Introduction to Quantum Computation

1
Introduction to Quantum Computation

• Neil Shenvi
• Department of Chemistry
• Yale University

2
Talk Outline

• Background
• What is Quantum Computation?
• Quantum Algorithms
• Decoherence and Noise
• Implementations
• Applications

3
Background Classical Computation
Input
Computation
Output
2 2
4
Hello World!
C\Hello.exe
What is the essence of computation?
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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
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Classical Complexity
Some problems are more difficult than others.
Polynomial hierarchy
All Turing machine-equivalent computers have an
identical hierarchy.
Require exponential(?) time to solve
Require exponential time to solve
Require polynomial time to solve
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Classical Complexity
Some important problems do not have known
classical polynomial algorithm and or a known
place in the hierarchy.
Polynomial hierarchy
Factoring
?
Best known algorithm to factor N-digit
number Time Exp(N1/3)
Graph Isomorphism
?
Best known algorithm to compare two N-node
graphs Time Exp(N)
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Classical Computation Theory
What kind of systems can perform universal
computation?
DNA
Billiard balls
Desktop computers
These can all be shown to be equivalent to each
other and to a Turing machine!
Cellular automata
The Big Question What next?
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Talk Outline

• Background
• What is Quantum Computation?
• Quantum Algorithms
• Decoherence and Noise
• Implementations
• Applications

9
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.
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The Bit
The basic component of a classical computer is
the bit, a single binary variable of value 0 or
1.
At any given time, the value of a bit is either
0 or 1.
0
1
The state of a classical computer is described by
some long bit string of 0s and 1s.
0001010110110101000100110101110110...
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The Qubit
A quantum bit, or qubit, is a two-state system
which obeys the laws of quantum mechanics.
Spin-½ particle
0?
1?
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?.
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Computation with Qubits
How does the use of qubits affect computation?
Classical Computation
Data unit bit
1
0
Valid states
x 0 or 1
x 1
x 0
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Computation with Qubits
How does the use of qubits affect computation?
Classical Computation
Operations logical
Valid operations
1 0
0 -1
in
0 1
1 0
0 1 0 1
1 0
1-bit
NOT
1 1
1 -1
0 i
-i 0
out
in
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
0 1 0 1
0 1 0 0
0 1 0 1
2-bit
AND
out
in
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Computation with Qubits
How does the use of qubits affect computation?
Classical Computation
Measurement deterministic
Result of measurement
State
0
x 0
1
x 1
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More than one qubit
Two qubits
Single qubit
00?,01?,10?,11?
0?,1?
0
1
0
0
0
0
1
0
0
0
0
1
1
0
0
0
Hilbert space
1
0
0
1
,
,
,
H2
H2 ?2 H2?H2
,
c1
c2
c3
c4
c1
c2
c1
c2
c3
c4
u11 u12 u13 u14
u21 u22 u23 u24
u31 u32 u33 u34
u41 u42 u43 u44
u11 u12
u21 u22
c1
c2
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Quantum Circuit Model
Example Circuit
Two-qubit operation
One-qubit operation
Measurement
1?
CNOT
0?
sx
0?
1?
0
0
0
1
0
0
0
1
0
0
1
0
1
0
0
0
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0
CNOT
sx ? I
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Quantum Circuit Model
Example Circuit
0? 1?
______
CNOT
sx
v2
0?
1
0
0
0
0
0
0
1
1/v2
0
0
1/v2
1/v2
0
1/v2
0
1/v2
0
1/v2
0
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Some Interesting Consequences
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Talk Outline

• Background
• What is Quantum Computation?
• Quantum Algorithms
• Decoherence and Noise
• Implementations
• Applications

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Quantum Algorithms What can quantum computers do?

• Grovers search algorithm
• Quantum random walk search algorithm
• Shors Factoring Algorithm

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Grovers 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 Im thinking of a number between 1 and
100. What is it?
Oracle
78
No
Oracle
3
Yes
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Grovers Search Algorithm
Classical computer
Oracle
1
No
Oracle
2
No
Oracle
3
Yes
...
The best a classical computer can do on average
is N/2 queries.
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Grovers 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

sz
0?
sz
Hd
Hd
Hd
Hd
Hd

0?
Hd
Hd
Hd
Hd
Hd
O
O






0?
Hd
Hd
Hd
Hd
Hd
The circuit is not complicated, but it doesnt
provide an immediately intuitive picture of how
the algorithm works. Are there any
more intuitive models for quantum search?
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Quantum Random Walk Search Algorithm
Idea extend classical random walk formalism to
quantum mechanics
Classical random walk
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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
1 -1 -1 -1
-1 1 -1 -1
-1 -1 1 -1
-1 -1 -1 1
-1 0 0 0
0 -1 0 0
0 0 -1 0
0 0 0 -1
1
C0
C1
2
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Quantum Random Walk Search Algorithm
Pros As general as Grovers search
algorithm. Cons Same complexity as Grovers
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 QRW- based algorithms?
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Shors Factoring Algorithm
Makes use of quantum Fourier Transform, which is
exponentially faster than classical FFT.
Find the factors of 57
Find the factors of 1623847601650176238761076269
17226121712398721039746218761871207362384612987398
2634897121861102379691863198276319276121
3 x 19
whimper
All known algorithms for factoring an n-bit
number on a classical computer take time
proportional to O(n!).
But Shors algorithm for factoring on a quantum
computer takes time proportional to O(n2 log n).
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Shors Factoring Algorithm
The details of Shors factoring algorithm are
more complicated than Grovers search algorithm,
but the results are clear
with a classical computer
bits 1024 2048 4096 factoring in 2006 105
years 5x1015 years 3x1029 years factoring in
2024 38 years 1012 years 7x1025 years factoring
in 2042 3 days 3x108 years 2x1022 years
with potential quantum computer (e.g., clock
speed 100 MHz)
bits 1024 2048 4096 qubits
5124 10244 20484 gates 3x109 2X1011 X1
012 factoring time 4.5 min 36 min 4.8 hours
R. J. Hughes, LA-UR-97-4986
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Talk Outline

• Background
• What is Quantum Computation?
• Quantum Algorithms
• Decoherence and Noise
• Implementations
• Applications

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Decoherence and Noise
What happens to a qubit when it interacts with an
environment?
Environment
Quantum computer

sN
s2
s3
s1
Quantum information is lost through decoherence.
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Types of Decoherence
T1 processes longitudinal relaxation, energy is
lost to the environment
V
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Effects of Environment on Quantum Memory
T1 timescale of longitudinal relaxation
T2 timescale of transverse relaxation
Fidelity of stored information decays with time.
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Effects of Environment on Quantum Algorithms
Ideal oracle
O
Grovers algorithm success rate
Noisy oracle
O
n of qubits
Errors accumulate, lowering success rate of
algorithm
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Suppressing Decoherence

1. Use decoherence free subspace (DFS)

4. Use pulse sequence to remove decoherence
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Talk Outline

• Background
• What is Quantum Computation?
• Quantum Algorithms
• Decoherence and Noise
• Implementations
• Applications

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Some Proposed Implementations for QC
NMR
Ion trap
B
Kane Proposal
Optical Lattice
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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).
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Solid State Electron Spin Qubit
Electron wavefunction
Phosphorus impurity
External Magnetic Field, B
Silicon lattice
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System Hamiltonian
Electron spin
N nuclear spins
1011 Hz / T
107 Hz / T
105 Hz
102 Hz
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Hyperfine-Induced Longitudinal Decay
Critical field for electron spin relaxation
For B gt Bc, T1 is infinite
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Hyperfine-Induced Transverse Decay
Free evolution
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Talk Outline

• Background
• What is Quantum Computation?
• Quantum Algorithms
• Decoherence and Noise
• Implementations
• Applications

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Applications

• Factoring RSA encryption
• Quantum simulation
• Spin-off technology spintronics, quantum
  cryptography
• Spin-off theory complexity theory, DMRG theory,
  N-representability theory

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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