singlequbit

A single qubit is the basic unit of quantum information, a two-level quantum system whose state lies in a two-dimensional Hilbert space. In standard usage the term is written as two words; the concatenated form "singlequbit" may appear in informal or shorthand contexts.

Any pure state can be written as |ψ⟩ = α|0⟩ + β|1⟩ with complex amplitudes α and β satisfying |α|^2 + |β|^2 = 1. Global phase e^{iγ} is physically unobservable, so states differing by a global phase are equivalent. When measured in the computational basis {|0⟩, |1⟩}, the probabilities of outcomes 0 and 1 are |α|^2 and |β|^2, and after measurement the qubit collapses to the corresponding basis state.

Single-qubit gates are 2×2 unitary operations acting on the state. Common gates include X (bit flip), Y and Z (phase and flip), H (Hadamard) which creates superposition, and the S and T (phase) gates. These gates form the building blocks of quantum circuits and are represented by rotations on the Bloch sphere.

For pure states, the state can also be represented on the Bloch sphere as a point on the unit sphere, with angles θ and φ related to α, β by α = cos(θ/2), β = e^{iφ} sin(θ/2). More generally, mixed states are described by the density operator ρ, where ρ can be written as (I + r·σ)/2 with a Bloch vector r. The magnitude of r indicates purity, with |r| = 1 for pure states.

During operation, interaction with the environment leads to decoherence, characterized by times T1 (relaxation) and T2 (dephasing). Qubits are typically initialized to |0⟩, and readout is performed by projective measurement in a chosen basis. Physical realizations include superconducting circuits, trapped ions, quantum dots, photonic qubits, and color-center spins such as NV centers.