Quantum Maxwell Erasure Decoder

• Quantum Maxwell Erasure Decoder is a tunable, symbolic decoding method for quantum LDPC codes that combines linear-time peeling with bounded guessing.
• It employs affine-form tracking and restrictive checks to handle stopping sets, enabling efficient near-ML decoding with controlled computational cost.
• Empirical results on BB and Tanner codes demonstrate robust asymptotic and practical error rate improvements suitable for real-time quantum erasure correction.

A Quantum Maxwell Erasure Decoder is a tunable, symbolic decoding algorithm designed for quantum low-density parity-check (qLDPC) codes to efficiently correct quantum erasures on the quantum erasure channel. It generalizes the classical Maxwell decoder to the quantum CSS code setting, interpolating smoothly between linear-time peeling and maximum-likelihood (ML) decoding via a guess-budget parameter. This approach enables practical, near-ML decoding performance at controlled computational cost, with strong asymptotic and empirical guarantees on quantum LDPC code families (Freire et al., 15 Jan 2026).

1. Decoding on the Quantum Erasure Channel: Classical and Quantum Context

In the quantum erasure channel, each qubit is erased (lost) independently with known locations. The physical effect is that an erased qubit is replaced by the maximally mixed state, which information-theoretically is equivalent to a random Pauli error at a known position. For CSS stabilizer codes defined by parity-check matrices $H_X, H_Z$, the decoder’s objective is, given the set of erased coordinates and measured syndromes, to reconstruct the error vector(s) up to stabilizer equivalence (Delfosse et al., 2017, Freire et al., 15 Jan 2026).

For surface codes, ML erasure decoding involves finding a matching error of minimal support consistent with the syndrome, which is facilitated by an efficient linear-time leaf-peeling algorithm exploiting the cycle structure of the code’s lattice (Delfosse et al., 2017). For general qLDPC codes, the Tanner graph may possess nontrivial stopping sets analogous to classical LDPCs, requiring more sophisticated strategies beyond peeling.

2. The Maxwell Decoder Framework and Stopping Sets

The classical peeling decoder solves erasure patterns by iteratively correcting variables adjacent to degree-one checks. Failure of peeling is characterized by the existence of a stopping set, i.e., an erasure pattern where each check neighbor is connected to at least two erased variables. The Maxwell decoder introduces a bounded-memory strategy: upon stalling, it makes binary guesses (pivots) on still-erased variables, continuing peeling, and reduces the guess count symbolically through dynamically imposed parity constraints (“restrictive checks”) (Freire et al., 15 Jan 2026).

In the CSS context, two instances of the decoding problem are solved independently (for $X$ and $Z$ errors) but on the same erasure set. Each is reduced to a system $H w = \sigma$ with variable domain restricted to the erased set. The decoder maintains symbolic affine forms to keep track of propagation and fix-up actions due to guesses.

3. Quantum Maxwell Erasure Decoder: Algorithm and Symbolic Structure

The Quantum Maxwell Erasure Decoder (QMED) operates as an “affine-form symbolic peeling decoder with bounded guessing.” Given parity-check matrix $H$, syndrome $\sigma$, erasure set $\mathcal{E}_0$, and guess budget $G_{\max}$, the MaxwellPeel algorithm proceeds in three nested stages (Freire et al., 15 Jan 2026):

1. Peeling loop: Processes all checks of degree one, fixes the associated variable, and propagates updates.
2. Restrictive-check loop: When a check becomes isolated but the syndrome is nonzero, imposes an affine constraint, solves for a pivot, substitutes across all forms, and removes one pivot (reimbursed guess).
3. Guessing: If peeling stalls and guess budget remains, selects a new variable (using a possible pivot-selection policy), assigns a new pivot, and continues.

Affine form tracking ensures that multiple guessed variables and subsequent constraints are managed efficiently, and guess “reimbursement” via restrictive checks reduces the effective branching factor. Termination with failure occurs if the guess budget is exceeded before completion.

Key equations:

• CSS syndrome equations: $\sigma_Z = H_Z e_X$, $\sigma_X = H_X e_Z$.
• Affine form: $w_j(\mathbf{x}) = a_0 + \sum_{p=1}^g a_p x_p$, with pivots $x_p$, tracked symbolically throughout.
• Restrictive checks: constraints of form $s_i(\mathbf{x}) = \sum_{p} b_p x_p + b_0 = 0$.

4. Complexity, Theoretical Guarantees, and ML Limit

QMED offers a performance-complexity continuum governed by $G_{\max}$:

• At $G_{\max}=0$, it reduces to pure peeling (strictly linear time but suboptimal for many qLDPCs).
• As $G_{\max}$ increases, the solution space coverage approaches that of ML decoding; for sufficiently large $G_{\max}$ (≤ number of stopping sets not containing logical operators), the decoder achieves ML performance (cubic time for general Gaussian elimination).

The runtime is $O(e\, d_v\, d_c\, G_{\max}^2)$ for a $d_v, d_c$-LDPC code with $e$ erasures. For constant $G_{\max}$ and bounded degrees, QMED is linear in the number of erasures (Freire et al., 15 Jan 2026).

The underlying theory establishes that if $s$ is the (minimal) stopping distance and $d$ is the minimum distance of the CSS code, then for guess budget $G_{\max}\ge d-s+1$, the QMED matches the ML failure exponent in the small erasure regime: $$p_L^{\rm QM}(G_{\max};\epsilon)\sim p_L^{\rm ML}(\epsilon)$$ as $\epsilon\to 0$. For practical regimes, even modest guess budgets can dramatically lower logical error rates by mitigating the impact of stopping sets (Freire et al., 15 Jan 2026).

5. Empirical Performance and Code Families

Empirical tests were conducted on bivariate-bicycle (BB) and quantum Tanner codes. For a $[360,12,\leq 24]$ BB code, MaxwellPeel with $G_{\max}=6$ approaches the ML performance benchmark; similar trends hold for quantum Tanner codes, though a residual performance gap remains for moderate $G_{\max}$.

Comparisons with cluster decoders (using cutoffs $C=20,100$) demonstrate that the MaxwellPeel approach is competitive in the waterfall regime at equivalent complexity. The decoder’s tunable nature enables system designers to match target performance and computational resources, and its symbolic structure is well-suited to implementations with lightweight pruning and redundant-check augmentation (Freire et al., 15 Jan 2026).

6. Practical Strategies for Parameter Selection

Guidelines for $G_{\max}$ selection include:

• Small constant values ($G_{\max}=1, 2, 3$) yield linear-time decoding with significant practical gains over peeling.
• To recover the ML exponent, the theoretical bound of $G_{\max}\ge d-s+1$ applies, but empirically, the needed guess budget may be lower due to guess reimbursement and favorable pivot-selection heuristics.
• Techniques such as depth-1 gauge-fixing (of fully erased stabilizers) and augmentation with redundant low-weight checks can further diminish the necessary guess budget (Freire et al., 15 Jan 2026).

Integration of QMED is particularly advantageous on hardware platforms dominated by detected erasures (e.g., photonic or hybrid matter–optics architectures), where low latency and real-time classical post-processing are required.

7. Relation to Surface-Code Erasure Decoding and Other Paradigms

The surface-code ML erasure decoder based on a linear-time spanning-forest peeling algorithm is a special case that exploits the absence of harmful stopping sets and the strict cycle structure of the surface lattice (Delfosse et al., 2017). For generic qLDPC codes, stopping sets are prevalent and prohibit pure peeling except for limited erasure patterns. QMED generalizes the approach, retaining fast operation on “peelable” patterns while extending success to broader regimes via bounded symbolic guessing.

In the continuous-variable (CV) optic setting, erasure correction is implemented via linear optical networks and syndrome extraction followed by either deterministic feedforward or probabilistic filtering; these architectures, while effective against photon loss, are structurally distinct from the LDPC-based QMED but address analogous decoding requirements under quantum erasure noise (Lassen et al., 2010).

A plausible implication is that similar symbolic guessing and pivot-elimination frameworks could be adapted to hybrid quantum codes or concatenated CV–qubit codes where stopping-like structures appear.

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References:

• (Delfosse et al., 2017) Delfosse and Zémor, "Linear-Time Maximum Likelihood Decoding of Surface Codes over the Quantum Erasure Channel"
• (Freire et al., 15 Jan 2026) "Quantum Maxwell Erasure Decoder for qLDPC codes"
• (Lassen et al., 2010) "Quantum optical coherence can survive photon losses: a continuous-variable quantum erasure correcting code"