Two-bit gates are universal for quantum computation.

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

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