The first two that you should study are:
The quantum NOT gate swaps the state of and , i.e. it maps:As a result, this gate also inverts the probability of measuring 0 or 1, e.g.
- if the old probability of 0 was 0, then it becomes 1
- if the old probability of 0 was 0.2, then it becomes 0.8
Equation 2.
Quantum NOT gate matrix
. 
The most common way to construct multi-qubit gates is to use single-qubit gates as part of a controlled quantum gate.
Controlled quantum gates are gates that have two types of input qubits:These gates can be understood as doing a certain unitary operation only if the control qubits are enabled or disabled.
- control qubits
- operand qubits (terminology made up by Ciro Santilli just now)
The CNOT gate is a controlled quantum gate that operates on two qubits, flipping the second (operand) qubit if the first (control) qubit is set.
Equation 1.
CNOT gate matrix
. 
CNOT gate symbol
. Source. The symbol follow the generic symbol convention for controlled quantum gates shown at Figure "Generic controlled quantum gate symbol", but replacing the generic "U" with the Figure "Quantum NOT gate symbol".On the standard basis:we see that this means that only and should be possible. Therefore, the state must be of the form:where and are two complex numbers such that
If we operate the CNOT gate on that state, we obtain:and so the input is unchanged as desired, because the control qubit is 0.
Therefore, in that case, what happened is that the probabilities of and were swapped from and to and respectively, which is exactly what the quantum NOT gate does.
So from this we understand more concretely what "the gate only operates if the first qubit is set to one" means.
Now go and study the Bell state and understand intuitively how this gate is used to produce it.
This gate set alone is not a set of universal quantum gates.
Notably, circuits containing those gates alone can be fully simulated by classical computers according to the Gottesman-Knill theorem, so there's no way they could be universal.
This means that if we add any number of Clifford gates to a quantum circuit, we haven't really increased the complexity of the algorithm, which can be useful as a transformational device.
Set of quantum logic gate composed of the Clifford gates plus the Toffoli gate. It forms a set of universal quantum gates.
- quantumtech.blog/2023/01/17/quantum-computing-with-neutral-atoms/ OK this one hits it:So we understand that it is truly like the classical computer analog vs digital case.
As Alex Keesling, CEO of QuEra told me, "... whereas in gate-based [digital] quantum computing the focus is on the sequence of the gates, in analog quantum processing it's more about the position of the atoms and where you place them so they can mirror real life problems. We arrange the atoms and define the forces that drive them and then measure the result... so it’s a geometric encoding of the problem itself."
- thequantuminsider.com/2022/06/28/why-analog-neutral-atoms-quantum-computing-is-a-promising-direction-for-early-quantum-advantage on The Quantum Insider useless article mostly by Pasqal
TensorFlow quantum by Masoud Mohseni (2020)
Source. At the timestamp, Masoud gives a thought experiment example of the perhaps simplest to understand analog quantum computer: chained double-slit experiments with carefully calculated distances between slits. Calulating the final propability distribution of that grows exponentially.It is also possible to carry out quantum computing without qubits using processes with a continuous spectrum of measurement.
As of 2020, these approaches seem less developed/promising, but who knows.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. Web editor. You can also edit articles on the Web editor without installing anything locally.Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact