Quantum logic gates are needed for physical implementation Updated 2025-07-16
One direct practical reason is that we need to map the matrix to real quantum hardware somehow, and all quantum hardware designs so far and likely in the future are gate-based: you manipulate a small number of qubits at a time (2) and add more and more of such operations.
While there are "quantum compilers" to increase the portability of quantum programs, it is to be expected that programs manually crafted for a specific hardware will be more efficient just like in classic computers.
TODO: is there any clear reason why computers can't beat humans in approximating any unitary matrix with a gate set?
This is analogous to what classic circuit programmers will do, by using smaller logic gates to create complex circuits, rather than directly creating one huge truth table.
The gates themselves are just unitary matrices that operate on the input qubits and produce the same number of output qubits.
For example, the matrix for the CNOT gate, which takes 2 qubits as input is:
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0The final question is then: if I have a 2 qubit gate but an input with more qubits, say 3 qubits, then what does the 2 qubit gate (4x4 matrix) do for the final big 3 qubit matrix (8x8)? In order words, how do we scale quantum gates up to match the total number of qubits?
The intuitive answer is simple: we "just" extend the small matrix with a larger identity matrix so that the sum of the probabilities third bit is unaffected.
More precisely, we likely have to extend the matrix in a way such that the partial measurement of the original small gate qubits leaves all other qubits unaffected.
For example, if the circuit were made up of a CNOT gate operating on the first and second qubits as in:
0 ----+----- 0
|
1 ---CNOT--- 1
2 ---------- 2TODO lazy to properly learn right now. Apparently you have to use the Kronecker product by the identity matrix. Also, zX-calculus appears to provide a powerful alternative method in some/all cases.
Quantum matter physics course of the University of Oxford Updated 2025-07-16
2011- professor: Steven H. Simon. His start date is given e.g. at: www-thphys.physics.ox.ac.uk/people/SteveSimon/condmat2012/LectureNotes2012.pdf which is presumably an older version of: www-thphys.physics.ox.ac.uk/people/SteveSimon/QCM2022/QuantumMatter.pdf
Notes/book: www-thphys.physics.ox.ac.uk/people/SteveSimon/QCM2022/QuantumMatter.pdf Marked as being for Oxford MMathPhys, so it appears that this is a 4th year course normally. TODO but where is it listed under the course list of MMapthPhys? mmathphys.physics.ox.ac.uk/course-schedule
Course page index: www-thphys.physics.ox.ac.uk/people/SteveSimon/
www-thphys.physics.ox.ac.uk/people/SteveSimon/QCM2023/quantummatter.html mentions it is given in Hilary term
2023 syllabus as per www-thphys.physics.ox.ac.uk/people/SteveSimon/QCM2023/quantummatter.html#Syllabus:
- Fermi Liquids
- Weakly Interacting Fermions
- Response Functions and Screening
- Thomas Fermi
- RPA
- Plasmons
- Landau Fermi Liquid Theory
- Superfluidity
- Two Fluid Model and Quantized Circulation
- Landau Criterion for Superfluidity
- Two Fluid Model for Superconductors
- London Theory
- Flux Vortices
- Type I and Type II superconductors
- Microscopic Superfluidity
- Coherent States
- Bose Condensation
- Gross Pitaevskii Equation
- Off Diagonal Long Range Order
- Feynman Theory of Superfluidity (in book, but will skip in lectures. Not examinable)
- Ginzburg Landau Theory of Superfluids
- BCS Theory of Superconductors
Quantum Mechanical View of Reality by Richard Feynman (1983) Updated 2025-07-16
Sample playlist: www.youtube.com/playlist?list=PLW_HsOU6YZRkdhFFznHNEfua9NK3deBQy
Basically the same content as: Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979), but maybe there is some merit to this talk, as it is a bit more direct in some points. This is consistent with what is mentioned at www.feynman.com/science/qed-lectures-in-new-zealand/ that the Auckland lecture was the first attempt.
Some more information at: iucat.iu.edu/iub/5327621
By Mill Valley, CA based producer "Sound Photosynthesis", some info on their website: sound.photosynthesis.com/Richard_Feynman.html
They are mostly a New Age production company it seems, which highlights Feynman's absolute cult status. E.g. on the last video, he's not wearing shoes, like a proper guru.
Feynman liked to meet all kinds of weird people, and at some point he got interested in the New Age Esalen Institute. Surely You're Joking, Mr. Feynman this kind of experience a bit, there was nude bathing on a pool that oversaw the sea, and a guy offered to give a massage to the he nude girl and the accepted.
youtu.be/rZvgGekvHest=5105 actually talks about spin, notably that the endpoint events also have a spin, and that the transition rules take spin into account by rotating thing, and that the transition rules take spin into account by rotating things.
Real coordinate space of dimension three Updated 2025-07-16
Real-time attack speedrun Updated 2025-07-16
A non-tool-assisted speedrun.
Ciro Santilli views humans as biological robots, and therefore RTA videos can be thought of as probabilistic TAS with human achievable reflex constraints.
This aspect is especially highlighted in "speed run record evolution videos", which can be quite fun, e.g. www.youtube.com/watch?v=pmS9e7kzgS4 Ocarina of Time - World Record History and Progression (Any% Speedrun, 1990s-2017) by retro (2017)
From a similar point of view, Ciro also sometimes watches/learns a bit about competitive PvP games from a "could a computer play this better than a human" point of view.
Ciro also likes to watch commented manual speedruns of games as a way of experiencing the game at a high level without spending too much time on it, often from Games Done Quick. Their format is good because it generally showcases one player focusing more on the gameplay, and three couch commentators to give context, that's a good setup.
It is a
Real time live plot from streaming data Updated 2025-07-16
E.g. showing live data from a scientific instrument! TODO:
- superuser.com/questions/825588/what-is-the-easiest-way-of-visualizing-data-from-stdout-as-a-graph
- unix.stackexchange.com/questions/190337/how-can-i-make-a-graphical-plot-of-a-sequence-of-numbers-from-the-standard-input
- stackoverflow.com/questions/44470965/how-can-you-watch-gnuplot-realtime-data-plots-as-a-live-graph-with-automatic-up
- stackoverflow.com/questions/14074790/plotting-a-string-of-csv-data-in-realtime-using-linux
- stackoverflow.com/questions/11874767/how-do-i-plot-in-real-time-in-a-while-loop-using-matplotlib
Real-time polymerase chain reaction Updated 2025-07-16
Also known as: Quantitative PCR (qPCR).
This describes one possible concentration detection method with fluorescent molecules that only become fluorescent when the DNA is double stranded (SYBR Green)
Polymerase Chain Reaction (PCR) - Quantitative PCR (qPCR) by Applied Biological Materials (2016)
Source. This allows you to predict the exact initial concentration by extrapolating the exponential curve backwards.
Representation theory Updated 2025-07-16
Basically, a "representation" means associating each group element as an invertible matrices, i.e. a matrix in (possibly some subset of) , that has the same properties as the group.
Or in other words, associating to the more abstract notion of a group more concrete objects with which we are familiar (e.g. a matrix).
This is basically what everyone does (or should do!) when starting to study Lie groups: we start looking at matrix Lie groups, which are very concrete.
Or more precisely, mapping each group element to a linear map over some vector field (which can be represented by a matrix infinite dimension), in a way that respects the group operations:
As shown at Physics from Symmetry by Jakob Schwichtenberg (2015)
- page 51, a representation is not unique, we can even use matrices of different dimensions to represent the same group
- 3.6 classifies the representations of . There is only one possibility per dimension!
- 3.7 "The Lorentz Group O(1,3)" mentions that even for a "simple" group such as the Lorentz group, not all representations can be described in terms of matrices, and that we can construct such representations with the help of Lie group theory, and that they have fundamental physical application
Bibliography:
- www.youtube.com/watch?v=9rDzaKASMTM "RT1: Representation Theory Basics" by MathDoctorBob (2011). Too much theory, give me the motivation!
- www.quantamagazine.org/the-useless-perspective-that-transformed-mathematics-20200609 The "Useless" Perspective That Transformed Mathematics by Quanta Magazine (2020). Maybe there is something in there amidst the "the reader might not know what a matrix is" stuff.
Resonance Updated 2025-07-16
Examples:
- mechanical resonance, notably:
- pipe instruments
- electronic oscillators, notably:
- LC oscillator, and notably the lossy version RLC circuit
Perhaps a key insight of resonance is that the reonant any lossy system tends to look like the resonance frequency quite quickly even if the initial condition is not the resonant condition itself, because everything that is not the resonant frequency interferes destructively and becomes noise. Some examples of that:
- striking a bell or drum can be modelled by applying an impuse to the system
- playing a pipe instrument comes down to blowing a piece that vibrates randomly, and then leads the pipe to vibrate mostly in the resonant frequency. Likely the same applies to bowed string instruments, the bow must be creating a random vibration.
- playing a plucked string instrument comes down to initializing the system to an triangular wave form and then letting it evolve. TODO find a simulation of that!
Another cool aspect of resonance is that it was kind of the motivation for de Broglie hypothesis, as de Broglie was kind of thinking that electroncs might show discrete jumps on atomic spectra because of constructive interference.
Richard Feynman's drug use Updated 2025-07-16
From Surely You're Joking, Mr. Feynman chapter O Americano, Outra Vez!:
The people from the airlines were somewhat bored with their lives, strangely enough, and at night they would often go to bars to drink. I liked them all, and in order to be sociable, I would go with them to the bar to have a few drinks, several nights a week.One day, about 3:30 in the afternoon, I was walking along the sidewalk opposite the beach at Copacabana past a bar. I suddenly got this treMENdous, strong feeling: "That's just what I want; that'll fit just right. I'd just love to have a drink right now!"I started to walk into the bar, and I suddenly thought to myself, "Wait a minute! It's the middle of the afternoon. There's nobody here, There's no social reason to drink. Why do you have such a terribly strong feeling that you have to have a drink?" - and I got scared.I never drank ever again, since then. I suppose I really wasn't in any danger, because I found it very easy to stop. But that strong feeling that I didn't understand frightened me. You see, I get such fun out of thinking that I don't want to destroy this most pleasant machine that makes life such a big kick. It's the same reason that, later on, I was reluctant to try experiments with LSD in spite of my curiosity about hallucinations.
One notable drug early teens Ciro consumed was Magic: The Gathering, see also: Section "Magic: The Gathering is addictive".
Riemann hypothesis Updated 2025-07-16
visualizing the Riemann hypothesis and analytic continuation by 3Blue1Brown (2016) is a good quick visual non-mathematical introduction is to it.
One of the Millennium Prize Problems and Hilbert's problems.
Ring (mathematics) Updated 2025-07-16
A Ring can be seen as a generalization of a field where:
- multiplication is not necessarily commutative. If this is satisfied, we can call it a commutative ring.
- multiplication may not have inverse elements. If this is satisfied, we can call it a division ring.
The simplest example of a ring which is not a full fledged field and with commutative multiplication are the integers. Notably, no inverses exist except for the identity itself and -1. E.g. the inverse of 2 would be 1/2 which is not in the set. More specifically, the integers are a commutative ring.
The simplest non-commutative, non-division is is the set of all 2x2 matrices of real numbers:Note that is not a ring because you can by addition reach the zero matrix.
- we know that 2x2 matrix multiplication is non-commutative in general
- some 2x2 matrices have a multiplicative inverse, but others don't
Saylor Academy Updated 2025-07-16
This is an interesting initiative which has some similarities to Ciro Santilli's OurBigBook project.
The fatal flaw of the initiative in Ciro Santilli's opinion is the lack of user-generated content. We will never get there without UGC and algorithms, never.
Also as of 2021, it mostly useless business courses: learn.saylor.org unfortunately.
But it has several redeeming factors which Ciro Santilli aproves of:
- exam as a service-like
- they have a GitHub: github.com/saylordotorgo
The founder Michael J. Saylor looks a bit crooked, Rich people who create charitable prizes are often crooked comes to mind. But maybe he's just weird.
Michael Saylor interview by Lex Fridman (2022)
Source. At the timestamp:What statement... maybe he's actually not crooked, maybe it was just an accounting mistake... God, why.
When I go, all my assets will flow into a foundation, and the foundation's mission is to make education free for everybody forever.
If only Ciro Santilli knew how to contact him and convince him that his current approach is innefective and that Ciro has something better! Michael, please Google into this page some day, Ciro Santilli needs funding for OurBigBook.com. A hopeless Tweet at: twitter.com/cirosantilli/status/1548350114623660035. Also tried to hit his
saylor@strategy.com. Scalable Vector Graphics Updated 2025-07-16
Companies have been really slow to support SVG features in their browsers, and that is very saddening: medium.com/@michaelmangial1/introduction-to-scalable-vector-graphics-6450c03e8d2e
You can't drop SVG support for
canvas until there's a way to run untrusted JavaScript on the browser!SVG does have some compatibility annoyances, notably SVG fonts. But we should as a society work to standardize and implement a fix those, the benefits of SVG are just too great!
Examples:
- svg/svg.svg a minimal somewhat sane SVG:
- if the
widthandheightproperties were not given, you get the default 300x150, which seems to be set in the SVG standard:
- if the
- how to add na SVG image to a HTML file:
- svg/svg.html: external image. The included file is svg/svg.svg.
- svg/inline.html: inline.
- svg/billion-laughs.svg
- svg/html.svg
- svg/triangle.svg
- svg/viewBox.svg: this attribute allows you to control the default SVG
svg width=andheight=while keeping the coordinates of the drawing untouched. If theviewBoxaspect ratio differs from the width/height ratio, you likely want to play withpreserveAspectRatio, otherwise you would get white spaces by default on the generated image - CSS with SVG:
- svg/style.svg: inline CSS
- svg/style-external.svg: external CSS with:
<?xml-stylesheet type="text/css" href="svg.css" ?>, see also: stackoverflow.com/questions/18434094/how-to-style-svg-with-external-css- svg/subdir/style-external.html: is the relative CSS relative to the HTML or to the SVG? Answer: to the SVG... OMG. So how to make it work reliably?
- svg/current-color.html and svg/current-color.svg: illustrates
fill="currentColor". Only works for inline SVG however... See also: stackoverflow.com/questions/13000682/how-do-i-have-an-svg-image-inherit-colors-from-the-html-document/13002311
- JavaScript with SVG:
- svg/defs.html hows how
defsworks- svg/defs-external.html tries to include external
defsfrom svg/defs.svg, but that fails like everything else related to external SVGs
- svg/defs-external.html tries to include external
Scanning electron microscope Updated 2025-07-16
The Scanning Electron Microscope by MaterialsScience2000 (2014)
Source. Shows operation of the microscope really well. Seems too easy, there must have been some extra setup before however. Impressed by how fast the image update, it is basically instantaneous. Produced by Prof. Dr.-Ing. Rainer Schwab from the Karlsruhe University of Applied Sciences.Mosquito Eye Scanning Electron Microscope Zoom by Mathew Tizard (2005)
Source. Video description mentions is a composite video. Why can't you do it in one shot? Schrödinger equation for a free one dimensional particle Updated 2025-07-16
Schrödinger equation for a one dimensional particle with . The first step is to calculate the time-independent Schrödinger equation for a free one dimensional particle
Then, for each energy , from the discussion at Section "Solving the Schrodinger equation with the time-independent Schrödinger equation", the solution is:Therefore, we see that the solution is made up of infinitely many plane wave functions.
Schrödinger equation for a one dimensional particle Updated 2025-07-16
We select for the general Equation "Schrodinger equation":giving the full explicit partial differential equation:
- , the linear cartesian coordinate in the x direction
- , which analogous to the sum of kinetic and potential energy in classical mechanics
Equation 1.
Schrödinger equation for a one dimensional particle
. The corresponding time-independent Schrödinger equation for this equation is:
Equation 2.
time-independent Schrödinger equation for a one dimensional particle
. Schrödinger equation solution for the helium atom Updated 2025-07-16
No closed form solution, but good approximation that can be calculated by hand with the Hartree-Fock method, see hartree-Fock method for the helium atom.
Bibliography:
Schrödinger picture Updated 2025-07-16
To better understand the discussion below, the best thing to do is to read it in parallel with the simplest possible example: Schrödinger picture example: quantum harmonic oscillator.
"Making a measurement" for an observable means applying a self-adjoint operator to the state, and after a measurement is done:Those last two rules are also known as the Born rule.
- the state collapses to an eigenvector of the self adjoint operator
- the result of the measurement is the eigenvalue of the self adjoint operator
- the probability of a given result happening when the spectrum is discrete is proportional to the modulus of the projection on that eigenvector.For continuous spectra such as that of the position operator in most systems, e.g. Schrödinger equation for a free one dimensional particle, the projection on each individual eigenvalue is zero, i.e. the probability of one absolutely exact position is zero. To get a non-zero result, measurement has to be done on a continuous range of eigenvectors (e.g. for position: "is the particle present between x=0 and x=1?"), and you have to integrate the probability over the projection on a continuous range of eigenvalues.In such continuous cases, the probability collapses to an uniform distribution on the range after measurement.The continuous position operator case is well illustrated at: Video "Visualization of Quantum Physics (Quantum Mechanics) by udiprod (2017)"
Self adjoint operators are chosen because they have the following key properties:
- their eigenvalues form an orthonormal basis
- they are diagonalizable
Perhaps the easiest case to understand this for is that of spin, which has only a finite number of eigenvalues. Although it is a shame that fully understanding that requires a relativistic quantum theory such as the Dirac equation.
The next steps are to look at simple 1D bound states such as particle in a box and quantum harmonic oscillator.
This naturally generalizes to Schrödinger equation solution for the hydrogen atom.
The solution to the Schrödinger equation for a free one dimensional particle is a bit harder since the possible energies do not make up a countable set.
This formulation was apparently called more precisely Dirac-von Neumann axioms, but it because so dominant we just call it "the" formulation.
Quantum Field Theory lecture notes by David Tong (2007) mentions that:
if you were to write the wavefunction in quantum field theory, it would be a functional, that is a function of every possible configuration of the field .
Science fiction film Updated 2025-07-16
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