  We present a scalable set of universal gates and multiply controlled gates in
a qudit basis through a bijective mapping from N qubits to qudits with D = 2^N
levels via rotations in U(2). For each of the universal gates (H, CNOT, and T),
as well as the NOT gate and multiply-controlled-Z gates, we describe a
systematic approach to identifying the set of U(2) rotations required to
implement each gate for any qudit of size N and with minimal use of an ancilla
level. The qudit gates are analyzed in terms of the total rotation count and
gate depth as the system scales with D. We apply the qudit-basis to Grover's
Algorithm and compare the circuit depth vs. system size to a qubit-based
circuit. The results show that there is a dramatic reduction in circuit depth
as the size of the system increases for the qudit circuit compared to qubit
circuit. In particular, multiply controlled gates are the driving factor in the
reduction of circuit complexity for qudit-based system scales with D.
