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A <qubit> has state $\alpha|0\rangle+\beta|1\rangle$ with $|\alpha|^2+|\beta|^2=1$. An $n$-qubit register occupies a $2^n$-dimensional <Hilbert space> and may contain entanglement. This large state space does not mean that a measurement reads out all its complex amplitudes: computational-basis outcomes occur with the <Born rule> probabilities.
A <quantum circuit> prepares a register, applies <quantum gates>, and measures selected outputs, possibly using classical feed-forward. Closed gates are <unitary operators> and therefore reversible; initialization, measurement and discarding can be treated using a larger system and environment. Single-qubit rotations together with an entangling two-qubit gate give a <universal quantum gate set>. The <CNOT gate> acts as $|x,y\rangle\mapsto|x,y\oplus x\rangle$.
For example, a <Hadamard gate> on the first qubit of $|00\rangle$ followed by a CNOT creates $(|00\rangle+|11\rangle)/\sqrt2$. A Hadamard produces amplitudes with relative signs, so later gates can make unwanted alternatives interfere destructively and useful alternatives reinforce. <Quantum algorithms> organize this interference to improve a specified computational task; superposition alone does not supply an arbitrary speedup or permit reading every branch.
A finite discrete universal set such as Clifford gates together with a non-Clifford phase gate approximates general unitaries to chosen precision. Runtime includes the number of gates, repetitions and classical processing. Measurement outcomes may require multiple runs to estimate a probability or amplify success.
<Quantum decoherence> and imperfect gates limit an implementation. <Quantum error correction> encodes logical information into several physical qubits and measures error syndromes without directly learning the encoded amplitudes. Under appropriate noise and control assumptions, fault tolerance permits reliable longer computations; simply maintaining an isolated ideal state is not a realistic account of a large processor.
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