INTRODUCTION TO QUANTUM MATHEMATICS 3

QUANTUM BITS AND VECTOR SPACES

In this opportunity, we revise the concepts of vector, and vector spaces which allow us to represent the Quantum Bits.

As we have seen in the last publications, we use the real numbers to measure different features in nature. However, it rapidly appears in problems for which more general tools are needed, for instance:

1. If you apply some force to move a box, how do we express the difference between the angles in which you act on the box?

2. If you try to go through a river, from side to side, what is the direction you should follow, if you want to get a specific point?

3. If you are studying a fluid, and you are interested in velocity at each point of the fluid, temperature, time, viscosity, how do you collect this information to be able to predict the state of the system?

These are just a few examples of problems that motivate the introduction of the concept of vector. Let us start with the simplest case, the two-dimensional real vectors. René Descartes had the idea of disposing two perpendicular numeric real lines and use them to ubicate the point in the plane containing the lines, in such a way that any point in the plane is described with two real numbers. A vector is then represented by an arrow joining two points in the plane. Arrows being parallel, following the same direction, and having the same length are all equivalent and represent the same vector. A typical notation for a vector joining the points

The sum between two vectors is naturally defined component by component, as you can see in the picture below:

As a consequence of the Pythagoras's Theorem, the length of a vector

In order to be mind of this concept inside our real world, we invite you to think that the vector

represents a force acting a body sited in the origin

in the direction of

with and intensity or magnitude of the force

Now, if we duplicate the intensity of this force but maintaining the direction, we can represent the new force vector as

This motivates to define multiplication by a scalar

Now, it is interesting to study the following expression:

Any force vector in the plane can be decomposed as a sum of two basis forces, one acting on the horizontal direction

and another acting on the vertical direction

This is awesome! But, why? Just because this simple but deep observation gives us, for example, the basics for our quantum computing at least, without counting the theories of electric fields, magnetic fields, gravitational fields among others which allow us to flight using an aircraft or to take a radiography to help in our health care.

Let us define the standard n-dimensional complex vector space:

where each vector is a linear combination of the elements of the standard basis whose elements are the vectors

where the one number is in the i-th component of the vector, that is to say

In general, it is possible to define an abstract complex vector space starting with arbitrary elements, that for sure have a good significance for us. Let´s say

is a set of basic elements which we want to be a basis for our abstract complex vector space. We identify any abstract linear combination of the elements of

with a vector containing the scalars in this way:

In this setting, the magnitude of a vector is given by

where for a complex number

its modulus

This point of view is very useful, as we can see right now by means of the introduction of the atoms of quantum computing.

Quantum Bits

In classical computation, the bits can have just two possible “states”, let´s write

The enlightening idea is to interpret the complex vector

as a “particle” that has a probability

of being measured in the state, and a probability

of being measured in state. So, we think of a q-bit as a powerful classical bit that has the special feature of being in both states “at the same time”! This is a crucial point in our travel through the Quantum Computing Universe. Several concepts need to be analyzed: What particles are we talking about? What means probability? What means measuring? And What means at the same time? These are several and not easy questions to solve, but we can discuss a little bit on it. At this time, focus on the probability question.

Imagine a coin. When you throw it, there are two (mainly) possible results, one for each side. If we denote these possible results as, in a conveniently way, side 0, and side 1. During the flight the coin is not over just one side, we can think that it is in the state

Which is interpreted as a state that after measurement, in this case it would be “after getting on the floor”, approximately 50 of each 100 flips should result in seeing side, and another approximately 50 of each 100 flips should result in side.

Finally, if we have two q-bits (think of them as coins mentioned above), then there are four possible results after flipping both: The complex abstract vector

Represents a quantum state in which after many flipping, we expect to see about 30% of the results 00, and about 50% of the results to be 10.

INTERESTING! We guess you don’t expect it. See you at the next delivery.

Bibliography

• The Atoms of Computation (qiskit.org) JUPITER BOOK COMMUNITY, IBM.
• Magnetic field - Wikipedia
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556

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