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This question is about identifying the correct and widely accepted terminology.

Consider a function $U: \mathbb{R}^m \to \mathcal{U}(2^n)$ mapping a vector of $m$ real numbers to an element of the unitary group of size $2^n$, i.e., a $2^n \times 2^n$ unitary matrix. Is it correct to name such an object a parameterized quantum circuit, and to name the elements of $\mathbf{x} \in \mathbb{R}^m$ its parameters?

Such an object has been primarily used (but not exclusively) in the field of quantum machine learning. Some of the most notable examples are data encoding unitaries $U : \mathbb{R}^m \to \mathcal{U}(2^n)$, $U(\mathbf{x}) \in \mathcal{U}(2^n)$, analogous to feature maps in classical kernels, and variational quantum circuits, $U: \mathbb{R}^m \to \mathcal{U}(2^n)$, $U(\boldsymbol{\theta}) \in \mathcal{U}(2^n)$; in the former, the parameters of the parameterized quantum circuit are the input of the problem, while in the latter, the parameters are freely trainable parameters of the underlying machine learning model. Is it correct to state that in both cases, $U$ can be referred to as a parameterized quantum circuit, regardless of its use, and to specify them more precisely as data encoding unitaries and variational form (*) when we know the use of such an object?

Can you bring any evidence in the literature that supports your claim?

(*) with variational quantum circuit identifying a machine learning model in the form $h(\mathbf{x}; \boldsymbol{\theta}) = \langle 0 | U^\dagger(\mathbf{x}; \boldsymbol{\theta}) O U(\mathbf{x}; \boldsymbol{\theta}) | 0\rangle$, with $|0\rangle$ initial state and $O$ observable, thus relying on a parametric quantum circuit whose parameters are, in part, the input feature vector $\mathbf{x}$ and, in part, the freely trainable parameters of the model $\boldsymbol{\theta}$.

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    $\begingroup$ Maybe someone will bring up some details on this, but at first sight, I'd say you seem to be correct! $\endgroup$
    – Tristan Nemoz
    Aug 30, 2023 at 21:29

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You have correctly identified that Supervised quantum machine learning models are kernel methods . Namely,

A lot of near-term and fault-tolerant quantum models can be replaced by a general support vector machine whose kernel computes distances between data-encoding quantum states.

In terms of words, it is always correct that $U$ is a parametrized quantum circuit, as it is a quantum circuit that depends on a set of parameters. How you interpret the parameters and what you classify as the input versus the computation versus the measurement part of your problem is up to you, so it is certainly true that there will be different names for essentially the same task. Calling something a data encoding unitary or a quantum kernel is almost always possible, even when you claim you've done a variational quantum circuit, because they share the same fundamental structure (and thus their potential advantages lie in the same places), as codified by the linked article.

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  • $\begingroup$ Imagine: an arxiv-only article with over 150 citations! Great resource $\endgroup$ Sep 6, 2023 at 19:00

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