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Classical computation processes information described by classical states and transition rules, in contrast with computations that use coherent quantum states. A digital classical model commonly represents data by bits; a Boolean circuit or Turing machine supplies rules for transforming those representations. A randomized algorithm adds random choices, while each particular execution still follows one sequence of classical states.

For example, an AND gate maps two input bits to one output bit, with output one only for input 11. A probabilistic selection between two inputs describes uncertainty about which classical state was used; it is not a coherent quantum superposition of those inputs. Stating the model separates the abstract information-processing rule from its physical implementation: ordinary electronic devices ultimately obey quantum physics even when their exposed computational behavior is classical.

A quantum algorithm specifies quantum state preparation, a sequence of quantum gates and quantum measurements, and any classical computation controlling them. For a pure quantum state $|\psi ⟩={\sum }_x{\alpha }_x|x⟩$, computational-basis measurement returns $x$ with probability $|{\alpha }_x{|}^2$, as prescribed by the Born rule. David Deutsch formulated a universal quantum computer using unitary operators and measurement.^[ref]

For example, one Hadamard gate prepares $H|0⟩=(|0⟩+|1⟩)/\sqrt{2}$. Computational-basis measurement gives either bit with probability $1/2$. Applying another Hadamard gate before measuring instead gives $H^2|0⟩=|0⟩$: the amplitudes for outcome one cancel by quantum interference. This illustrates why manipulating probability amplitudes matters; merely preparing a superposition does not reveal every component.

An analysis of algorithms must state the counted resources: quantum gate count, number of qubits, quantum state preparation cost and repetitions needed for the required probability of a correct result. Quantum query complexity counts quantum oracle calls separately; one such call need not be one elementary quantum gate.