  We simulate the implementation of a T-gate, or $\frac{\pi}{8}$-gate, for a
[7,1,3] encoded logical qubit in a non-equiprobable error environment. We
demonstrate that the use of certain non-fault tolerant methods in the
implementation may nevertheless enable reliable quantum computation while
reducing basic resource consumption. Reliability is determined by calculating
gate fidelities for the one-qubit logical gate. Specifically, we show that
despite using a non-fault tolerant procedures in constructing a logical zero
ancilla to implement the T-gate the gate fidelity of the logical gate, after
perfect error correction, has no first order error terms. Meaning, any errors
that may have occurred during implementation are `correctable' and fault
tolerance may still be achieved.
