
Measurement: deterministic
x = ‘0’
State Result of measurement
‘0’
x = ‘1’ ‘1’
Quantum Computation
Measurement: stochastic
| = |0
| = |0- |1
State Result of measurement
| = |1
2
‘0’
‘1’
‘0’ 50%
‘1’ 50%
More than one qubit
1000
u11 u12
u21 u22
Single qubit
c1
c2
c1
c2
Two qubits
H2 = 10
01,
|0,|1
H2 2 = H2H2 = ,
|00,|01,|10,|110100
,
0010
,
0001
c1
c2
c3
c4
c1
c2
c3
c4
u11 u12 u13 u14
u21 u22 u23 u24
u31 u32 u33 u34
u41 u42 u43 u44
Hilbertspace
U| = U| =Operator
| = c1|0 + c2|1 = | c1|00 + c2|01 +c3|10 + c4|11
==Arbitrarystate
Quantum Circuit Model
1000
0 0 1 00 0 0 11 0 0 00 1 0 0
σx I =
0010
1 0 0 00 1 0 00 0 0 10 0 1 0
CNOT =
0001
0001
|0
|0
|1
|0
|1
|1
‘1’
‘1’
Example Circuit
σx
One-qubit operation
CNOT
Two-qubit operation Measurement
Quantum Circuit Model
1/√2 01/√2 0
1000
σx CNOT
|0 + |1
|0
Example Circuit
√2______
1/√2 01/√2 0
1/√2 0 01/√2
0001
|0 + |1
|0√2
______ ‘0’
‘0’or
‘1’
‘1’
or
50% 50%
Separable state:can be written astensor product
| = | |
Entangled state:cannot be written as tensor product
| ≠ | |
??
Some Interesting Consequences
Quantum SuperordinacyAll classical quantum computations can be performed by a quantumcomputer. U
No cloning theoremIt is impossible to exactly copy an unknown quantum state
||0
||
ReversibilitySince quantum mechanics is reversible (dynamics are unitary),quantum computation is reversible.
|00000000 | |00000000
Talk Outline
BackgroundWhat is Quantum Computation?Quantum AlgorithmsDecoherence and NoiseImplementationsApplications
Quantum Algorithms: What can quantum computers do?
Grover’s search algorithmQuantum random walk search algorithmShor’s Factoring Algorithm
Grover’s Search Algorithm
Imagine we are looking for the solution to a problem withN possible solutions. We have a black box (or ``oracle”) that can check whether a given answer is correct.
78
Question: I’m thinking of a number between 1 and 100. What is it?
Oracle No
3 Oracle Yes
Grover’s Search Algorithm
The best a classical computer can do on average is N/2 queries.
1 Oracle No
...
2 Oracle No
3 Oracle Yes
Classical computer
Oracle1+2+3+... No+No+Yes+No+...
Quantum computer
Using Grover’s algorithm, a quantum computer can find the answer in N queries!
Superposition over all N possible inputs.
Grover’s Search Algorithm
Pros:Can be used on any unstructured search problem, evenNP-complete problems.Cons:Only a quadratic speed-up over classical search.
The circuit is not complicated, but it doesn’t provide an immediatelyintuitive picture of how the algorithm works. Are there any moreintuitive models for quantum search?
O
σz
O
σz
…
……
…
|0|0
|0
O(N) iterations
Hd
Hd
Hd
…
Hd
Hd
Hd…
Hd
Hd
Hd
…Hd
Hd
Hd
…
Hd
Hd
Hd
Quantum Random Walk Search Algorithm
Idea: extend classical random walk formalism to quantum mechanics
A
tp
1tp
Classical random walk:
C S
| t 1| t
Quantum random walk:
1| |t tU
U S C Moves walkers based on coin
Flips coin
Pr( )ijA j i 1t tp A p
Quantum Random Walk Search Algorithm
To obtain a search algorithm, we use our “black box” to apply a differenttype of coin operator, C1, at the marked node
C0
C1
1 -1-1 -1-1 1 -1 -1-1 -1 1 -1-1 -1-1 1
C0=12 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 implementationSlightly more memory used
Interesting Feature: Search algorithm flows naturallyout of random walk formalism. Motivation for new QRW-based algorithms?
Shor’s Factoring Algorithm
Find the factors of: 57
3 x 19
Find the factors of: 1623847601650176238761076269172261217123987210397462187618712073623846129873982634897121861102379691863198276319276121
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).
Makes use of quantum Fourier Transform, which is exponentiallyfaster than classical FFT.
# bits 1024 2048 4096factoring in 2006 105 years 5x1015 years 3x1029 yearsfactoring in 2024 38 years 1012 years 7x1025 yearsfactoring in 2042 3 days 3x108 years 2x1022 years
with a classical computer
# bits 1024 2048 4096# qubits 5124 10244 20484# gates 3x109 2X1011 X1012
factoring time 4.5 min 36 min 4.8 hours
with potential quantum computer (e.g., clock speed 100 MHz)
R. J. Hughes, LA-UR-97-4986
Shor’s Factoring Algorithm
The details of Shor’s factoring algorithm are more complicated thanGrover’s search algorithm, but the results are clear:
Talk Outline
BackgroundWhat is Quantum Computation?Quantum AlgorithmsDecoherence and NoiseImplementationsApplications
Decoherence and Noise
What happens to a qubit when it interacts with an environment?
0
0 1,
1
z
j jj
H H V
H B
V A
Quantum computer Environment
V
Quantum information is lost through decoherence.
σ1σ2 σ3
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
Fidelity of stored information decays with time.
T1 – timescale oflongitudinal relaxation
T2 – timescale oftransverse relaxation
Effects of Environment on Quantum Algorithms
Errors accumulate, lowering success rate of algorithm
Gro
ver’
s a
lgo
rith
m s
ucc
ess
rat
e
n = # of qubits
O
O
Idealoracle
Noisyoracle
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
B
E
|0
|1 When E >> V, decoherenceis smallE
3. Use decoherence free subspace (DFS)
4. Use pulse sequence to remove decoherence
Talk Outline
BackgroundWhat is Quantum Computation?Quantum AlgorithmsDecoherence and NoiseImplementationsApplications
Some Proposed Implementations for QC
NMR
B
Ion trap
Optical Lattice
Kane Proposal
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
Silicon lattice
Phosphorus impurity
Electron wavefunction
Si28 (no spin)
Si29 (spin ½)
External MagneticField, B
Hyperfine couplingDipolar coupling
System Hamiltonian
Electronspin
N nuclearspins
( , )S z I jz j j jk j k
j j j k
H BS BI A S I b I I
Hyperfine coupling Dipolar coupling
~105 Hz ~102 Hz~107 Hz / T~1011 Hz / T
Hyperfine-Induced Longitudinal Decay
21( ) 82
cz
BS t
B
For B > Bc, T1 is infinite
jjc
S I
AB
Critical field for electronspin relaxation:
Hyperfine-Induced Transverse Decay
Free evolution Spin echo pulse sequence
Spin echo pulse sequence removes nearly all dephasing!
Talk Outline
BackgroundWhat is Quantum Computation?Quantum AlgorithmsDecoherence and NoiseImplementationsApplications
Applications
Factoring – RSA encryptionQuantum simulationSpin-off technology – spintronics, quantum
cryptographySpin-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 WhaleyDr. Christina ShenviDr. John Tully and the Tully Group