One possibly interesting and possibly obvious point of view, is that a quantum computer is an experimental device that executes a quantum probabilistic experiment for which the probabilities cannot be calculated theoretically efficiently by a nuclear weapon.
This is how quantum computing was originally theorized by the likes of Richard Feynman: they noticed that "Hey, here's a well formulated quantum mechanics problem, which I know the algorithm to solve (calculate the probability of outcomes), but it would take exponential time on the problem size".
The converse is then of course that if you were able to encode useful problems in such an experiment, then you have a computer that allows for exponential speedups.
This can be seen very directly by studying one specific quantum computer implementation. E.g. if you take the simplest to understand one, photonic quantum computer, you can make systems for which you need exponential time to calculate the probabilities that photons will exit through certain holes and not others.
The obvious aspect of this idea is by coming from quantum logic gates are needed because you can't compute the matrix explicitly as it grows exponentially: knowing the full explicit matrix is impossible in practice, and knowing the matrix is equivalent to knowing the probabilities of every outcome.
These are two conflicting constraints:
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?
As of 2022, this tends to be the more "default" when you talk about a quantum computer.
But there are some serious analog quantum computer contestants in the field as well.
Quantum circuits are the prevailing model of quantum computing as of the 2010's - 2020's
Sample implementations:
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: that is the same as .
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 qubit.
And the outcome still has the second qubit as always 0, because we haven't made them interact.
The quantum state 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.
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,
Another consequence of this is that programming quantum computers does not look like programming the more "common" procedural programming languages such as C or Python, since those fundamentally rely on processor register / memory state all the time.
Quantum programmers can however use classic languages to help describe their quantum programs more easily, for example this is what happens in Qiskit, where you write a Python program that makes Qiskit library calls that describe the quantum program.
At Section "Quantum computing is just matrix multiplication" we saw that making a quantum circuit actually comes down to designing one big unitary matrix.
We have to say though that that was a bit of a lie.
Quantum programmers normally don't just produce those big matrices manually from scratch.
Instead, they use quantum logic gates.
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 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 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 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)
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.
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 most commonly considered quantum gates take 1, 2, or 3 qubits as input.
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 0
The 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 ---------- 2
then we would just extend the 2x2 CNOT gate to:
TODO 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.
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.
The first two that you should study are:
The Hadamard gate takes or (quantum states with probability 1.0 of measuring either 0 or 1), and produces states that have equal probability of 0 or 1.
Figure 1. Hadamard gate symbol. Source.
The quantum NOT gate swaps the state of and , i.e. it maps:
As a result, this gate also inverts the probability of measuring 0 or 1, e.g.
  • if the old probability of 0 was 0, then it becomes 1
  • if the old probability of 0 was 0.2, then it becomes 0.8
Figure 1. Quantum NOT gate symbol. Source.
The most common way to construct multi-qubit gates is to use single-qubit gates as part of a controlled quantum gate.
Controlled quantum gates are gates that have two types of input qubits:
These gates can be understood as doing a certain unitary operation only if the control qubits are enabled or disabled.
The first example to look at is the CNOT gate.
Figure 1. Generic controlled quantum gate symbol. Source.
The black dot means "control qubit", and "U" means an arbitrary Unitary operation.
When the operand has a conventional symbol, e.g. the Figure "Quantum NOT gate symbol" for the quantum NOT gate to form the CNOT gate, that symbol is used in the operand instead.
Some authors use the convention of:
  • filled black circle: conventional controlled quantum gate, i.e. operate if control qubit is active
  • empty (White) circle: operarate if control qubit is inactive
The CNOT gate is a controlled quantum gate that operates on two qubits, flipping the second (operand) qubit if the first (control) qubit is set.
This gate is the first example of a controlled quantum gate that you should study.
Figure 1. CNOT gate symbol. Source. The symbol follow the generic symbol convention for controlled quantum gates shown at Figure "Generic controlled quantum gate symbol", but replacing the generic "U" with the Figure "Quantum NOT gate symbol".
To understand why the gate is called a CNOT gate, you should think as follows.
First let's produce a generic quantum state vector where the control qubit is certain to be 0.
On the standard basis:
we see that this means that only and should be possible. Therefore, the state must be of the form:
where and are two complex numbers such that
If we operate the CNOT gate on that state, we obtain:
and so the input is unchanged as desired, because the control qubit is 0.
If however we take only states where the control qubit is for sure 1:
Therefore, in that case, what happened is that the probabilities of and were swapped from and to and respectively, which is exactly what the quantum NOT gate does.
So from this we understand more concretelly what "the gate only operates if the first qubit is set to one" means.
Now go and study the Bell state and understand intuitively how this gate is used to produce it.
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.
A popular set of universal quantum gates derived from Clifford gates is Clifford plus T.
Set of quantum logic gate composed of the Clifford gates plus the Toffoli gate. It forms a set of universal quantum gates.
Video 1. 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.
TODO synonym to analog quantum computer?
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.
These computers can be seen as analogous to classical non-quantum analog computers.
Video 1. How To Build A Quantum Computer by Lukas's Lab (2023) Source.
Super quick overview of the main types of quantum computer physical implementations, so doesn't any much to a quick Google.
He sais he's going to make a series about it, so then something useful might actually come out. The first one was: Video "How to Turn Superconductors Into A Quantum Computer by Lukas's Lab (2023)", but it is still too basic.
The author's full name is Lukas Baker,, found with Google reverse image search, even though the LinkedIn image is very slightly different from the YouTube one.
As of 2023 he was a PhD student at NYU.
5 Companies Working With Diamond NV Quantum Computing Technology.
Video 1. Architecture All Access: Quantum Computing by James Clarke (2021) Source.
Based on the Josephson effect. Yet another application of that phenomenal phenomena!
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 there are actually a few different types of superconducting qubits:
  • flux
  • charge
  • phase
and hybridizations of those such as:
  • microwave radiation to excite circuit, or do nothing and wait for it to fall to 0 spontaneously
  • interaction: TODO
  • readout: TODO
Video 1. Quantum Computing with Superconducting Qubits by Alexandre Blais (2012) Source.
Video 2. Quantum Transport, Lecture 16: Superconducting qubits by Sergey Frolov (2013) Source. describes several possible realizations: charge, flux, charge/flux and phase.
Video 3. Building a quantum computer with superconducting qubits by Daniel Sank (2019) Source. Daniel wears a "Google SB" t-shirt, which either means shabi in Chinese, or Santa Barbara. Google Quantum AI is based in Santa Barbara, with links to UCSB.
Video 4. A Brief History of Superconducting Quantum Computing by Steven Girvin (2021) Source.
Video 5. Superconducting Qubits I Part 1 by Zlatko Minev (2020) Source.
The Q&A in the middle of talking is a bit annoying.
Video 6. Superconducting Qubits I Part 2 by Zlatko Minev (2020) Source.
Video 7. How to Turn Superconductors Into A Quantum Computer by Lukas's Lab (2023) Source. This video is just the introduction, too basic. But if he goes through with the followups he promisses, then something might actually come out of it.
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: Video "Superconducting Qubits I Part 1 by Zlatko Minev (2020)".
This is unlike atomic systems like trapped ion quantum computers, where each atom is necessarily exactly the same as the other.
Superconducting qubits are regarded as promising because superconductivity is a macroscopic quantum phenomena of Bose Einstein condensation, and so as a macroscopic phenomena, it is easier to control and observe.
This is mentioned e.g. in this relatively early: While most quantum phenomena is observed at the atomic scale, superconducting qubits are micrometer scale, which is huge!
Physicists are comfortable with the use of quantum mechanics to describe atomic and subatomic particles. However, in recent years we have discovered that micron-sized objects that have been produced using standard semiconductor-fabrication techniques – objects that are small on everyday scales but large compared with atoms – can also behave as quantum particles.
Atom-based qubits like trapped ion quantum computers have parameters fixed by the laws of physics.
However superconducting qubits have a limit on how precise their parameters can be set based on how well we can fabricate devices. This may require per-device characterisation.
In Ciro's ASCII art circuit diagram notation, it is a loop with three Josephson junctions:
|          |
|          |
|          |
Video 1. Superconducting Qubit by NTT SCL (2015) Source.
Offers an interesting interpretation of superposition in that type of device (TODO precise name, seems to be a flux qubit): current going clockwise or current going counter clockwise at the same time.
Their circuit is a loop with three Josephson junctions, in Ciro's ASCII art circuit diagram notation:
|          |
|          |
|          |
They name the clockwise and counter clockwise states and (named for Left and Right).
When half the magnetic flux quantum is applied as microwaves, this produces the ground state:
where and cancel each other out. And the first excited state is:
Then he mentions that:
  • to go from 0 to 1, they apply the difference in energy
  • if the duration is reduced by half, it creates a superposition of .
Used e.g. in the Sycamore processor.
The most basic type of transmon is in Ciro's ASCII art circuit diagram notation, an LC circuit e.g. as mentioned at from Video "The transmon qubit by Leo Di Carlo (2018)":
| Island 1 |
   |   |
   X   C
   |   |
| Island 2 |
+----------+ from Video "Superconducting Qubits I Part 1 by Zlatko Minev (2020)" describes a (possibly simplified) physical model of it, as two superconducting metal islands linked up by a Josephson junction marked as X in the diagram as per-Ciro's ASCII art circuit diagram notation:
+-------+       +-------+
|       |       |       |
| Q_1() |---X---| Q_2() |
|       |       |       |
+-------+       +-------+
The circuit is then analogous to a LC circuit, with the islands being the capacitor. The Josephson junction functions as a non-linear inductor.
Others define it with a SQUID device instead: from Video "The transmon qubit by Leo Di Carlo (2018)". He mentions that this allows tuning the inductive element without creating a new device.
Video 1. The superconducting transmon qubit as a microwave resonator by Daniel Sank (2021) Source.
Video 2. Calibration of Transmon Superconducting Qubits by Stefan Titus (2021) Source. Possibly this Keysight which would make sense.
EdX course. Meh! Just give me the YouTube list!!
But seriously, this is a valuable little list.
The course is basically exclusively about transmons.
Video 1. The transmon qubit by Leo Di Carlo (2018) Source. Via QuTech Academy.
Video 2. Circuit QED by Leo Di Carlo (2018) Source. Via QuTech Academy.
Video 3. Measurements on transmon qubits by Niels Bultink (2018) Source. Via QuTech Academy. I wish someone would show some actual equipment running! But this is of interest.
Video 4. Single-qubit gate by Brian Taraskinki (2018) Source. Good video! Basically you make a phase rotation by controlling the envelope of a pulse.
Video 5. Two qubit gates by Adriaan Rol (2018) Source.
Video 6. Assembling a Quantum Processor by Leo Di Carlo (2018) Source. Via QuTech Academy.
Funding rounds:
  • March 2022: 27M Euros
About their qubit:
Google's quantum hardware/software effort.
The AI is just prerequisite buzzword of the era for any project.
According to job postings such as: their center is in Goleta, California, near Santa Barbara. Though Google tends to promote it more as Santa Barbara, see e.g. Daniel's t-shirt at Video "Building a quantum computer with superconducting qubits by Daniel Sank (2019)".
Video 1. Control of transmon qubits using a cryogenic CMOS integrated circuit (QuantumCasts) by Google (2020) Source. Fantastic video, good photos of the Google Quantum AI setup!
Started at Google Quantum AI in 2014.
Has his LaTeX notes at: One day he will convert to Interesting to see that he is able to continue his notes despite being at Google.
This is a good read: May 14, 2021. Their topology is so weird, not just a rectangle, one wonders why! You get different error rates in different qubits, it's mad.
Figure 1. Google Sycamore Weber quantum computer connectivity graph. Weber is a specific processor of the Sycamore family. From this we see it clearly that qubits are connected to at most 4 other qubits, and that the full topology is not just a simple rectangle.
The term "IBM Q" has been used in some promotional material as of 2020, e.g.: though the fuller form "IBM Quantum Computing" is somewhat more widely used.
They also internally named an division as "IBM Q":
Open source superconducting quantum computer hardware design!
Video 1. OpenSuperQ intro by Quantum Flagship (2021) Source.
Their main innovation seems to be their 3D design which they call "Coaxmon".
Video 1. The Coaxmon by Oxford Quantum Circuits (2022) Source.
Founding CEO of Oxford Quantum Circuits.
As mentioned at she is not the original tech person:
she was finally headhunted by Oxford Science and Innovation to become the founding CEO of OQC. The company was spun out of Oxford University's physics department in 2017, at which point Wisby was handed "a laptop and a patent".
Did they mean Oxford Sciences Enterprises? There's nothing called "Oxford Science and Innovation" on Google. Yes, it is just a typo says it clearly:
I was headhunted by Oxford Sciences Enterprises to be the founding CEO of OQC. mentions that the core patent was by Dr. Peter Leek:
Video 1. Forest: an Operating System for Quantum Computing by Guen Prawiroatmodjo (2017) Source. The title of the talk is innapropriate, this is a very basic overview of the entire Rigetti Computing stack. Still some fine mentions. Her name is so long, TODO origin? She later moved to Microsoft Quantum:
Video 1. Topological Quantum Computer by Professor John Preskill. Source.
Video 2. Topological Quantum Computation by Jason Alicea (2021) Source.
Video 3. Anyons by Yuly Billig (2022) Source.
TODO understand.
Video 1. Trapping Ions for Quantum Computing by Diana Craik (2019) Source.
A basic introduction, but very concrete, with only a bit of math it might be amazing:
Sounds complicated, several technologies need to work together for that to work! Videos of ions moving are from
A major flaw of this presentation is not explaining the readout process.
Video 2. How To Trap Particles in a Particle Accelerator by the Royal Institution (2016) Source. Demonstrates trapping pollen particles in an alternating field.
Video 3. Ion trapping and quantum gates by Wolfgang Ketterle (2013) Source.
Video 4. Introduction to quantum optics by Peter Zoller (2018) Source. THE Zoller from Cirac–Zoller CNOT gate talks about his gate.
Trapped ion people acknowledge that they can't put a million qubits in on chip (TODO why) so they are already thinking of ways to entangle separate chips. Thinking is maybe the key word here. One of the propoesd approaches inolves optical links. Universal Quantum for example explicitly rejects that idea in favor of electric field link modularity.
Video 1. Quantum Simulation and Computation with Trapped Ions by Christopher Monroe (2021) Source.
Video 2. Quantum Computing with Trapped Ions by Christopher Monroe (2018) Source. Co-founder of IonQ. Cool dude. Starts with basic background we already know now. Mentions that there is some relationship between atomic clocks and trapped ion quantum computers, which is interesting. Then he goes into turbo mode, and you get lost unless you're an expert! Video 1. "Quantum Simulation and Computation with Trapped Ions by Christopher Monroe (2021)" is perhaps a better watch.
Video 1. Quantum Computing with Networked Ion traps by NQIT (2018) Source. The video is a bit useless. But it does show the networked approach proposal a little bit. Universal Quantum's homepage particularly rejects that.
This job announcement from 2022 gives a good idea about their tech stack: Notably, they use ARTIQ.
Merger between Cambridge Quantum Computing, which does quantum software, and Honeywell Quantum Solutions, which does the hardware.
In 2015, they got a 50 million investment from Grupo Arcano, led by Alberto Chang-Rajii, who is a really shady character who fled from justice for 2 years:Merged into Quantinuum later on in 2021.
TODO vs all the others?
As of 2021, their location is a small business park in Haywards Heath, about 15 minutes north of Brighton[ref]
Funding rounds:
Homepage says only needs cooling to 70 K. So it doesn't work with liquid nitrogen which is 77 K?
Homepage points to foundational paper:
Video 1. Universal Quantum emerges out of stealth by University of Sussex (2020) Source. Explains that a more "traditional" trapped ion quantum computer would user "pairs of lasers", which would require a lot of lasers. Their approach is to try and do it by applying voltages to a microchip instead.
Video 2. Quantum Computing webinar with Sebastian Weidt by Green Lemon Company (2020) Source. The sound quality is to bad to stop and listen to, but it presumaby shows the coding office in the background.
Video 3. Fireside Chat with with Sebastian Weidt by Startup Grind Brighton (2022) Source. Very basic target audience:
These people are cool.
They use optical tweezers to place individual atoms floating in midair, and then do stuff to entangle their nuclear spins.
Uses photons!
The key experiment/phenomena that sets the basis for photonic quantum computing is the two photon interference experiment.
The physical representation of the information encoding is very easy to understand:
  • input: we choose to put or not photons into certain wires or no
  • interaction: two wires pass very nearby at some point, and photons travelling on either of them can jump to the other one and interact with the other photons
  • output: the probabilities that photos photons will go out through one wire or another
Video 1. Jeremy O'Brien: "Quantum Technologies" by GoogleTechTalks (2014) Source. This is a good introduction to a photonic quantum computer. Highly recommended.
Good talk by CEO before starting the company which gives insight on what they are very likely doing: Video "Jeremy O'Brien: "Quantum Technologies" by GoogleTechTalks (2014)"
PsiQuantum appears to be particularly secretive, even more than other startups in the field.
They want to reuse classical semiconductor fabrication technologies, notably they have close ties to GlobalFoundries.
So he went to the US and raised N times more from the American military-industrial complex.