Addition, subtraction, multiplication and division are multi-bit operations that are implemented as a series of lower level logical operations, both on a classical computer and a quantum computer. Many of the logical operations you can do on a classical computer can be done in an analog fashion on a quantum computer, but not all. Most notably, while you can always copy a classical bit, it is not always possible with a quantum bit.

On a classical computer you can perform a set of logical operations to convert a bit string into another. On a quantum computer the operations you can performed are called ''unitary operations''. If you start with a string of quantum bits in some state |00101> (corresponding to the classical bit string 00101), a unitary operation can do everything to the string that a classical computer can, but in addition it can take the quantum bits to a superposition of many strings. Unitary operations are reversible, so they can also take superpositions back to ordinary strings. Since they are reversible they cannot delete any information (like after an AND gate you don't know whether the input was 00, 01 or 10 if the output was 0), but you can get around this by having spare qubits where you can store the missing information.

As an example there are only four possible one-bit logical operations: leave it alone, flip the bit, output 0, output 1 (only the two first are reversible). One of the most common quantum one-qubit operations is a hadamard gate, which takes |0> to |0>+|1> and |1> to |0>-|1> (ignoring a normalization factor). The hadamard is its own inverse, applying it twice does nothing to the qubit.

Points of interest: [http://www.quantiki.org/wiki/index.php/Circuit_model Circuit model], [http://www.quantiki.org/wiki/index.php/Qubit Qubit], [http://www.quantiki.org/wiki/index.php/Hadamard Hadamard], [http://www.quantiki.org/wiki/index.php/What_is_Quantum_Computation%3F What is Quantum Computation?]