QUANTUM COMPUTING: SHORS FACTORING ALGORITHM A MATHEMATICAL - - PowerPoint PPT Presentation
QUANTUM COMPUTING: SHORS FACTORING ALGORITHM A MATHEMATICAL DESCRIPTION OF THE FUNCTION OF SHORS ALGORITHM AND THE BASIC PRINCIPLES OF QUANTUM COMPUTATION A HIGH-LEVEL SUMMARY Eero Jskelinen Mathematics MOTIVATION Quantum
QUANTUM COMPUTING: SHOR’S FACTORING ALGORITHM
A MATHEMATICAL DESCRIPTION OF THE FUNCTION OF SHOR’S ALGORITHM AND THE BASIC PRINCIPLES OF QUANTUM COMPUTATION
Eero Jääskeläinen Mathematics
A HIGH-LEVEL SUMMARY
MOTIVATION
Quantum computation is at the heart of the intersection of physics and mathematics.
Quantum computation provides solutions to real problems
AIMS AND SCOPE Not…
The development of a new method or invention.
But instead…
A complete introduction to a revolutionary algorithm in a way that provides new intuitions.
Including original proofs and discussion.
A focus on the practical implementation of the algorithm.
A mathematical description of the function of the quantum process.
Highlights the novelty, peculiarity and elegance of QC.
A light or popularized introduction to quantum computing.
Detailed description including proofs, but largely self-contained.
MATHEMATICAL BACKGROUND
Quantum computation is described in the language of complex linear algebra
Dirac bra-ket notation useful in representing quantum states
Unitary operations represent quantum state evolution, i.e. quantum gates
n
v v v =
n
w w w =
1 1 1 1 1 2 U = −
Modular arithmetic is also utilized in the discussion.
MATHEMATICAL BACKGROUND
The Euclidean algorithm is a method of computing the greatest common divisor of two integers.
1 1 1 2 2 1 2 3 3 1 2 k k k
A BQ R B R Q R R R Q R R R Q
+ +
= + = + = + =
It is shown that, for the sequence above,
This algorithm is efficient: It can be shown that the value of R at least halves in two iterations. Hence the number of iterations necessary is at most
1 2 2
gcd( , ) gcd( , )
k k k k
A B R R Q Q
+ + +
= =
2
log 2 B +
REFORMULATION OF FACTORING N
To take advantage of the properties of quantum computation, a corollary problem is developed:
,
(mod )
x a N
f x a N =
Define Find r such that
, , ,
1(mod ) ( ) ( )
a N a N a N
f r N f x f x r = +
Then
1
r
N a − ∣
If r is even
2 2
1 1
r r
N a a − + ∣
Obtain Factor of N:
2
gcd , 1
r
N a
THE QUANTUM FRAMEWORK
Qubits are the basic units of information.
States 1, 0, and superpositions of these
1 , 1 , 1 = = =
Quantum registers are systems of multiple qubits.
Numbers are stored in binary, like in classical computation.
Quantum gates are interactions that transform the states of qubits
Represented by linear matrices (operators), satisfying requirements based on the principles of quantum physics.
Drawn with Qiskit-library in python.
THE INVERSE QUANTUM FOURIER TRANSFORM AND MEASUREMENT
After some manipulation of qubits, we obtain the superposition state 1
1
t jr
− =
= +
A Periodic Superposition of States with Period r
Inverse Quantum Fourier Transform:
‘A sequence of gates that changes the state to relate to the frequency of the states.’
Measure to get an integer z.
OBTAINING A RESULT: CONTINUED FRACTIONS
It is shown that, with probability
1 2 2 2
m m
z k r −
2
4 0.40828 P =
Measured value. Some integer. Required period. Number
(2m>N2)
takes the nearest possible value to
2m z k r
for some integer k.
OBTAINING A RESULT: CONTINUED FRACTIONS
Resulting from the Euclidean algorithm, we can construct continued fraction representations of rationals:
1 2 1 2 3
1 , , , , 1 1 1 , for 1,2, ,
m j m
a j m A a a a a a a B a a a a = + + + = + +
can take on any non-zero value for some real x.
1 x
Continued fraction if
𝐵 𝐶 = 𝑏0, 𝑏1, 𝑏2, … , 𝑏𝑛
1 2
, , , , , ,
m
C a a a a c c =
For all and C,
1 c
OBTAINING A RESULT: CONTINUED FRACTIONS
Given the condition
1 2 2 2
m m
z k r −
it is proven that if
1 2
, , , ,
j
k a a a a r =
1 2
, , , , , 2
j m
z a a a a c =
then where c>1. This implies that is one convergent of (Assuming k and r are coprime so that the fraction is in lowest terms.)
𝑙 𝑠 𝑨 2𝑛
Period r found in a way that is known to be efficient computationally.
CONTRIBUTIONS
Self-contained description of a complex
and valuable algorithm.
Including introduction to the
mathematical framework, justification
fractions.
Insight into the Elegance of Quantum
Computing and its mathematics.
Inspiration to the future from
understanding a non-trivial process.
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REFERENCES (2)
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