Two-bit gates are universal for quantum computation

A proof is given, which relies on the commutator algebra of the unitary Lie, that quantum gates operating on just two bits at a time are sufficient construct a general quantum circuit. The best previous result had shown the of three-bit gates, by analogy to the universality of the Toffoli-bit gate of classical reversible computing. Two-bit quantum gates may be by magnetic resonance operations applied to a pair of electronic or spins. A ``gearbox quantum computer'' proposed here, based on the of atomic force microscopy, would permit the operation of such-bit gates in a physical system with very long phase breaking (i. e. , quantum coherence) times. Simpler versions of the gearbox computer could be used do experiments on Einstein-Podolsky-Rosen states and related entangled states.