THE QUANTUM APPLICATIONS

OF ABSTRACT MULTI-LINEAR

ALGEBRA

GYULA RÁBAI

SUMMARY

DID YOU CATCH THAT?

SUMMARY

FROM SET TO MAGMA.

There was a list of things. [Set]

*Start to all of Maths BTW

FROM SET TO MAGMA.

There was a list of things

called M.

Each pair of things from M

could be ‘composed’ to form

another thing from M.

[Set]

[Total Binary Closure]

*Start to all of Abstract Algebra BTW

MAGMA = SET + (BIN-ARITY +

TOTALITY + CLOSURE)

There was a list of things (such as a and b) called M.[Set]

All possible pairs of things from M can be

‘composed’ to form something else from M. [Total Binary Closure]

ABELIAN GROUP = MAGMA + A LOT OF

RULES There was a list of things (such as a and b) called M.[Set]

All possible pairs of things from M can be

‘composed’ to form something else from M. [Total Binary Closure]

The following is true for all possible a, b or

c in M

(a ◦b) ◦c = a ◦(b ◦c) [Associativity]

There is an e for all possible a where a ◦e = a [Identity]

There is an ~a for all possible a where a ◦~a = e[Inversibility]

a ◦b = b ◦a [Commutativity]

COMMUTATIVE/ABELIAN GROUP

[Inversibility]

GROUP EXAMPLE - INTEGERS

For any integers a, b and c:

All integers can add to make another integer [Total binary closure]

a + 0 = a [Identity]

a + (-a) = 0 [Inversibility]

a + b = b + a [Commutativity]

a + (b + c) = (a + b) + c [Associativity]

COMMUTATIVE/ABELIAN GROUP

[Inversibility]

QUOTIENTS

Let there be an equivalence where for any integer a:

a ~ a + 2

QUOTIENTS

Let there be an equivalence where for any integer a:

a ~ a + 2

Here, the only distinction between numbers is even or odd.

QUOTIENTS

Let there be an equivalence where for any integer a:

a ~ a + 2

Here, the only distinction between numbers is even or odd.

These are called equivalence classes and are put in [] generally.

QUOTIENTS

Let there be an equivalence where for any integer a:

a ~ a + 2

Here, the only distinction between numbers is even or odd.

These are called equivalence classes and are put in [] generally.

GENERAL QUOTIENTS

For any* equivalence, you can get the set of the equivalence classes

over a set using the mod operator: /

(Note this is not the standard notation for enforcing equivalences. But

we never go quantum if we stay bogged down in the notation)

* You generally need to keep the algebraic structure the same:

don’t break the rules you have already set up.

FIELD SUMMARY

You can add and minus (as per a group)

You can multiply and divide (as per a group)

AND

a(b + c) = ab + ac [Distributivity]

E.g. like a complex or real number

A VECTOR SPACE

You can add vectors (as per a group)

You can scale vectors with a member of a field.

Vectors are distributive, inversible … with respect to scalar

multiplication.

SO WHAT ABOUT QUANTUM?

PHOTON

PHOTON FROM THE FRONT

POSSIBLE POLARIZATIONS

POLARISATION IS A VECTOR

ALL VECTORS ARE THE LINEAR

COMBINATION OF BASES

ALL VECTORS ARE THE LINEAR

COMBINATION OF BASES

ALL VECTORS ARE THE LINEAR

COMBINATION OF BASES

ALL VECTORS ARE THE LINEAR

COMBINATION OF BASES

BASES

SUPERPOSITION

2 PLACES AT ONCE?

Superpose being here and being there. Each one has a

probability. Unfortunately, the more mass the lower the

probability that you are elsewhere and not here:

Conclusion: Obesity saves you from quantum teleporting to

Mars.

POLAROIDS EXPERIMENT:

•Amplitude is meaningless (experimentally).

•Probability

REMOVE MAGNITUDE

CIRCULAR IS AN IMAGINARY POLARIZATION

WHEN YOU MULTIPLY BY

•You get the same polarization because | |=1

REMOVE GLOBAL PHASE

QUANTUM COMPUTING

You can let up be 1 and down be 0. You have a qubit!!!

QUANTUM COMPUTING

You can let up be 1 and down be 0. You have a qubit!!!

Qubits cannot be cloned --> safe way to transfer encryption

keys.

MULTIPLE CLASSICAL BITS

You take the ‘direct sum’ of the vector spaces representing each qubits

states e.g.

V represents qubit 1’s state, W represents qubit 2’s state

This means that we add together the linear combination representing v

and the one representing w:

The new vector has 4 new bases instead of 2:

MULTIPLE QUBITS

•You take the tensor product.

•A tensor is a vector space combined of at least 2 other vector

spaces defined over the same field as follows

•Let us call the resulting vector space T. We introduce the

following rules:

BASES FOR MULTIPLE QUBITS

In quantum computing, you take the ‘tensor product’ and not the

‘direct sum’ of the vector spaces to get the multi-qubit state-

space.

If we try 3 qubits:

It works out to be according to our rules about a tensor space T

(last slide) and its distributivity:

*Taken from Quantum Computing: A Gentle Introduction

COMPARISON

•3 classical bits gives 3x2=6 bases. 3 qubits gives 2^3=8 bases

•n qubits gives classical bits worth of possibilities.

•The difference in the number of possibilities as n grows:

SOME STATES CANNOT BE REPRESENTED AS

THE SUM OF 3 INDIVIDUAL QUBITS

•The tensor product over is the outer product.

•That results in all

•Most matrices cannot be made

with an outer product but are still

REMEMBER!