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.
Era of quantum computing before we reach physical errors small enough to do perfect quantum error correction as demonstrated by the quantum threshold theorem.
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.
One of their learning sites: www.qutube.nl/
The educational/outreach branch of QuTech.
Funding:
Quantum computing is hard because we want long coherence but fast control by 
Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16Mentioned e.g. at:
These are two conflicting constraints:
- long coherence times: require isolation from external world, otherwise observation destroys quantum state
- fast control and readout: require coupling with external world
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.
- quantumtech.blog/2023/01/17/quantum-computing-with-neutral-atoms/ OK this one hits it:So we understand that it is truly like the classical computer analog vs digital case.
As Alex Keesling, CEO of QuEra told me, "... whereas in gate-based [digital] quantum computing the focus is on the sequence of the gates, in analog quantum processing it's more about the position of the atoms and where you place them so they can mirror real life problems. We arrange the atoms and define the forces that drive them and then measure the result... so it’s a geometric encoding of the problem itself."
- thequantuminsider.com/2022/06/28/why-analog-neutral-atoms-quantum-computing-is-a-promising-direction-for-early-quantum-advantage on The Quantum Insider useless article mostly by Pasqal
TensorFlow quantum by Masoud Mohseni (2020)
Source. At the timestamp, Masoud gives a thought experiment example of the perhaps simplest to understand analog quantum computer: chained double-slit experiments with carefully calculated distances between slits. Calulating the final propability distribution of that grows exponentially.It is also possible to carry out quantum computing without qubits using processes with a continuous spectrum of measurement.
As of 2020, these approaches seem less developed/promising, but who knows.
Architecture All Access: Quantum Computing by James Clarke (2021)
Source. 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:
- flux
- charge
- phase
Input:
Quantum Computing with Superconducting Qubits by Alexandre Blais (2012)
Source. - youtu.be/uPw9nkJAwDY?t=293 superconducting qubits are good because superconductivity is macroscopic. Explains how in non superconducting metal, each electron moves separatelly, and can hit atoms and leak vibration/photos, which lead to observation and quantum error
- youtu.be/uPw9nkJAwDY?t=429 made of aluminium
- youtu.be/uPw9nkJAwDY?t=432 shows the circuit diagram, and notes that the thing is basically a LC circuitusing the newly created just now Ciro's ASCII art circuit diagram notation. Note that the block on the right is a SQUID device.
+-----+ | | | +-+-+ | | | C X X | | | | +-+-+ | | +-----+ - youtu.be/uPw9nkJAwDY?t=471 mentions that the frequency between states 0 and 1 is chosen to be 6 GHz:This explains why we need to go to much lower temperatures than simply the superconducting temperature of aluminum!
- higher frequencies would be harder/more expensive to generate
- lower frequencies would mean less energy according to the Planck relation. And less energy means that thermal energy would matter more, and introduce more noise.6 GHz is aboutFrom the definition of the Boltzmann constant, the temperature which has that average energe of particles is of the order of:
- youtu.be/xjlGL4Mvq7A?t=138 superconducting quantum computer need non-linear components (too brief if you don't know what he means in advance)
- youtu.be/xjlGL4Mvq7A?t=169 quantum computing is hard because we want long coherence but fast control
Non-linearity is needed otherwise the input energy would just make the state go to higher and higher energy levels, e.g. from 1 to 2. But we only want to use levels 0 and 1.
The way this is modelled in by starting from a pure LC circuit, which is an harmonic oscillator, see also quantum LC circuit, and then replacing the linear inductor with a SQUID device, e.g. mentioned at: youtu.be/eZJjQGu85Ps?t=1655 Video "Superconducting Qubits I Part 1 by Zlatko Minev (2020)".
- requires intense refrigeration to 15mK in dilution refrigerator. Note that this is much lower than the actual superconducting temperature of the metal, we have to go even lower to reduce noise enough, see e.g. youtu.be/uPw9nkJAwDY?t=471 from Video "Building a quantum computer with superconducting qubits by Daniel Sank (2019)"
- less connectivity, normally limited to 4 nearest neighbours, or maybe 6 for 3D approaches, e.g. compared to trapped ion quantum computers, where each trapped ion can be entangled with every other on the same chip
Superconducting qubits are bad because it is harder to ensure that they are all the same by Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16This is unlike atomic systems like trapped ion quantum computers, where each atom is necessarily exactly the same as the other.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact