Quantum circuits vs classical circuits Updated 2025-07-16
Just like a classic programmer does not need to understand the intricacies of how transistors are implemented and CMOS semiconductors, the quantum programmer does not understand physical intricacies of the underlying physical implementation.
The main difference to keep in mind is that quantum computers cannot save and observe intermediate quantum state, so programming a quantum computer is basically like programming a combinatorial-like circuit with gates that operate on (qu)bits:
For this reason programming a quantum computer is much like programming a classical combinatorial circuit as you would do with SPICE, verilog-or-vhdl, in which you are basically describing a graph of gates that goes from the input to the output
For this reason, we can use the words "program" and "circuit" interchangeably to refer to a quantum program
Also remember that and there is no no clocks in combinatorial circuits because there are no registers to drive; and so there is no analogue of clock in the quantum system either,
Universal quantum gates Updated 2025-07-16
Just like as for classic gates, we would like to be able to select quantum computer physical implementations that can represent one or a few gates that can be used to create any quantum circuit.
Unfortunately, in the case of quantum circuits this is obviously impossible, since the space of N x N unitary matrices is infinite and continuous.
Therefore, when we say that certain gates form a "set of universal quantum gates", we actually mean that "any unitary matrix can be approximated to arbitrary precision with enough of these gates".
Or if you like fancy Mathy words, you can say that the subgroup of the unitary group generated by our basic gate set is a dense subset of the unitary group.
Clifford gates Updated 2025-07-16
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.
Toffoli gate Updated 2025-07-16
List of quantum logic gates Updated 2025-07-16
BB84 Updated 2025-07-16
Does not require entangled particles, unlike E91 which does.
en.wikipedia.org/w/index.php?title=Quantum_key_distribution&oldid=1079513227#BB84_protocol:_Charles_H._Bennett_and_Gilles_Brassard_(1984) explains it well. Basically:
- Alice and Bob randomly select a measurement basis of either 90 degrees and 45 degrees for each photon
- Alice measures each photon. There are two possible results to either measurement basis: parallel or perpendicular, representing values 0 or 1. TODO understand better: weren't the possible results supposed to be pass or non-pass? She writes down the results, and sends the (now collapsed) photons forward to Bob.
- Bob measures the photons and writes down the results
- Alice and Bob communicate to one another their randomly chosen measurement bases over the unencrypted classic channel.This channel must be authenticated to prevent man-in-the-middle. The only way to do this authentication that makes sense is to use a pre-shared key to create message authentication codes. Using public-key cryptography for a digital signature would be pointless, since the only advantage of QKD is to avoid using public-key cryptography in the first place.
- they drop all photons for which they picked different basis. The measurements of those which were in the same basis are the key. Because they are in the same basis, their results must always be the same in an ideal system.
- if there is an eavesdropper on the line, the results of measurements on the same basis can differ.Unfortunately, this can also happen due to imperfections in the system.Alice and Bob must decide what level of error is above the system's imperfections and implies that an attacker is listening.
E91 Updated 2025-07-16
Requires entangled particles, unlike BB84 which does not.
Fractional quantum Hall effect Updated 2025-07-16
TODO original experiment?
Laughlin paper: 1981 Quantized Hall conductivity in two dimensions.
Spin Hall effect Updated 2025-07-16
String theory Updated 2025-07-16
Quantum field Updated 2025-07-16
Mathematical formulation of quantum field theory Updated 2025-07-16
The Dirac equation, OK, is a partial differential equation, so we can easily understand its definition with basic calculus. We may not be able to solve it efficiently, but at least we understand it.
But what the heck is the mathematical model for a quantum field theory? TODO someone was saying it is equivalent to an infinite set of PDEs somehow. Investigate. Related:
The path integral formulation might actually be the most understandable formulation, as shown at Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979).
Quantum electrodynamics by Lifshitz et al. 2nd edition (1982) chapter 1. "The uncertainty principle in the relativistic case" contains an interesting idea:
The foregoing discussion suggests that the theory will not consider the time dependence of particle interaction processes. It will show that in these processes there are no characteristics precisely definable (even within the usual limitations of quantum mechanics); the description of such a process as occurring in the course of time is therefore just as unreal as the classical paths are in non-relativistic quantum mechanics. The only observable quantities are the properties (momenta,
polarizations) of free particles: the initial particles which come into interaction, and the final particles which result from the process.
Quantum electrodynamics Updated 2025-07-16
Theory that describes electrons and photons really well, and as Feynman puts it "accounts very precisely for all physical phenomena we have ever observed, except for gravity and nuclear physics" ("including the laughter of the crowd" ;-)).
While Ciro acknowledges that QED is intrinsically challenging due to the wide range or requirements (quantum mechanics, special relativity and electromagnetism), Ciro feels that there is a glaring gap in this moneyless market for a learning material that follows the Middle Way as mentioned at: the missing link between basic and advanced. Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979) is one of the best attempts so far, but it falls a bit too close to the superficial side of things, if only Feynman hadn't assumed that the audience doesn't know any mathematics...
The funny thing is that when Ciro Santilli's mother retired, learning it (or as she put it: "how photons and electrons interact") was also one of her retirement plans. She is a pharmacist by training, and doesn't know much mathematics, and her English was somewhat limited. Oh, she also wanted to learn how photosynthesis works (possibly not fully understood by science as that time, 2020). Ambitious old lady!!!
Combines special relativity with more classical quantum mechanics, but further generalizing the Dirac equation, which also does that: Dirac equation vs quantum electrodynamics. The name "relativistic" likely doesn't need to appear on the title of QED because Maxwell's equations require special relativity, so just having "electro-" in the title is enough.
Before QED, the most advanced theory was that of the Dirac equation, which was already relativistic but TODO what was missing there exactly?
As summarized at: youtube.com/watch?v=_AZdvtf6hPU?t=305 Quantum Field Theory lecture at the African Summer Theory Institute 1 of 4 by Anthony Zee (2004):
- classical mechanics describes large and slow objects
- special relativity describes large and fast objects (they are getting close to the speed of light, so we have to consider relativity)
- classical quantum mechanics describes small and slow objects.
- QED describes objects that are both small and fast
That video also mentions the interesting idea that:Therefore, for small timescales, energy can vary a lot. But mass is equivalent to energy. Therefore, for small time scale, particles can appear and disappear wildly.
- in special relativity, we have the mass-energy equivalence
- in quantum mechanics, thinking along the time-energy uncertainty principle,
QED is the first quantum field theory fully developed. That framework was later extended to also include the weak interaction and strong interaction. As a result, it is perhaps easier to just Google for "Quantum Field Theory" if you want to learn QED, since QFT is more general and has more resources available generally.
Like in more general quantum field theory, there is on field for each particle type. In quantum field theory, there are only two fields to worry about:
- photon field
- electromagnetism field
Lecture 01 | Overview of Quantum Field Theory by Markus Luty (2013)
Source. This takes quite a direct approach, one cool thing he says is how we have to be careful with adding special relativity to the Schrödinger equation to avoid faster-than-light information. Weak interaction Updated 2025-10-14
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. Quantum chromodynamics Updated 2025-07-16
Quarks, Gluon flux tubes, Strong Nuclear Force, & Quantum Chromodynamics by Physics Videos by Eugene Khutoryansky (2018)
Source. Some decent visualizations of how the field lines don't expand out like they do in electromagnetism, suggesting color confinement. Quantum field theory simulations Updated 2025-07-16
TODO why is it so hard to find anything non perturbative :-(
- www.youtube.com/channel/UCPHFUHiwbpMqC8ONxEICCiQ NanoNebula using raw Perl PDFL en.wikipedia.org/wiki/Perl_Data_Language (the Perl NumPy)
- www.youtube.com/watch?v=9TJe1Pr5c9Q "Interplay of Quantum Electrodynamics and Quantum Chromodynamics in the Nontrivial Vacuum" by CSSM Visualisation (2019)
On a quantum computer...:
- www.cornell.edu/video/john-preskill-simulating-quantum-field-theory-with-quantum-computer Simulating Quantum Field Theory with a Quantum Computer by John Preskill (2019)
- www.youtube.com/watch?v=Lln-C21u0U8 Quantum Simulation from Quantum Chemistry to Quantum Field Theory by Peter Love (2019)
Infinities in quantum field theory Updated 2025-07-16
TODO concrete example, please...
Internal and spacetime symmetries Updated 2025-07-16
The different only shows up for field, not with particles. For fields, there are two types of changes that we can make that can keep the Lagrangian unchanged as mentioned at Physics from Symmetry by Jakob Schwichtenberg (2015) chapter "4.5.2 Noether's Theorem for Field Theories - Spacetime":
- spacetime symmetry: act with the Poincaré group on the Four-vector spacetime inputs of the field itself, i.e. transforming into
- internal symmetry: act on the output of the field, i.e.:
From the spacetime theory alone, we can derive the Lagrangian for the free theories for each spin:Then the internal symmetries are what add the interaction part of the Lagrangian, which then completes the Standard Model Lagrangian.
Quantum field theory bibliography Updated 2025-07-16
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)
There are unlisted articles, also show them or only show them.