Quantum computing
Alex Karassev
Quantum Computer
Quantum computer uses properties of elementary
particle that are predicted by quantum mechanics
Usual computers: information is stored in bits
Quantum Computers: information is stored in qubits
Theoretical part of quantum computing is developed
substantially
Practical implementation is still a big problem
What is a quantum computer good for?
Many practical problems require too much time if we
attempt to solve them on usual computers
 It takes more then the age of the Universe to factor a
1000-digits number into primes!
The increase of processor speed slowed down
because of limitations of existing technologies
Theoretically, quantum computers can provide
"truly" parallel computations and operate with huge
data sets
Probability questions
How many times (in average) do we need to toss a
coin to get a tail?
How many times (in average) do we need to roll a
die to get a six?
Loaded die: alter a die so that the probability of
getting 6 is 1/2.
Quantum computers and probability
When the quantum computer gives you the
result of computation, this result is correct
only with certain probability
Quantum algorithms are designed to "shift" the
probability towards correct result
Running the same algorithm sufficiently many
times you get the correct result with high
probability, assuming that we can verify
whether the result is correct or not
The number of repetition is much smaller then
for usual computers
Short History
1970-е: the beginning of quantum information theory
1980: Yuri Manin set forward the idea of quantum
computations
1981: Richard Feynman proposed to use quantum
computing to model quantum systems. He also
describe theoretical model of quantum computer
1985: David Deutsch described first universal
quantum computer
1994: Peter Shor developed the first algorithm for
quantum computer (factorization into primes)
Short History
1996: Lov Grover developed an algorithm for search
in unsorted database
1998: the first quantum computers on two qubits,
based on NMR (Oxford; IBM, MIT, Stanford)
2000: quantum computer on 7 qubits, based on NMR
(Los-Alamos)
2001: 15 = 3 x 5 on 7- qubit quantum comp. by IBM
2005-2006: experiments with photons; quantum
dots; fullerenes and nanotubes as "particle traps"
2007: D-Wave announced the creation of a quantum
computer on 16 qubits
Quantum system
Quantum system is a system of elementary particles
(photons, electrons, or nucleus) governed by the
laws of quantum mechanics
Parameters of the system may include positions of
particles, momentum, energy, spin, polarization
The quantum system can be characterized by its
state that is responsible for the parameters
The state can change under external influence
 fields, laser impulses etc.
 measurements
Some quantum mechanics
Superposition: if a system can be in either of two
states, it also can be in superposition of them
Some parameters of elementary particles are discrete
(energy, spin, polarization of photons)
Changes are reversible
The parameters are undetermined before
measurements
The original state is destroyed after measurement
No Cloning Theorem: it is impossible to create a
copy of unknown state
Quantum entanglement and quantum teleportation
Qubit
Qubit is a unit of quantum information
In general, one qubit simultaneously "contains"
two classical bits
Qubit can be viewed as a quantum state of one
particle (photon or electron)
Qubit can be modeled using polarization, spin, or
energy level
Qubit can be measured
As the result of measurement, we get one classical
bit: 0 or 1
A model of qubit
|ψ⟩ = a0 |0⟩ + a1 |1⟩
or
vector (a0,a1 )
a0 и a1 are complex numbers such that |a0|2 + |a1 |2 =1
|ψ⟩ is a superposition of basis states |0⟩ и |1⟩
The choice of basis states is not unique
The measurement of ψ⟩ results
in 0 with probability |a0|2 and in 1 with probability |a1|2
After the measurement the qubit collapses into the
basis state that corresponds to the result
Example:
1
3
0 
1
2
2
1/4
0
3/4
1
Several qubits
The system of n qubits "contain" 2n classical bits
(basis states)
Thus the potential of a quantum computer grows
exponentially
We can measure individual qubits in the multi-qubit
system
 For example, in a two-qubit system we can measure
the state of first or second qubit, or both
The results of measurement are probabilistic
After the measurement the system collapses in the
corresponding state
Example: two qubits
|ψ⟩ = a0 |00⟩ + a1 |01⟩ + a2 |10⟩ + a3 |11⟩
Let's measure the first bit:
1
3
   
1 2
3
1
2
1 2
3
00  13 01  23 10 
0

00 
1
2

2 2
3
probability
01
11
1
result
2
9
3
3
2
7

 
10 
3
3
3
7
2

11
The coefficients changes so that the ratio is the same
7
9
Independent qubits
A system of two independent qubits
(two non-interacting particles):
1
 23 00  12 
5
3
01 
00 
5
6
01 
2
3
0 
5
3
1
3
2
 23 10 
3
2

3
3
10 
11
=
1
2
0 
3
2
1
2
1
3
15
6
5
3
11
Entangled states
There is no qubits
The value of
second bit with
100% probability
a0 |0⟩ + a1 |1⟩
b0 |0⟩ + b1 |1⟩
s.t. the state
11
22
01
01 
11
22
0
10
10
|01⟩ 1
measure the
first bit
could be represented as
1
|10⟩ 0
a0b0 |00⟩ + a0 b1 |01⟩ + a1 b0 |10⟩ + a1 b1 |11⟩
Examples
Maximally entangled states (Bell's basis)
1
2
1
2
01 
1
2
10
00 
1
2
11
Is the following state entangled?
1
3
00  01  10 
1
3
2
3
3
3
11
Quantum Teleportation
Entangled qubits A and B
A
C
1
2
00 
1
2
qubit with unknown state
that Alice wants to send to Bob
11
B
Now Bob knows
the state of B
makes А and C
entangled
makes B into C
some
transformations
measures C
Now Bob has qubit C
Operations on bits
NOT: NOT(0) =1, NOT(1)=0
OR: 0 OR 0 = 0, 1 OR 0 = 0 OR 1 = 1 OR 1 = 1
AND: 0 AND 0 = 1 AND 0 = 0 AND 1 = 0, 1 AND 1 = 1
XOR (addition modulo two):
0 ⊕ 0 = 1 ⊕ 1 = 0, 0 ⊕ 1 = 1 ⊕ 0 = 1
What is NOT ( x OR y)?
What is NOT (x AND y)?
NOT (x OR y) = NOT (x) AND NOT (y)
NOT (x AND y) = NOT (x) OR NOT (y)
Classical and quantum computation
Operations AND and OR are not invertible: even if
we know the value of one of two bits and the result
of the operation we still cannot restore the value of
the other bit
 Example: suppose x AND y = 0 and y = 0
 what is x?
Because of the laws of quantum mechanics
quantum computations must be invertible (since the
changes of the quantum system are reversible)
Are there such operations?
Yes! E.g. XOR (addition modulo two)
Linearity and parallel computations
Example: let F be a quantum operation that
correspond to a function f(x,y) = (x',y'). Then:
F a0 00  a1 01  a2 10  a3 11  
 a0 f (00)  a1 f (01)  a2 f (10)  a3 f (11)
Thus one application of F gives a system that
contains the results of f on all inputs!
It is enough to know the results on basis states
Matrix representation
Invertibility
Some matrices…
A matrix is a table of numbers, e.g.
We can multiply matrices by vectors:
1 2 3
1 42 53 6 1   11 
4  5 6   2   4 1 
 1  1 0  
1  1 0  1
2  2  1 (1)   4 
5  2  6  (1)   8 
11  (1)  2  0  (1)  1
Moreover, we even can multiply matrices!
Operations on one qubit
Quantum NOT
NOT( a0 |0⟩ + a1 |1⟩) = a0 |1⟩ + a1 |0⟩
0 1
 a 0  0 1 a 0 
NOT   
 

 a1  1 0  a1 
1  0
Hadamard gate
H( a0 |0⟩ + a1 |1⟩) = 1/√2 [ (a0 + a1)|0⟩ + (a0 - a1)|0⟩ ]
0 
1
2
0 
1
2
1
1 
1
2
1 
1
2
0
 a 0  1 1 1   a 0 
H  
  


2 1 - 1  a1 
 a1 
Two qubits: controlled NOT (CNOT)
CNOT (x,y) = (x, x XOR y)= (x, x⊕y)
0⊕0=1⊕1=0, 0⊕1=1⊕0=1
CNOT( a0|00⟩+a1|01⟩+a2|10⟩+a3|11⟩ ) = a0|00⟩+a1|01⟩+a3|11⟩+a2|10⟩
00  00
01  01
10  11
11  10
 a 0  1
  
 a 1  0
CNOT  
0
a2
  
a  0
 3 
0 0 0  a 0 



1 0 0  a 1 

0 0 1  a 2 
  
0 1 0  a 3 
How quantum computer works
The routine
 Initialization (e.g. all qubits are in state |0⟩
 Quantum computations
 Reading of the result (measurement)
"Ideal" quantum computer:
 must be universal (capable of performing arbitrary
quantum operations with given precision)
 must be scalable
 must be able to exchange data
Quantum algorithms
Shor's algorithm
 Factorization into primes
 Work in polynomial time with respect to the number
of digits in the representation of an integer
 Can be used to break RSA encryption
Grover's algorithm
 Database search
 "Brute force": about N operations where N is the
number of records in the database
 Grover's algorithm: about
N operations
Problems
Decoherence
 Quantum system is extremely sensitive to external
environment, so it should be safely isolated
 It is hard to achieve the decoherence time that is
more than the algorithm running time
Error correction (requires more qubits!)
Physical implementation of computations
New quantum algorithms to solve more problems
Entangled states for data transfer
Practical Implementations
The use of nucleus spins and NMR
Electrons spins and quantum dots
Energy level of ions and ion traps
Use of superconductivity
Adiabatic quantum computers
D-Wave: quantum computer Orion
January 19, 2007: D-Wave Systems (Burnaby, British
Columbia) announced a creation of a prototype of commercial
quantum computer, called Orion
According to D-Wave, adiabatic quantum computer Orion uses
16 qubits and can solve quite complex practical problems (e.g.
search a database and solve Sudoku puzzle)
Unfortunately, D-Wave did not disclose any technical details of
their computer
This caused a significant criticism among specialists
Recently, the company received 17 millions investments
Homework
Is the following state entangled?
1
2
00  12 01  12 10  12 11
What happens if we apply twice
 negation?
 Hadamard gate?
Thank You!
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