Game-Theoretic Discovery of Quantum Error-Correcting Codes Through Nash Equilibria

TL;DR

A game-theoretic framework recasting code optimization as strategic interactions between competing objectives, where Nash equilibria systematically generate codes with desired properties is introduced, providing systematic, interpretable frameworks for quantum system design.

Abstract

Quantum error correction code discovery has relied on algebraic constructions with predetermined structure or computational search lacking mechanistic interpretability. We introduce a game-theoretic framework recasting code optimization as strategic interactions between competing objectives, where Nash equilibria systematically generate codes with desired properties. We validate the framework by demonstrating it rediscovers the optimal $[\![15,7,3]\!]$ quantum Hamming code (Calderbank-Shor-Steane 1996) from competing objectives without predetermined algebraic structure, with equilibrium analysis providing transparent mechanistic insights into why this topology emerges. Applied across six objectives -- distance maximization, hardware adaptation, rate-distance optimization, cluster-state generation, surface-like topologies, and connectivity enhancement -- the framework generates distinct code families through objective reconfiguration rather than algorithm redesign. Scalability to hardware-relevant sizes is demonstrated at $n=100$ qubits, discovering codes including $[\![100,50,4]\!]$ with distance-4 protection and 50\% encoding rate, with tractable $O(n^3)$ per-iteration complexity enabling discovery in under one hour. This work opens research avenues at the intersection of game theory and quantum information, providing systematic, interpretable frameworks for quantum system design.

Figures (3)

• Figure 1: Framework Generativity and Validation Through Optimal Code Rediscovery. Panels A--F show code parameter distributions (distance $d$ vs. rate $k/n$) discovered for: (A) distance optimization achieving $d=6$ at $k/n=0.20$, (B) hardware adaptation systematically rediscovering the optimal $[\![15,7,3]\!]$ quantum Hamming code (Calderbank-Shor-Steane 1996) with $d=3$ at $k/n=0.47$, (C) rate-distance tradeoff exploring Pareto frontier, (D) cluster-state search generating regular bipartite graphs, (E) surface-like topologies with $\delta_{\text{avg}} \approx 4$, (F) connectivity optimization maximizing $\kappa(G)$. Summary table lists best codes per objective. Bottom: Evolution of code distance over iterations demonstrates convergence to distinct equilibria from identical initialization, with each objective trajectory (colored lines with shaded confidence bands from 20 independent trials) separating by iteration 10, validating that objective reconfiguration alone generates diverse code families.
• Figure 2: Mechanistic Insight Through Equilibrium Analysis. (A) Strategic evolution timeline for distance optimization showing code distance $d$ (blue circles, left axis) and total reward (red squares, right axis) over 22 iterations. Three phases emerge: exploration (iterations 0--10, blue shading) with rapid distance growth, competition (iterations 10--18, gray shading) with reward oscillations as players balance objectives, and convergence (iterations 18--22, green shading) to Nash equilibrium at $[\![15,3,6]\!]$. (B) Final equilibrium graph topology visualization showing 15 vertices with high connectivity. Each vertex's stabilizer $K_v = X_v \bigotimes_{u \in N(v)} Z_u$ corresponds to its neighborhood (edges shown). Inset: transparent circuit construction $|G\rangle = \prod \text{CZ}$ requires 43 CZ gates, one per graph edge, enabling immediate experimental implementation.
• Figure 3: Practical Advantages at Hardware Scales. (A) Logical error rate $\varepsilon_L$ versus physical error rate $p$ for discovered codes (solid lines with circles/diamonds) and surface code baselines (dashed lines with squares) at distances $d=3,5$. Error bars show $\pm 1\sigma$ statistical uncertainty from Monte Carlo simulation with $10^5$ syndrome measurement cycles. Gray dashed line shows uncorrected error rate for reference. Discovered codes achieve comparable error suppression with threshold behavior $\varepsilon_L \propto p^{(d+1)/2}$. (B) Encoding rate comparison: the rediscovered $[\![15,7,3]\!]$ quantum Hamming code achieves $k/n=0.47$ (47% encoding rate), substantially higher than distance-3 surface codes at $k/n \approx 0.11$, providing resource efficiency in overhead-constrained regimes. (C) Discovery time scaling with system size $n$. Game-theoretic approach (blue solid, $O(n^3)$ per iteration) remains tractable through $n \approx 100$ (green-shaded region), crossing exhaustive search complexity (red dashed, $O(2^{n^2})$) at $n=7$ with over 6 orders of magnitude advantage at $n=20$.