WHY complex numbers? Because interference (the source of quantum speedup) needs phase. Two paths with amplitudes +21 and −21 cancel; only complex/signed amplitudes let this happen. Probabilities alone (always positive) cannot cancel.
WHY:α,β are 2 complex numbers = 4 real numbers. Two constraints kill two:
Normalization∣α∣2+∣β∣2=1 removes one real parameter.
Global phase is unobservable (measurement depends only on ∣α∣2,∣β∣2, and physical predictions are invariant under ∣ψ⟩→eiγ∣ψ⟩) removes another.
So 4−2=2 real parameters remain → a point on a sphere. Write:
∣ψ⟩=cos2θ∣0⟩+eiϕsin2θ∣1⟩
Why θ/2 and not θ? So that θ ranges over [0,π] maps the full sphere: θ=0→∣0⟩ (north pole), θ=π→∣1⟩ (south pole). Orthogonal states sit at opposite poles, 180° apart physically though orthogonal in Hilbert space.
WHY so cold / so isolated? The two levels differ by energy ΔE. Thermal noise excites transitions when kBT≳ΔE. For superconducting qubits ΔE/h∼5 GHz, so we need T with kBT≪h(5 GHz), i.e. tens of mK.
WHY unitary? Because quantum evolution must preserve total probability (∣α∣2+∣β∣2=1). Only norm-preserving (unitary) maps do this. Non-unitary = measurement/loss = decoherence.
Single-qubit example — the Hadamard:
H=21(111−1),H∣0⟩=2∣0⟩+∣1⟩
This creates superposition — the "start the coin spinning" operation. Two-qubit entangling gates (CNOT, CZ, Mølmer–Sørensen) are needed for universality.
Readout HOW: superconducting qubits use a coupled resonator whose frequency shifts depending on the qubit state (dispersive readout); measuring the microwave transmission reveals ∣0⟩ vs ∣1⟩.
Recall Feynman: explain to a 12-year-old
Imagine a magic spinning coin. While it spins, it's both heads and tails, and you can gently tilt HOW it spins. If you have many such coins that can feel each other (entanglement), you can set up a clever spin so that all the wrong answers cancel out like waves, and only the right answer stays loud when the coins finally fall. The hard part is: air, heat, and touching all knock the coins flat. So we build them in a super-cold, super-quiet freezer and only touch them with perfectly-timed puffs (microwaves or lasers). We must finish the trick before the coins fall flat — that "falling flat" time is called coherence.
Dekho, classical bit ek switch hai — ON ya OFF, bas. Lekin qubit ek ghoomti hui coin jaisa hai: jab tak spin ho raha hai, wo heads aur tails dono ek saath hai. Isko hum likhte hain ∣ψ⟩=α∣0⟩+β∣1⟩, aur measure karne par ∣0⟩ milne ki probability ∣α∣2 hoti hai. Yahan magic interference ka hai — complex amplitudes ke wajah se galat answers ek dusre ko cancel kar dete hain, jaise waves. Isiliye phase (complex number) zaroori hai, sirf probability se yeh cancel nahi hota.
Hardware ka pura khel yeh hai: coin ko ghoomta hua rakhna kitni der tak. Yeh time hai coherence — T1 (energy leak hoke coin flat gir jaana) aur T2 (spin toh chal raha, par direction ka pata kho jaana). Ek important rule: T2≤2T1, kyunki relaxation khud phase ko bhi tod deta hai. Gate karne me jitna time lagta hai (tg), uske comparison me T2 bada hona chahiye — roughly T2/tg gates tak hi algorithm chal sakta hai.
Superconducting qubits ko 15 milliKelvin tak thanda karte hain. Kyun? Boltzmann formula P1/P0=e−ΔE/kBT ke hisaab se, agar garam hua toh qubit apne aap excite ho jaayega aur initialize (known state se shuru karna) fail ho jaayega. Room temperature par toh coin poori tarah scramble ho jaati hai. Isiliye cold, quiet, isolated environment — aur touch bhi sirf perfectly-timed microwave/laser pulses se, jo Bloch sphere par rotation (unitary gate) dete hain.
Ek galatfehmi mat paalna: zyada qubits ka matlab automatic speed nahi. Agar qubits noisy hain toh error correction me hazaron physical qubits ek logical qubit banate hain. Quality (fidelity, coherence) > quantity. Aur 2n amplitudes storage ke liye nahi, interference-based computation ke liye powerful hain — measure karoge toh sirf n classical bits milenge.
