3-qubit code that corrects single phase-flip (Z) errors by encoding in the X basis.

3-qubit repetition code that encodes 1 logical qubit and corrects single bit-flip (X) errors.

Generate 4 truly random bits using quantum superposition and measurement.

Reversible quantum half-adder: computes sum = a XOR b and carry = a AND b for 1-bit inputs.

Estimate the eigenphase φ=1/8 of the T gate: U|ψ⟩=e^{i2πφ}|ψ⟩ using 3 counting qubits.

Find the hidden period s='11' of a 2-to-1 function f(x) = f(x⊕s) with exponential quantum speedup.

Fair coin flipping over a quantum channel: Alice and Bob each contribute randomness via entanglement.

Demonstrate the phase kickback trick: a CNOT with ancilla in |−⟩ imprints a phase on the control qubit.

W state: symmetric 3-qubit entangled state (|001⟩ + |010⟩ + |100⟩)/√3.

One first-order Trotter step simulating the 1D transverse-field Ising model on 4 qubits (J=1, h=0.5, dt=0.1).

Quadratic speedup for unstructured search: finds |11⟩ among 4 basis states with one Grover iteration.

Count the number of solutions to a 2-qubit oracle (marks |11⟩) using QPE applied to Grover's operator.

Core period-finding subroutine of Shor's algorithm: find the period r of f(x) = 7^x mod 15.

Estimate Re(⟨ψ|U|ψ⟩) for U=Z gate and |ψ⟩=|+⟩ using a single ancilla qubit.

Greenberger–Horne–Zeilinger (GHZ) state: maximally entangled 3-qubit state (|000⟩ + |111⟩)/√2.

Measure |⟨ψ|φ⟩|² between two quantum states using a Fredkin (controlled-SWAP) gate.

First quantum error-correcting code: encodes 1 logical qubit into 9 physical qubits, correcting any single-qubit error.

Detect (not correct) any single-qubit error using 4 physical qubits encoding 2 logical qubits.

Estimate the amplitude a = sin²(π/8) ≈ 0.146 using 3 precision qubits via quantum phase estimation.

Preparation of the maximally entangled two-qubit Bell state |Φ⁺⟩ = (|00⟩ + |11⟩)/√2.