Quantum computing - A Compilation of Concepts

QUANTUM
COMPUTING
A N E X P LO R AT I O N T H R O U G H E X P E R I M E N T S

INTRODUCTION
 “I think I can safely say that nobody
understands quantum mechanics” - Feynman
 1982 - Feynman proposed the idea of creating
machines based on the laws of quantum
mechanics instead of the laws of classical
physics.
 1985 - David Deutsch developed the quantum turing
machine, showing that quantum circuits are universal.
 1994 - Peter Shor came up with a quantum
algorithm to factor very large numbers in polynomial
time.
1997 - Lov Grover develops a quantum search
algorithm with O(√N) complexity

TALK OUTLINE
• Background
• What is Quantum Computation?
• Quantum Algorithms
• Decoherence and Noise
• Implementations
• Applications
Quantum
Random
Walks
O
Noise in Grover’s
Algorithm
Decoherence in Spin
Systems

QUANTUM COMPUTING AN INTRODUCTION

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?

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 value
of 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 the
Basis vectors |0 and |1.

HOW TO PROGRAM A QUANTUM COMPUTER

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’

0 1
1 0
Operations: logical
Valid 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 =
√2
1
1-qubit
2-qubit

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
1
0
0
0
u11 u12
u21u22
Single qubit
c1
c2
c1
c2
Two qubits
H2 =
1
0
0
1,
|0,|1
H2
2 = H2H2 = ,
|00,|01,|10,|11
0
1
0
0
,
0
0
1
0
,
0
0
0
1
c1
c2
c3
c4
c1
c2
c3
c4
u11 u12 u13 u14
u21 u22 u23 u24
u31 u32 u33 u34
u41 u42 u43 u44
Hilbert
space
U| = U| =Operator
| = c1|0 + c2|1 = |
c1|00 + c2|01 +
c3|10 + c4|11
==
Arbitrary
state

QUANTUM CIRCUIT MODEL
1
0
0
0
0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0
σx  I =
0
0
1
0
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
CNOT =
0
0
0
1
0
0
0
1
|0
|0
|1
|0
|1
|1
‘1’
‘1’
Example Circuit
σx
One-qubit
operation
CNOT
Two-qubit
operation Measurement

QUANTUM CIRCUIT MODEL
1/√2
0
1/√2
0
1
0
0
0
σx CNOT
|0 + |1
|0
Example Circuit
√2
______
1/√2
0
1/√2
0
1/√2
0
0
1/√2
0
0
0
1
|0 + |1
|0
√2
______
‘0’
‘0’
or
‘1’
‘1’
or
50% 50%
Separable state:
can be written as
tensor product
| = |  |
Entangled state:
cannot be written
as tensor product
| ≠ |  |
?
?

SOME INTERESTING CONSEQUENCES
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
|
|
Reversibility
Since quantum mechanics is reversible (dynamics are unitary),
quantum computation is reversible.
|00000000 | |00000000

REPRESENTATION OF DATA - QUBITS
A bit of data is represented by a single atom that is in one of
two states denoted by |0> and |1>. A single bit of this form is
known as a qubit
A physical implementation of a qubit could use the two energy
levels of an atom. An excited state representing |1> and a
ground state representing |0>.
Excited
State
Ground
State
Nucleus
Light pulse of
frequency  for
time interval t
Electron
State |0> State |1>

REPRESENTATION OF DATA - SUPERPOSITION
A single qubit can be forced into a superposition of the two states
denoted by the addition of the state vectors:
|> =  |0> +  |1>
Where  and  are complex numbers and | | + |  | = 1
1 2
1 2 1 2
2 2
A qubit in superposition is in both of the
states |1> and |0 at the same time

REPRESENTATION OF DATA - SUPERPOSITION
Light pulse of
frequency  for time
interval t/2
State |0> State |0> + |1>
Consider a 3 bit qubit register. An equally weighted
superposition of all possible states would be denoted by:
|> = |000> + |001> + . . . + |111>
1
√8
1
√8
1
√8

DATA RETRIEVAL
 In general, an n qubit register can represent the numbers 0
through 2^n-1 simultaneously.
Sound too good to be true?…It is!
 If we attempt to retrieve the values represented within a
superposition, the superposition randomly collapses to
represent just one of the original values.
In our equation: |> =  |0> +  |1> ,  represents the
probability of the superposition collapsing to |0>. The ’s
are called probability amplitudes. In a balanced
superposition,  = 1/√2 where n is the number of qubits.
1 2 1
n

RELATIONSHIPS AMONG DATA - ENTANGLEMENT
Entanglement is the ability of quantum systems to exhibit
correlations between states within a superposition.
Imagine two qubits, each in the state |0> + |1> (a superposition
of the 0 and 1.) We can entangle the two qubits such that the
measurement of one qubit is always correlated to the
measurement of the other qubit.

Due to the nature of quantum physics, the destruction of
information in a gate will cause heat to be evolved which can
destroy the superposition of qubits.
OPERATIONS ON QUBITS - REVERSIBLE LOGIC
A B C
0 0 0
0 1 0
1 0 0
1 1 1
Input Output
A
B
C
In these 3 cases,
information is
being destroyed
Ex.
The AND Gate
This type of gate cannot be used. We must use
Quantum Gates.

QUANTUM GATES
 Quantum Gates are similar to classical gates, but do not have
a degenerate output. i.e. their original input state can be derived
from their output state, uniquely. They must be reversible.
This means that a deterministic computation can be performed
on a quantum computer only if it is reversible. Luckily, it has
been shown that any deterministic computation can be made
reversible.(Charles Bennet, 1973)

QUANTUM GATES - HADAMARD
Simplest gate involves one qubit and is called a Hadamard
Gate (also known as a square-root of NOT gate.) Used to put
qubits into superposition.
H
State
|0>
State
|0> + |1>
H
State
|1>
Note: Two Hadamard gates used in
succession can be used as a NOT gate

QUANTUM GATES - CONTROLLED NOT
A gate which operates on two qubits is called a Controlled-
NOT (CN) Gate. If the bit on the control line is 1, invert
the bit on the target line.
A - Target
B - Control
A B A’ B’
0 0 0 0
0 1 1 1
1 0 1 0
1 1 0 1
Input Output
Note: The CN gate has a similar
behavior to the XOR gate with some
extra information to make it reversible.
A’
B’

EXAMPLE OPERATION - MULTIPLICATION BY 2
Carry Bit
Carry
Bit
Ones
Bit
Carry
Bit
Ones
Bit
0 0 0 0
0 1 1 0
Input Output
Ones Bit
 We can build a reversible logic circuit to calculate multiplication
by 2 using CN gates arranged in the following manner:
0
H

QUANTUM GATES - CONTROLLED CONTROLLED NOT (CCN)
A - Target
B - Control 1
C - Control 2
A B C A’ B’ C’
0 0 0 0 0 0
0 0 1 0 0 1
0 1 0 0 1 0
0 1 1 1 1 1
1 0 0 1 0 0
1 0 1 1 0 1
1 1 0 1 1 0
1 1 1 0 1 1
Input Output
A’
B’
C’
A gate which operates on three qubits is called a
Controlled Controlled NOT (CCN) Gate. Iff the bits on
both of the control lines is 1,then the target bit is inverted.

A UNIVERSAL QUANTUM COMPUTER
 The CCN gate has been shown to be a universal reversible
logic gate as it can be used as a NAND gate.
A - Target
B - Control 1
C - Control 2
A B C A’ B’ C’
0 0 0 0 0 0
0 0 1 0 0 1
0 1 0 0 1 0
0 1 1 1 1 1
1 0 0 1 0 0
1 0 1 1 0 1
1 1 0 1 1 0
1 1 1 0 1 1
Input OutputA’
B’
C’
When our target input is 1, our target
output is a result of a NAND of B and C.

QUBITS
• A Quantum Bit
(Qubit) is a two-level
quantum system.
• We can label the
states |0> and |1>.
• In principle, this
could be any two-
level system.
|1>
|0>

QUBITS
• Unlike a classical bit, which is definitely in either state, the state of a Qubit is in general
a mix of |0> and |1>.
• We assume a normalized state: 10 10 cc 
1
2
1
2
0  cc

QUBITS
• For convenience, we will use the matrix representation













1
0
1
0
1
0

QUANTUM GATE
• A Quantum Logic Gate is an operation that we perform on one or more Qubits that
yields another set of Qubits.
• We can represent them as linear operators in the Hilbert space of the system.

QUANTUM NOT GATE
• As in classical computing, the NOT gate returns a 0 if the input is 1 and a 1 if the input
is 0.
• The matrix representation is






01
10

OTHER QUANTUM GATES
• Other gates include the Hadamard-Walsh matrix:
• And Phase Flip operation:






11
11
2
1






i
e0
01

MULTIPLE QUBITS
• Any useful classical computer has more than one bit.
Likewise, a Quantum Computer will probably consist of
multiple qubits.
• A system of n Qubits is called a Quantum Register of length n.
• To represent that Qubit 1 has value b1, Qubit 2 has value b2,
etc., we will use the notation:
nnbbb 2211

MULTIPLE QUBITS
• For n Qubits, the vector representing the state is a 2n column vector.
• The operations are then 2n x 2n matrices.
• For n = 2, we use the representations




























































1
0
0
0
11
0
1
0
0
01
0
0
1
0
10
0
0
0
1
00 21212121

QUANTUM CNOT GATE
• An important Quantum Gate for n = 2 is the conditional not gate.
• The conditional not gate flips the second bit if and only if the first bit is on.
Input Output
Qubit 1 Qubit 2 Qubit 1 Qubit 2
0 0 0 0
0 1 0 1
1 0 1 1
1 1 1 0














0100
1000
0010
0001

REVERSIBILITY AND NO-CLONING
• In Quantum Computing, we use unitary operations (U*U = 1).
• This ensures that all of the operations that we perform are reversible.
• This fact is important, because there is no way to perfectly copy a state in Quantum
Computing (No-Cloning Theorem).

NO-CLONING THEOREM
• That is, the No-Cloning Theorem says that there is no linear operation that copy an
arbitrary state to one of the basis states:
• We can get around this if we are only interested in copying basis vectors, though.
 ie

ENTANGLEMENT
• In Quantum Mechanics, it sometimes occurs that a measurement of one particle will
effect the state of another particle, even though classically there is no direct
interaction. (This is a controversial interpretation).
• When this happens, the state of the two particles is said to be entangled.

ENTANGLEMENT: FORMALISM
• More formally, a two-particle state is entangled if it cannot be written as a product of
two one-particle states.
• If a state is not entangled, it is decomposable.
 2121
1100
2
1

 
   2211
21212121
10
2
1
10
2
1
11100100
2
1



ENTANGLEMENT: EXAMPLE
• The state of two spinors is prepared such that the z-
component of the spin is zero.
• If we measure m = +1/2 for one particle, then the other
particle must have m =-1/2.
• The measurement performed on one particle resulted in the
collapse of the wavefunction of the other particle.

UNIVERSAL GATE SETS
• It would be convenient if there was a small set of operations
from which all other operations could be produced.
• That is, a set of operators {U1,…,Un} such that any other
operator W could be written W = UiUj…Uk.
• Such a set of operators in the context of computation is called
a universal gate set.

CLASSICAL NAND GATE
• One universal set for Classical Computation consists of only the NAND gate which
returns 0 only if the two inputs are 1.
NAND
Input 1 Input B Output
0 0 1
0 1 1
1 0 1
1 1 0
)),(),,((),(
)),(),,((),(
),()(
QQNANDPPNANDNANDQPOR
QPNANDQPNANDNANDQPAND
PPNANDPNOT




QUANTUM UNIVERSAL GATE SET
• There are a few universal sets in Quantum Computing.
• Two convenient sets:
• CNOT and single Qubit Gates
• CNOT, Hadamard-Walsh, and Phase Flips
• Having such a set could greatly simplify implementation and design of Quantum
Algorithms.

SOME PROPOSED IMPLEMENTATIONS FOR QC
NMR
B
Ion trap
Optical Lattice
Kane
Proposal

PHYSICAL IMPLEMENTATION
• Any physical implementation of a quantum computer must have the following
properties to be practical(DiVincenzo)
• The number of Qubits can be increased
• Qubits can be arbitrarily initialized
• A Universal Gate Set must exist
• Qubits can be easily read
• Decoherence time is relatively small

DECOHERENCE
• As the number of Qubits increases, the influence of external environment perturbs the
system.
• This causes the states in the computer to change in a way that is completely
unintended and is unpredictable, rendering the computer useless.
• This is called decoherence.

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.
78
Question: I’m thinking of a number between 1 and 100. What is it?
Oracle No
3 Oracle Yes

The best a classical computer
can do on average is N/2 queries.
1 Oracle
No
...
2 Oracle
No
3 Oracle
Yes
Classical computer
Oracle
1+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.

Pros:
Can be used on any unstructured search problem, even
NP-complete problems.
Cons:
Only a quadratic speed-up over classical search.
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?
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
ALGORITHMIdea: extend classical random walk formalism to quantum mechanics
A
tp
r
1tp 
r
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  
r r

ALGORITHM
To obtain a search algorithm, we use our “black box” to apply a different
type 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=
1
2 C1=
-1 0 0 0
0 -1 0 0
0 0 -1 0
0 0 0 -1

ALGORITHMPros:
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 QRW-
based algorithms?

SHOR’S FACTORING ALGORITHM
Find the factors of: 57
3 x 19
Find the factors of:
16238476016501762387610762691722612171239872103974621876187
12073623846129873982634897121861102379691863198276319276121
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 exponentially
faster than classical FFT.

SHOR’S ALGORITHM
• A Quantum Algorithm, due to P. W. Shor (1994) allows for very fast factoring of
numbers.
• The algorithm uses other algorithms: the Quantum Fourier Transform, and Euclid’s
Algorithm.
• It also relies on elements of group theory.

SHOR’S ALGORITHM
• Because of the unpredictability of Quantum Mechanics, it only
gives the correct answer to within a certain probability.
• Multiple runs can be performed to increase the probability
that the answer is correct. This increases the complexity to
• A Quantum Computer with 7 Qubits was developed in 2001
to implement Shor’s algorithm to factor 15.
 nn 2
3
log

# 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 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 than
Grover’s search algorithm, but the results are clear:

SHOR’S ALGORITHM
Shor’s algorithm shows (in principle,) that a quantum
computer is capable of factoring very large numbers in
polynomial time.
The algorithm is dependant on
Modular Arithmetic
Quantum Parallelism
Quantum Fourier Transform

SHOR’S ALGORITHM - PERIODICITY
 Choose N = 15 and x = 7 and we get the following:
7 mod 15 = 1
7 mod 15 = 7
7 mod 15 = 4
7 mod 15 = 13
7 mod 15 = 1
0
1
2
3
4
 An important result from Number Theory:
F(a) = x mod N is a periodic functiona
.
.
.

SHOR’S ALGORITHM - IN DEPTH ANALYSIS
To Factor an odd integer N (Let’s choose 15) :
1. Choose an integer q such that N < q < 2N let’s pick 256
2. Choose a random integer x such that GCD(x, N) = 1 let’s pick 7
3. Create two quantum registers (these registers must also be
entangled so that the collapse of the input register corresponds to
the collapse of the output register)
• Input register: must contain enough qubits to represent
numbers as large as q-1. up to 255, so we need 8 qubits
• Output register: must contain enough qubits to represent
numbers as large as N-1. up to 14, so we need 4 qubits
2 2

SHOR’S ALGORITHM - PREPARING DATA
4. Load the input register with an equally weighted
superposition of all integers from 0 to q-1. 0 to 255
5. Load the output register with all zeros.
The total state of the system at this point will be:
1
√256
∑ |a, 000>
a=0
255
Input
Register
Output
Register
Note: the comma here
denotes that the
registers are entangled

SHOR’S ALGORITHM - MODULAR ARITHMETIC
6. Apply the transformation x mod N to each number in
the input register, storing the result of each computation
in the output register.
a
Input Register 7 Mod 15 Output Register
|0> 7 Mod 15 1
|1> 7 Mod 15 7
|2> 7 Mod 15 4
|3> 7 Mod 15 13
|4> 7 Mod 15 1
|5> 7 Mod 15 7
|6> 7 Mod 15 4
|7> 7 Mod 15 13
a
0
1
7
6
5
4
3
2
Note that we are using decimal
numbers here only for simplicity.
.
.

SHOR’S ALGORITHM - SUPERPOSITION COLLAPSE
7. Now take a measurement on the output register. This will
collapse the superposition to represent just one of the results
of the transformation, let’s call this value c.
Our output register will collapse to represent one of
the following:
|1>, |4>, |7>, or |13
For sake of example, lets choose |1>

SHOR’S ALGORITHM - ENTANGLEMENT
8. Since the two registers are entangled, measuring the output
register will have the effect of partially collapsing the input
register into an equal superposition of each state between 0
and q-1 that yielded c (the value of the collapsed output
register.)
Now things really get interesting !
Since the output register collapsed to |1>, the input register
will partially collapse to:
|0> + |4> + |8> + |12>, . . .
The probabilities in this case are since our register is
now in an equal superposition of 64 values (0, 4, 8, . . . 252)
1
√64
1
√64
1
√64
1
√64
1
√64

SHOR’S ALGORITHM - QFT
We now apply the Quantum Fourier transform on the
partially collapsed input register. The fourier transform has
the effect of taking a state |a> and transforming it into a
state given by:
1
√q
∑ |c> * e
c=0
q-1
2iac / q

1
√256
∑ |c> * e
c=0
255
2iac / 256
1
√64
∑ |a> , |1>
a  A
Note: A is the set of all values that 7 mod 15 yielded 1.
In our case A = {0, 4, 8, …, 252}
So the final state of the input register after the QFT is:
a
1
√64
∑ , |1>
a  A
1
√256
∑ |c> * e
c=0
255
2iac / 256

The QFT will essentially peak the probability amplitudes at
integer multiples of q/4 in our case 256/4, or 64.
|0>, |64>, |128>, |192>, …
So we no longer have an equal superposition of states, the
probability amplitudes of the above states are now higher
than the other states in our register. We measure the register,
and it will collapse with high probability to one of these
multiples of 64, let’s call this value p.
With our knowledge of q, and p, there are methods of
calculating the period (one method is the continuous fraction
expansion of the ratio between q and p.)

SHOR’S ALGORITHM - THE FACTORS :)
10. Now that we have the period, the factors of N can be
determined by taking the greatest common divisor of N
with respect to x ^ (P/2) + 1 and x ^ (P/2) - 1. The idea
here is that this computation will be done on a classical
computer.
We compute:
Gcd(7 + 1, 15) = 5
Gcd(7 - 1, 15) = 3
We have successfully factored 15!
4/2
4/2

SHOR’S ALGORITHM - PROBLEMS
 The QFT comes up short and reveals the wrong period. This
probability is actually dependant on your choice of q. The
larger the q, the higher the probability of finding the correct
probability.
 The period of the series ends up being odd
If either of these cases occur, we go back to
the beginning and pick a new x.

NMR IMPLEMENTATION
• Vandersypen, et al. used
an NMR computer to
implement Shor’s
algorithm.
• We can consider two
different Qubits as two
different nuclei in the
magnetic field, oriented in
slightly different
directions, so that the
energy splitting is
different between them.
|1>1
|0>1
|1>2
|0>2

NMR IMPLEMENTATION
• Since the energy splittings are different, we can control each Qubit independently by
using different frequencies of radiation.
• The two Qubits will also interact slightly due to their spins. This allows for the
implementation of a CNOT gate.

OTHER IMPLEMENTATIONS
• There are other possible ways to produce quantum computers:
• Quantum dots
• Superconductors
• Lasers acting on ion traps
• Molecular magnetic computers

DECOHERENCE AND NOISE
What happens to a qubit when it interacts with an environment?
0
0 1,
1
z
j j
j
H H V
H B
V A

 
 

 
r r
Quantum computer Environment
V
Quantum information is lost through decoherence.
σ1
σ2 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 of
longitudinal relaxation
T2 – timescale of
transverse relaxation

EFFECTS OF ENVIRONMENT ON QUANTUM ALGORITHMS
Errors accumulate, lowering success rate of algorithm
Grover’salgorithmsuccessrate
n = # of qubits
O
O
Ideal
oracle
Noisy
oracle

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, decoherence
is smallE
3. Use decoherence free subspace (DFS)
4. Use pulse sequence to remove decoherence

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 Magnetic
Field, B
Hyperfine couplingDipolar coupling

SYSTEM HAMILTONIAN
Electron
spin
N nuclear
spins
( , )
S z I jz j j jk j k
j j j k
H BS BI A S I b I I        
r r r r
Hyperfine coupling Dipolar coupling
~105 Hz ~102 Hz
~107 Hz / T
~1011 Hz / T

HYPERFINE-INDUCED
LONGITUDINAL DECAY
2
1( ) 8
2
c
z
B
S t
B
 
   
 
For B > Bc, T1 is infinite
jj
c
S I
A
B
 


h
Critical field for electron
spin relaxation:

HYPERFINE-INDUCED TRANSVERSE
DECAY
Free evolution Spin echo pulse sequence
Spin echo pulse sequence removes nearly all dephasing!

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

FUTURE PROSPECTS
• Currently, research in Quantum Computing is more based on proof-of-principle rather
than research into practical applications.
• The infancy of the science is a significant inhibitor. In the future, decoherence may be
a serious issue.

FUTURE PROSPECTS
• Although many Quantum Algorithms seem to threaten
classical computing (such as RSA-encryption), Classical
Computers will be significantly larger than Quantum
Computers for the foreseeable future.
• Kurzweil, for example, suggests that practical quantum
computing will be achieved at approximately the same time
humanity achieves immortality (before 2099).

CONCLUDING REMARKS
• Quantum Computing could provide a radical change in the
way computation is performed.
• The unit of information in Quantum Computing is the Qubit,
which is a two state-system. Basic operations are unitary
operators on the Hilbert space of this system.
• The advantages of Quantum Computing lie in the aspects of
Quantum Mechanics that are peculiar to it, most notably
entanglement.
• Practical Quantum Computers are a significant ways off.

Quantum computing - A Compilation of Concepts

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