3. Content we'll see over
● Motivation and goals
● Introduction to quantum computing
● Quantum computing basis
● Postulates overview
● Shor's algorithm
● Introduction to quantum cryptography and BB84
● BB84 protocol functioning
● Example of the protocol
● Explanation of BB-84 Web Simulator with a demonstration
● Future of the project
● Conclusions
4. Motivation and goals
Motivation Goals
➢ Discover a little bit more ➢ Comprehend the
the quantum world principles of quantum
computing
➢ Make out how is evolving
the quantum computing ➢ Understand its most
from a computer powerful algorithm
engineer view ➢ Interpret the future
cryptography which is
based on quantum
principles
Practical goals
➢ Implementation of Shor's algorithm in QCL
➢ Development of a Web simulator based on BB-84 protocol
5. Origins of quantum computing
● In 1980's decade several physicists, such as Paul Benioff and
Richard Feynman, proposed the idea of using quantum laws in
the field of computation in order to simulate quantum problems
(or dilemmas)
● Moreover in a few years we will hit natural limits on the size of
transistors
• Max out on the number of transistors per chip
• Reach the minimum size for transistors
6. What is quantum computing?
● Definition: Use of quantum phenomenon to perform
computational operations
● Operations are done at an atomic level
● It allows the encoding of bits as physical states
– Direction of spin of an electron
– The direction of polarization of a photon
– The energy level of an excited atom
7. What is a qubit?
● A qubit is a bit represented by a quantum system is governed
by the laws of quantum physics.
● So a qubit (an electron, commonly used as a qubit, is always in
one of two spin states) can take two distinct values:
– State 0 goes with the spin up
– State 1 goes with the spin down
8. Quantum vs. Transistor
● In conventional computing, a bit is 0 or 1. Equivalent to
switching a light switch on and off.
● But under a quantum basis, a qubit is 0 and/or 1.
● This superposition of states is what makes quantum computing
fundamentally different and so powerful.
9. Postulates
● A set of 4 mathematical postulates describes quantum
computing:
– State Space
– Evolution
– Measurement
– Composite systems
10. State Space Postulate
● Associated with any isolated physical system is a complex
vector space with inner product (a Hilbert space) known as the
state space of the system. The physical state of the system is
completely described by its state vector, which is a unit vector
in the system’s state space.
● The simplest quantum mechanical system is the qubit, which
has a 2-d state space. With an orthonormal basis {|0⟩,|1⟩} the
most general state vector of a qubit can be expressed as
|Ψ⟩ = α|0⟩ + β|1⟩,
where α,β Є C and IαI²+IβI² = 1.
11. Evolution Postulate
● The evolution of a closed quantum system is described by a
unitary transformation. The state vector |Ψ⟩ of a system at time
t₁ is related to its state vector |Ψ'⟩ at time t₂ by a unitary
operator U which depends on t₁, t₂,
|Ψ'⟩ = U|Ψ⟩.
● The Pauli matrices and the Hadamard gate (H) are all 2x2
unitary matrices, and so describe valid qubit transformations.
12. Measurement Postulate
● We cannot know the exact state of a quantum system because
measuring implies disrupting.
● Regardless of a certain superposition α|0⟩ + β|1⟩, we get |0⟩
with a probability ||α||² or |1⟩ with a probability ||β||² as a
result of the measure.
● Let's recall the first postulate where IαI²+IβI² = 1, which
means the quantum state is unitary (normalized).
13. System Composition Postulate
● The state |Ψ⟩ of a composite system is the tensor product of the quantum
sub-systems:
U|Ψ⟩⊗U|Ψ⟩ = (U⊗U)(|Ψ⟩⊗|Ψ⟩)
● A 2-qubit system example with |Ψ⟩=α|0⟩+β|1⟩ and |Ψ⟩=γ|0⟩+δ|1⟩.
14. Bloch Sphere
The Bloch Sphere is a
representation of a qubit, the
fundamental building block of
quantum computers.
The most general state
vector of a qubit can be
expressed as
where θ is the polar angle
and φ is the az-imuthal angle
of the unit vector in 3-d real
vector space.
15. Important as well...
Quantum computing offers:
reversible computing which means we can recover the input
data with the output data.
a framework based on quantum gates (similar to the logic
gates) in order to design algorithms.
a Quantum Fourier Transform which is much more powerful
than the classical Fast Fourier Transform. (Essential maths
for factorizing)
16. Quantum vs. Fast
● The inner implementation of the QFT requires profound math
concepts and prolonged understanding.
● QFT is vital to factorize numbers in Shor's algorithm in the
same way as FFT in classical computing algorithms.
● Factorizing an N-bit number implies a 2^N-bit register on which
a Fourier Transform is applied.
● Cost analysis:
– Classic way →
– Quantum way →
17. Quantum vs. Fast (II)
As a sample of quantum
computing potential, on
the right and bottom are
shown the cost
representations of
factorizing an RSA code
of 170 decimal digits (563
bits):
18. Quantum Computer Language
● QCL is the most advanced implemented quantum programming
language.
● Its syntax resembles syntax of the C programming language
and classical data types are similar to data types in C.
● A quantum programme example:
qureg x1[2]; // 2-qubit quantum register x1
qureg x2[2]; // 2-qubit quantum register x2
H(x1); // Hadamard operation on x1
H(x2[1]); // Hadamard operation on the first qubit of the register x1
20. Shor's algorithm (N)
I. Pick a random number a < N.
II. Compute gcd(a, N) – greatest common divisor.
III. If gcd(a, N) ≠ 1, then there is a non-trivial factor of N, so we're done.
IV. Otherwise, use the period-finding function to find r, the period of the following
function: f(x) = a^x mod N.
V. If r is odd, go back to step I.
VI. If a^(r /2) ≡ −1 (mod N), go back to step I.
VII. gcd(a^(r/2) ± 1, N) returns a nontrivial factor of N. We are done.
For instance: gcd(4 ± 1, 15) → 3 (factor of N)
21. Quantum Cryptography
As of this slide, a slight hop in the same quantum world is done,
from computing to cryptography.
22. BB-84 Protocol
● History:
– Published in 1984
– Charles Bennett and Gilles Brassard
● Resources:
– Same quantum mechanics postulates
– Utility of light particles (photons)
– Taking advantage of optical fiber
23. BB-84 Essential Tips
● Each photon represents a bit of information
● Alphabet of communication extended:
Basis 0 1
+ ↑ →
x ↘ ↗
● Two actors intervene in the protocol: Alice and Bob
● A third actor might be a spy
24. BB-84 Step by Step
1. Alice generates a random key
2. Alice codifies each bit into a qubit and then sends these over to
Bob
3. She uses two alphabets to codify the key
0 1
Rectilinear basis 0º → |0⟩ 90º ↑ |1⟩
Diagonal basis 45º ↗ |0⟩+|1⟩ 135º ↘ |0⟩-|1⟩
4. Bob chooses randomly the alphabet to measure
25. BB-84 Step by Step
5. Bob results
See whether basis coincides or not. In other words, check compatibility.
26. BB-84 Estimation Error
To clarify how much error Bob produces to the global key:
Bob chooses qubit basis randomly
50% 50%
Wrong Correct
Reads qubit state Reads qubit state
50% 50% 100%
Wrong Correct Correct
It is estimated that 25% of the key is wrong
27. Example
Alice's bit 0 1 1 0 1 0 0 1
Alice's basis + + x + x x x +
Alice's polarization ↑ → ↘ ↑ ↘ ↗ ↗ →
Bob's basis + x x x + x + +
Bob's measurement ↑ ↗ ↘ ↗ → ↗ → →
Public discussion
Shared Secret Key 0 1 0 1
28. BB-84 Security
● If Eve does the same as Bob:
Eve knows 75% of the bits Alice
sent.
Bob knows 75% of the bits Alice
sent.
● Nonetheless Alice and Bob
are able to detect Eve.
● Eve causes an error of 25%
in Bob's key.
29. BB-84 Web Simulator
● Concept
– As BB-84 is quite unknown, it's been developed a Web
simulator based on its cryptographic idea.
● Technology
– Ruby on Rails (Back-end)
– Twitter Bootstrap (Front-end)
– PostgreSQL (Data Base)
31. Demonstration
➔ Simulate a 32-bit key distribution between Alice
and Bob
● Addition of a 10% communication error due to the
channel physics
● Setting 50% digit proportion by which Eve is
going to be detected (when simulating with an
spy in the middle of the channel)
Let's start
32. Time Planification
Research on Quantum Computing - 200 h
Understanding and implementation of Shor's
algorithm - 100 h
Research on Quantum Cryptography - 100 h
Learning Ruby on Rails - 100 h
Web Simulator development - 100 h
Composition of report - 200 h
33. Future of the project
● Detailing the Discrete Fourier Transform (QFT)
● Go deeper about the postulates
● Implement Deutsch and Grover's algorithms
● Explanation of the different technological attempts to build a
quantum computer
34. Conclusions
● Long experience on computer science is reflected to the
evolution of the research on quantum computing. Many errors
haven't been made and won't be made again.
● Quantum computers won't come true to a few years more. But
quantum cryptography has been proved in several countries
efficiently, which means that it won't take that long.
● Ruby on Rails is an excellent technology to work with. It permits
build up consistent web applications in a few time once one
knows how it works.
