  We present a novel Clifford+T decomposition of a Toffoli gate. Our
decomposition requires no SWAP gates in order to be implemented on 2D square
lattices of qubits. This decomposition enables shallower, more fault-tolerant
quantum computations on both NISQ and error-corrected architectures. We present
the derivation of the circuit, and illustrate the qubit mapping on a
Sycamore-like architecture.
