Foundations — Quantum computing hardware basics

This page assumes you have seen nothing. We collect every symbol the parent note uses and build each from a plain-words meaning, a picture, and the reason the topic cannot do without it. Read top to bottom — each block only uses ideas from the blocks above it.

1. The state ket $|⟩$ — a labelled arrow

Picture it (Figure s01): an arrow that can point anywhere on a globe. In the figure, the red arrow is $|\psi ⟩$ — trace it from the centre out to the sphere's surface; that single arrow is the whole state. Notice the two black dots: the top dot is labelled $|0⟩$ and the bottom dot $|1⟩$. The red arrow is caught mid-tilt between them — that "in between" is the whole point.

We give two special directions their own tags:

• $|0⟩$ — the arrow pointing straight up (the classical "0").
• $|1⟩$ — the arrow pointing straight down (the classical "1").

WHY the topic needs it: a classical bit needs one letter (0 or 1). A qubit points between those, so we need a name for the in-between arrow — that name is $|\psi ⟩$.

See Classical Bits vs Qubits for the switch-vs-spinning-coin contrast, and Bloch Sphere for the globe this arrow lives on.

2. Amplitudes $\alpha ,\beta$ and the complex numbers $\mathbb{C}$

Picture it: mixing two paints. $\alpha$ is "how much white (up)" and $\beta$ is "how much black (down)." A pure up arrow has $\alpha =1,\beta =0$; a 50/50 tilt has $\alpha =\beta =\frac{1}{\sqrt{2}}$.

Why the plus sign? The "+" means superposition — being both at once, not "up OR down" but "up AND down mixed." This is the spinning coin, mathematically.

Picture it: an ordinary number sits on a line (left–right). A complex number sits on a plane — it has a length and a direction (an angle). That angle is called the phase.

WHY the topic needs complex numbers: interference. If two ways of getting an answer have amplitudes $+\frac{1}{\sqrt{2}}$ and $-\frac{1}{\sqrt{2}}$, they cancel to zero — that is how quantum computers erase wrong answers. Plain probabilities are always positive and can never cancel. Only signed/complex amplitudes can. Phase = the direction of the arrow-in-the-plane = the thing that lets waves add or cancel.

See Superposition and Entanglement.

3. Modulus-squared $|\alpha {|}^2$ — turning amplitude into odds

Picture it: the amplitude is a paint mix; squaring its length is the recipe that turns "how much white paint" into "how likely you see white when you look."

Picture it (Figure s02): the red arrow's tip sits exactly on the circle — it never lands inside or outside. Look at the two dotted black lines dropping from the tip: the horizontal one has length $|\alpha |$, the vertical one has length $|\beta |$. They are the two legs of a right triangle whose hypotenuse (the red arrow) is fixed at length 1. Normalization is simply "the red arrow's length is 1."

WHY the topic needs it: it is why quantum gates must be unitary (Section 8) — they are allowed to rotate the arrow but never stretch it off the sphere.

4. From four numbers to two angles — global phase and the Bloch sphere

Two complex amplitudes $\alpha ,\beta$ are four real numbers ($\alpha =a_1+a_{2}i$, $\beta =b_1+b_{2}i$). Yet the parent page describes the state with only two angles $\theta ,\varphi$. Two numbers must disappear. Here is exactly why.

Picture it: turning both hands of a clock by the same amount changes nothing about the time-difference you read between them. Only the relative phase between $\alpha$ and $\beta$ matters. So we are free to "use up" the global phase to make $\alpha$ a plain positive real number — that removes the fourth free number.

$4\text{ real numbers}-\underset{\text{normalization}}{\underbrace{1}}-\underset{\text{global phase}}{\underbrace{1}}=2\text{ real numbers}\Rightarrow \text{a point on a sphere.}$

Picture it (Figure s03): the red arrow is the state. Follow it up to the vertical axis — the angle it opens away from the top pole $|0⟩$ is $\theta$ (the "tilt" knob). Now look down at the flat equatorial plane: the small black arc there is $\varphi$, how far the arrow's shadow has swung around (the "spin" knob). Two knobs, one point — nothing more is needed.

The parent writes $|\psi ⟩=\cos \frac{\theta }{2}|0⟩+e^{i\varphi }\sin \frac{\theta }{2}|1⟩.$

• $\cos \frac{\theta }{2}$ and $\sin \frac{\theta }{2}$ are the cosine and sine — from a right triangle, cosine is the adjacent side and sine the opposite side. Here they split the arrow's length-1 between "up" and "down": at the top ($\theta =0$) cosine is 1 (all up), at the bottom ($\theta =\pi$) sine is 1 (all down). Because ${\cos }^2+{\sin }^2=1$ automatically, normalization is built in for free.
• $e^{i\varphi }$ is the surviving relative-phase factor (a unit complex number). Its length is 1 (it changes no probability) but its direction stores the "which way is it spinning" information — the interference knob.

Deeper: Bloch Sphere.

5. The exponential $e^x$ — the natural language of decay and phase

Picture it: a hot cup cooling, or radioactive atoms decaying — each second the same fraction is lost, giving a smooth curve $e^{-t/\tau }$ that halves, halves, halves. Three different exponentials appear on the parent page:

• $e^{-\Delta E/k_{B}T}$ — Boltzmann factor (Section 6), how likely the environment kicks the qubit up.
• $e^{-t/T_2}$ — how coherence fades with time (Section 7).
• $e^{i\varphi }$ — with an imaginary exponent, $e^{i\varphi }$ does not shrink at all; it rotates. This is the phase from Section 4.

WHY the topic needs it: every "how fast does X leak away" question in quantum hardware is answered by an exponential, and every interference question by $e^{i\varphi }$.

6. Energy symbols: $\Delta E$, $k_B$, $T$, $h$

Picture it: $\Delta E$ is the height of a step you must climb from the ground floor $|0⟩$ to the first floor $|1⟩$. $k_{B}T$ is how hard the random thermal "wind" pushes. If the wind is as strong as the step is tall, the qubit gets randomly blown upstairs and your "start in 0" fails.

WHY the topic needs it: this single formula is why the fridge exists. See Boltzmann Distribution and Cryogenics and Dilution Refrigerators.

7. Time symbols: $T_1$, $T_2$, $t_g$

Picture it: $T_1$ = the arrow dropping toward the north pole; $T_2$ = the arrow's longitude $\varphi$ smearing into a random blur around the globe. Both are "$1/e$ times" — the time for the quantity to shrink to $1/e\approx 37\%$ of its start, read straight off the exponential $e^{-t/T}$.

WHY the topic needs them: the ratio $T_2/t_g$ is how many nudges you get before the coin dies — the true measure of how deep an algorithm can run.

8. Operators, unitarity, matrices, the dagger $†$

Picture it: a unitary is a rotation of the globe — it can point the arrow anywhere but never stretches or shrinks it (length stays 1). That is exactly what normalization (Section 3) demands.

WHY the topic needs it: every gate must be unitary because quantum evolution can never lose or create total probability. See Unitary Operators and Reversible Computing.

9. The pieces that build on all the above

• Josephson junction — the special circuit element that makes a superconducting qubit's energy ladder uneven, so the bottom two rungs $|0⟩,|1⟩$ can be addressed without accidentally climbing higher. See Josephson Junction.
• Coherence — the umbrella word for "the coin is still meaningfully spinning" (both $T_1$ and $T_2$ intact).
• Quantum error correction — because even a great coin eventually falls, we spread one logical coin across many physical ones. See Quantum Error Correction.

Prerequisite map

Plain-text fallback (read "→" as "feeds into"):

• Classical bit (0 or 1) → the state ket $|\psi ⟩$ → amplitudes $\alpha ,\beta$.
• Complex numbers (length + phase) → amplitudes $\alpha ,\beta$.
• Amplitudes → probability via modulus-squared → normalization (sums to 1).
• Normalization + global-phase invariance → the two Bloch angles $\theta ,\varphi$.
• Exponential function → both Boltzmann kicks ($P_1/P_0$) and coherence decay ($T_1,T_2$).
• Energy gap, $k_{B}T$, $h$ → Boltzmann kicks.
• Normalization + Bloch sphere → unitary gates $U$.
• All of {Bloch angles, Boltzmann kicks, coherence decay, unitary gates} → qubit hardware.

Equipment checklist

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