Magic-State Injection

The $T$ gate is what makes Clifford+ $T$ universal — but $T$ is not transversal for the standard fault-tolerant codes, so we cannot apply it directly the way we apply Clifford gates. The standard fault-tolerant solution is magic-state injection: prepare a special resource state ahead of time, then consume it with only Clifford operations to apply a logical $T$ to live data. This routes the hard, non-transversal work entirely into off-line state preparation.

The magic state

The relevant resource for the $T$ gate is

|A\rangle = T|+\rangle = \frac{|0\rangle + e^{i\pi/4}\,|1\rangle}{\sqrt{2}} .

Equivalently $|A\rangle = T H |0\rangle$ . It is called a magic state because, although it is just a fixed single-qubit state, combining it with (free) Clifford gates lifts the Clifford group all the way to universality — Bravyi and Kitaev's foundational result.

A noisy supply of approximate $|A\rangle$ states can be purified by magic-state distillation: consuming many low-fidelity copies and Clifford operations to output fewer high-fidelity copies. Distillation is the dominant cost of fault-tolerant computing, which is exactly why we count $T$ gates so carefully.

Gate teleportation: injecting T

Given one copy of $|A\rangle$ , a logical $T$ on a data qubit $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ is applied by gate teleportation:

Prepare the ancilla in $|A\rangle$ .
Apply $\mathrm{CNOT}$ from the data qubit (control) to the ancilla (target).
Measure the ancilla in the computational basis.
If the outcome is $0$ , the data qubit holds $T|\psi\rangle$ — done. If the outcome is $1$ , it holds $S^\dagger T|\psi\rangle$ up to a global phase, so apply the Clifford correction $S$ to recover $T|\psi\rangle$ (since $S\,S^\dagger = I$ ).

Every step except the magic-state preparation uses only Clifford gates, a measurement, and a classically-controlled Clifford correction — all of which are fault-tolerant. The single non-transversal operation has been pushed into preparing $|A\rangle$ , which can be done off-line and purified by distillation.

Why measurement-and-correct is fault-tolerant

The byproduct operator after teleportation is always a Clifford (a Pauli times possibly $S$ ), and Cliffords can be tracked classically (Gottesman–Knill) or applied transversally. So even though the circuit contains a mid-circuit measurement and a conditional gate, no single fault spreads uncontrollably: the data qubit interacts with the ancilla through exactly one CNOT, satisfying the "one fault, one error per block" rule.

Try it

The simulator here does not perform the mid-circuit measurement and conditional correction, so we focus on the part that carries the "magic": preparing the resource state $|A\rangle = T H |0\rangle$ on a single qubit. The grader checks the state vector: amplitude $1/\sqrt{2}$ on $|0\rangle$ and $e^{i\pi/4}/\sqrt{2}$ on $|1\rangle$ .

Run your code to see the quantum state.