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.

Quantum computing is hard because we want long coherence but fast control

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

Quantum computer type

Model of quantum computing

Measurement and circuit based quantum computers

Circuit-based quantum computer

Synonym to gate-based quantum computer/digital quantum computer?

Measurement-based quantum computer (One-way 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 Circuit-based quantum computer where you measure all output qubits at once.

TODO what is the advantage?

Analog and digital quantum computers

Digital quantum computer (Gate-based quantum computer)

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 circuit

A quantum circuit is a graph of quantum logic gates.

Quantum circuits are the prevailing model of quantum computing as of the 2010's - 2020's

Quantum state vector

Sample implementations:

Tensor product in quantum computing

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:

∣ 00 ⟩

that is the same as

∣ 0 ⟩ \otimes ∣ 0 ⟩

The tensor product is called a "product" because it distributes over addition.

E.g. consider:

(\frac{∣ 0 ⟩ + ∣ 1 ⟩}{2}) \otimes ∣ 0 ⟩ = \frac{∣ 0 ⟩ \otimes ∣ 0 ⟩ + ∣ 1 ⟩ \otimes ∣ 0 ⟩}{2} = \frac{∣ 0 0 ⟩ + ∣ 1 0 ⟩}{2}

(1)

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

\frac{∣ 0 0 ⟩ + ∣ 1 0 ⟩}{2}

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:

\frac{∣ 0 0 ⟩ + ∣ 1 1 ⟩}{2} = \frac{∣ 0 ⟩ \otimes ∣ 0 ⟩ + ∣ 1 ⟩ \otimes ∣ 1 ⟩}{2}

(2)

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.

Quantum circuits vs classical circuits

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.

Quantum logic gate

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:

Quantum logic gates are needed because you can't compute the matrix explicitly as it grows exponentially

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

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 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.

Quantum logic gates are needed for physical implementation

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:

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.

Bibliography:

quantumcomputing.stackexchange.com/questions/2299/how-to-interpret-a-quantum-circuit-as-a-matrix

Universal quantum gates

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.

Single-qubit gate

The first two that you should study are:

Hadamard gate

The Hadamard gate takes

∣ 0 ⟩

∣ 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 = \frac{1}{2} [11 1 - 1]

(1)

Equation 1. Hadamard gate matrix.

Pauli gate

Pauli-X gate (Quantum NOT gate)

The quantum NOT gate swaps the state of

∣ 0 ⟩

and

∣ 1 ⟩

, i.e. it maps:

x ∣ 0 ⟩ + y ∣ y ⟩ \to y ∣ 0 ⟩ + x ∣ y ⟩

(1)

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

[0110]

(2)

Equation 2. Quantum NOT gate matrix.

Figure 1. Quantum NOT gate symbol. Source.

Phase shift gate

Multi-qubit gate

The most common way to construct multi-qubit gates is to use single-qubit gates as part of a controlled quantum gate.

Controlled quantum gate

Controlled quantum gates are gates that have two types of input qubits:

control qubits
operand qubits (terminology made up by Ciro Santilli just now)

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.

Empty circle control qubit notation

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

CNOT gate (Controlled NOT gate)

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.

⎣ ⎢ ⎢ ⎢ ⎡ 1000010000010010 ⎦ ⎥ ⎥ ⎥ ⎤

(1)

Equation 1. CNOT gate matrix.

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:

∣ 00 ⟩ ∣ 01 ⟩ ∣ 10 ⟩ ∣ 11 ⟩

(2)

we see that this means that only

∣ 00 ⟩

and

∣ 01 ⟩

should be possible. Therefore, the state must be of the form:

⎣ ⎢ ⎢ ⎢ ⎡ x y 00 ⎦ ⎥ ⎥ ⎥ ⎤

(3)

where

x

and

y

are two complex numbers such that

∣ x ∣ + ∣ y ∣ = 1.0

If we operate the CNOT gate on that state, we obtain:

⎣ ⎢ ⎢ ⎢ ⎡ 1000010000010010 ⎦ ⎥ ⎥ ⎥ ⎤ \times ⎣ ⎢ ⎢ ⎢ ⎡ x y 00 ⎦ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎡ x y 00 ⎦ ⎥ ⎥ ⎥ ⎤

(4)

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:

⎣ ⎢ ⎢ ⎢ ⎡ 1000010000010010 ⎦ ⎥ ⎥ ⎥ ⎤ \times ⎣ ⎢ ⎢ ⎢ ⎡ 00 x y ⎦ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎡ 00 y x ⎦ ⎥ ⎥ ⎥ ⎤

(5)

Therefore, in that case, what happened is that the probabilities of

∣ 10 ⟩

and

∣ 11 ⟩

were swapped from

x

and

y

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.

Toffoli gate

Clifford gates

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.

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.

Gottesman-Knill theorem

List of quantum logic gates

Analog quantum computer

quantumtech.blog/2023/01/17/quantum-computing-with-neutral-atoms/ OK this one hits it:
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."
So we understand that it is truly like the classical computer analog vs digital case.
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

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.

Continuous-variable quantum information

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.

Quantum computer physical implementation

Lists of the most promising implementations:

As of 2020, the hottest by far are:

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, www.linkedin.com/in/lukasbaker1331/, 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.

Diamond vacancy quantum computer

thequantuminsider.com/2022/03/31/5-quantum-computing-companies-working-with-nv-centre-in-diamond-technology/ on The Quantum Insider

5 Companies Working With Diamond NV Quantum Computing Technology.

N-V center quantum computer

Nitrogen-vacancy center

Electron on helium quantum computer (eHe quantum computer)

Organization developing electron on helium quantum computer

EeroQ

eeroq.com/

Quantum dot quantum computer

Organization developing quantum dot quantum computer

Quantum Motion (2017)

Funding:

2023: £42m (~$50m) www.uktech.news/deep-tech/quantum-motion-raises-42m-20230221

Intel quantum computer

Video 1. Architecture All Access: Quantum Computing by James Clarke (2021) Source.

Superconducting quantum computing

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:

transmon

Input:

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. youtu.be/Kz6mhh1A_mU?t=1171 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.

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 circuit
```
+-----+
|     |
|   +-+-+
|   |   |
C   X   X
|   |   |
|   +-+-+
|     |
+-----+
```
using the newly created just now Ciro's ASCII art circuit diagram notation. Note that the block on the right is a SQUID device.
youtu.be/uPw9nkJAwDY?t=471 mentions that the frequency between states 0 and 1 is chosen to be 6 GHz:
- 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 about $6^{9} \times h = 6 \times 1 0^{9} \times 6.62 \times 1 0^{- 34} \approx 4 \times 1 0^{- 24} J$
  From the definition of the Boltzmann constant, the temperature which has that average energe of particles is of the order of:
  $T = E / k_{b} = 4 \times 1 0^{- 24} / 1.38 \times 1 0^{- 23} \approx 0.3 K$
  (1)
This explains why we need to go to much lower temperatures than simply the superconducting temperature of aluminum!

Video 4. A Brief History of Superconducting Quantum Computing by Steven Girvin (2021) Source.

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

Video 5. Superconducting Qubits I Part 1 by Zlatko Minev (2020) Source.

The Q&A in the middle of talking is a bit annoying.

youtu.be/eZJjQGu85Ps?t=2443 the first actually useful part, shows a transmon diagram with some useful formulas on it

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.

Superconducting quantum computer need non-linear components

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)".

Superconducting qubit

Pros and cons of superconducting qubits

Con of superconducting qubits

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

This is unlike atomic systems like trapped ion quantum computers, where each atom is necessarily exactly the same as the other.

Pro of superconducting qubits

Superconducting qubits are good because superconductivity is macroscopic

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/superconducting-quantum-bits/. 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.

Superconducting qubits are bad because of fabrication variation

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.

Superconducting qubit type

Flux qubit

In Ciro's ASCII art circuit diagram notation, it is a loop with three Josephson junctions:

+----X-----+
|          |
|          |
|          |
+--X----X--+

https://upload.wikimedia.org/wikipedia/en/0/04/Flux_Qubit_-_Holloway.jpg

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:

+----X-----+
|          |
|          |
|          |
+--X----X--+

They name the clockwise and counter clockwise states

∣ L ⟩

and

∣ R ⟩

(named for Left and Right).

When half the magnetic flux quantum is applied as microwaves, this produces the ground state:

∣ 0 ⟩ = ∣ L ⟩ + ∣ R ⟩

(1)

where

L

and

R

cancel each other out. And the first excited state

∣ 1 ⟩

is:

∣ 0 ⟩ = ∣ L ⟩ - ∣ R ⟩

(2)

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 $∣ 0 ⟩ + ∣ 1 ⟩$ .

Transmon (2007)

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 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: 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.

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.

Rabi cycle (Rabi oscillations)

The Hardware of a Quantum Computer by TU Delft

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.

Organization developing superconducting quantum computer

Alice&Bob

Funding rounds:

March 2022: 27M Euros

About their qubit:

alice-bob.com/2023/02/15/computing-256-bit-elliptic-curve-logarithm-in-9-hours-with-126133-cat-qubits/ Computing 256-bit elliptic curve logarithm in 9 hours with 126,133 cat qubits (2023). This describes their "cat qubit".

Cat qubit

Google Quantum AI

quantumai.google/hardware/our-lab

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 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!

Daniel Sank

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.

Sycamore processor

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.

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.

IBM Quantum Computing (IBM Q)

The term "IBM Q" has been used in some promotional material as of 2020, e.g.: www.ibm.com/mysupport/s/topic/0TO50000000227pGAA/ibm-q-quantum-computing?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/ibm-thinks-ready-turn-quantum-050100574.html

IBM quantum computer

OpenSuperQ

Open source superconducting quantum computer hardware design!

Video 1. OpenSuperQ intro by Quantum Flagship (2021) Source.

youtu.be/2cU2PA4Q0Bw?t=61: Dilution refrigerator by Bluefors

Oxford Quantum Circuits (2017, OQC)

oxfordquantumcircuits.com/

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/deep-tech/oqc-funding-japan-20230203
2022: $47m (38M pounds) techcrunch.com/2022/07/04/uks-oxford-quantum-circuits-snaps-up-47m-for-quantum-computing-as-a-service/
2017: $2.7m globalventuring.com/university/oxford-quantum-calculates-2-7m/

Video 1. The Coaxmon by Oxford Quantum Circuits (2022) Source.

Ilana Wisby

Founding CEO of Oxford Quantum Circuits.

As mentioned at www.investmentmonitor.ai/tech/innovation/in-conversation-with-oxford-quantum-circuits-ilana-wisby 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/meet-the-founder-ilana-wisby-ceo-of-oxford-quantum-circuits/ 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/peter-leek-00954b62/

Rigetti Computing (2017, OQC)

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: www.linkedin.com/in/gueneverep/.

Topological quantum computer

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.

Trapped ion quantum computer

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:

youtu.be/j1SKprQIkyE?t=217 you need ultra-high vacuum
youtu.be/j1SKprQIkyE?t=257 you put the Calcium on a "calcium oven", heat it up, and make it evaporates a little bit
youtu.be/j1SKprQIkyE?t=289 you need lasers. You shine the laser on the calcium atom to eject one of the two valence electrons from it. Though e.g. Universal Quantum is trying to do away with them, because alignment for thousands or millions of particles would be difficult.
youtu.be/j1SKprQIkyE?t=518 keeping all surrounding electrodes positive would be unstable. So they instead alternate electrode quickly between plus and minus
youtu.be/j1SKprQIkyE?t=643 talks about the alternative, of doing it just with electrodes on a chip, which is easier to manufacture. They fly at about 100 microns above the trap. And you can have multiple ions per chip.
youtu.be/j1SKprQIkyE?t=1165 using microwaves you can flip the spin of the electron, or put it into a superposition. From more reading, we understand that she is talking about a hyperfine transition, which often happen in the microwave area.
youtu.be/j1SKprQIkyE?t=1210 talks about making quantum gates. You have to put the ions into a magnetic field at one of the two resonance frequencies of the system. Presumably what is meant is an inhomogenous magnetic field as in the Stern-Gerlach experiment.
This is the hard and interesting part. It is not clear why the atoms become coupled in any way. Is it due to electric repulsion?
She is presumably describing the Cirac–Zoller CNOT gate.

Sounds complicated, several technologies need to work together for that to work! Videos of ions moving are from www.physics.ox.ac.uk/research/group/ion-trap-quantum-computing.

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.

youtu.be/lJOuPmI--5c?t=1601 Cirac–Zoller CNOT gate was the first 2 qubit gate. Explains it more or less.

Video 4. Introduction to quantum optics by Peter Zoller (2018) Source. THE Zoller from Cirac–Zoller CNOT gate talks about his gate.

www.youtube.com/watch?v=W3l0QPEnaq0&t=427s shows that the state is split between two options: center of mass mode (ions move in same direction), and strechmode (atoms move in opposite directions)
youtu.be/W3l0QPEnaq0?t=658 shows a schematic of the experiment

Cirac–Zoller controlled-NOT gate (1995)

Ion trap

Modular trapped ion quantum computer

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.

Organization developing trapped ion quantum computer

IonQ (2015)

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.

youtu.be/9aOLwjUZLm0?t=1216 superconducting qubits are bad because it is harder to ensure that they are all the same
youtu.be/9aOLwjUZLm0?t=1270 our wires are provided by lasers. Gives example of ytterbium $^{+ 1}$ , which has nice frequencies for practical laser choice. Ytterbium ends in 6s2 5d1, so they must remove the 5d1 electron? But then you are left with 2 electrons in 6s2, can you just change their spins at will without problem?
youtu.be/9aOLwjUZLm0?t=1391 a single atom actually reflects 1% of the input laser, not bad!
youtu.be/9aOLwjUZLm0?t=1475 a transition that they want to drive in Ytterbium has 355 nm, which is easy to generate TODO why.
youtu.be/9aOLwjUZLm0?t=1520 mentions that 351 would be much harder, e.g. as used in inertially confied fusion, takes up a room
youtu.be/9aOLwjUZLm0?t=1539 what they use: a pulsed laser. It is made primarily for photolithography, Coherent, Inc. makes 200 of them a year, so it is reliable stuff and easy to operate. At www.coherent.com/lasers/nanosecond/avia-nx we can see some of their 355 offers. archive.ph/wip/JKuHI shows a used system going for 4500 USD.
youtu.be/9aOLwjUZLm0?t=1584 Cirac and Zoller proposed the idea of using entangled ions soon after they heard about Shor's algorithm in 1995
youtu.be/9aOLwjUZLm0?t=1641 you use optical tweezers to move the pairs of ions you want to entangle. This means shining a laser on two ions at the same time. Their movement depends on their spin, which is already in a superposition. If both move up, their distance stats the same, so the Coulomb interaction is unchanged. But if they are different, then one goes up and the other down, distance increases due to the diagonal, and energy is lower.
youtu.be/9aOLwjUZLm0?t=1939 S. Debnah 2016 Nature experiment with a pentagon. Well, it is not a pentagon, they are just in a linear chain, the pentagon is just to convey the full connectivity. Maybe also Satanism. Anyways. This point also mentions usage of an acousto-optic modulator to select which atoms we want to act on. On the other side, a simpler wide laser is used that hits all atoms (optical tweezers are literally like tweezers in the sense that you use two lasers). Later on mentions that the modulator is from Harris, later merged with L3, so: www.l3harris.com/all-capabilities/acousto-optic-solutions
youtu.be/9aOLwjUZLm0?t=2119 Bernstein-Vazirani algorithm. This to illustrate better connectivity of their ion approach compared to an IBM quantum computer, which is a superconducting quantum computer
youtu.be/9aOLwjUZLm0?t=2354 hiden shift algorithm
youtu.be/9aOLwjUZLm0?t=2740 Zhang et al. Nature 2017 paper about a 53 ion system that calculates something that cannot be classically calculated. Not fully controllable though, so more of a continuous-variable quantum information operation.
youtu.be/9aOLwjUZLm0?t=2923 usage of cooling to 4 K to get lower pressures on top of vacuum. Before this point all experiments were room temperature. Shows image of refrigerator labelled Janis cooler, presumably something like: qd-uki.co.uk/cryogenics/janis-recirculating-gas-coolers/
youtu.be/9aOLwjUZLm0?t=2962 qubit vs gates plot by H. Neven
youtu.be/9aOLwjUZLm0?t=3108 modular trapped ion quantum computer ideas. Mentions experiment with 2 separate systems with optical link. Miniaturization and their black box. Mentions again that their chip is from Sandia. Amazing how you pronounce that.

NQIT

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.

Oxford Ionics (2017, OQC)

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:

2023: $36m www.forbes.com/sites/gilpress/2023/01/09/36-million-oxford-ionics-funding-to-jump-start-quantum-computing-in-2023/?sh=6af75e7a6ccb

Quantinuum

Merger between Cambridge Quantum Computing, which does quantum software, and Honeywell Quantum Solutions, which does the hardware.

Quantinuum hardware

Quantinuum H1

Quantinuum H1-2

E.g.: www.quantinuum.com/pressrelease/demonstrating-benefits-of-quantum-upgradable-design-strategy-system-model-h1-2-first-to-prove-2-048-quantum-volume from 2021.

Cambridge Quantum Computing

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.

tket

github.com/CQCL/tket

TODO vs all the others?

Honeywell Quantum Solutions

Universal Quantum

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/deep-tech/universal-quantum-german-contract-20221102 Both co-founders 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/universal-quantum-wins-67m-contract-to-build-the-fully-scalable-trapped-ion-quantum-computer-16eba31b869e
2021: $10M (7.5M GBP) grant from the British Government: www.uktech.news/news/brighton-universal-quantum-wins-grant-20211105
This 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 Rolls-Royce, quantum developer Riverlane, and world-class 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 end-user Rolls-Royce 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/riverlane-joins-7-5-million-consortium-to-build-error-corrected-quantum-processor/ 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/50-million-in-funding-for-uk-quantum-industrial-projects/
2020: $4.5M (3.5M GBP) www.crunchbase.com/organization/universal-quantum. Just out of stealth.

Co-founders:

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 punk-rock 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

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.

youtu.be/rYe9TXz35B8?t=127 shows some 3D models. It shows how piezoelectric actuators are used to align or misalign some plates, which presumably then determine conductivity

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:

youtu.be/SwHaJXVYIeI?t=680 we are not at a point where you can buy victory. There is too much uncertainty involved across different approaches.
youtu.be/SwHaJXVYIeI?t=949 his background
youtu.be/SwHaJXVYIeI?t=1277 difference between venture capitalists in different countries
youtu.be/SwHaJXVYIeI?t=1535 they are 33 people now. They've just setup their office in Haywards Heath, north of Bristol.

Cirac–Zoller controlled-NOT gate

Neutral atom quantum computer

quantumtech.blog/2023/01/17/quantum-computing-with-neutral-atoms/

2023: $100m techcrunch.com/2023/01/23/pasqal-raises-100m-to-build-a-neutral-atom-based-quantum-computer/

Photonic quantum computer (Linear optical quantum computing)

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.