Elementary gates for quantum computation.

We show that a set of gates that consists of all one-bit quantum gates (U(2)) and the two-bit exclusive-or gate (that maps Boolean values (x, y) to (x, x⊕y)) is universal in the sense that all unitary operations on arbitrarily many bits n (U(2n)) can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical AND of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of twoand three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary n-bit unitary operations. PACS numbers: 03.65.Ca, 07.05.Bx, 02.70.Rw, 89.80.+h ∗ Clarendon Laboratory, Oxford OX1 3PU, UK; [email protected]. Yorktown Heights, New York, NY 10598, USA; bennetc/[email protected]. Department of Computer Science, Calgary, Alberta, Canada T2N 1N4; [email protected]. § Laboratory for Computer Science, Cambridge MA 02139 USA; [email protected]. ¶ Murray Hill, NJ 07974 USA; [email protected]. ‖ Physics Dept., New York, NY 10003 USA; [email protected]. ∗∗ Physics Dept., Los Angeles, CA 90024; [email protected]. (and IBM Research.) †† Inst. for Exptl. Physics, A-6020 Innsbruck, Austria; [email protected].