Quantum computing and ambiguity

Question

I do a bit of hobby programming and I often search the internet for little oddities that are fun to ponder over. I have read a few passages that try to explain quantum computing to the layman like myself. I have read of the Qubit, the more 'power' version of the bit, and its bad habit of being in superposition. This, to me, sounds as if it sits halfway between 1 and 0.

So, I reason that one can create a qu-binary number with these; something resembling a ternary number, made from 0's, 1's and 1/2's (or Q's). I have read that a quantum computer has more 'power' when it comes to computation because one qu-value is a possibility between at most n^2 regular values in n bits. I have constructed a little problem with this value when you try to store a specific set of regular values in a qu-value.

Imagine a value is a superposition between 2 and 3. In qu-binary, I would write "10 or 11 -> 1Q", as the last bit is "both". OK, so this works. But what about real values 2, 3 and 4 in superposition? in my ternary notation "QQQ" is potentially any of the possibilities 0 through 7, and so actually represents a whole lot more values than I want!?

My question is, how does it really work? Am I thinking about it all wrong? Because this is how the whole subject of Quantum computing looks like from the outside. Or is this an example of quantum computing's non-determinism? I assume all bits are completely isolated from one another and have no qu-knowledge of any other. Maybe something obscure like quantum gates sharing information between bits could explain the problem; or if the bits represent continuous probabilities. I don't know. Could someone explain it for me?

Answer 1

You need to be a bit (pardon the pun) more strict about the size of the (Hilbert) space you're playing with. A qu_bit can be in a superposition of two (pure) states, but not more. For this reason, "real values 2, 3 and 4 in superposition" doesn't make sense. To draw an analogy to the binary system you mentioned, it's as if you're trying to stuff large numbers into a bit.

This restriction appears more clearly in the visual representation of a pure state, the Bloch sphere.

Secondly, you need to be careful when drawing analogies between bits and qubits. For example, your statement "all bits are completely isolated from one another and have no qu-knowledge of any other" is wrong in the general case when there may be entanglement between qubits.

I started learning quantum computing with some CS knowledge, and this was a very helpful reference. I think it'll get you started on the right path.

Answer 2

A qubit is a (complex) vector, with $2$ complex coordinates, you may write it :

$$\vec q = \alpha \vec 0 + \beta \vec 1 \tag{1}$$ Here $\alpha$ and $\beta$ are complex quantities and $\vec 0, \vec 1$ is a basis, so you have $\vec 0.\vec 0= \vec 1.\vec 1=1$, and $\vec 1.\vec 0 = \vec 0.\vec 1=0$. The qbit is generally normed to $1$, so you have $|\alpha|^2+|\beta|^2=1$. A standard notation, instead of writing $\vec q$ is to note it $|q\rangle$ (ket). An other notation is to write the complex conjugate vector $(\vec q)^*$ as $\langle q|$ (bra). A last useful notion is the "hermitian product" : $\langle q'|q\rangle = (\vec q')^*.\vec q$. The orthonormality of the basis could be written now : $\langle 0|0\rangle = \langle 1|1\rangle = 1$, $\langle 0|1\rangle = \langle 1|0\rangle = 0$

So you see that this has nothing to do with classical bits.

The most interesting consequence is that you may study $2$ entangled qbits, where the vectorial nature of the qbits have important consequences on quantum correlations, which cannot be reproduced by classical correlations.