Team Q-gar | iQuHack 2026

1. The Challenge

Objective: Maximize total utility qubits by expanding a connected quantum subgraph, starting from a single node.

Physical Constraint

Fidelity ≥ Threshold

Economic Constraint

Finite Bell Pair Budget

Each edge represents a noisy Bell-pair channel. To claim it, players must distill sufficient fidelity using LOCC operations, consuming limited entanglement resources.

Theory: LOCC Distillation

Distillation consumes multiple noisy pairs to produce one higher-fidelity Bell pair.

\rho^{\otimes n} \xrightarrow{\text{LOCC}} |\Phi^+\rangle

Mechanism: We use recurrence protocols: pairs are consumed via bilateral CNOTs and parity measurements.

Match: keep (distilled)
Mismatch: discard (error detected)

Approach: split the problem in two: (1) characterize edge noise to choose the best distillation circuit, (2) treat the game board as a weighted graph to decide where to expand for maximum ROI.

2. Physics Stream

Methodology: Fidelity Probing

We reconstructed edge noise by probing fidelity under local basis rotations, allowing identification of the underlying output state.

F(U) = \langle\Phi^+|U \rho_{\text{out}} U^\dagger|\Phi^+\rangle

Difficulty levels correspond to distinct quantum noise channels, not scalar noise.

Noise Characterization (D1–D5)

D1: Bit Flip dominant: X

$\rho \approx 0.85|\Phi^+\rangle + 0.15|\Psi^+\rangle$

D2: Phase Flip dominant: Z

$\rho \approx 0.80|\Phi^+\rangle + 0.20|\Phi^-\rangle$

D3: Mixed (X-biased) harder than D1

$\rho \approx 0.75|\Phi^+\rangle + 0.25|\Psi^+\rangle$

D4: Mixed (Z-biased) harder than D2

$\rho \approx 0.72|\Phi^+\rangle + 0.28|\Phi^-\rangle$

D5: Depolarizing X + Y + Z

$\rho \approx 0.7056|\Phi^+\rangle + 0.1344|\Phi^-\rangle + 0.1344|\Psi^+\rangle + 0.0256|\Psi^-\rangle$

Unified Circuit Strategy

A single optimized recurrence circuit handles all error types via basis rotation. For Z-dominant noise (D2/D4), we conjugate the protocol with Hadamards to convert phase flips into bit flips.

# Basis-Rotated Distillation
if noise == "PHASE_FLIP":
H ∘ protocol ∘ H # Z → X

Validation: Predicted success probability and output fidelity matched game outputs across D1–D5, letting the bot treat distillation outcomes as reliable edge weights.

3. Strategy & Results

The "Auto Bot" Target Score

Node selection balances short-term reward with long-term network leverage.

\text{Score} = \frac{\text{Pts}}{\text{Cost}+1} \times (1 + \text{Centrality})

Hybrid Logic: Quantum physics determines edge cost (expected Bell pair consumption), while classical graph algorithms optimize global expansion.

# Core loop (conceptual)
centrality = betweenness(G)
costs = dijkstra_costs(G, start, weight="difficulty")

score(node) = points(node)/(costs[node]+1) * (1 + centrality[node])
pick argmax score(node)

> PRIORITY: ROI × CENTRALITY
> AVOID: HIGH-COST D5 CUTS

Fig. 1 — Strategic visualization of a graph cluster.

Results so far: Integrating noise-adaptive circuits into the bot improved point acquisition and reduced wasted budget on poorly chosen paths. We also corrected a floating-point threshold edge case (e.g. 0.9 → 0.89999) using a small numerical epsilon.

Targets high-density utility clusters (centrality)
Navigates around “expensive” D5 edges
Uses N=3 recurrence for reliable D3/D4 distillation

Conclusion Progress in the challenge comes from hybrid design: physics ensures link quality (fidelity), while graph theory ensures network efficiency (topology). Our agent makes near-deterministic decisions by treating distillation statistics as calibrated edge weights.

MIT IQUHACK 2026 • TEAM Q-GAR

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