Quantum computing basics

Computer Science and Engineering (B.E./B.Tech) · Emerging areas · free study notes for exams, placements and interviews

In short

Quantum computing processes information in qubits, two-level quantum systems that can hold a superposition of 0 and 1 and can be entangled with one another, so an n-qubit register is described by 2ⁿ complex amplitudes that evolve together under unitary gates. This matters because a few problems — integer factoring, discrete logarithms, unstructured search, simulation of molecules and materials — have quantum algorithms that are provably faster than the best known classical ones, which is why RSA/ECC are being replaced by post-quantum cryptography. In Indian universities it appears in AICTE/CBCS electives on quantum computing, quantum information and emerging technologies, and in GATE CS/PH-style questions on state vectors, gate matrices, measurement probabilities and algorithm complexity. Industrially it is a research-and-skills area: the National Quantum Mission (Union Cabinet approval, April 2023; outlay ₹6,003.65 crore for 2023-24 to 2030-31), four thematic hubs (quantum computing at IISc Bengaluru, communication at IIT Madras, sensing and metrology at IIT Bombay, materials and devices at IIT Delhi), plus quantum teams at TCS, Infosys, Wipro, HCLTech, QNu Labs, QpiAI, DRDO, ISRO and C-DAC. Hardware numbers in this field change every few months, so treat specific qubit counts as snapshots, not fixed facts.

Explained

Qubit, superposition and the Bloch sphere

A classical bit is 0 or 1. A qubit is a unit vector in a two-dimensional complex Hilbert space: |ψ⟩ = α|0⟩ + β|1⟩ with α, β ∈ ℂ and |α|² + |β|² = 1.
|0⟩ and |1⟩ (the computational basis) are column vectors [1 0]ᵀ and [0 1]ᵀ. α and β are amplitudes, not probabilities; probabilities are their squared magnitudes.
Dirac notation: |ψ⟩ is a ket (column), ⟨ψ| its conjugate transpose (bra, row), ⟨φ|ψ⟩ an inner product (a number), |ψ⟩⟨φ| an outer product (a matrix).
Global phase is unobservable: |ψ⟩ and e^{iγ}|ψ⟩ give identical measurement statistics. Relative phase between α and β is physical and is what interference exploits.
Bloch sphere: any pure qubit state maps to a point on a unit sphere, |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ}sin(θ/2)|1⟩. North pole = |0⟩, south = |1⟩, equator = equal superpositions such as |+⟩ = (|0⟩+|1⟩)/√2 and |−⟩ = (|0⟩−|1⟩)/√2. Gates are rotations of this sphere; mixed states (noisy qubits) lie inside it, described by a density matrix ρ.
Superposition is not "both values at once" in a classical sense — it is a single definite quantum state whose basis amplitudes interfere. A superposition cannot be read out; only one basis outcome is obtained per measurement.

Many qubits, entanglement and measurement

Combining systems uses the tensor product: 2 qubits span a 4-dimensional space {|00⟩, |01⟩, |10⟩, |11⟩}; n qubits span 2ⁿ dimensions and need 2ⁿ−1 independent complex numbers to specify. This exponential state space is the source of both the hoped-for power and the difficulty of classical simulation.
A product (separable) state factorises, e.g. |+⟩⊗|0⟩. An entangled state does not: the four Bell states, e.g. |Φ⁺⟩ = (|00⟩+|11⟩)/√2, cannot be written as a product of single-qubit states.
Measurement postulate: measuring in the basis {|x⟩} gives outcome x with probability P(x) = |⟨x|ψ⟩|², and the state collapses to |x⟩. Measurement is irreversible, probabilistic and destroys superposition.
In |Φ⁺⟩, each qubit alone is random (50/50), but the two results always agree — correlation without a local stored value. Bell/CHSH test: any local hidden-variable theory obeys |S| ≤ 2, quantum mechanics reaches 2√2 ≈ 2.828; experiments agree with quantum mechanics (2022 Nobel Prize, Aspect-Clauser-Zeilinger).
No-cloning theorem: no unitary can copy an arbitrary unknown state (|ψ⟩|0⟩ → |ψ⟩|ψ⟩ is impossible for all |ψ⟩), proved from linearity. Consequences: no quantum backups, no signal-amplifier repeaters, no fan-out of a wire — but it also makes eavesdropping detectable (QKD).
Entanglement does not transmit information faster than light. Teleportation moves one qubit's state using one shared entangled pair (1 ebit) plus 2 classical bits; superdense coding sends 2 classical bits with 1 qubit plus 1 ebit. The Holevo bound limits n qubits to carrying at most n classical bits of accessible information.

Quantum gates and circuits

Every closed-system operation is a unitary matrix U with U†U = I — hence reversible, length-preserving and square. Measurement is the only non-unitary step.
Single-qubit gates (matrices below): X (bit flip, NOT), Y, Z (phase flip), H (Hadamard — creates superposition), S = √Z, T = √S (π/8 gate), and rotations Rx(θ), Ry(θ), Rz(θ).
Two- and three-qubit gates: CNOT (flips the target if the control is |1⟩ — the standard entangler), CZ, SWAP, Toffoli (CCNOT) and Fredkin (CSWAP), which make classical logic reversible.
Universality: {H, T, CNOT} is universal — any unitary can be approximated to arbitrary accuracy (Solovay-Kitaev). {H, S, CNOT} alone (the Clifford group) is *not* universal and is classically simulable in polynomial time (Gottesman-Knill); the T gate supplies the non-Clifford power and is the expensive resource in fault-tolerant designs.
A quantum circuit reads left to right: wires are qubits, boxes are gates, the meter symbol is measurement. Standard pattern: initialise to |0…0⟩ → create superposition with H → apply an oracle/unitary → interfere so wrong answers cancel → measure. Depth = number of sequential layers, width = qubits; both must fit inside the coherence time.
Reversibility means classical irreversible gates (AND, OR) are embedded with ancilla and "garbage" qubits; Landauer's principle links irreversible bit erasure to a minimum heat kT ln2.
Hardware has a native gate set and limited connectivity, so a compiler (Qiskit transpiler) decomposes gates and inserts SWAPs — this inflates real circuit depth well beyond the textbook circuit.

Algorithms: where the speed-up is, and where it is not

Deutsch-Jozsa: decides whether an n-bit oracle function is constant or balanced in 1 query against 2ⁿ⁻¹+1 classical worst-case queries — the teaching example of exponential query advantage.
Bernstein-Vazirani (find a hidden string in 1 query) and Simon's algorithm (hidden XOR period, exponential separation) are the usual follow-ups.
Grover's search: finds a marked item among N unsorted items in O(√N) oracle calls instead of O(N) — a quadratic speed-up, proved optimal. Works by amplitude amplification (oracle phase flip + diffusion operator). Over-iterating *reduces* success probability, because it is a rotation.
Quantum Fourier transform (QFT): the quantum analogue of the DFT on 2ⁿ amplitudes using O(n²) gates. It powers phase estimation, the engine of period finding.
Shor's algorithm: factors an n-bit integer in roughly O(n³) time by reducing factoring to finding the period r of a^x mod N with the QFT, then taking gcd(a^{r/2}±1, N). It breaks RSA, Diffie-Hellman and ECC (an exponential speed-up); AES and SHA are only weakened quadratically by Grover, so doubling symmetric key length (AES-256) suffices.
Simulation and optimisation: Hamiltonian simulation for chemistry and materials (Feynman's original motivation), VQE and QAOA — hybrid quantum-classical loops suited to today's noisy machines, with speed-ups that are heuristic, not proved. HHL for linear systems and quantum machine learning carry heavy data-loading caveats.
Complexity: quantum polynomial time is BQP, with P ⊆ BQP ⊆ PSPACE. BQP is *not* known to contain NP-complete problems — quantum computers are not believed to solve the travelling salesman or SAT exactly in polynomial time. "Quantum parallelism" is constrained by measurement: only interference that makes wrong amplitudes cancel yields an advantage.

Noise, decoherence and quantum error correction

Decoherence is the loss of quantum behaviour through uncontrolled coupling to the environment. T1 = energy relaxation (|1⟩ → |0⟩) time; T2 = dephasing time (loss of relative phase), with T2 ≤ 2·T1.
Typical error channels: bit flip (X), phase flip (Z), depolarising, amplitude damping, leakage out of the computational subspace, crosstalk, and SPAM (state-preparation and measurement) error. Readout error is usually the largest single contributor.
Order of magnitude today: superconducting single-qubit error ~10⁻⁴-10⁻³, two-qubit error ~10⁻³-10⁻², coherence tens to hundreds of microseconds against gate times of tens of nanoseconds; trapped ions have far longer coherence (seconds) but slower gates (microseconds). Figures improve continuously.
Quantum error correction (QEC) must work without measuring the data (collapse) and without copying it (no-cloning). It encodes one logical qubit in many physical qubits and measures only syndromes via ancillas.
Standard codes: 3-qubit repetition code (bit-flip only), Shor's 9-qubit code (arbitrary single-qubit error), Steane [[7,1,3]], [[5,1,3]] (smallest for one arbitrary error), and the surface code — a 2-D nearest-neighbour code that is the leading practical choice, with a threshold near 1% physical error.
Threshold theorem: if physical error per gate is below the code threshold, logical error can be suppressed exponentially by increasing code distance d. Price: one useful logical qubit may need 10³-10⁴ physical qubits, which is why cryptographically relevant Shor runs are estimated to need millions of physical qubits.
Error mitigation (zero-noise extrapolation, measurement calibration, dynamical decoupling) is not correction — it improves estimates of expectation values without making long circuits reliable.
NISQ (Noisy Intermediate-Scale Quantum) describes present machines: tens to a few hundred noisy qubits, no full error correction. Benchmarks: quantum volume, randomised benchmarking, CLOPS (speed), and "random circuit sampling" claims of quantum advantage (Google 2019 and later), several of which have been partly matched by improved classical simulation — advantage claims should be read with that caveat.

Hardware platforms and the DiVincenzo criteria

DiVincenzo criteria for a physical implementation: (1) scalable well-characterised qubits, (2) ability to initialise to a fiducial state, (3) coherence time ≫ gate time, (4) a universal gate set, (5) qubit-specific measurement.
Superconducting qubits (transmons; IBM, Google, Rigetti): lithographic circuits with Josephson junctions, ~5 GHz transitions, microwave control, cooled to ~10-20 mK in a dilution refrigerator. Fast gates, scalable fabrication; short coherence, needs cryogenics.
Trapped ions (IonQ, Quantinuum): hyperfine states of ions in RF traps, laser-driven gates. Excellent fidelity and all-to-all connectivity; slow gates, harder to scale.
Photonic (Xanadu, PsiQuantum): single photons, beam splitters, squeezed light; room temperature and natural for communication, but deterministic two-photon gates are difficult.
Spin qubits in silicon, neutral atoms in optical tweezers, topological qubits (Majorana-based, still unproven), and NMR (historically the first demonstrations, including early Indian work at IISc/TIFR; not scalable).
Full-stack view: cryostat and control electronics → calibration/pulse layer (OpenPulse) → transpiler → SDK (Qiskit, Cirq, PennyLane, Q#) → cloud access (IBM Quantum, AWS Braket, Azure Quantum) → application. Most Indian student work happens at the SDK and simulator level, including MeitY's QSim toolkit (C-DAC, IISc, IIT Roorkee) and the Amazon/IBM cloud back ends used by Indian campuses.

Quantum communication, cryptography and the Indian ecosystem

QKD distributes symmetric keys with security based on physics, not computational hardness. BB84 (Bennett-Brassard 1984) uses four polarisation states in two conjugate bases; sifting, error estimation, error reconciliation and privacy amplification follow. E91 uses entanglement; B92 uses two states; decoy states defeat photon-number-splitting attacks.
Any eavesdropper disturbs the states (no-cloning plus the uncertainty principle), raising the QBER; above roughly 11% for BB84 the key is discarded. QKD needs an authenticated classical channel, so it does not remove the need for signatures.
Post-quantum cryptography (PQC) replaces vulnerable public-key maths with lattice, hash and code-based schemes. NIST standardised ML-KEM (FIPS 203, from Kyber), ML-DSA (FIPS 204, from Dilithium) and SLH-DSA (FIPS 205, from SPHINCS+) in August 2024, with HQC selected later as a backup KEM. Migration is urgent because of "harvest now, decrypt later" — traffic captured today can be decrypted after a large quantum computer exists.
Indian milestones (verify current status before quoting): DST's QuEST programme (2019); ISRO's free-space QKD demonstration between two buildings at SAC Ahmedabad (2021); DRDO-IIT Delhi fibre QKD over ~100 km on the Prayagraj-Vindhyachal link (2022); C-DAC/IISc/IIT Roorkee QSim; QNu Labs (Bengaluru) commercial QKD and QRNG products; QpiAI's indigenous superconducting system announced in 2025; state initiatives such as the Amaravati Quantum Valley; I-HUB QTF at IISER Pune.
Policy and standards touchpoints: National Quantum Mission under DST, MeitY/C-DAC programmes, CERT-In advisories and the DPDP Act 2023 (reasonable security safeguards — a reason to plan crypto-agility), TEC/TSDSI work on QKD interfaces, and ITU-T Y.3800-series QKD-network recommendations.

Key points

Qubit - a two-level quantum system; state α|0⟩+β|1⟩ with |α|²+|β|²=1, where amplitudes are complex.
Superposition - interference-capable combination of basis states; destroyed by measurement, never directly observable.
n qubits - 2ⁿ amplitudes; classical simulation cost grows exponentially (≈16·2ⁿ bytes for a state vector in double precision).
Entanglement - a joint state that does not factorise; the resource behind teleportation, dense coding and E91 QKD.
Measurement - P(x) = |⟨x|ψ⟩|², then collapse; probabilistic and irreversible.
Unitarity - U†U = I makes all gates reversible; no erasure or copying inside a circuit.
No-cloning - an unknown state cannot be duplicated; blocks quantum amplifiers but enables eavesdropper detection.
Hadamard - makes |0⟩ → |+⟩; H² = I, so two Hadamards restore the input.
CNOT - the standard two-qubit entangler; H on the control followed by CNOT produces a Bell state.
Universal set - {H, T, CNOT}; Clifford-only circuits are classically simulable (Gottesman-Knill).
Grover - O(√N) search, quadratic and provably optimal; iterate ≈ (π/4)√N times, no more.
Shor - polynomial-time factoring and discrete log via QFT period finding; breaks RSA/ECC, not AES.
BQP - quantum polynomial time; P ⊆ BQP ⊆ PSPACE, and NP-complete problems are not known to be in BQP.
T1 / T2 - relaxation and dephasing times, with T2 ≤ 2T1; they cap usable circuit depth.
Surface code - practical 2-D QEC with ~1% threshold; thousands of physical qubits per logical qubit.
NISQ - today's noisy machines without full QEC; VQE/QAOA hybrids suit them, with heuristic speed-ups only.
PQC - ML-KEM/ML-DSA/SLH-DSA standardised by NIST in 2024; the practical defence against quantum attack.
National Quantum Mission - ₹6,003.65 crore, 2023-31, four hubs (IISc, IIT Madras, IIT Bombay, IIT Delhi).

Formulas and examples

Qubit state: |ψ⟩ = α|0⟩ + β|1⟩, normalisation |α|² + |β|² = 1 (α, β dimensionless complex amplitudes)
Bloch form: |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ} sin(θ/2)|1⟩, θ ∈ [0, π] polar, φ ∈ [0, 2π) azimuthal (radians or degrees)
Measurement: P(x) = |⟨x|ψ⟩|²; expectation ⟨A⟩ = ⟨ψ|A|ψ⟩
Register size: n qubits → 2ⁿ basis states; classical state-vector memory ≈ 2ⁿ × 16 bytes (two doubles per amplitude)
Unitarity: U†U = UU† = I; evolution |ψ'⟩ = U|ψ⟩; Schrödinger: iħ d|ψ⟩/dt = H|ψ⟩ (ħ = 1.0546×10⁻³⁴ J·s)
Grover iterations: k ≈ (π/4)√(N/M) for M marked items out of N; classical average ≈ (N+1)/(M+1) queries
Shor: period r of f(x) = aˣ mod N, then factors = gcd(a^{r/2} ± 1, N); runtime ≈ O((log N)³)
Circuit success (independent errors): F ≈ (1 − ε)^g, ε = error per gate, g = gate count
Qubit frequency vs temperature: E = hf, T_eq = hf/k (h = 6.626×10⁻³⁴ J·s, k = 1.381×10⁻²³ J/K)
Fidelity of pure states: F = |⟨φ|ψ⟩|²; CHSH bound: |S| ≤ 2 classical, |S| ≤ 2√2 quantum

X = [0 1; 1 0]      Y = [0 -i; i 0]     Z = [1 0; 0 -1]
H = (1/√2)[1 1; 1 -1]   S = [1 0; 0 i]   T = [1 0; 0 e^{iπ/4}]
CNOT (control q0, target q1) = [1 0 0 0; 0 1 0 0; 0 0 0 1; 0 0 1 0]

Example 1 - Normalise and find probabilities. |ψ⟩ = 2|0⟩ + 3i|1⟩ (unnormalised).

Norm = √(|2|² + |3i|²) = √(4 + 9) = √13 ≈ 3.6056.
Normalised: |ψ⟩ = (2/√13)|0⟩ + (3i/√13)|1⟩.
P(0) = 4/13 = 0.3077 (30.77%), P(1) = 9/13 = 0.6923 (69.23%); sum = 1. ✓
Bloch angles: cos(θ/2) = 0.5547 → θ/2 = 56.31°, θ ≈ 112.6°; the phase of β is +i, so φ = 90°.

Example 2 - Hadamard and interference. |ψ⟩ = 0.6|0⟩ + 0.8|1⟩, apply H.

New amplitudes: (0.6 + 0.8)/√2 = 1.4/1.41421 = 0.98995 for |0⟩; (0.6 − 0.8)/√2 = −0.2/1.41421 = −0.14142 for |1⟩.
P(0) = 0.98995² = 0.98, P(1) = 0.14142² = 0.02. Phases shifted the odds from 36/64 to 98/2 — this is interference, not classical mixing.
Check H² = I: applying H again returns exactly 0.6|0⟩ + 0.8|1⟩.

Example 3 - Bell state and correlation. Start |00⟩, apply H to qubit 0, then CNOT(0→1).

After H: (|00⟩ + |10⟩)/√2. After CNOT: (|00⟩ + |11⟩)/√2 = |Φ⁺⟩.
Each outcome 00 and 11 has probability (1/√2)² = 0.5; 01 and 10 have probability 0.
Individually, qubit 0 is 50/50 random, yet the two readings always match — and the state cannot be written as a product, so it is entangled.

Example 4 - Grover versus brute force. Search one marked entry in a database of N = 2²⁰ = 1,048,576 records.

√N = 1,024; iterations k ≈ (π/4) × 1,024 = 0.7854 × 1,024 ≈ 804 oracle calls.
Classical average = N/2 = 524,288 calls → speed-up ≈ 524,288/804 ≈ 652× (quadratic, not exponential).
With M = 4 marked items: k ≈ (π/4)√(1,048,576/4) = 0.7854 × 512 ≈ 402 calls.

Example 5 - Why the fridge runs at millikelvin. Transmon transition frequency f = 5 GHz.

E = hf = 6.626×10⁻³⁴ × 5×10⁹ = 3.313×10⁻²⁴ J.
Equivalent temperature T = E/k = 3.313×10⁻²⁴ / 1.381×10⁻²³ = 0.240 K = 240 mK.
At 15 mK, thermal excitation ≈ e^{−240/15} = e^{−16} ≈ 1.1×10⁻⁷ — the qubit stays in |0⟩. At 1 K it would be e^{−0.24} ≈ 0.79, i.e. almost fully scrambled.

Example 6 - Noise budget and classical simulation limit.

Circuit with g = 200 two-qubit gates at ε = 0.5% error: F ≈ 0.995²⁰⁰ = e^{200 ln 0.995} = e^{−1.0025} ≈ 0.367 (37%). At ε = 0.1%: 0.999²⁰⁰ ≈ 0.819 (82%).
State vector for n = 50 qubits: 2⁵⁰ × 16 bytes = 2⁵⁴ bytes = 16 PiB (≈18 petabytes) — beyond any single machine, which is why ~30 qubits is the practical laptop limit and ~45 the supercomputer limit.

Example 7 - Shor on N = 15, a = 7. 7¹ = 7, 7² = 49 mod 15 = 4, 7³ = 28 mod 15 = 13, 7⁴ = 91 mod 15 = 1 → period r = 4.

gcd(7² − 1, 15) = gcd(48, 15) = 3; gcd(7² + 1, 15) = gcd(50, 15) = 5. Factors 3 × 5 = 15. ✓ The quantum part only finds r; the gcd steps are classical.

Aspect	Classical bit / CPU	Qubit / QPU
State	0 or 1	α	0⟩+β	1⟩, continuum of states
n units describe	one n-bit value	2ⁿ amplitudes simultaneously
Operations	AND, OR, NOT (mostly irreversible)	Unitary, always reversible
Copying	Free	Impossible for unknown states
Readout	Deterministic	Probabilistic, collapses the state
Errors	Bit flip; cheap ECC	Bit + phase flip; QEC needs 10³-10⁴:1 overhead
Best at	Everything in daily computing	Factoring, search, quantum simulation

Platform	Qubit	Temperature	Strength	Weakness
Superconducting	Transmon	~10-20 mK	Fast gates, chip fabrication	Short coherence, cryogenics
Trapped ion	Ion hyperfine levels	Room temp, UHV	High fidelity, all-to-all	Slow gates, scaling
Photonic	Single photon	Room temp	Natural for networks	Probabilistic 2-qubit gates
Silicon spin	Electron spin	<1 K	CMOS-compatible	Early maturity, crosstalk
Neutral atom	Rydberg atom	µK (tweezers)	Large arrays, reconfigurable	Atom loss, gate fidelity

Where it is used

IT services and consulting (TCS, Infosys, Wipro, HCLTech, Accenture, Capgemini India): quantum readiness assessments, crypto-inventory and PQC migration for banks and telcos, Qiskit/PennyLane proof-of-concepts in finance and logistics.
Cybersecurity and telecom: QNu Labs QKD and quantum random number generators, BSNL/telco pilot links, CERT-In-driven crypto-agility reviews; "harvest now, decrypt later" risk assessment for long-lived data under the DPDP Act 2023.
Defence, space and government labs: DRDO and ISRO QKD over fibre and free space, C-DAC's QSim simulator, National Quantum Mission hub projects at IISc and the IITs, I-HUB QTF (IISER Pune) startup support.
Hardware and deep-tech startups: QpiAI, QuBeats, BosonQ Psi, Qulabs — roles in cryogenics, RF/microwave control electronics, FPGA control stacks, device fabrication and calibration software.
Pharma, chemicals and materials R&D: molecular ground-state estimation with VQE for catalysts and battery chemistry; energy and EV companies run exploratory projects with cloud QPUs.
Job roles and exams: quantum software engineer, quantum algorithms researcher, cryogenic/control engineer, PQC security architect, and quantum-computing teaching posts; also GATE CS/PH electives, CSIR-NET physics, and entrance tests for M.Tech/PhD in quantum technology at IITs and IISc.

Interview questions

Q: How is a qubit different from a classical bit? A: A bit holds 0 or 1; a qubit holds a normalised complex superposition α|0⟩+β|1⟩, can be entangled with other qubits, and yields 0 or 1 only probabilistically on measurement.
Q: Does an n-qubit computer try 2ⁿ answers in parallel? A: No — it holds 2ⁿ amplitudes, but a measurement returns one outcome. An algorithm is useful only if interference cancels the wrong amplitudes, which is why speed-ups exist for few problems.
Q: Why must quantum gates be unitary? A: Closed-system quantum evolution preserves total probability and is reversible; unitarity (U†U = I) guarantees both, so no gate may erase or duplicate information.
Q: State the no-cloning theorem and one consequence. A: No single unitary can copy an arbitrary unknown state; it follows from linearity. Consequence: an eavesdropper cannot tap a quantum channel without disturbing it, which is what makes BB84 secure.
Q: What exactly does H on |0⟩ produce, and what if you apply H twice? A: H|0⟩ = (|0⟩+|1⟩)/√2 = |+⟩, measuring 0 or 1 with probability 0.5 each; H² = I, so a second H returns |0⟩ deterministically — proof that superposition is not just randomness.
Q: How do you build a Bell state, and why is it not just correlated randomness? A: H on the control then CNOT gives (|00⟩+|11⟩)/√2. Classical correlation would satisfy |S| ≤ 2 in a CHSH test; entanglement reaches 2√2, which experiments confirm.
Q: Grover gives what kind of speed-up, and does it break AES? A: Quadratic, O(√N) — it halves effective symmetric key strength, so AES-128 behaves like 64-bit security against an ideal quantum attacker; moving to AES-256 restores the margin.
Q: Why does Shor's algorithm break RSA but not AES? A: RSA/ECC rest on factoring and discrete logarithms, which reduce to period finding that the QFT solves in polynomial time; AES has no such algebraic structure, so only generic Grover search applies.
Q: Can a quantum computer solve NP-complete problems efficiently? A: Not as far as is known — NP-complete problems are not believed to lie in BQP; only specific structured problems and quantum simulation show large proven advantage.
Q: Difference between T1 and T2, and which limits circuit depth? A: T1 is energy relaxation, T2 is dephasing (T2 ≤ 2T1); the shorter of the two against gate duration sets how many sequential gates can run before the state is lost — usually T2.
Q: Why can't you just use a repetition code for qubits? A: You cannot copy an unknown state and you cannot look at the data without collapsing it; QEC instead encodes into entangled blocks and measures ancilla syndromes, and must handle phase errors as well as bit flips.
Q: What is NISQ, and what should a company do today? A: Noisy Intermediate-Scale Quantum — tens to hundreds of qubits with no full error correction, suited to small hybrid VQE/QAOA experiments. The actionable work today is PQC migration and crypto-agility, not production quantum workloads.

Common mistakes

Treating α and β as probabilities and writing α + β = 1 instead of |α|² + |β|² = 1; forgetting to normalise before computing probabilities.
Ignoring relative phase, or assuming a global phase factor is measurable — the first changes interference, the second never shows up.
Saying a qubit "is 0 and 1 at the same time" and concluding a quantum computer brute-forces all passwords; it does not, and this answer is a standard interview red flag.
Claiming entanglement allows faster-than-light signalling, or forgetting that teleportation needs 2 classical bits.
Running Grover for too many iterations (it is a rotation that overshoots), or quoting its gain as exponential rather than quadratic.
Multiplying gate matrices in the wrong order (circuits read left to right, matrices compose right to left: U = U₃U₂U₁) and mixing up qubit ordering conventions, which flips Qiskit's bit labels.
Equating physical and logical qubits in a hardware claim, and so expecting a 100-qubit NISQ machine to run Shor on RSA-2048 — that needs millions of physical qubits with error correction.