I'm trying to understand the definition of a Deutsch gate.

In particular what does $\theta$ mean in its presentation? Is it derived from the coefficients of the input state or a free parameter or something else?

A Deutsch gate is given below.

$$ |a, b, c \rangle \mapsto i \cos(\theta) |a, b, c\rangle + \sin(\theta) |a, b, 1-c\rangle \;\; \text{when a = b = 1} \\ |a, b, c \rangle \mapsto |a, b, c\rangle \;\; \text{otherwise} $$

I think that $a, b, c$ range over $\{0, 1\}$, so this thing could theoretically be expanded into an 8 by 8 matrix and presented that way.

I am totally mystified about the meaning of $\theta$ though. What does $\theta$ mean? Does it have a closed form in terms of the coefficients of the basis elements or is it something else?

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    $\begingroup$ Welcome to QCSE! It is not clear to me what the question is. One interpretation would be that you are asking whether Deutsch gate is an infinite family of gates parametrized by $\theta$. In this case the answer is yes, but then what is the relevance of Turing machines and the Hadamard gate to the question? Another interpretation would be that you are asking whether one can build a univeral(?) quantum Turing machine from Deutsch gate. In this case, I think the answer is yes as long as $\theta$ is an irrational multiple of $\pi$, but then what is the relevance of the discussion of Hadamard gate? $\endgroup$ Sep 16 '21 at 5:43
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    $\begingroup$ Please clarify the question and maybe remove parts that are not relevant. Also, note that each post should be a separate question, so if you do have multiple questions (which is fine!) please post each separately. $\endgroup$ Sep 16 '21 at 5:44
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    $\begingroup$ @AdamZalcman, That helps. Thanks for your patience. Including the other stuff was supposed to provide context for "where the question comes from". I've removed everything that isn't specifically about the definition of the Deutsch gate. Your comment answers my question, actually. It sounds like $\theta$ is a free parameter, but as long as it isn't a rational multiple of $\pi$ the gate is universal. $\endgroup$ Sep 16 '21 at 13:33

As @AdamZalcman pointed out on the comments, $\theta$ is a parameter that can take in any value from $0$ to $2\pi$. This gate is a way of generalizing the three-bit Toffoli gate which is universal for classical boolean logic, into a gate which is universal for quantum logic. The similarity between both gates can be seen easier with its matrix representation (and you can see $\text{D}(\pi/2) = \text{CCNOT}$):

$$ \text{D}(\theta) = \begin{bmatrix} 1 \\ & 1 \\ & & 1 \\ & & & 1 \\ & & & & 1 \\ & & & & & 1 \\ & & & & & & i \cos(\theta) & \sin(\theta) \\ & & & & & & \sin(\theta) & i\cos(\theta) \end{bmatrix} $$

You may find the start of section II.B. Universal Quantum Logic Gates from Classical and Quantum Logic Gates: An Introduction to Quantum Computing useful. Finally, as Adam pointed out and it is stated on the paper linked: "due to the property that $\text{D}(\theta) \text{D}(\theta^\prime) = i \text{D}(\theta + \theta^\prime)$, the parameter $\theta$ can be fixed as an irrational multiple of $\pi$, isolating one gate that can generate the whole family of gates, $\text{D}(\theta)$, asymptotically"


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