Used e.g. by Oxford Quantum Circuits, www.linkedin.com/in/john-dumbell-627454121/ mentions:

Presumably the point of it is to allow simulation in classical computers?

The unique group referred to at: every Lie algebra has a unique single corresponding simply connected Lie group.

The third one of the University of California after UC Berkeley and UCLA.

Technique that uses multiple non-ideal qubits (physical qubits) to simulate/produce one perfect qubit (logical).

One is philosophically reminded of classical error correction codes, where we also have multiple input bits per actual information bit.

TODO understand in detail. This appears to be a fundamental technique since all physical systems we can manufacture are imperfect.

Part of the fundamental interest of this technique is due to the quantum threshold theorem.

For example, when PsiQuantum raised 215M in 2020, they announced that they intended to reach 1 million physical qubits, which would achieve between 100 and 300 logical qubits.

Video "Jeremy O'Brien: "Quantum Technologies" by GoogleTechTalks (2014)" youtu.be/7wCBkAQYBZA?t=2778 describes an error correction approach for a photonic quantum computer.

Bibliography:

This theorem roughly states that states that for every quantum algorithm, once we reach a certain level of physical error rate small enough (where small enough is algorithm dependant), then we can perfectly error correct.

This algorithm provides the conceptual division between noisy intermediate-scale quantum era and post-NISQ.

A quantum algorithm that is thought to be more likely to be useful in the NISQ era of quantum computing.

TODO clear example of the computational problem that it solves.

This is a term "invented" by Ciro Santilli to refer to quantum compilers that are able to convert non-specifically-quantum (functional, since there is no state in quantum software) programs into quantum circuit.

The term is made by adding "quantum" to the more "classical" concept of "high-level synthesis", which refers to software that converts an imperative program into register transfer level hardware, typicially for FPGA applications.

One of their learning sites: www.qutube.nl/

The educational/outreach branch of QuTech.

Synonym to gate-based quantum computer/digital quantum computer?

TODO confirm: apparently in the paradigm you can choose to measure only certain output qubits.

This makes things irreversible (TODO what does reversibility mean in this random context?), as opposed to Circuit-based quantum computer where you measure all output qubits at once.

TODO what is the advantage?

We don't need to understand a super generalized version of tensor products to know what they mean in basic quantum computing!

Intuitively, taking a tensor product of two qubits simply means putting them together on the same quantum system/computer.

When we write the bra-ket notation: $|00⟩$ that is the same as $|0⟩\otimes |0⟩$.

The tensor product is called a "product" because it distributes over addition.

E.g. consider:

Intuitively, in this operation we just put a Hadamard gate qubit together with a second pure $|0⟩$ qubit.

And the outcome still has the second qubit as always 0, because we haven't made them interact.

The quantum state $\frac{|00⟩+|10⟩}{\sqrt{2}}$ is called a separable state, because it can be written as a single product of two different qubits. We have simply brought two qubits together, without making them interact.

If we then add a CNOT gate to make a Bell state:we can now see that the Bell state is non-separable: we've made the two qubits interact, and there is no way to write this state with a single tensor product. The qubits are fundamentally entangled.

One key insight, is that the matrix of a non-trivial quantum circuit is going to be huge, and won't fit into any amount classical memory that can be present in this universe.

This is because the matrix is exponential in the number qubits, and $2^{100}$ is more than the number of atoms in the universe!

Therefore, off the bat we know that we cannot possibly describe those matrices in an explicit form, but rather must use some kind of shorthand.

But it gets worse.

Even if we had enough memory, the act of explicitly computing the matrix is not generally possible.

This is because knowing the matrix, basically means knowing the probability result for all possible $2^N$ outputs for each of the $2^N$ possible inputs.

But if we had those probabilities, our algorithmic problem would already be solved in the first place! We would "just" go over each of those output probabilities (OK, there are $2^N$ of those, which is also an insurmountable problem in itself), and the largest probability would be the answer.

So if we could calculate those probabilities on a classical machine, we would also be able to simulate the quantum computer on the classical machine, and quantum computing would not be able to give exponential speedups, which we know it does.

To see this, consider that for a given input, say 000 on a 3 qubit machine, the corresponding 8-sized quantum state looks like:and therefore when you multiply it by the unitary matrix of the quantum circuit, what you get is the first column of the unitary matrix of the quantum circuit. And 001, gives the second column and so on.

As a result, to prove that a quantum algorithm is correct, we need to be a bit smarter than "just calculate the full matrix".

Which is why you should now go and read: Section "Quantum algorithm".

This type of thinking links back to how physical experiments relate to quantum computing: a quantum computer realizes a physical experiment to which we cannot calculate the probabilities of outcomes without exponential time.

So for example in the case of a photonic quantum computer, you are not able to calculate from theory the probability that photons will show up on certain wires or not.

The Hadamard gate takes $|0⟩$ or $|1⟩$ (quantum states with probability 1.0 of measuring either 0 or 1), and produces states that have equal probability of 0 or 1.