  Quantum computation is traditionally expressed in terms of quantum bits, or
qubits. In this work, we instead consider three-level qu$trits$. Past work with
qutrits has demonstrated only constant factor improvements, owing to the
$\log_2(3)$ binary-to-ternary compression factor. We present a novel technique
using qutrits to achieve a logarithmic depth (runtime) decomposition of the
Generalized Toffoli gate using no ancilla--a significant improvement over
linear depth for the best qubit-only equivalent. Our circuit construction also
features a 70x improvement in two-qudit gate count over the qubit-only
equivalent decomposition. This results in circuit cost reductions for important
algorithms like quantum neurons and Grover search. We develop an open-source
circuit simulator for qutrits, along with realistic near-term noise models
which account for the cost of operating qutrits. Simulation results for these
noise models indicate over 90% mean reliability (fidelity) for our circuit
construction, versus under 30% for the qubit-only baseline. These results
suggest that qutrits offer a promising path towards scaling quantum
computation.
