Binary math is at the heart of most computing, in large part because of the ease with which two energy states can be achieved.  I have always thought that having more states could improve computing power (e.g. using a trit instead of a bit), but there is a seeming lack of attention being paid to the problem.Some work has been done in the quantum computing area using qubits to achieve more than the normal two energy levels (see, for example, the recent effort at UCSB).What advances have been made toward having ternary (or greater) computers, and what are the primary the implications of the extra states?
Ternary (and beyond) computation and quantum computing?
quantum computing
Computing with trits instead of bits is like computing with Turing machines that have an alphabet size of 3 instead of an alphabet size of 2.  Increasing the alphabet size like that (not necessarily 2 to 3, but 2 to larger) can permit a speedup in running time, and a compression in the use of space.  (See http://en.wikipedia.org/wiki/Linear_speedup_theorem for some initial discussion.)  Note that these improvements are fairly minimal -- they won't make an intractable problem tractable.  (If you use an intractably large alphabet instead of computing over bits, that's cheating, because then you are just transferring the hardness from one part of the computation to another.)Qubits at least appear to have a fundamentally different character from bits or trits.  The state of a qubit (or, more generally, a quDit, quantum digit) is a collection of probability amplitudes that associates the qubit to some extent with each possible state.  Bits, trits, etc., are crisply in one state, the end.  In particular, if Factoring is not in polynomial time, then there exists no ptime algorithm over trits that factor integers, while there does exist a quantum algorithm over qudits that factors integers.http://en.wikipedia.org/wiki/Qubit