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.
Mentioned 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
Synonym to gatebased quantum computer/digital quantum computer?
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 Circuitbased 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.
The tensor product is called a "product" because it distributes over addition.
E.g. consider:
$(2 ∣0⟩+∣1⟩ )⊗∣0⟩=2 ∣0⟩⊗∣0⟩+∣1⟩⊗∣0⟩ =2 ∣00⟩+∣10⟩ $
Intuitively, in this operation we just put a Hadamard gate qubit together with a second pure $∣0⟩$ qubit.
And the outcome still has the second qubit as always 0, because we haven't made them interact.
The quantum state $2 ∣00⟩+∣10⟩ $ 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 nonseparable: 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.
$2 ∣00⟩+∣11⟩ =2 ∣0⟩⊗∣0⟩+∣1⟩⊗∣1⟩ $
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 combinatoriallike 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, verilogorvhdl, 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.
The following are the main reasons for that:
One key insight, is that the matrix of a nontrivial 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 $2_{100}$ 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 $2_{N}$ outputs for each of the $2_{N}$ 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 $2_{N}$ 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 8sized 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.
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 gatebased: 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, zXcalculus appears to provide a powerful alternative method in some/all cases.
Bibliography:
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 $∣0⟩$ or $∣1⟩$ (quantum states with probability 1.0 of measuring either 0 or 1), and produces states that have equal probability of 0 or 1.
$H=2 1 [11 1−1 ]$
The quantum NOT gate swaps the state of $∣0⟩$ and $∣1⟩$, i.e. it maps:
As a result, this gate also inverts the probability of measuring 0 or 1, e.g.
$x∣0⟩+y∣y⟩→y∣0⟩+x∣y⟩$
 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
$[01 10 ]$
The most common way to construct multiqubit gates is to use singlequbit 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.
 control qubits
 operand qubits (terminology made up by Ciro Santilli just now)
The first example to look at is the CNOT gate.
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.
$⎣⎢⎢⎢⎡ 1000 0100 0001 0010 ⎦⎥⎥⎥⎤ $
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 $∣00⟩$ and $∣01⟩$ should be possible. Therefore, the state must be of the form:
where $x$ and $y$ are two complex numbers such that $∣x∣+∣y∣=1.0$
$∣00⟩∣01⟩∣10⟩∣11⟩$
$⎣⎢⎢⎢⎡ xy00 ⎦⎥⎥⎥⎤ $
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.
$⎣⎢⎢⎢⎡ 1000 0100 0001 0010 ⎦⎥⎥⎥⎤ ×⎣⎢⎢⎢⎡ xy00 ⎦⎥⎥⎥⎤ =⎣⎢⎢⎢⎡ xy00 ⎦⎥⎥⎥⎤ $
If however we take only states where the control qubit is for sure 1:
$⎣⎢⎢⎢⎡ 1000 0100 0001 0010 ⎦⎥⎥⎥⎤ ×⎣⎢⎢⎢⎡ 00xy ⎦⎥⎥⎥⎤ =⎣⎢⎢⎢⎡ 00yx ⎦⎥⎥⎥⎤ $
Therefore, in that case, what happened is that the probabilities of $∣10⟩$ and $∣11⟩$ were swapped from $x$ and $y$ to $y$ and $x$ 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 GottesmanKnill 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.
Set of quantum logic gate composed of the Clifford gates plus the Toffoli gate. It forms a set of universal quantum gates.
 quantumtech.blog/2023/01/17/quantumcomputingwithneutralatoms/ OK this one hits it:
As Alex Keesling, CEO of QuEra told me, "... whereas in gatebased [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."
So we understand that it is truly like the classical computer analog vs digital case.  thequantuminsider.com/2022/06/28/whyanalogneutralatomsquantumcomputingisapromisingdirectionforearlyquantumadvantage on The Quantum Insider useless article mostly by Pasqal
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 nonquantum analog computers.
Lists of the most promising implementations:
As of 2020, the hottest by far are:
thequantuminsider.com/2022/03/31/5quantumcomputingcompaniesworkingwithnvcentreindiamondtechnology/ on The Quantum Insider
5 Companies Working With Diamond NV Quantum Computing Technology.
Funding:
 2023: £42m (~$50m) www.uktech.news/deeptech/quantummotionraises42m20230221
Based on the Josephson effect. Yet another application of that phenomenal phenomena!
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
and hybridizations of those such as:
Input:
 microwave radiation to excite circuit, or do nothing and wait for it to fall to 0 spontaneously
 interaction: TODO
 readout: TODO
Nonlinearity 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
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: physicsworld.com/a/superconductingquantumbits/. 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 micronsized objects that have been produced using standard semiconductorfabrication techniques – objects that are small on everyday scales but large compared with atoms – can also behave as quantum particles.
Atombased 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 perdevice characterisation.
In Ciro's ASCII art circuit diagram notation, it is a loop with three Josephson junctions:
+X+
 
 
 
+XX+
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 youtu.be/cb_f9KpYipk?t=180 from Video "The transmon qubit by Leo Di Carlo (2018)":
++
 Island 1 
++
 
X C
 
++
 Island 2 
++
youtu.be/eZJjQGu85Ps?t=2443 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
The circuit is then analogous to a LC circuit, with the islands being the capacitor. The Josephson junction functions as a nonlinear inductor.
X
in the diagram as perCiro's ASCII art circuit diagram notation:
++ ++
   
 Q_1() X Q_2() 
   
++ ++
Others define it with a SQUID device instead: youtu.be/cb_f9KpYipk?t=328 from Video "The transmon qubit by Leo Di Carlo (2018)". He mentions that this allows tuning the inductive element without creating a new device.
But seriously, this is a valuable little list.
The course is basically exclusively about transmons.
Funding rounds:
 March 2022: 27M Euros
About their qubit:
 alicebob.com/2023/02/15/computing256bitellipticcurvelogarithmin9hourswith126133catqubits/ Computing 256bit elliptic curve logarithm in 9 hours with 126,133 cat qubits (2023). This describes their "cat 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: archive.ph/wip/Fdgsv their center is in Goleta, California, near Santa Barbara. Though Google tends to promote it more as Santa Barbara, see e.g. Daniel's tshirt at Video "Building a quantum computer with superconducting qubits by Daniel Sank (2019)".
Cool dude. Uses Stack Exchange: physics.stackexchange.com/users/31790/danielsank
Started at Google Quantum AI in 2014.
Has his LaTeX notes at: github.com/DanielSank/theory. One day he will convert to OurBigBook.com. Interesting to see that he is able to continue his notes despite being at Google.
This is a good read: quantumai.google/hardware/datasheet/weber.pdf 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.
The term "IBM Q" has been used in some promotional material as of 2020, e.g.: www.ibm.com/mysupport/s/topic/0TO50000000227pGAA/ibmqquantumcomputing?language=en_US though the fuller form "IBM Quantum Computing" is somewhat more widely used.
They also internally named an division as "IBM Q": sg.news.yahoo.com/ibmthinksreadyturnquantum050100574.html
Open source superconducting quantum computer hardware design!
Their main innovation seems to be their 3D design which they call "Coaxmon".
Funding:
 2023: $1m (869,000 pounds) for Japan expansion: www.uktech.news/deeptech/oqcfundingjapan20230203
 2022: $47m (38M pounds) techcrunch.com/2022/07/04/uksoxfordquantumcircuitssnapsup47mforquantumcomputingasaservice/
 2017: $2.7m globalventuring.com/university/oxfordquantumcalculates27m/
Founding CEO of Oxford Quantum Circuits.
As mentioned at www.investmentmonitor.ai/tech/innovation/inconversationwithoxfordquantumcircuitsilanawisby 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 oxfordscienceenterprises.com/news/meetthefounderilanawisbyceoofoxfordquantumcircuits/ says it clearly:
I was headhunted by Oxford Sciences Enterprises to be the founding CEO of OQC.
oxfordquantumcircuits.com/story mentions that the core patent was by Dr. Peter Leek: www.linkedin.com/in/peterleek00954b62/
TODO understand.
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.
This job announcement from 2022 gives a good idea about their tech stack: web.archive.org/web/20220920114810/https://oxfordionics.bamboohr.com/jobs/view.php?id=32&source=aWQ9MTA%3D. Notably, they use ARTIQ.
Funding:
Merger between Cambridge Quantum Computing, which does quantum software, and Honeywell Quantum Solutions, which does the hardware.
E.g.: www.quantinuum.com/pressrelease/demonstratingbenefitsofquantumupgradabledesignstrategysystemmodelh12firsttoprove2048quantumvolume from 2021.
In 2015, they got a 50 million investment from Grupo Arcano, led by Alberto ChangRajii, 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:
 2022:
 67m euro contract with the German government: www.uktech.news/deeptech/universalquantumgermancontract20221102 Both cofounders are German. They then immediatly announced several jobs in Hamburg: apply.workable.com/universalquantum/?lng=en#jobs so presumably linked to the Hamburg University of Technology campus of the German Aerospace Center.
 medium.com/@universalquantum/universalquantumwins67mcontracttobuildthefullyscalabletrappedionquantumcomputer16eba31b869e
 2021: $10M (7.5M GBP) grant from the British Government: www.uktech.news/news/brightonuniversalquantumwinsgrant20211105This grant is very secretive, very hard to find any other information about it! Most investment trackers are not listing it.The article reads:
Universal Quantum will lead a consortium that includes RollsRoyce, quantum developer Riverlane, and worldclass researchers from Imperial College London and The University of Sussex, among others.
Interesting!A but further down the article gives some more information of partners, from which some of the hardware vendors can be deduced:The consortium includes enduser RollsRoyce supported by the Science and Technology Facilities Council (STFC) Hartree Centre, quantum software developer Riverlane, supply chain partners Edwards, TMD Technologies (now acquired by Communications & Power Industries (CPI)) and Diamond Microwave
 Edwards is presumably Edwards Vacuum, since we know that trapped ion quantum computers rely heavily on good vacuum systems. Edwards Vacuum is also located quite close to Universal Quantum as of 2022, a few minutes drive.
 TMD Technologies is a microwave technology vendor amongst other things, and we know that microwaves are used e.g. to initialize the spin states of the ions
 Diamond Microwave is another microwave stuff vendor
The money comes from UK's "Industrial Strategy Challenge Fund".www.riverlane.com/news/2021/12/riverlanejoins75millionconsortiumtobuilderrorcorrectedquantumprocessor/ gives some more details on the use case provided by Rolls Royce:The work with Rolls Royce will explore how quantum computers can develop practical applications toward the development of more sustainable and efficient jet engines.This starts by applying quantum algorithms to take steps to toward a greater understanding of how liquids and gases flow, a field known as 'fluid dynamics'. Simulating such flows accurately is beyond the computational capacity of even the most powerful classical computers today.This funding was part of a larger quantum push by the UKNQTP: www.ukri.org/news/50millioninfundingforukquantumindustrialprojects/  2020: $4.5M (3.5M GBP) www.crunchbase.com/organization/universalquantum. Just out of stealth.
Cofounders:
 Sebastian Weidt. He is German, right? Yes at youtu.be/SwHaJXVYIeI?t=1078 from Video 3. "Fireside Chat with with Sebastian Weidt by Startup Grind Brighton (2022)". The company was founded by two Germans from Essex!
 Winfried Hensinger: if you saw him on the street, you'd think he plays in a punkrock band. That West Berlin feeling.
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: www.science.org/doi/10.1126/sciadv.1601540
These people are cool.
They use optical tweezers to place individual atoms floating in midair, and then do stuff to entangle their nuclear spins.
Funding:
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
 2022: $15 million www.orcacomputing.com/blog/orcacomputingcompletes15millionseriesafundinground
 2021: $14.5 million for an Innovate UK project
CEO: Jeremy O'Brien
Raised 215M in 2020: www.bloomberg.com/news/articles/20200406/quantumcomputingstartupraises215millionforfasterdevice
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.