# Quantum bits

1. Sep 30, 2007

### Levi Porter

Are quantum bits just a form of a ternary numeral system?

If something can be 0, 1, or both simultaneously, isn't the superposition just another equal value?

If the superposition of 0 and 1 were literally individual, combined, and possibly something different simultaneously, then it seems that a quantum bit could be an operational function of a numeral system base of 3, 4 or 5, I would guess.

Is there a way to translate quantum bits in a traditional numerological system?

2. Sep 30, 2007

### Hurkyl

Staff Emeritus
Individual q-bits are sphere-valued. If you choose an axis, then 0 and 1 are the north and south pole of the sphere.

3. Sep 30, 2007

### Levi Porter

Thanks for the response Hurkyl.

Does sphere-valued mean that information, pulses, or values occupy the entire volume of a sphere?

If the poles are just a predetermined and or adjustable locational points of reference including superposition, then it still seems that it could be correspondent to a numeral system?

Last edited: Sep 30, 2007
4. Sep 30, 2007

### Hurkyl

Staff Emeritus
The problem is that, unlike a classical bit, you cannot directly observe its value: measuring a qubit along some axis turns it into a classical bit. If the value was originally near the north pole, then the qubit is more likely to turn into a 1. If it was near the south pole, then it was more likely to turn into a 0.

(Incidentally, all orientations I have used are an arbitrary choice)

The art of quantum computing is to get your qubits to interact without measuring them while the computer is running -- you only want to measure them at the very end, at which point you are very likely to see the answer you wanted.

Of course, when writing your algorithm, you would design it to use the full set of values. But I have no idea how it would correspond to a numeral system.

Incidentally, I don't see how classical bits correspond to a numeral system either.

Last edited: Sep 30, 2007
5. Sep 30, 2007

### Levi Porter

I could be misinterpreting...anyway, my understanding was that conventional bits are generated by electric pulses of either on or off, translated into or as a 1 or 0, and used as a code assigned to all digital information originating from a functionality of two possibilities referred to as the binary numeral system or base-2 number system.

At least this is my understanding from http://en.wikipedia.org/wiki/Binary_numeral_system

6. Sep 30, 2007

### -Job-

Not really, for one because upon measurement a qubit will always have the value 0 or 1. For example, if i have 4 bits then i can store $2^{4}$ states, and pass any of them to you and you'd be able to recognize which state i had passed you (0100, 0011, etc). On the other hand if i have 4 qubits, then any of the $3^{4}$ quantum states i send you will always collapse to one of $2^{4}$ states, so although we can interpret a qubit as a base-3 digit, we can't use it as base-3 digit for storing and passing information.

In quantum computation we also use more information for each qubit than can be stored ina base-3 digit. For example rather than having the three states:
0, 1, 01
In quantum computation you might have:
[x] 1, [y] 0
Where $x^{2}$ gives the probability of the qubit being 1 and $y^{2}$ gives the probability of it being 0.

Therefore a qubit can have any number of states defined by [x y], where $$x^{2} + y^{2} = 1$$.

In operation, a quantum machine isn't equivalent to a base-3 machine either (which is actually equivalent to a base-2 machine, as they both can be modeled by a turing machine). When a quantum computer with n qubits performs an operation, it's performing an operation on 2^5 states (the number of states that the system of 5 qubits is in a superposition of). An operation changes the probabilities of each possible state, the one with the highest probability being the most likely to be revealed upon measurement.

This is something that a system of conventional bits would need years to model for even some small number of qubits.

Scott Aaronson has in his blog a pretty good explanation of Peter Shor's quantum factoring algorithm using simple terms:
http://scottaaronson.com/blog/?p=208

I particularly like one of the explanations given in one of the comments:
http://scottaaronson.com/blog/?p=208#comment-10026

7. Oct 2, 2007