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I am a quantum computing enthusiast, and recently I stumbled upon this the following two propositions:

$$ \alpha|1\rangle + \beta|0\rangle$$

What does this mean?

My understanding of this is that: the two bits, 1 and 0 are represented in a state of superposition, hence the bra-ket notation (which is commonly used for quantum mechanics), i.e., this is a qubit.

Or is there a more concise explanation of this?


$$(\alpha|1\rangle + \beta|0\rangle)^N$$

What does it mean to raise this quantity (of superimposed bits, i.e., qubit) to the $N$th degree? If we take $2^N$ where $N$ is the number of qubits then this tells us the number of bits in the desired number of qubits.

Is what I have stated in this post, generally correct?

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1 Answer

The expression \begin{align} \alpha|1\rangle+\beta|0\rangle \end{align} is the state of a single qubit written as a linear combination of the state $|1\rangle$ and the state $|0\rangle$. If you were to make a measurement on this qubit, then you would either return $1$ or $0$ with probabilities $|\alpha|^2$ and $|\beta|^2$ respectively.

The expression \begin{align} (\alpha|1\rangle + \beta|0\rangle)^N \end{align} is probably a shorthand for \begin{align} \underbrace{(\alpha|1\rangle + \beta|1\rangle)\otimes\cdots \otimes(\alpha|1\rangle + \beta|0\rangle)}_{N-\text{factors}}, \end{align} namely the $N$-fold tensor product of the state $\alpha|1\rangle + \beta|1\rangle$ with itself. This represents the state of $N$ qubits.

If you make a measurement on a system of $N$ such qubits, then you will obtain one of $2^N$ possibilities, namely the $2^N$ distinct sequences of $1$'s and $0$'s obtained by expanding out the product. The probability of obtaining such a sequence is its associated coefficient. In fact, in this case, the probability of measuring a single such sequence is $|\alpha|^n|\beta|^{N-n}$ where $n$ is the number of $1$'s in the sequence, and therefore $N-n$ is the number of $0$'s in the sequence. So, for example, the state \begin{align} |1\rangle|1\rangle\underbrace{|0\rangle\cdots |0\rangle}_{N-2\,\text{factors}} \end{align} has associated probability $|\alpha|^2|\beta|^{N-2}$ of measurement.

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