EchoRoute

Our team account is **Kevin Zhu** on BlueQubit Challenge Leaderboards

Inspiration

The challenge gave us hidden OTOC(2) echo circuits, $U, X_b, U^\dagger, Z_q, U, X_b, U^\dagger$, that we could only run by ID on BlueQubit's simulators: exact statevector (up to 34 qubits), MPS (bond dimension $\chi$) and Pauli-path propagation (PPS, threshold $\varepsilon$). Raising precision and hoping for the best works for some circuits and fails badly for others. We wanted to know, for each circuit, which simulator to trust and where extra compute actually helps.

What it does

EchoRoute is a decision framework for estimating OTOCs on a hosted simulator:

Route by structure before tuning precision. $n \le 34$: exact statevector. 1-D chain: MPS. Mostly Clifford: PPS. 2-D Haar grid at scrambling depth: the scrambling branch.
Test the representation before the precision. The transpilation level decides whether PPS sees anything at all. Levels 2/3 erased the signal on S3 at $\varepsilon = 0.01$, and on G3 level 2 returned zeros while level 1 kept it.
Refine only rows that move between precision levels (G2: $\chi$ 512 on 23 rows, 2048 on one row).
Scrambling branch (G4): PPS at transpilation level 3 is the baseline. High-$\chi$ MPS refines rows near the butterfly. The low-$\chi$ MPS directions shown to carry no signal are subtracted, and the refined vectors are mixed.
Keep only rules that hold on a known answer. We validated each one on exact proxies built like the hidden circuits (up to 32 qubits).
Audit every cloud job: fresh quote → budget gate → receipt → ledger.

How we built it

Python with Qiskit, Qiskit Aer, quimb and the BlueQubit SDK. We wrote local exact, MPS and coefficient-cutoff Pauli-propagation engines for calibration, plus an audited cloud workflow (quotes, budget and GPU-quota gates, receipts, coherent single-configuration cohorts). We also built about 30 exact proxies of the G4 circuit family (16–32 qubits) at $0. All-Z observable lists return $\langle Z_j\rangle$ for all 38 qubits fromone row.

Results

Problem	Route	Score
S1–S3	exact / exact / local PPS	100 each
G1 (30-qubit chain)	exact statev
G2 (36-qubit chain)	MPS, $\chi$ refined on moving rows	581/600
G3 (36-qubit Clifford-heavy grid) n = 3\times10^{-4}\)	773/800
G4 (38-qubit Haar grid, depth 16)	PPS baseline → MPS refinement → ensemble	284/1000

On G4, MPS alone gave nothing usable. PPS at $\varepsilon = 10^{-4}$, level 3 was the baseline (133). Adding MPS $\chi=256$ valy brought it to 165, and removing the uninformative MPS directions gave 199. On a 32-qubit circuit with a known exact answer, the same subtraction rule raised agreement wito 424 (out of 1000). The final ensemble scored 284; one of its ingredients was fitted to results of earlier uploads rather than simulated, as its method note states

Challenges

Agreement is not accuracy. On G4, MPS at $\chi = 32$ and $\chi = 64$ agreed closely with each other and were both wrong. A tro well and drifts toward +1.
PPS accuracy is not monotonic in $\varepsilon$. On a sample row, $10^{-4}$ was worse than $10^{-3}$.
Extrapolation and fusion failed held-out tests, so we switched them off.
Budget: about $7.80 of GPU time billed cost, so we rebuilt the GPU gate around observed runtimes.

What we learned

In the scrambling regime most rows se floor. The signal lives on the rows the butterfly reaches last, and MPS gives you geometry rather than values there. Cheap exact proxies of the same circuit family are the bg what to trust.

What's next

PPS at $\varepsilon = 3\times10^{-5}$ on G4's 22 late-contact rows. On a 24-qubit testbed, one $\varepsilon$ decade on those roexact answer from 213 to 327 (out of 1000).