Quantum computing hardware basics — Hinglish notes

6.5.13 · Hardware › Advanced & Emerging Architectures

WHAT is a qubit?

WHY complex numbers? Kyunki interference (quantum speedup ka source) ko phase chahiye. Do paths jinke amplitudes $+ \frac{1}{2}$ aur $- \frac{1}{2}$ hain, cancel ho jaate hain; sirf complex/signed amplitudes hi yeh hone dete hain. Akeli probabilities (hamesha positive) cancel nahi ho saktin.

Deriving the Bloch sphere

WHY: $α, β$ 2 complex numbers hain = 4 real numbers. Do constraints do ko khatam karte hain:

Normalization $∣ α ∣^{2} + ∣ β ∣^{2} = 1$ ek real parameter hatata hai.
Global phase unobservable hota hai (measurement sirf $∣ α ∣^{2}, ∣ β ∣^{2}$ par depend karta hai, aur physical predictions $∣ ψ ⟩ \to e^{iγ} ∣ ψ ⟩$ ke under invariant hain) ek aur hatata hai.

Toh $4 - 2 = 2$ real parameters bachte hain → ek sphere par ek point. Likho: $∣ ψ ⟩ = cos \frac{θ}{2} ∣0 ⟩ + e^{i ϕ} sin \frac{θ}{2} ∣1 ⟩$

$θ /2$ kyun, $θ$ kyun nahi? Taaki $θ$ ka range $[0, π]$ puri sphere ko map kare: $θ = 0 \to ∣0 ⟩$ (north pole), $θ = π \to ∣1 ⟩$ (south pole). Orthogonal states opposite poles par baithe hain, physically $180°$ apart, haalaanki Hilbert space mein orthogonal hain.

Figure — Quantum computing hardware basics

HOW do we physically build one? (The main platforms)

WHY itna thanda / itna isolated? Do levels mein energy ka fark $Δ E$ hota hai. Thermal noise tab transitions excite karta hai jab $k_{B} T ≳ Δ E$ . Superconducting qubits ke liye $Δ E / h \sim 5 GHz$ hai, toh hamen chahiye $T$ aisa ki $k_{B} T ≪ h (5 GHz)$ , yani tens of mK.

The DiVincenzo criteria (WHAT hardware must deliver)

Hardware engineers jo bhi ladte hain, sab in sab ko ek doosre ke against trade karne mein reduce ho jaata hai.

Coherence: WHY qubits die

Gates and readout

WHY unitary? Kyunki quantum evolution total probability preserve karni chahiye ( $∣ α ∣^{2} + ∣ β ∣^{2} = 1$ ). Sirf norm-preserving (unitary) maps hi yeh karte hain. Non-unitary = measurement/loss = decoherence.

Single-qubit example — Hadamard: $H = \frac{1}{2} (11 1 - 1), H ∣0 ⟩ = \frac{∣0 ⟩ + ∣1 ⟩}{2}$ Yeh superposition create karta hai — "coin ko ghoomna shuru karo" wali operation. Two-qubit entangling gates (CNOT, CZ, Mølmer–Sørensen) universality ke liye zaroori hain.

Readout HOW: superconducting qubits ek coupled resonator use karte hain jiska frequency qubit state ke hisaab se shift hoti hai (dispersive readout); microwave transmission measure karne se $∣0 ⟩$ vs $∣1 ⟩$ pata chalta hai.

Common mistake Steel-manned misconceptions

"Ek qubit infinite information store karta hai kyunki $α, β$ continuous hain." Kyun sahi lagta hai: state IS ek continuous point on a sphere. Fix: tum woh continuum read out nahi kar sakte — measurement ek classical bit par collapse ho jaati hai. $n$ qubits computation ke liye $2^{n}$ amplitudes rakhte hain, lekin reading sirf $n$ bits deti hai. Power interference mein hai, storage mein nahi.

"Zyada qubits = automatically faster." Kyun sahi lagta hai: Hilbert space $2^{n}$ ki tarah badhta hai. Fix: bina low error rates ke bekar hai. Noisy qubits ko error correction chahiye; ~1000 physical qubits se 1 logical qubit ban sakta hai. Quality (fidelity, $T_{2} / t_{g}$ ) raw count se zyada important hai.

" $T_{2}$ , $T_{1}$ se bahut bada ho sakta hai." Kyun sahi lagta hai: dephasing energy loss se alag lagta hai. Fix: derivation force karta hai $T_{2} \leq 2 T_{1}$ — relaxation khud hi phase destroy karta hai.

"Thanda karna sirf ek engineering nicety hai." Fix: yeh fundamental hai — $P_{1} / P_{0} = e^{- Δ E / k_{B} T}$ se, warmth literally initial state ko re-scramble kar deti hai (criterion 2 fail ho jaata hai).

Recall Feynman: explain to a 12-year-old

Ek magic spinning coin imagine karo. Jab tak woh ghoomti hai, woh dono heads aur tails hai, aur tum gently tilt kar sakte ho KI WOH KAISE GHOOMTI HAI. Agar tumhare paas bahut si aisi coins hain jo ek doosre ko feel kar sakti hain (entanglement), toh tum ek clever spin set up kar sakte ho taaki saare galat answers cancel ho jaayein jaise waves, aur jab coins finally girein tab sirf sahi answer loud bache. Mushkil wala part yeh hai: hawa, heat, aur chhona sab coins ko flat kar dete hain. Toh hum unhe ek super-thande, super-shaant freezer mein banate hain aur unhe sirf perfectly-timed puffs (microwaves ya lasers) se touch karte hain. Hamen yeh trick coins ke girne se pehle poori karni hogi — woh "girne" wala time coherence kehlata hai.

Connections

Superposition and Entanglement
Bloch Sphere
Josephson Junction
Cryogenics and Dilution Refrigerators
Quantum Error Correction
Boltzmann Distribution
Unitary Operators and Reversible Computing
Classical Bits vs Qubits

Flashcards

Qubit kya hota hai?

Ek two-level quantum system superposition mein

α ∣0 ⟩ + β ∣1 ⟩

jisme

∣ α ∣^{2} + ∣ β ∣^{2} = 1

ho.

Probabilities ki jagah complex amplitudes kyun?

Phase destructive interference allow karta hai (galat answers cancel hote hain); positive probabilities cancel nahi ho saktin.

∣0 ⟩

measure hone ki probability?

∣ α ∣^{2}

Bloch parametrization mein

θ /2

kyun?

Taaki

θ \in [0, π]

puri sphere cover kare aur orthogonal states

∣0 ⟩, ∣1 ⟩

opposite poles par ho.

Global phase kyun hatate hain?

Yeh physically unobservable hai — measurement predictions

∣ ψ ⟩ \to e^{iγ} ∣ ψ ⟩

ke under invariant hain.

Transmon mein Josephson junction kya provide karta hai?

Anharmonicity, taaki lowest do levels uniquely qubit ki tarah addressable hon.

Superconducting qubits ko ~15 mK tak kyun thanda karte hain?

k_{B} T ≪ Δ E

banane ke liye taaki thermal excitation

P_{1} / P_{0} = e^{- Δ E / k_{B} T}

negligible ho.

Thermal excitation ratio ka formula?

P_{1} / P_{0} = e^{- Δ E / k_{B} T}

(Boltzmann).

DiVincenzo criteria list karo.

Scalable qubits; initialization; long coherence; universal gate set; qubit-specific measurement.

T_{1}

kya hai?

Energy relaxation time; excited-state probability

1/ e

tak decay hoti hai.

T_{2}

kya hai?

Dephasing/coherence time; phase (off-diagonal) coherence

1/ e

tak decay hoti hai.

T_{1}, T_{2}

ke beech relation?

1/ T_{2} = 1/ (2 T_{1}) + 1/ T_{ϕ}

, toh

T_{2} \leq 2 T_{1}

Gates unitary kyun hone chahiye?

Total probability (norm) preserve karne ke liye — quantum evolution reversible hoti hai.

Hadamard gate

∣0 ⟩

ke saath kya karta hai?

Equal superposition

(∣0 ⟩ + ∣1 ⟩) / 2

create karta hai.

Decoherence dominate karne se pehle roughly kitne gates?

Lagbhag

T_{2} / t_{g}

(coherence time divided by gate time).

Superconducting qubit ko read out kaise karte hain?

Dispersive readout — ek coupled resonator ki frequency qubit state ke saath shift hoti hai, microwave transmission mein dikhayi deti hai.

n

qubits =

2^{n}

classical bits of storage kyun nahi?

Measurement sirf

n

classical bits par collapse hoti hai;

2^{n}

amplitudes interference power dete hain, readout nahi.