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In quantum computation, a common operation performed between two quantum states is the tensor product, which allows us to create a new and higher-dimensional state from two lower-dimensional states. The tensor product is usually denoted by the symbol $\otimes$. So, if $\lvert \psi\rangle = \alpha \lvert 0\rangle + \beta \lvert 1\rangle \in \mathbb{C}^n$ and $\lvert \phi\rangle = \gamma \lvert 0\rangle + \delta \lvert 1\rangle \in \mathbb{C}^n$ are the states of two qubits, then $\lvert \psi\rangle \otimes \lvert \phi\rangle \in \mathbb{C}^{n^2}$ is their tensor product.

I have just come across the notation $\lvert \mathbf{x}, 0\rangle$ while reading section $1$ of this paper. What does it mean? I understood $\mathbf{x}$ is a (classical) bit string of length $n$ (and I think this the reason it's in bold).

I am aware of the fact that the tensor product of two vectors $\lvert \psi\rangle$ and $\lvert \phi\rangle$ can be shortened as follows $\lvert \psi\rangle \lvert \phi\rangle$. So, I don't think $\lvert \mathbf{x}, 0\rangle$ is also a shorthand for $\lvert \mathbf{x} \rangle \otimes\lvert 0\rangle$.

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Yes, $|\mathbf{x},0\rangle$ is a shorthand for $|\mathbf x\rangle\otimes |0\rangle$.

Note that $|\mathbf x\rangle$ itself, with $\mathbf x = x_1x_2\dots x_N$ a bit string, is just a shorthand for $$ |\mathbf x\rangle \equiv |x_1\rangle \otimes |x_2\rangle \otimes\cdots |x_N\rangle\ . $$

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  • $\begingroup$ If that's the case, what's the meaning of applying a tensor product between a classical $n$ bit string $\lvert \mathbf{x} \rangle$ and a $2$-dimensional vector (or computational basis state) $\lvert 0 \rangle \in \mathbb{C}^2$? Why would one do that? Is $\lvert \mathbf{x} \rangle$ really a classical $n$ bit string? Why do we use the notation $\lvert \rangle$ then? $\endgroup$ – nbro Apr 26 '18 at 9:56
  • $\begingroup$ Updated. And: Why would one do that? To get classical information into a quantum computer. $\endgroup$ – Norbert Schuch Apr 26 '18 at 11:21
  • $\begingroup$ In the case of section 1 of your paper the idea is to use $|0\rangle$ as an ancillary qubit in the expression $|\mathbf{x},0\rangle$. An ancilla qubit is a qubit that is needed for computation in an intermediate step. Note the intermediate step in Fig. 1 where the ancilla stores the value of f(x) as $|f(x)\rangle$. Subsequently, an $R_X$ is applied to the ancilla (see circuit diagram) and then the ancilla is uncomputed by the second application of $U_f$. $\endgroup$ – Mark Fingerhuth Apr 26 '18 at 15:27
  • $\begingroup$ So, is the notation $\lvert \mathbf{x}, 0 \rangle$ (instead of $\lvert \mathbf{x} 0 \rangle$ or $\lvert \mathbf{x}\rangle \otimes \lvert 0 \rangle$ ) used because $\lvert 0\rangle$ is an ancilla qubit or because $\lvert \mathbf{x} \rangle$ is the quantum state associated with a binary string $\mathbf{x}$ (or both)? $\endgroup$ – nbro Apr 26 '18 at 15:53

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