 Solution

A qubit has state $\alpha |0⟩+\beta |1⟩$ 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 $|x,y⟩\mapsto |x,y\oplus x⟩$.

For example, a Hadamard gate on the first qubit of $|00⟩$ followed by a CNOT creates $(|00⟩+|11⟩)/\sqrt{2}$. 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.