STORY / FULL STACK

How a quantum computer works

Follow one program down the layers of a quantum computer, into the refrigerator and onto the chip, then watch qubits being created, entangled and read out.

Scroll down, or press Play and lean back. Back and Next step through the story, and so do the arrow keys. Choose how deep to go.

Chapter 1 of 9

The stack

A quantum computer is mostly ordinary computing. Only the chip at the bottom is quantum. The layers above it work on quantum programs, but they run on ordinary computers and electronics.

  1. Quantum algorithm
  2. Programming paradigm and languagesProgramming languages
  3. Quantum arithmeticArithmeticRuntimeCompiler
  4. Quantum instruction setInstruction set
  5. Microarchitecture
  6. Quantum to classical
  7. Quantum chip
  8. Error correction

Ten parts, seven layers

Six layers of ordinary computing.
One quantum chip.

Hover or tap a layer to read what it does.

Goal

Link two qubits so their answers agree

OPENQASM 3.0;
include "stdgates.inc";
qubit[2] q;
bit[2] c;
h q[0];
cx q[0], q[1];
c = measure q;

Compiler

What we wrote
What the chip runs

Instructions and pulses

startinstructionlasts
0 nssx q0, sx q130 ns
30 nscz q0, q160 ns
90 nssx q130 ns
120 nsmeasure q0, q1
q0 driveq1 driveCZ linereadout03090120200ns

Microwaves to the chip

30 ns

A real pulse holds about 150 wiggles; ten are drawn.

q0q1

Back up

  1. Algorithm0048.7%010.0%100.0%1151.3%
  2. Runtimeafter 1000 runs: 00 seen 487 times, 11 seen 513 times
  3. Microarchitectureeach echo becomes a bit pair, 00 or 11
  4. Quantum to classicala faint echo per qubit, turned into a voltage
Quantum algorithm
The recipe: a step-by-step method that uses qubits to answer a question, such as the lowest energy of a molecule.
Programming paradigm and languages
How people write the recipe down, in a language such as OpenQASM or a Python library.
Quantum arithmetic
Ready-made building blocks, such as adders, so nobody has to build everything from single gates.
Runtime
Software that runs the job: it sends circuits to the machine, repeats them thousands of times and gathers the answers.
Compiler
Translates the code into the few operations the chip can do, and places program qubits on real qubits that are wired together.
Quantum instruction set
The list of instructions the machine understands, each with an exact start time, like the instruction set of a normal processor.
Microarchitecture
Fast digital hardware that turns each instruction into the right pulse shape at the right nanosecond.
Quantum to classical
Electronics that turn numbers into microwave pulses, and the faint echoes coming back into numbers again.
Quantum chip
The only quantum layer: the qubits themselves, cooled to a few thousandths of a degree above absolute zero.
Quantum error correction
Spread over the middle layers: many physical qubits are combined into one sturdier logical qubit, and errors are found and fixed while the program runs.

1.1

Seven layers, one machine

Think of a quantum computer as a stack of layers. The top is the problem you want solved. The bottom is a chip with qubits. Every layer in between translates the problem one step further. Some of them have quantum in their names because they handle quantum programs, yet they are ordinary software and electronics. Hover or tap a layer to see what it does.

The layers follow Bertels et al., 2019, who treat the quantum chip as an accelerator beside a normal processor, like a graphics card. Error correction spans the middle layers.

1.2

Start with a goal and a little code

Our example program has one goal: link two qubits into a Bell pair, so that their answers agree when both are measured. The code needs three steps: a Hadamard gate on the first qubit, a controlled-NOT gate (CNOT) on both, then measure. Chapter 7 shows why agreement alone is not yet the whole story.

The listing is OpenQASM 3 (Cross et al., 2022), an open language for quantum circuits.

1.3

The compiler translates

The chip cannot run a Hadamard or a CNOT directly. It knows only a few native operations. The compiler rewrites our two gates as ten native operations in seven time slots: turns about the Z axis (RZ), half flips (SX) and one controlled phase (CZ). Only four of them take time: three SX pulses and the CZ. The result is the same.

CZ, RZ, SX and X are the native set of IBM Heron processors (IBM documentation). Both rewrites below are checked in this page's unit tests.

H≃RZ(π2) X RZ(π2)CNOT=(I⊗H) CZ (I⊗H)\begin{gathered}H \simeq R_Z(\tfrac{\pi}{2})\,\sqrt{X}\,R_Z(\tfrac{\pi}{2}) \\[4pt] \mathrm{CNOT} = (I\otimes H)\,\mathrm{CZ}\,(I\otimes H)\end{gathered}

Exactly, the native product equals e−iπ/4 H, a global phase that no measurement can see; the unit tests check the phase too. A real compiler also merges neighbouring Z turns and routes qubits to the coupling map.

RZ(π2) X RZ(π2)=e−iπ/4 HRZ(θ)=(e−iθ/200eiθ/2)\begin{gathered}R_Z(\tfrac{\pi}{2})\,\sqrt{X}\,R_Z(\tfrac{\pi}{2}) = e^{-i\pi/4}\,H \\[4pt] R_Z(\theta) = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}\end{gathered}

1.4

Every instruction gets a time

Next, each operation gets a start time, to the nanosecond. The microarchitecture turns every instruction into a pulse shape on the right line. Turns about Z take no time at all: the electronics simply shift the phase of the pulses that follow.

Z turns are frame changes (McKay et al., 2017); timed instructions follow eQASM (Fu et al., 2019). The timing uses a 30 ns single-qubit pulse and a two-qubit gate twice as long, the example values of Krinner et al., 2019.

1.5

Microwaves reach the chip

Electronics at room temperature turn the pulse shapes into real microwaves, billions of wiggles per second. They travel down cables into the refrigerator and reach the qubits. The pulses do the computing.

Qubits sit at 4 to 6 GHz and readout resonators at 4 to 8 GHz (Krinner et al., 2019). A 30 ns pulse at 5 GHz holds 150 cycles; the drawing shows fewer.

1.6

Answers travel back up

Reading out gives a faint microwave echo for each qubit. It is amplified and turned into a voltage, then into a bit, 0 or 1. The program runs again and again. The counts become a histogram: about half 00 and half 11, never 01 or 10.

The histogram comes from the ideal model in this page. Real hardware adds some wrong answers, which error correction and mitigation try to remove.

Chapter 2 of 9

Into the refrigerator

The chip hangs at the bottom of a dilution refrigerator, a few thousandths of a degree above absolute zero.

2.1

Colder at every stage

The refrigerator is a stack of metal plates, each colder than the one above. In one empty refrigerator the plates below room temperature measured 35 K, 2.85 K, 0.88 K and 82 mK. The bottom plate, the mixing chamber, reached 6 mK, six thousandths of a degree above absolute zero. The chip hangs just below it.

The labels give measured plate temperatures of an empty Bluefors XLD400 (Krinner et al., 2019, Table 1). The same paper estimates that its wiring could run 50 qubits with the mixing chamber at about 14 mK.

2.2

Why so cold?

A qubit's two levels are close in energy, so heat alone can lift it from 0 to 1. At room temperature it would be found in 1 almost half the time; at the mixing chamber, almost never. The numbers on the right are the ideal chance of finding a 5 GHz qubit in 1 at each plate. Real qubits stay a bit warmer than the plate: one carefully measured qubit was still found in 1 about 1 time in 1000 (Jin et al., 2015).

This is the ideal thermal-equilibrium estimate for two levels, not a measured value. For 5 GHz, hf/k is 0.24 K. Real qubits run warmer than their plate: in one study the excited population stopped falling near 0.1%, as if the qubit were at 35 mK (Jin et al., 2015).

P(1)=11+exp⁡(hf/kBT)P(1) = \frac{1}{1 + \exp\left(hf/k_B T\right)}

The lines are budgeted the same way: Krinner et al., 2019 attenuate until the Bose-Einstein occupation of the cable, carried down from 300 K, falls to about 10−3 photons at the chip.

nBE(T,f)=1exp⁡(hf/kBT)−1n_{\mathrm{BE}}(T, f) = \frac{1}{\exp\left(hf/k_B T\right) - 1}

2.3

Signals go down and get quieter

Control pulses enter at the top. On the way down they pass attenuators, small parts that throw away 99% of the power each time. That sounds wasteful. It also throws away the heat noise from the warmer cables, so what reaches the chip is a clean whisper.

Three 20 dB attenuators on the 4 K plate, the cold plate and the mixing chamber, 60 dB in total, keep thermal photons near 10−3 at the chip. A 30 ns π pulse, a full flip, needs a peak of about −66 dBm at the chip in the design of Krinner et al. (2019); a half flip needs a quarter of that power.

2.4

The answer comes back up, amplified

The readout echo is fainter still, a very weak signal. It climbs a separate line. Isolators and a circulator work like one-way valves, so noise cannot run down to the chip. An amplifier at the bottom and a transistor amplifier at 4 K make the signal strong enough to record.

Krinner et al., 2019: a travelling-wave parametric amplifier (TWPA) with 20 to 30 dB of gain and two isolators on the mixing chamber, a circulator on the cold plate and a HEMT amplifier, a high-electron-mobility transistor, with about 40 dB of gain at 4 K.

Chapter 3 of 9

The layers of different qubits

There are several ways to build a qubit, each a different stack of materials. The drawings are not to scale; the labels give real sizes.

Not to scale. The labels give real sizes.

Superconducting transmon
Silicon spin qubits
Trapped ions
Neutral atoms
Photons on a chip
Topological qubits

3.1

Superconducting circuits

A superconducting qubit is a small electrical circuit on a chip. Two metal pads hold charge. Between them sits a Josephson junction: two aluminium films with an oxide only a few atoms thick. It makes the energy steps uneven, so two levels can serve as 0 and 1. A resonator beside it is used for readout. Chips connect their qubits in different patterns. Rigetti chips and IBM Nighthawk use a square lattice, where qubits away from the edge have four neighbours; IBM Heron uses heavy-hex, where each qubit has two or three (Rigetti, IBM Nighthawk, IBM processor types). The superconducting lab draws both lattices.

Tantalum films on sapphire gave transmon lifetimes above 0.3 ms (Place et al., 2021). The junction barrier is 1 to 2 nm of aluminium oxide (Zeng et al., 2015).

  1. Josephson junction, aluminium, a 1 to 2 nm oxide barrier, aluminium
  2. Superconducting film, tantalum or niobium, cut into two pads
  3. Readout resonator, a strip of film ringing near 7 GHz
  4. Sapphire or silicon wafer, 0.53 mm thick

Demonstrated, as of October 2026: processors with more than 100 qubits and error correction below threshold (Google Quantum AI, 2025).

Open the superconducting lab

3.2

Silicon spin qubits

Here the qubit is the spin of one electron, trapped in a thin layer of silicon a few tens of nanometres below the surface. Metal gates on top squeeze it into place. A tiny magnet gives each electron its own frequency, so microwaves can address them one at a time.

Layer thicknesses, the 90 nm dot pitch and the cobalt micromagnet are those of the six-qubit device in Philips et al., 2022.

  1. Cobalt micromagnet, 200 nm thick
  2. Gate electrodes, 3 layers of Ti:Pd, 20 to 30 nm each
  3. Electrons in an 8 nm silicon well, 30 nm under the surface, dots 90 nm apart
  4. Relaxed SiGe buffer, 70% silicon, 30% germanium

Demonstrated, as of October 2026: six-qubit processors in silicon (Philips et al., 2022).

Open the silicon spin lab

3.3

Trapped ions

An ion is an atom with one electron removed, so electric fields can hold it. Gold electrodes on a chip form a trap, and the ion floats about 40 micrometres above the surface. Laser beams set its state, entangle it with its neighbours and read it out.

The first microfabricated surface trap held magnesium ions about 40 µm above gold electrodes (Seidelin et al., 2006).

  1. Laser beams, set, entangle and read the qubit
  2. One ion, the qubit, about 40 µm above the gold
  3. Radio-frequency field, holds the ion in a pocket
  4. Surface-electrode trap, gold electrodes on a chip

Demonstrated, as of October 2026: processors with tens of qubits (Moses et al., 2023).

Open the trapped-ion lab

3.4

Neutral atoms

Neutral atoms are held by light. A microscope objective focuses laser beams into spots about 1 micrometre wide, called optical tweezers, each holding one atom a few micrometres from the next, inside a glass vacuum cell. Another laser swells pairs of atoms into huge Rydberg states to entangle them.

Tweezer spots near 1 µm and spacings of a few µm follow Browaeys and Lahaye (2020).

  1. Rydberg laser, swells two atoms to entangle them
  2. Atoms, one per tweezer, a few µm apart
  3. Optical tweezers, light focused to about 1 µm
  4. Microscope objective, focuses the light, collects the glow

Demonstrated, as of October 2026: arrays of hundreds of atoms with logical qubits (Bluvstein et al., 2024).

Open the neutral-atom lab

3.5

Photons on a chip

On this chip, a photonic qubit is one particle of light in one of two waveguides, thin silicon channels. Other designs use polarization or arrival time. Where two guides run close, light can hop across: a beam splitter. At the end, superconducting nanowires click when a photon arrives. One photon entering an ideal, balanced beam splitter makes either detector click, 50% each.

The benchmarked chips guide light in silicon-on-insulator waveguides; silicon nitride is previewed for the next generation. Reported fidelities such as 99.98% for state preparation and measurement do not count photon loss (PsiQuantum, 2025).

  1. Two waveguides, a photon in one of the two is the qubit
  2. Beam splitter, guides run close, light hops across
  3. Single-photon detectors, NbN wire 90 nm wide, about 5 nm thick, near 2 K
  4. Silicon core, light is guided inside it; silicon nitride is next
  5. Oxide cladding on silicon

Demonstrated, as of October 2026: chip components made in a commercial foundry (PsiQuantum, 2025).

Open the photonic lab

3.6

Topological qubits

A topological qubit would store information in four Majorana zero modes, at the ends of two tiny wire sections. This is called a tetron. Each pair of modes shares one charge that is either even or odd. The information is spread out, where many kinds of local noise cannot easily disturb it. Whether these modes are really there is still debated (Legg, 2025).

Strip sizes and the four-mode linear tetron follow Aghaee et al. (2023). A June 2026 preprint swaps aluminium for lead and reports that a wire keeps its even or odd charge for about 20 s before a stray charge flips it (Aghaee et al., 2026). That flip is local noise reaching the modes, so a long time is the goal.

  1. Four Majorana zero modes, claimed, at the ends of two wire sections
  2. Gates, squeeze the electrons into a wire
  3. Superconducting strip, Al, 10 µm long, under 120 nm wide, under 10 nm thick
  4. InAs quantum well, a thin wire of electrons forms under the strip
  5. InP wafer with buffer

Claimed, not independently confirmed: Microsoft calls its Majorana 1 chip topological (Microsoft, February 2025). Its Nature paper reports a parity measurement and, as the journal notes, is not evidence of Majorana zero modes (Microsoft Azure Quantum, 2025). A 2026 preprint reports 20 s parity lifetimes (Aghaee et al., 2026). None of this is independently confirmed, as of October 2026.

Open the braiding lab Inside a topological computer

Chapter 4 of 9

Create a qubit

Every computation starts the same way: each qubit is reset to 0.

Energy levels
|1⟩0.0%
about 5 GHz apart
|0⟩100.0%
Qubit
|0⟩|1⟩
arrow length 1.00
|0⟩1.000100.0%
|1⟩00.0%

4.1

Start from zero

A qubit has a lowest energy level, called 0, and the next one up, called 1. To reset it we let it give its energy away as a microwave photon, so it drops to 0. In between, the arrow shrinks through the centre: for a moment the qubit is a blend, not one definite state.

Energy relaxation: the excited population decays as exp(−t/T1) and the Bloch vector moves along the z axis. Fast reset schemes measure and flip, or drive the energy out.

The drawn blend is a diagonal density matrix with p falling close to exponentially; its Bloch vector is (0, 0, 1 − 2p), so its length passes through 0 when p = 1/2.

ρ(t)=(1−p) ∣0⟩⟨0∣+p ∣1⟩⟨1∣r⃗=(0, 0, 1−2p)\begin{gathered}\rho(t) = (1-p)\,|0\rangle\langle 0| + p\,|1\rangle\langle 1| \\[4pt] \vec r = (0,\,0,\,1-2p)\end{gathered}

4.2

The Bloch sphere

Picture the qubit's state as an arrow inside a ball. Up is 0, down is 1, and every other direction is a superposition. The arrow's length tells how pure the state is. Full length means one definite quantum state; shorter means blended or entangled.

∣ψ⟩=cos⁡θ2 ∣0⟩+eiφsin⁡θ2 ∣1⟩|\psi\rangle = \cos\tfrac{\theta}{2}\,|0\rangle + e^{i\varphi}\sin\tfrac{\theta}{2}\,|1\rangle

Chapter 5 of 9

Superposition

One shaped microwave pulse turns a 0 into an even superposition of 0 and 1.

The pulse
about 30 ns

Carrier at the qubit's frequency, near 5 GHz, drawn slower

Energy levels
|1⟩0.0%
about 5 GHz apart
|0⟩100.0%
Qubit
|0⟩|1⟩
arrow length 1.00
|0⟩1.000100.0%
|1⟩00.0%

5.1

A shaped microwave pulse

We send one carefully shaped microwave pulse, about 30 nanoseconds long, tuned to the qubit's own frequency. It turns the arrow a quarter turn, from the top down to the equator. Watch the bars: the chance of reading 1 rises from 0% to 50%.

Drawn as Ry(θ) with θ from 0 to π/2. Starting from |0⟩, Ry(π/2) and the Hadamard give the same state. Pulse length from Krinner et al., 2019; pulse shaping is reviewed in Krantz et al. (2019).

5.2

Amplitudes, then odds

A superposition is one definite state with two amplitudes. Here each amplitude is 0.707. Square an amplitude and you get the chance of that answer: 0.707 × 0.707 = 0.5. So a measurement gives 0 or 1, each 50.0% of the time.

∣+⟩=12 ∣0⟩+12 ∣1⟩P(k)=∣⟨k∣+⟩∣2=12\begin{gathered}|{+}\rangle = \tfrac{1}{\sqrt{2}}\,|0\rangle + \tfrac{1}{\sqrt{2}}\,|1\rangle \\[4pt] P(k) = |\langle k|{+}\rangle|^2 = \tfrac12\end{gathered}

Chapter 6 of 9

Entangle two qubits

A Hadamard and a CNOT turn two separate qubits into one entangled pair.

q0|0⟩q1|0⟩H
Qubit q0
|0⟩|1⟩
arrow length 1.00
Qubit q1
|0⟩|1⟩
arrow length 1.00
|00⟩1.000100.0%
|01⟩00.0%
|10⟩00.0%
|11⟩00.0%
Link meter concurrence0.00

6.1

Two qubits, four amplitudes

Two qubits have four possible answers: 00, 01, 10 and 11. Their state has one amplitude for each. We start in 00 for certain. The circuit at the top reads from left to right, and the gold line marks where we are.

6.2

A Hadamard on the first

The pulse puts the first qubit into superposition. Now 00 and 10 each have amplitude 0.707. Each qubit still has its own full-length arrow: the first on the equator, the second at the top. Nothing is entangled yet.

6.3

CNOT links them

CNOT flips the second qubit only when the first is 1. So 10 becomes 11, while 00 stays. Watch both arrows shrink as the link meter rises. Neither qubit keeps a direction of its own.

Each arrow is the Bloch vector of one qubit's reduced density matrix. For a pure two-qubit state its length is √(1 − C²), where C is the concurrence. The frames in between follow exp(iπt(I − CNOT)/2), which ends exactly at CNOT.

Along that path the concurrence is |sin(πt/2)| and both reduced states lose purity together; at t = 1 each is the maximally mixed I/2.

ρA=TrB ∣ψ⟩⟨ψ∣=12(I+r⃗A⋅σ⃗)∣r⃗A∣=1−C2\begin{gathered}\rho_A = \mathrm{Tr}_B\,|\psi\rangle\langle\psi| = \tfrac12\left(I + \vec r_A\cdot\vec\sigma\right) \\[4pt] |\vec r_A| = \sqrt{1 - C^2}\end{gathered}

6.4

Entangled

The pair is now (00 + 11)/√2, a Bell state. Both arrows have length 0 and the link meter, the concurrence, reads 1, the maximum. Each qubit on its own is a coin toss. Together they always agree.

∣Φ+⟩=12(∣00⟩+∣11⟩)C=2 ∣a00a11−a01a10∣=1\begin{gathered}|\Phi^+\rangle = \tfrac{1}{\sqrt2}(|00\rangle + |11\rangle) \\[4pt] C = 2\,|a_{00}a_{11} - a_{01}a_{10}| = 1\end{gathered}

Chapter 7 of 9

Read out

Measuring gives one random answer per run. Many runs reveal the pattern.

The readout resonator's note
probe frequency, near 7 GHzsignal getting throughqubit 0qubit 1probe tone
Qubit is0echo phase

Bell pair, then measure straightH on both, then measure schematic clouds

q0 readout
reads 0reads 1
q1 readout
reads 0reads 1

State before the next shot ideal

0050.0%010.0%100.0%1150.0%

Last shot: none yet New pair made for the next shot.

Shots so far 0

0000.0%
0010.0%
0100.0%
0110.0%

dashed: what two glued coins would give

7.1

Listening to a resonator

We never look at a qubit directly. Each qubit is coupled to a resonator, a circuit that rings at about 7 GHz. A qubit in 1 pulls that note slightly down. A short probe tone between the two notes comes back changed, and the change tells 0 from 1.

Dispersive readout: the resonator sits at ωr − χ or ωr + χ depending on the qubit (Blais et al., 2021). For a transmon below its resonator χ is negative, so a qubit in 1 pulls the resonator down (Krantz et al., 2019). Readout in 80 ns with 97% average assignment has been shown (Heinsoo et al., 2018).

In the dispersive limit, with detuning much larger than the coupling, the Jaynes-Cummings model reduces to a qubit-state-dependent resonator frequency (Blais et al., 2021); the transmitted probe acquires a phase of opposite sign for the two states. Here σz = −1 for |0⟩ and +1 for |1⟩, and χ < 0.

H/ℏ≈ωr a†a+ωq2 σz+χ a†a σzH/\hbar \approx \omega_r\, a^\dagger a + \tfrac{\omega_q}{2}\,\sigma_z + \chi\, a^\dagger a\, \sigma_z

7.2

One shot

Each measurement is called a shot. It gives one point per qubit in these planes, and the cloud the point lands in gives the bit. The clouds are drawn schematically. A shot collapses both qubits at once: if the first reads 1, so does the second. The first shot fires by itself; press 1 shot for more.

7.3

Many shots

Now run the program many times. The histogram fills up: about 50% 00 and 50% 11, and never 01 or 10. Each single answer is random, but the two qubits always agree. Agreement alone could come from two coins glued together, though. The dashed outlines show what glued coins would give: the same. The next step tells them apart.

The clouds are schematic. Shots are sampled from the ideal Bell state with a seeded random generator; real readout adds noise and a few misread shots.

7.4

Turn both, then measure

Now give each qubit an H gate, which swaps the up-down question for a left-right one, just before measuring. Two glued coins would now agree only half the time: 25% for each answer, the dashed outlines. The Bell pair still gives only 00 and 11. Two separate qubits cannot agree perfectly in both tests. Passing both shows the pair is entangled.

Turning both with H swaps the question from 0 or 1 to plus or minus. The Bell state is unchanged by H on both qubits, so the answers still agree. The glued-coin mixture, half 00 and half 11, spreads evenly over all four answers.

(H⊗H) ∣Φ+⟩=∣Φ+⟩ρcoins=12(∣00⟩⟨00∣+∣11⟩⟨11∣)→14 each\begin{gathered}(H\otimes H)\,|\Phi^+\rangle = |\Phi^+\rangle \\[4pt] \rho_{\mathrm{coins}} = \tfrac12\left(|00\rangle\langle 00| + |11\rangle\langle 11|\right) \to \tfrac14 \text{ each}\end{gathered}

Any separable state obeys |⟨XX⟩| + |⟨ZZ⟩| ≤ 1; the Bell pair gives 2. This entanglement witness trusts quantum mechanics for each qubit. A Bell test, which does not, needs more measurement angles.

Chapter 8 of 9

Try it yourself

Build a two-qubit circuit and watch the same pictures respond.

Qubit q0
|0⟩|1⟩
arrow length 1.00
Qubit q1
|0⟩|1⟩
arrow length 1.00
|00⟩1.000100.0%
|01⟩00.0%
|10⟩00.0%
|11⟩00.0%
Link meter concurrence0.00

Start: both qubits in 0.

8.1

Build your own circuit

Pick a qubit, then add gates. Each gate animates the arrows, amplitudes, odds and link meter. Try H on q0, then CNOT. Then add S or T to q0 and see which numbers change and which do not.

The circuit builder needs JavaScript.

Qubit
One qubit
Two qubits
Steps
Try

Chapter 9 of 9

Sources, assumptions and limits

What on this page is exact, what is typical, and where the numbers come from.

9.1

What is exact and what is not

The quantum states on this page are computed exactly for an ideal two-qubit model. Hardware numbers are typical values from the papers below, not one particular machine.

Assumptions and limits

  • Ideal model: the two-qubit states, chances, arrow lengths and concurrence are exact for perfect gates with no noise.
  • Frames between two steps follow a smooth path that ends exactly at each gate. Only the end points are gate results.
  • Hardware numbers are typical values from the cited papers, not the specification of one machine.
  • Drawings are schematic and not to scale. Labels give real sizes.
  • Readout clouds are schematic. Shots are sampled from the ideal state with a seeded random generator, so a replay gives the same sequence.
  • The chances of finding a qubit in 1 at each plate assume a two-level qubit in thermal equilibrium at the plate temperature. They are estimates, not measurements.

Status words, as of October 2026

Demonstrated
shown in published, peer-reviewed experiments.
Claimed
reported by the builders, not yet confirmed by others.
Proposed
a design on paper or in simulation.
Hypothetical
not yet known to exist.

Sources

  1. Bertels et al., Quantum computer architecture: towards full-stack quantum acceleratorsarXiv:1903.09575 (2019). The layered stack in chapter 1 follows their figure.
  2. Cross et al., OpenQASM 3: a broader and deeper quantum assembly languageACM Transactions on Quantum Computing 3, 12 (2022). The language of the code listing.
  3. IBM Quantum, Native gates and operationsHeron processors run CZ, RZ, SX and X natively. Checked October 2026.
  4. McKay et al., Efficient Z gates for quantum computingPhys. Rev. A 96, 022330 (2017). Z turns as zero-duration frame changes.
  5. Fu et al., eQASM: an executable quantum instruction set architectureHPCA 2019. Timed instructions executed by a quantum microarchitecture.
  6. Krinner et al., Engineering cryogenic setups for 100-qubit scale superconducting circuit systemsEPJ Quantum Technology 6, 2 (2019). Plate temperatures (Table 1), 3 × 20 dB drive attenuation, TWPA, isolators, circulator, HEMT, 4 to 6 GHz qubits, 30 ns pulses; a 30 ns π pulse needs about −66 dBm peak at the chip, a π/2 pulse a quarter of that.
  7. Krantz et al., A quantum engineer's guide to superconducting qubitsApplied Physics Reviews 6, 021318 (2019). Transmons, pulses and dispersive readout; Eq. 145: for a qubit below its resonator the dispersive shift is negative.
  8. Blais et al., Circuit quantum electrodynamicsRev. Mod. Phys. 93, 025005 (2021). The dispersive shift used for readout.
  9. Jin et al., Thermal and residual excited-state population in a 3D transmon qubitPhys. Rev. Lett. 114, 240501 (2015). Below 35 mK the excited population stops falling, near 0.1%: an effective qubit temperature of 35 mK.
  10. Heinsoo et al., Rapid high-fidelity multiplexed readout of superconducting qubitsPhys. Rev. Applied 10, 034040 (2018). Readout in 80 ns with 97% average assignment.
  11. Place et al., New material platform for superconducting transmon qubits with coherence times exceeding 0.3 millisecondsNature Communications 12, 1779 (2021). Tantalum on 0.53 mm sapphire, Al/AlOx/Al junctions of about 0.03 µm², resonators at 6.8 to 7.3 GHz.
  12. Zeng et al., Direct observation of the thickness distribution of ultra thin AlOx barriers in Al/AlOx/Al Josephson junctionsJ. Phys. D 48, 395308 (2015). Barriers 1 to 2 nm thick.
  13. Rigetti Computing, Novera QPUChecked October 2026. Built on the Ankaa-class architecture, "a square lattice of qubits and tunable couplers"; the 9-qubit chip is a 3×3 array with a tunable coupler between each adjacent pair.
  14. IBM Quantum, Nighthawk r231 August 2026. A square lattice in which most qubits connect to four nearest neighbours, against two or three in earlier IBM processors.
  15. IBM Quantum, Processor typesChecked October 2026. Heron r2: 156 qubits in a heavy-hexagonal lattice; Nighthawk: a grid topology.
  16. Google Quantum AI, Quantum error correction below the surface code thresholdNature 638, 920 (2025). A 105-qubit superconducting processor.
  17. Philips et al., Universal control of a six-qubit quantum processor in siliconNature 609, 919 (2022). 8 nm 28Si well, 30 nm SiGe spacer, three Ti:Pd gate layers with 5 nm Al2O3, 200 nm cobalt micromagnet, 90 nm dot pitch.
  18. Seidelin et al., A microfabricated surface-electrode ion trap for scalable quantum information processingPhys. Rev. Lett. 96, 253003 (2006). Gold electrodes, ion about 40 µm above the surface.
  19. Moses et al., A race-track trapped-ion quantum processorPhys. Rev. X 13, 041052 (2023). A 32-qubit trapped-ion processor.
  20. Browaeys and Lahaye, Many-body physics with individually controlled Rydberg atomsNature Physics 16, 132 (2020). Tweezers focused to about 1 µm, atoms a few µm apart.
  21. Bluvstein et al., Logical quantum processor based on reconfigurable atom arraysNature 626, 58 (2024). Up to 280 atom qubits and logical qubits.
  22. PsiQuantum team, A manufacturable platform for photonic quantum computingarXiv:2404.17570 (2024), later published in Nature (2025). Silicon-on-insulator waveguides with NbN detectors about 5 nm thick and 90 nm wide at about 2 K; silicon nitride waveguides previewed for the next generation.
  23. Aghaee et al., InAs-Al hybrid devices passing the topological gap protocolPhys. Rev. B 107, 245423 (2023). InAs quantum well on InP; Al strip 10 µm long, under 120 nm wide, under 10 nm thick. Fig. 22: the linear tetron, two topological sections with four Majorana zero modes.
  24. Microsoft, Microsoft's Majorana 1 chip carves new path for quantum computingCompany announcement, 19 February 2025. The claim of a topological core; a claim, not a peer-reviewed result.
  25. Microsoft Azure Quantum, Interferometric single-shot parity measurement in InAs-Al hybrid devicesNature 638, 651 (2025). The journal noted that the paper is not evidence of Majorana zero modes.
  26. H. F. Legg, Comment on "InAs-Al hybrid devices passing the topological gap protocol"arXiv:2502.19560 (2025). Argues that the topological gap protocol is not a reliable test and that later work built on it should be revisited.
  27. Aghaee et al., 20 second parity lifetime in an InAs-Pb tetron devicearXiv:2606.03884 (June 2026). Lead replaces aluminium; a preprint, not yet independently confirmed.

The layered stack in chapter 1 is redrawn after the figure in Bertels et al., arXiv:1903.09575. The quantum states come from this page's own simulator, checked by test/how-it-works.test.mjs.