Heterogeneous quantum error-correcting codes

Abstract

We introduce heterogeneous quantum error-correcting codes composed of qubit types with distinct error channels and study their performance in the code-capacity regime using maximum-likelihood tensor network decoding. In the regime where both qubit types share the same noise bias but differ in physical error rate, placing noisier qubits in the bulk -- where each error triggers more syndrome bits -- and cleaner qubits on the boundary yields thresholds exceeding 0.4 (compared to ~0.2 for the reverse placement) and improvements exceeding three orders of magnitude in logical error rate at high bias, with the advantage growing exponentially with code distance. In the regime where both types share the same error rate but differ in bias, the optimal strategy reverses: placing high-bias (more predictable) qubits on the boundary increases the threshold from 0.292(5) to 0.360(9) at a bias ratio of 100, and from 0.29(1) to 0.398(4) at a bias ratio of 1000. We also observe a striking bias-inversion property: the logical error channel becomes strongly XX X- and YY Y-biased despite the physical noise being ZZ Z-biased. We propose a stabilizer-ratio hypothesis that provides a unified information-theoretic explanation for both placement rules and predicts even larger advantages for code families such as color codes.

Figures (5)

• Figure 1: Rotated XY surface codes with Bulk-Noisy placement (top row, a--c) and Boundary-Noisy placement (bottom row, d--f) for distances 5, 7, and 9. Green (red) circles denote quiet/high-bias (noisy/low-bias) qubits. The number of each qubit type is approximately equal.
• Figure 2: Logical failure rates for Regime A with $\eta = 10$ and $p_{\text{noisy}}/p_{\text{quiet}} = 10$, for (a) Bulk-Noisy (distances 5, 7, 9) and (b) Boundary-Noisy (distances 5, 7, 9, 11). The crossing of the distance curves indicates the threshold.
• Figure 3: Ratio of logical error rates $p_{L,1}/p_{L,0}$ between Boundary-Noisy and Bulk-Noisy as a function of code distance, for $p_{\text{noisy}} = 0.30$ and $p_{\text{noisy}}/p_{\text{quiet}} = 10$, at three bias values. At $\eta = 100$ the advantage exceeds $10^3$ by distance 9. The $\eta = 0.5$ control (depolarizing channel) confirms that the placement effect requires alignment between the noise bias and the Clifford deformation.
• Figure 4: Logical failure rates for Regime B with Bulk-Noisy placement, $\eta_{\text{low}} = 10$, $\eta_{\text{high}} = 100$, for distances 5, 7, and 9. The crossing near $p \approx 0.36$ confirms the threshold estimate in Table \ref{['table:thresholds_variable_eta']}.
• Figure 5: Disaggregated logical Pauli error rates for Boundary-Noisy, with $p_{\text{noisy}} = 0.2$, $p_{\text{noisy}}/p_{\text{quiet}} = 10$, $\eta = 100$, for distances 5--11. Despite the physical noise being strongly $Z$-biased, the logical channel is dominated by $X_L$ and $Y_L$ errors (which nearly overlap), with a logical bias $\eta_L \approx 4 \times 10^{-3}$. The $Z_L$ error rate is suppressed by over two orders of magnitude.