  We show the applicability of the Cartan decomposition of Lie algebras to
quantum circuits. This approach can be used to synthesize circuits that can
efficiently implement any desired unitary operation. Our method finds explicit
quantum circuit representations of the algebraic generators of the relevant Lie
algebras allowing the direct implementation of a Cartan decomposition on a
quantum computer. The construction is recursive and allows us to expand any
circuit down to generators and rotation matrices on individual qubits, where
through our recursive algorithm we find that the generators themselves can be
expressed with controlled-not (CNOT) and SWAP gates explicitly. Our approach is
independent of the standard CNOT implementation and can be easily adapted to
other cross-qubit circuit elements. In addition to its versatility, we also
achieve near-optimal counts when working with CNOT gates, achieving an
asymptotic cnot cost of $\frac{21}{16}4^n$ for $n$ qubits.
