This paper presents a universal set of quantum gates for quantum computation. It shows that a set of gates consisting of all one-bit quantum gates (U(2)) and the two-bit exclusive-or gate (which maps Boolean values (x,y) to (x,x⊕y)) is universal in the sense that any unitary operation on arbitrarily many bits can be expressed as compositions of these gates. The paper investigates the number of these gates required to implement other gates, such as generalized Deutsch-Toffoli gates, which 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. The paper derives upper and lower bounds on the exact number of elementary gates required to build up a variety of two- and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and makes some observations about the number required for arbitrary n-bit unitary operations. The paper discusses the background of quantum computation, including the concept of reversible computation and the role of quantum mechanics in computation. It introduces the concept of quantum gate arrays and their relation to classical computational complexity theory. The paper also discusses the properties of quantum gates, including the matrix properties of quantum gates and the simulation of general Λ₁(U) gates. It presents various results on the simulation of two-bit and three-bit gates, including the simulation of general Λ₂(U) gates and the construction of efficient gate arrays for n-bit operations. The paper also discusses the asymptotic growth rate of simulations with respect to n and shows that the number of basic operations required for simulating general Λₙ₋₁(U) gates is quadratic in the general case and linear in many cases of interest. The paper concludes with a discussion of linear approximate simulations of general Λₙ₋₁(U) gates and linear simulations in special cases.This paper presents a universal set of quantum gates for quantum computation. It shows that a set of gates consisting of all one-bit quantum gates (U(2)) and the two-bit exclusive-or gate (which maps Boolean values (x,y) to (x,x⊕y)) is universal in the sense that any unitary operation on arbitrarily many bits can be expressed as compositions of these gates. The paper investigates the number of these gates required to implement other gates, such as generalized Deutsch-Toffoli gates, which 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. The paper derives upper and lower bounds on the exact number of elementary gates required to build up a variety of two- and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and makes some observations about the number required for arbitrary n-bit unitary operations. The paper discusses the background of quantum computation, including the concept of reversible computation and the role of quantum mechanics in computation. It introduces the concept of quantum gate arrays and their relation to classical computational complexity theory. The paper also discusses the properties of quantum gates, including the matrix properties of quantum gates and the simulation of general Λ₁(U) gates. It presents various results on the simulation of two-bit and three-bit gates, including the simulation of general Λ₂(U) gates and the construction of efficient gate arrays for n-bit operations. The paper also discusses the asymptotic growth rate of simulations with respect to n and shows that the number of basic operations required for simulating general Λₙ₋₁(U) gates is quadratic in the general case and linear in many cases of interest. The paper concludes with a discussion of linear approximate simulations of general Λₙ₋₁(U) gates and linear simulations in special cases.