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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. Wedemonstrate that the use of certain non-fault tolerant methods in theimplementation may nevertheless enable reliable quantum computation whilereducing basic resource consumption. Reliability is determined by calculatinggate fidelities for the one-qubit logical gate. Specifically, we show thatdespite using a non-fault tolerant procedures in constructing a logical zeroancilla to implement the T-gate the gate fidelity of the logical gate, afterperfect error correction, has no first order error terms. Meaning, any errorsthat may have occurred during implementation are `correctable' and faulttolerance may still be achieved.

Non-Fault Tolerant T-Gates for the [7,1,3] Quantum Error Correction Code