Inspiration

Shuffle a deck thoroughly, flip one card, then un-shuffle exactly. Without the flip you get the original deck back. With it, the un-shuffle does not come out clean, and the damage spreads from that card. That is the quantum butterfly effect, and the OTOC measures how disturbed each card ended up.

This is the quantity behind Google's Quantum Echoes experiment and their quantum-advantage claim. We are on the classical side of it.

What it does

Estimate out-of-time-order correlators (OTOCs) classically, for circuits that are too big to simulate exactly. S1, S2, and S3 are open; G1, G2, G3, G4 are completely hidden.

How we built it

The scoring is the thing to understand. Squared correlation means every value can be ten times too small and still score perfectly. That sounds forgiving, and it is, but it has a sharp edge. A flat vector scores zero, and a negative correlation scores zero.

In the Heisenberg picture a Clifford gate maps one Pauli operator to exactly one other. No branching, no truncation loss. Only the non-Clifford rotations split a term in two. So the cost is exponential in the non-Clifford count, not in the total gate count.

Challenges we ran into

We chased accuracy when the score didn't care about it- since the challenge only rewards getting the overall pattern right, we spent effort on aspects that weren't worth points.

The platform glitched on us- 38 jobs in a row came back empty.

Accomplishments that we're proud of

G1 at 30 qubits fits the free exact simulator, so a 300-point graded problem was a free perfect score. That is easy to miss if you assume every graded circuit must be hard.

A chain is the one geometry MPS handles well, and G2 is 84 percent non-Clifford so Pauli path was ruled out before we spent anything.

What we learned

The ordering proof matters more than it sounds. Pairing the right value with the wrong ID silently zeroes a problem, and it is the single highest-risk bug in the challenge. We verified it once in the first half hour and never touched the convention again.

We can tell how good our hidden answers are without knowing the answers. Run a circuit twice at two quality settings and compare the two. How much they disagree tells us roughly how far we are from the truth, which gives us a live score estimate for a problem nobody can check. That's the number we optimize and report.