Ciro Santilli @cirosantilli  40

This LibreTexts book does have some interest!

100 10-20 minute videos properly split by topic, good resource!

Instructor: Barton Zwiebach.

Free material from university courses:

Author: David Tong.

Summary:

Looks very impressive! Last update marked 2011 as of 2020.

Goes up to "A.15 quantum field theory in a Nanoshell", Ciro have to review it to see if there's anything worthwhile in that section.

Personal page says he retired as of 2020: www.eng.fsu.edu/~dommelen/ But hopefully he has more time for these notes!

And he appears to have his own lightweight markup language that transpiles to LaTeX called l2h: www.eng.fsu.edu/~dommelen/l2h/

For the UCSD Physics 130 course.

Last updated: 2013.

Very good! Goes up to the Dirac equation.

There were apparently some lecture videos at: web.archive.org/web/20030604194654/http://physicsstream.ucsd.edu/courses/spring2003/physics130a/ as pointed out by Matthew Heaney^[ref], .mov files can be found at: web.archive.org/web/*/http://physicsstream.ucsd.edu/courses/spring2003/physics130a/*, but we were yet unable to open them, related:

Basically a precise statement of "quantum entanglement is spooky".

TODO what's the largest molecule done on a classical computer?

This is a quick tutorial on how a quantum computer programmer thinks about how a quantum computer works. If you know:a concrete and precise hello world operation can be understood in 30 minutes.

Although there are several types of quantum computer under development, there exists a single high level model that represents what most of those computers can do, and we are going to explain that model here. This model is the is the digital quantum computer model, which uses a quantum circuit, that is made up of many quantum gates.

Beyond that basic model, programmers only may have to consider the imperfections of their hardware, but the starting point will almost always be this basic model, and tooling that automates mapping the high level model to real hardware considering those imperfections (i.e. quantum compilers) is already getting better and better.

The way quantum programmers think about a quantum computer in order to program can be described as follows:

To operate a quantum computer, you follow the step of operation of a quantum computer:

Each time you do this, you are literally conducting a physical experiment of the specific physical implementation of the computer:and each run as the above can is simply called "an experiment" or "a measurement".

The output comes out "instantly" in the sense that it is physically impossible to observe any intermediate state of the system, i.e. there are no clocks like in classical computers, further discussion at: quantum circuits vs classical circuits. Setting up, running the experiment and taking the does take some time however, and this is important because you have to run the same experiment multiple times because results are probabilistic as mentioned below.

Unlike in a classical computer, the output of a quantum computer is not deterministic however.

But the each output is not equally likely either, otherwise the computer would be useless except as random number generator!

This is because the probabilities of each output for a given input depends on the program (unitary matrix) it went through.

Therefore, what we have to do is to design the quantum circuit in a way that the right or better answers will come out more likely than the bad answers.

We then calculate the error bound for our circuit based on its design, and then determine how many times we have to run the experiment to reach the desired accuracy.

The probability of each output of a quantum computer is derived from the input and the circuit as follows.

First we take the classic input vector of dimension N of 0's and 1's and convert it to a "quantum state vector" $\overrightarrow{q_{in}}$ of dimension $2^n$:

We are after all going to multiply it by the program matrix, as you would expect, and that has dimension $2^n\times 2^n$!

Note that this initial transformation also transforms the discrete zeroes and ones into complex numbers.

For example, in a 3 qubit computer, the quantum state vector has dimension $2^3=8$ and the following shows all 8 possible conversions from the classic input to the quantum state vector:

This can be intuitively interpreted as:

Now that we finally have our quantum state vector, we just multiply it by the unitary matrix $Q$ of the quantum circuit, and obtain the $2^n$ dimensional output quantum state vector $\overrightarrow{q_{out}}$:

And at long last, the probability of each classical outcome of the measurement is proportional to the square of the length of each entry in the quantum vector, analogously to what is done in the Schrödinger equation.

For example, suppose that the 3 qubit output were:

Then, the probability of each possible outcomes would be the length of each component squared:i.e. 75% for the first, and 25% for the third outcomes, where just like for the input:

All other outcomes have probability 0 and cannot occur, e.g.: 001 is impossible.

Keep in mind that the quantum state vector can also contain complex numbers because we are doing quantum mechanics, but we just take their magnitude in that case, e.g. the following quantum state would lead to the same probabilities as the previous one:

This interpretation of the quantum state vector clarifies a few things:

As we could see, this model is was simple to understand, being only marginally more complex than that of a classical computer, see also: quantumcomputing.stackexchange.com/questions/6639/is-my-background-sufficient-to-start-quantum-computing/14317#14317 The situation of quantum computers today in the 2020's is somewhat analogous to that of the early days of classical circuits and computers in the 1950's and 1960's, before CPU came along and software ate the world. Even though the exact physics of a classical computer might be hard to understand and vary across different types of integrated circuits, those early hardware pioneers (and to this day modern CPU designers), can usefully view circuits from a higher level point of view, thinking only about concepts such as:as modelled at the register transfer level, and only in a separate compilation step translated into actual chips. This high level understanding of how a classical computer works is what we can call "the programmer's model of a classical computer". So we are now going to describe the quantum analogue of it.

Bibliography:

This is the true key question: what are the most important algorithms that would be accelerated by quantum computing?

Some candidates:

Do you have proper optimization or quantum chemistry algorithms that will make trillions?

Maybe there is some room for doubt because some applications might be way better in some implementations, but we should at least have a good general idea.

However, clear information on this really hard to come by, not sure why.

Whenever asked e.g. at: physics.stackexchange.com/questions/3390/can-anybody-provide-a-simple-example-of-a-quantum-computer-algorithm/3407 on Physics Stack Exchange people say the infinite mantra:

Lists:

Software that maps higher level languages like Qiskit into actual quantum circuits.

It is hard to beat the list present at Quantum computing report: quantumcomputingreport.com/players/.

The much less-complete Wikipedia page is also of interest: en.wikipedia.org/wiki/List_of_companies_involved_in_quantum_computing_or_communication It has the merit of having a few extra columns compared to Quantum computing report.

Also of interest: quantumzeitgeist.com/interactive-map-of-quantum-computing-companies-from-around-the-globe/

Based on the Josephson effect. Yet another application of that phenomenal phenomena!

Philosophically, superconducting qubits are good because superconductivity is macroscopic.

It is fun to see that the representation of information in the QC basically uses an LC circuit, which is a very classical resonator circuit.

As mentioned at en.wikipedia.org/wiki/Superconducting_quantum_computing#Qubit_archetypes there are actually a few different types of superconducting qubits:

and hybridizations of those such as:

Input:

Other good lists:

Bibliography:

In the context of quantum computing of the 2020's, a "classical computer" is a computer that is not "quantum", i.e., the then dominating CMOS computers.

As en.wikipedia.org/w/index.php?title=ZX-calculus&oldid=1071329204#Diagram_rewriting tries to explain but fails to deliver as usual consider the GHZ state represented as a quantum circuit.

How can we easily prove that that quantum circuit equals the state:?

The naive way would be to just do the matrix multiplication as explained at Section "Quantum computing is just matrix multiplication".

However, ZX-calculus provides a simpler way.

And even more importantly, sometimes it is the only way, because in a real circuit, we would not be able to do the matrix multiplication

What we do in ZX-calculus is we first transform the original quantum circuit into a ZX graph.

This is always possible, because we can describe how to do the conversion simply for any of the Clifford plus T gates, which is a set of universal quantum gates.

Then, after we do this transformation, we can start applying further transformations that simplify the circuit.

It has already been proven that there is no efficient algorithm for this (TODO source, someone said P-sharp complete best case)

But it has been proven in 2017 that any possible equivalence between quantum circuits can be reached by modifying ZX-calculus circuits.

There are only 7 transformation rules that we need, and all others can be derived from those, universality.

So, we can apply those rules to do the transformation shown in Wikipedia:

and one of those rules finally tells us that that last graph means our desired state:because it is a Z spider with $m=3$ and $n=1$.

Bibliography: