They say superposition enables qubit to live in linear superposition of two states. I.e. \begin{equation} |\phi\rangle = a|0\rangle + b |1\rangle \end{equation}
Why are we interested only in linear combination of the two states ?
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$\begingroup$ $a,b$ are complex numbers, so for instance $\sqrt{\frac{3}{4}}|0 \rangle + \sqrt{\frac{1}{4}}i|1 \rangle$ means that there is 75% chance when we measure the qubit to find it in the state $|0\rangle$ and 25% chance to find it in the state $|1\rangle$. There are only these 2 possible results, because we never measure and see a composite state of a single qubit like $|0\rangle|1\rangle$ or $|0\rangle + |1\rangle$, but always a pure state of either $|0\rangle$ or $|1\rangle$. $\endgroup$– JamesCommented May 31, 2023 at 14:28
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$\begingroup$ I wouldn't say we are "only" interested in linear combinations. If we don't consider superpositions then we're really just doing classical logic (fliping 0's and 1's). quantum mechanics allows these superposition states. Therefore we can use quantum gate operations to manipulate these superposition states. In certain specific cases doing a calculation with quantum gates can be shown to be advantageous compared to other approaches. $\endgroup$– CallumCommented May 31, 2023 at 14:49
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$\begingroup$ The states $|0\rangle$ (b=0) and $|1\rangle$ (a=0) are perfectly valid states for a qubit to be in. But in quantum computing we can manipulate superposition states as well. $\endgroup$– CallumCommented May 31, 2023 at 15:09
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A qubit is defined as living in a 2-dimensional Hilbert space. This means one cannot define more than 2 linearly independent basis states. Since $|0\rangle$ and $|1\rangle$ are linearly independent (they are actually orthogonal), any other state must be in their span, so written as a linear combination $a|0\rangle+b|1\rangle$. This is a fact from linear algebra, not from quantum mechanics. (Mixed states are a bit more general, but same idea of 2 linearly independent basis states.)
Now of course one can define another quantum system that lives in an $n$-dimensional Hilbert space! Then we will talk about linear combinations of $n$ different basis states, like $a|0\rangle+b|1\rangle+\cdots + z|n-1\rangle$. These are still interesting for quantum mechanics! It just so happens that most quantum computational algorithms start with qubits as their basic ingredients, in analogy with bits for classical computation, but there is no physical reason to be restricted to such. In fact, one can even talk about infinite dimensional Hilbert spaces; people do this all the time.
answered May 31, 2023 at 17:25