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Quantum Computing
Nature and the way things are
Applications
Moore's Law:
Rapid Feedback
Learning a larger number of concurrent patterns decreasing the time needed when learning it by a sequential approach
Quantum Computers
CAN
Break the “unbreakable” classical ciphers with seconds.
Generate “unbreakable” quantum ciphers.
basic unit
of information in a quantum computer
Data registeries: every qubit holds only 1 registry of information
Atomic or subatomic particle
A unit vector in a two-dimensional complex vector space defined with a fixed orthonormal basis.
& are complex and represent the square root of the probability of the states
Orthonormality
qubits are in superposition of &
represent vector states visually
Any vector is represented by the two angles and on the surface of the sphere
In general
Number of basis vectors:
, where b is the number of basis vectors in a single qubit, and n is the number of qubits in the system.
Basis vectors
We use tensor product which gives us all the possible values for the states of the qubits in the system.
Basis vectors
the basis vectors are:
n=3
Basis vectors
the minimum number of real numbers required to fully describe the state of that system
We have 4 number: a,b,c,d
From orthonormality:
Non physical sate representatoin uniqueness:
We have 2 constraints
DOF: 4-2=2
Classically
Combine 2 systems
New System
DOF-System1=2
DOF-NewSytem=2+2=4
DOF-System2=2
Qubits
4 complex numbers: 8 real numbers
2 constraints
DOF: 8-2=6 > 4 ?
spooky action at a distance
Two quantum systems connected to each other, then any measurement on one system affects the other system instantly even if they were separated light years distances.
The states of these two qubits are shared and dependent on each other which adds more degrees of freedom
Probability of qubit 1 being measured 0, and qubit 2 being measured 0: 0.5
Probability of qubit 1 being measured 0, and qubit 2 being measured 1: 0
Probability of qubit 1 being measured 1, and qubit 2 being measured 0: 0
Probability of qubit 1 being measured 0, and qubit 2 being measured 0: 0.5
Probability of the qubits being measured with same values: 0.5 + 0.5 = 1
Probability of the qubits being measured with different values: 0 + 0 = 0
the probability of the first qubit being measured 0 is 0.5, but the probability of the second qubit being measured 0 is either 0 or 1 and it totally depends on the first measurement.
Test For Entanglement (Entanglement Criteria):
If a state vector of a multi-qubit system can be expressed as a tensor product of single-qubit states, then it is not entangled.
i.e
A B is always unentangled!
Test of Entanglement
Find such that:
A contracidction
what Alice does with her particle affects Bob's measurement on his particle
what Bob does with his particle affects Alices's measurement on her particle
There is some hidden state which we cannot detect and this hidden state is local to each particle. The behavior of the entangled pair is determined by these hidden local states.
No local state can predict the results of an experiment and state has to be shared.
The probability of the qubit (S) being measured in alignment with an apparatus (A):
The magnetic property of electrons
Spin is measured to be up or down
The initial spin of the electron??
P
st
1 measurement
nd
2 measurement
rd
3 measurement
P(aligned)
P(anti-aligned)
electromagnetic wave
The electric and magnetic oscillations are at right-angles to the direction of propagation of the wave
Each photon that is directed along the z-axis has its own angle, which is called the angle of polarization
plastic filter of light polarization.
Polarizing sheet:
Complete Measurement
P(00) =
P(10) =
P(11) =
P(01) = 0
Partial Measurement
We measure only one of the qubits
If qubit 1
P(0) =
:
P(0) =
:
1- Measurement changes the state of the system.
2-Measurements cause the superposition to collapse.
3- The first measurement might be probabilistic.
4- The first measurement changes the state of the qubit so that subsequent measurements are deterministic.
5- If the apparatus used is aligned with the qubit’s state, then the result is deterministic.
6- Random behaviour appears only when we measure the state in a direction that is not aligned (or anti-aligned) with the state.
7- The randomness caused by measurement is irreversible.
8- When partially measuring a system, all the system is affected and inconsistent states in the superposition disappear.
Quantum gate is any operation that changes the state of the qubit.
Physically
Microwave pulses carefully calibrated and in resonance with the qubit’s transition frequency
Mathematically
Reversible matrix transformations .
Unitary Matrix:
No-Cloning Theorem:
Unkown systems cannot cloned.
entangled qubits have a shared state
Charles Bennett and Gilles Brassard in 1984
Single Use Shared-Secret
Securing the Single Use Shared-Secret
Vertically polarized photon
Bit: 1
Horizontally polarized photon
Bit: 2
Quantum Channel
Bob
Alice
Aligned
Anti-aligned
Eve
Cryptography with entanglement
P(Alice measuring 1 & Bob measuring 1 ) = 1
P(Alice measuring + & Bob measuring +) = 1
P(Alice measuring 1 & Bob measuring +) = 1/2
P(Alice measuring + & Bob measuring 1) = 1/2
Mathematical trick to encode 2 bits of information onto a single qubit.
CNOT
4 Possibiliets:
To Bob
Hadamard
The process of transferring an unknown state of a qubit to another qubit
No qubits are transferred, Alice only sends information.
Alice Partailly measures her qubits.
result
Bob
Ability to be in superposition
n
H generates a superposition of all 2 possible combinations of the inputs
n
2 function values f(x) along with their corresponding input value x in one cycle
First Algorithm By David Deutsch in 1985
If a function determine whether is constant or balanced.
constant
balanced
:
Multiple bit generalization for Deutsch's problem
:
for
i=1:
00
0
Classically:
01
i=2:
0
i=3:
10
?
Quantum:
designed by Peter Shor in 1994 to solve the
integer factorization problem
The fundamental theorem of arithmetic
Every positive integer greater than one can be written uniquely as a product of primes
Used in Most protected Protocols (RSA): HTTPS, SSL ...
Quantum Algorithm for Period Finding
STEP-1: Initialize both registers to
STEP-2: Apply QFT on the first register :
M
STEP-3: Apply the modular exponentiation function U :
f (x)
a, N
STEP-4: Paritally measure the second register
The first register:
-1
STEP-5: Apply (QFT ) on the first register:
M
Superposition states with period , with no offset.
A random multiple of
STEP-6: Measure the first register :
gcd(k, ,M)=
Whenever gcd(k, )=1
takes an M-dimensional complex vector to another M-dimensional complex vector with modified amplitude.
th
the n roots of unity are the complex solutions of the equation: .
Phase Estimation
Applying Wlash Transformation:
Classical Algorithm for integer factorization
STEP-1: Pick any integer a < N
STEP-2:
A- If a is not relatively prime to N. Then a is a common factor of N, a=p or a=q. #DONE
B- If a is relatively prime to N, that is gcd(a.N)=1. Then proceed to Step-3.
x
Step-3: Compute the period r of f (x) = a mod(N).
a,N
STEP-4: If r is odd, go back to STEP-1. If r is even, proceed to STEP-5.
r/2
STEP-5: If a + 1 = 0 mod(N), go back to STEP-1. Otherwise, proceed
to STEP-6.
r
STEP-6: a=1 mod(N), so a - 1 = 0 mod(N). Then, there exists a natural
number k, such that: a - 1 = k.N, where r is even.
r
r/2
So, (a - 1 )(a + 1) = k.N = k.p.q
r/2
STEP-7: Let p = gcd ((a - 1), N), and q= gcd((a + 1), N). #DONE
STEP-3
Integer Factorization
Period Finding
r
x
for the function f (x) = a mod(N), gcd(a,N)=1
a, N
x
f (x)=3 mod(10)
3, 10
r = 4
field sieve algorithm
Could Quantum Supremacy be any time soon?
Counter-intuitive!
Noise and coherence
Error-Correction
To successfully simulate the quantum world, we need to use a machine that obeys the quantum laws of physics