Entropic Rigidity in Quantum Memories: How Geometry and Algebra Control the Onset of Degeneracy Corrections

Abstract: Maximum-probability (MP) decoding selects the most probable microscopic error, whereas degenerate maximum-likelihood (MLD) decoding includes the configurational entropy of an entire logical sector. Using code-capacity Pauli noise to isolate rigidity intrinsic to the code, we determine the first physical error weight $m$ at which their logical winner sets become disjoint, even under optimal MP tie resolution. Code distance imposes the universal bound $m\geq h=\lceil d/2\rceil$. We define the entropic rigidity depth $r$ through $m=h+r$ and certify a three-level hierarchy: $r=0$ for planar surface codes and two concatenated families, $r=1$ for odd-distance square toric codes and the Gross $[[144,12,12]]$ quantum low-density-parity-check code, and $r=2$ for a separable family with hypergraph product and bivariate bicycle descriptions. The onset fixes the leading operational failure gap, proportional to the $m$th power of the physical noise strength. Geometry and algebra therefore provide quantifiable controls of configurational entropy and an exact benchmark for low-noise decoder selection.

• The paper introduces the strict disagreement valuation m=h+r, proving that code distance sets the earliest possible MP–MLD split while entropic rigidity depth r measures delays from geometry and algebra.
• Exact spectral expansions and machine-verifiable certificates show immediate disagreement (r=0) in concatenated and planar families, but delayed onsets in toric and qLDPC codes, including r=1 or 2.
• The results connect decoding performance to degeneracy: the MP failure gap begins at K_m t^m, providing deterministic witness tests for whether a decoder captures leading-order logical-sector multiplicity.

The paper develops a framework for determining, exactly and without sampling, the first physical error weight at which maximum-probability (MP) decoding and degenerate maximum-likelihood decoding (MLD) produce disjoint logical winner sets for a stabilizer quantum memory under code-capacity Pauli noise. MP selects the most probable single microscopic error consistent with a syndrome; degenerate MLD sums probabilities over an entire logical sector and compares partition functions. The central object is the strict disagreement valuation $m$: the smallest minimum-error weight $w_s$ of a syndrome whose MP and MLD winner sets are disjoint for all sufficiently small noise strength $t$, under any tie-breaking convention. The main structural result is that $m = h + r$, where $h = \lceil d/2 \rceil$ is a universal lower bound imposed by code distance, and the code-dependent integer $r$—the entropic rigidity depth—counts how many consecutive excitation layers beginning at $h$ remain entropy neutral before configurational multiplicity can overturn the ground-state decision.

Universal obstruction and the spectral criterion

The half-distance bound follows from an elementary argument: at the onset weight $m$, both the MP-winning and MLD-winning sectors contain weight-$m$ representatives with the same syndrome but different logical labels, so their product is a nontrivial logical Pauli of weight at most $2m$. Hence $w_s$0, i.e., $w_s$1. This "energy locking" holds for every interior Pauli-noise direction, including arbitrarily anisotropic ones.

To decide whether the split occurs exactly at $w_s$2 or is delayed, the paper expands each logical sector's probability mass along the ray $w_s$3 into composition-weighted multiplicities $w_s$4, forming a finite lexicographic spectrum. For small $w_s$5, MLD selects lexicographic maximizers while MP maximizes the tropical peak monomial $w_s$6. A strict eventual split occurs precisely when these winner sets are disjoint—a criterion that retains higher-order coefficients needed to resolve ties invisible at minimum weight alone.

A layer $w_s$7 is entropy neutral if exact certificates (composition-preserving pairings, classwise-singleton purity, invariant renormalization group returns, or algebraic packing arguments) show intersecting winner sets for all syndromes at that weight. The rigidity depth $w_s$8 counts consecutive neutral layers from $w_s$9 up to the first strict witness at $t$0. All neutrality claims are established by exact certificates rather than sampling.

Recursive mechanisms: concatenated families

Two inequivalent renormalization group (RG) mechanisms certify $t$1—immediate saturation of the distance bound—for self-concatenated codes. For the $t$2 family, contraction induces a four-phase parity cycle on minimum sector weights that closes under the exact RG map; twelve finite recipes cover all depths within a certified anisotropic noise cell. For the cyclic perfect $t$3 code, a ten-state alphabet (identity, three overlap states, six directed resolutions) closes under a five-child substitution, propagating a strict half-distance split witness to every depth $t$4. Both results hold throughout explicit open semialgebraic regions of the Pauli simplex with order-one normalized witness margins—for instance, the five-qubit region contains a rational rectangle with worst-case margins as small as $t$5 yet strictly positive. The certificates use rational Farkas decompositions over invariant cones, so no floating-point comparison enters any proof.

Boundary topology as an entropic valve

For rotated planar surface codes of odd distance $t$6, open boundaries permit corridor proliferation at the first allowed layer: an analytic reflection-principle enumeration shows the peak-favoring and mass-favoring classes differ whenever $t$7, $t$8, and $t$9, giving $m = h + r$0 and thus $m = h + r$1.

Periodic closure shuts this valve. On the square torus, two same-syndrome weight-$m = h + r$2 errors in competing classes must be complementary segments of one noncontractible cycle, forcing each logical class to have exactly one minimum representative—entropy neutral. A transverse excursion requires each competitor to gain one error, producing the first imbalance at $m = h + r$3: two representatives in class $m = h + r$4 against one in class $m = h + r$5, with spectra $m = h + r$6 and $m = h + r$7. Hence $m = h + r$8, so odd-distance toric codes exhibit $m = h + r$9. The contrast between boundary conditions is sharp:

Family	Distance	$h = \lceil d/2 \rceil$0	Exact onset
$h = \lceil d/2 \rceil$1	$h = \lceil d/2 \rceil$2	0	$h = \lceil d/2 \rceil$3
$h = \lceil d/2 \rceil$4	$h = \lceil d/2 \rceil$5	0	$h = \lceil d/2 \rceil$6
Rotated planar (odd $h = \lceil d/2 \rceil$7)	$h = \lceil d/2 \rceil$8	0	$h = \lceil d/2 \rceil$9
Square toric (odd $r$0)	$r$1	1	$r$2
Gross $r$3	12	1	7
HGP/BB $r$4	$r$5	2	$r$6

Algebraic rigidity in qLDPC memories

Sparse parity-check algebra imposes the strongest protection. For the separable hypergraph-product/bivariate-bicycle family $r$7 (encoding efficiency $r$8), a Pauli systolic gap of four excludes low-weight nontrivial centralizers beyond the pure minimum lines, forcing all competitors at weights $r$9 and $h$0 into matched compositions. At $h$1, a wraparound cap saturates a packing of disjoint parity detectors and produces the first imbalanced fiber: two representatives in class $h$2 versus singletons in classes $h$3 and $h$4, giving $h$5 for every $h$6 on the cone $h$7. This is a theorem for all sizes, not an extrapolation from finite instances.

For the Gross $h$8 bivariate bicycle code, an exhaustive SAT/UNSAT certificate over 24 translation/Pauli sectors proves that every minimum (weight-12) logical is Pauli-pure. Purity forces complementary half-distance halves to share composition, making all weight-6 fibers entropy neutral ($h$9). Combined with a proof that no centralizer exists at weight 13, and an exhaustive fixed-syndrome enumeration yielding exactly three weight-7 representatives—one in class $m$0 (composition $m$1) and two in class $m$2 ($m$3)—this establishes $m$4 exactly, throughout $m$5. The authors note explicitly that this is an instance-specific result, not a statement about BB codes in general.

Operational failure gap

The onset translates directly into decoder performance. Giving MP its optimal tie rule, the total failure-probability gap satisfies

$m$6

with every per-syndrome contribution nonnegative, so the leading coefficient cannot cancel even under best-tie MP. This quantifies the leading penalty of ground-state decoding relative to coset summation. The framework applies only to decoders whose decisions remain within the exact MP-optimal set: generic MWPM and BP-OSD incur additional algorithm-dependent gaps not bounded here. The certified witnesses also serve as deterministic tests of degeneracy awareness: any decoder claiming to capture leading-sector multiplicity must reproduce the MLD winner at each witness syndrome.

Finite-$m$7 calibration via exact coset sums (independent Stim-based orbit sums at $m$8, cross-checked against factor-graph contractions) confirms the predicted planar $m$9 and toric $m$0 onsets. The raw ten-percent validity window for $m$1 contracts roughly with code area (planar $m$2: $m$3), but retaining the identity factor $m$4 removes the dominant volume dependence and enlarges windows substantially (planar $m$5: $m$6), with residual variation governed by the discrete local spectrum rather than monotonicity in $m$7.

Limitations and open questions

The paper is careful about scope. Code-capacity Pauli noise isolates intrinsic rigidity; circuit-level extensions require reformulating the atlas on detector fault hypergraphs, where additivity of fault weights conditionally yields $m$8 with detector distance $m$9 generally distinct from code distance. The resulting circuit-level formula is presented as conditional structure, not a proved theorem—temporal boundaries, hook propagation, and correlated hyperedge probabilities can alter both quantities, and their scheduling dependence is deferred to separate work. The Gross result is a single-instance certificate; no all-size onset theorem exists for general constant-rate qLDPC families, where the authors conjecture that $2m$0 is jointly governed by Tanner graph expansion, Pauli-composition spectra, and low-excess centralizer growth. The operational gap coefficient bounds only the entropic contribution for decoders confined to the MP-optimal set, and the finite-noise validity windows are witness-specific diagnostics, not thresholds.

Conclusion

This work converts the abstract distinction between microscopic energy and configurational entropy into a computable integer invariant. Distance fixes when entropy may matter; geometry (open corridors versus periodic homology) and algebra (Pauli purity, systolic gaps, detector packings) determine how many additional layers it must wait, yielding the certified hierarchy $2m$1 across six code families with exact onsets $2m$2. Because each claim rests on machine-verifiable exact certificates—symbolic identities, rational Farkas multipliers, exhaustive SAT instances—the classification provides a rigorous, hardware-independent baseline for evaluating when degeneracy-aware decoding becomes necessary and for auditing specific decoders against known entropic failure modes.