A *fault-tolerant* computer encodes each logical qubit in many physical ones and corrects errors faster than they accumulate. On the codes in use, the Clifford gates (`h`, `s`, `cx` and their products) are cheap to apply to encoded qubits. Clifford gates alone are not enough: a Clifford circuit is easy to simulate classically (the engine's stabilizer tableau does it), so it cannot do anything a classical computer cannot.

Adding the T gate, a rotation by π/4 about Z, makes the set universal. But on a surface code a T gate cannot be applied directly. It consumes a *magic state*, made by *distillation* in a factory that takes many physical qubits and much time. So the cost of a fault-tolerant program is, to a first approximation, its **T count**.

The Toffoli gate is the workhorse of arithmetic. Written in Clifford+T it needs seven T gates; the critique counts them (`wai-quantum critique`), and the estimate in the last lesson prices them.