Similar to quantum supremacy, but add the goal that the computation must be useful, i.e. make money or solve some open mathematical problem, Ciro Santilli's wife was quite excited about the possibility of finding some counter examples in number theory with quantum computers.
Quantum logic gates are needed because you can't compute the matrix explicitly as it grows exponentially by 
Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16One 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 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.
This is because knowing the matrix, basically means knowing the probability result for all possible outputs for each of the 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 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 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
000 on a 3 qubit machine, the corresponding 8-sized quantum state looks like:000 -> 1000 0000 == (1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)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.
Explains beta decay. TODO why/how.
Maybe a good view of why this force was needed given beta decay experiments is: in beta decay, a neutron is getting split up into an electron and a proton. Therefore, those charges must be contained inside the neutron somehow to start with. But then what could possibly make a positive and a negative particle separate?
- the electromagnetic force should hold them together
- the strong force seems to hold positive charges together. Could it then be pushing opposite-charges apart? Why not? In any case this force doesn't seem to act on electrons, only quarks.
- gravity is too weak
www.thestargarden.co.uk/Weak-nuclear-force.html gives a quick and dirty:Also interesting:
Beta decay could not be explained by the strong nuclear force, the force that's responsible for holding the atomic nucleus together, because this force doesn't affect electrons. It couldn't be explained by the electromagnetic force, because this does not affect neutrons, and the force of gravity is far too weak to be responsible. Since this new atomic force was not as strong as the strong nuclear force, it was dubbed the weak nuclear force.
While the photon 'carries' charge, and therefore mediates the electromagnetic force, the Z and W bosons are said to carry a property known as 'weak isospin'. W bosons mediate the weak force when particles with charge are involved, and Z bosons mediate the weak force when neutral particles are involved.
Weak Nuclear Force and Standard Model of particle physics by Physics Videos by Eugene Khutoryansky (2018)
Source. Some decent visualizations of the field lines.Bibliography of the biliograpy:
- physics.stackexchange.com/questions/8441/what-is-a-complete-book-for-introductory-quantum-field-theory "What is a complete book for introductory quantum field theory?"
- www.quora.com/What-is-the-best-book-to-learn-quantum-field-theory-on-your-own on Quora
- www.amazon.co.uk/Lectures-Quantum-Field-Theory-Ashok-ebook/dp/B07CL8Y3KY
Recommendations by friend P. C.:
- The Global Approach to Quantum Field Theory
- Lecture Notes | Geometry and Quantum Field Theory | Mathematics ocw.mit.edu/courses/mathematics/18-238-geometry-and-quantum-field-theory-fall-2002/lecture-notes/
- Towards the mathematics of quantum field theory (Frederic Paugam)
- Path Integrals in Quantum Mechanics (J. Zinn–Justin)
- (B.Hall) Quantum Theory for Mathematicians (B.Hall)
- Quantum Field Theory and the Standard Model (Schwartz)
- The Algebra of Grand Unified Theories (John C. Baez)
- quantum Field Theory for The Gifted Amateur by Tom Lancaster (2015)
The most comprehensive list is the amazing curated and commented list of quantum algorithms as of 2020.
Appears in the Schrödinger equation.
If we pick k elements of the set, the stabilizer subgroup of those k elements is a subgroup of the given permutation group that keeps those elements unchanged.
Note that an analogous definition can be given for non-finite groups. Also note that the case for all finite groups is covered by the permutation definition since all groups are isomorphic to a subgroup of the symmetric group
TODO existence and uniqueness. Existence is obvious for the identity permutation, but proper subgroup likely does not exist in general.
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