Quantum basics

A qubit is the quantum version of a bit. Instead of being only 0 or 1, it can be in a superposition of both until it is measured.

The state of one qubit is a unit vector in a two-dimensional complex space. We write the two basis states as |0⟩ and |1⟩, and a general state as:

∣ ψ ⟩ = α ∣0 ⟩ + β ∣1 ⟩, ∣ α ∣^{2} + ∣ β ∣^{2} = 1

α and β are complex amplitudes. |α|² is the probability of measuring 0, |β|² the probability of measuring 1.

Every single-qubit state can be drawn as a point on the Bloch sphere. The north pole is |0⟩, the south pole is |1⟩, and points on the equator are equal superpositions with different phases:

∣ ψ ⟩ = cos \frac{θ}{2} ∣0 ⟩ + e^{i φ} sin \frac{θ}{2} ∣1 ⟩

θ is the angle from the north pole, φ the angle around the Z axis. In the State tab, QubitLab shows this vector for every qubit.

Two or more qubits live in the tensor product space. Two qubits have four basis states |00⟩, |01⟩, |10⟩, |11⟩. Qiskit orders bits with qubit 0 on the right, so |10⟩ means q1 = 1 and q0 = 0.

∣ Φ^{+} ⟩ = \frac{1}{2} (∣00 ⟩ + ∣11 ⟩)

The Bell state: an entangled pair. Neither qubit has a state of its own, so both Bloch arrows shrink to zero length.

Measurement in the computational basis gives a classical bit: outcome x with probability |amplitude of x|². After measuring, the state collapses to that outcome. Running the same circuit many times (shots) and counting outcomes estimates the probabilities.

p (x) = ∣ ⟨ x ∣ ψ ⟩ ∣^{2}, \overset{p}{^} (x) = \frac{counts ( x )}{shots}

Gates are unitary matrices U (U†U = I). A circuit is a product of gates applied in time order, so the final state is U_n … U_2 U_1 |0…0⟩. The Compile step in QubitLab shows this matrix for circuits up to 3 qubits.

Why OpenQASM 3? Every circuit sent to a simulator or real hardware is OpenQASM 3: a safe description of a circuit that the server can parse and check before anything executes. Python you write in the Playground is different: it runs in an isolated sandbox with no network, no files and no secrets (see Python in the Playground).

OpenQASM 3 cheat sheet