Fault-tolerant quantum computers build every gate from a "Clifford + T" set: Clifford gates (H, S, CNOT) can be corrected cheaply by an error-correcting code, but the T gate cannot — its noisy physical version must be purified before use. Magic state distillation does this by consuming many noisy copies of the T-state |A⟩ = (|0⟩ + eiπ/4|1⟩)/√2 to produce fewer, much cleaner ones.
The classic Bravyi–Kitaev 15-to-1 protocol feeds 15 noisy magic states, each with error probability εin, through a [[15,1,3]] Reed–Muller code circuit. If the syndrome measurement succeeds, the single surviving output state has a leading-order error rate:
ε_out ≈ 35 · ε_in³ (one round)
after k chained rounds: ε_k ≈ 35^((3^k − 1)/2) · ε_in^(3^k)
Note on the chained-round formula: expanding the one-round recursion εk = 35·εk−1³ gives exponent ek on 35 that satisfies ek = 3·ek−1 + 1, e₀ = 0 — solved by ek = (3k − 1)/2, i.e. 35 raised to a positive power. This view's round-by-round simulation (and the readouts above) always uses the exact one-round recursion, so it is unaffected either way; the closed form above is written with the corrected sign.
- εin slider — the per-state error rate coming off the physical device (typical superconducting/trapped-ion hardware sits around 0.1%–3%; the slider spans a wider range to make the cubic suppression visually obvious).
- Target rounds — chaining distillation: the purified output of round 1 becomes one of the 15 inputs to round 2, and so on. Each extra round costs a factor of 15 more physical T-states but cubes the error suppression again.
- Run Distillation Round — animates 15 noisy input states (colored by fidelity) converging into one purified output state at the centre of the ring, then updates ε and the qubit-cost readouts using the recursion above. The log-scale trace in the corner plots every ε value reached so far.
- Drag the diagram to pan, scroll/pinch to zoom.
This ε³ scaling is why magic state distillation is both essential and expensive: reaching the ε ≈ 10⁻¹⁰–10⁻¹⁵ logical error rates a large algorithm needs from εin ≈ 1% typically takes 2–3 rounds, consuming 15²–15³ physical T-states per usable logical T gate — a major share of the total qubit budget in every fault-tolerant architecture proposal (surface code included).