Quantum amplitude estimation challenge

Every algorithm estimates a = sin θ ∼ U(0.1, 0.9) from Grover-depth-n measurements with p = cos²((2n+1)θ). The score is a constant: the error percentile times the resources used. Lower is better, and the frontier is what counts, not a single number.

CδQ = εδ · ⟨total queries⟩ C̃δ = εδ² · ⟨sequential depth⟩ · ⟨total queries⟩ sequential depth = Grover depth run in series: nmax for one parallel round, the sum of per-round depths when adaptive εδ = δ-th percentile of |â − a|, δ ∈ {68, 95, 99} cost(n=0) = 1, cost(n≥1) = n nmax ∈ {…} max circuit depth ≤ nmax for every track
Trade-off frontier: sequential-depth constant vs total queries
Lower-left is better. C̃ charges wall-clock depth: adaptive rounds run one after another, a non-adaptive schedule runs once. It is invariant under shot count, CQ is not, so a method cannot buy a better position by taking more shots. Dashed: Pareto frontier over all entries; green: the Cramér–Rao floor of the flagship ladder, a reference for that schedule only.
Total queries · C68Q → C95Q → C99Q
Sorted by C95Q. ★ = Pareto-optimal across the three percentiles. Dashed: Cramér–Rao floor of the flagship ladder at each percentile.
Sequential depth · C̃68 → C̃95 → C̃99
Sorted by C̃95. Error per unit of sequential depth × queries. One parallel round keeps the depth at nmax; adaptive methods pay for every round in series. ★ = Pareto-optimal across the three percentiles.
C68 C95 C99 ◉ = challenge submission · Log axes · hover a row for values · click to pin
All entries
Click a header to sort, a row to expand. Q / D badges mark Pareto-optimal entries in the query and sequential-depth frontiers. † = literature method run at a matched target error, not depth-capped (hover for the share of trials above nmax).
# Algorithm Track Rounds Deepest rung Seq. depth Total queries CQ Sequential depth C̃ Pareto
C68C95C99 C̃68C̃95C̃99
Depth-limited regimeseparate metric, not ranked above
Depolarizing-noise regimeseparate metric, not ranked above

How to read the frontier

An entry is Pareto-optimal in a frontier when no other entry is at least as good on all three percentiles and strictly better on one. A method can win on tails (C99) while losing on the bulk (C68), so a single rank hides most of the story.

Why C̃ and not ε · depth: multiplying every shot count by k divides ε by √k and multiplies queries by k, so ε · depth can be driven down at will while ε · queries goes up. The product ε² · depth · queries does not move. The depth that enters is the sequential one, the Grover depth a single processor must run in series: a non-adaptive schedule runs all its circuits in one parallel round, so its sequential depth is its deepest rung, while an adaptive method waits for each round before choosing the next and pays the sum. (The paper's parallel constant ε95 nmax ≈ 0.21 is the same idea at one shot on the deepest rung.)

Reference floors

The dashed green lines are the Cramér–Rao floor of the flagship ladder itself, , averaged over the scales. They bound what any estimator can do on that schedule (the paper's exact ML sits within a few percent of them) and say nothing about other schedules: an adaptive method that spends fewer queries on disambiguation can legitimately go below them. No universal floor for the query constant is drawn.

Depth is capped at nmax for every track, so a method cannot lower its query constant by running deeper circuits than the schedules it is compared with. Sequential rounds are not capped; they show up in C̃.

Certification

Two kinds of rows. Literature baselines are reference methods from published papers, regenerated by leaderboard/populate_baselines.py on the public holdout seed. Challenge submissions arrive as pull requests under leaderboard/submissions/, pass the sandboxed verifier, and are certified by CI on a blind HMAC-derived seed, . Both kinds compete on the same frontier; the Source filter and the tag under each name tell them apart.