We establish a precise mathematical correspondence between the classification of quantum CSS stabilizer codes and the Kodaira-Neron classification of elliptic curves over p-adic fields. The stabilizer lattice of a CSS code defines a filtered (phi,N)-module over Q_p whose reduction type determines the code's error-correcting properties. We verify this classification across 7 code families (100% agreement on KN Propositions 3.1-3.7), present the first explicit construction of a semistable non-crystalline code from the Shor 9-qubit stabilizer monodromy, and catalogue 26 conjectures spanning code distance estimates via Hasse-Weil zeta functions, ultrametric tree structures in error syndromes, and Yamada (1969) weight calibration.
We present an algorithm that takes a CSS stabilizer code as input, and outputs another CSS stabilizer code such that the stabilizer generators all have weights O(1) and such that O(1) generators act on any given qubit. The number of logical qubits is unchanged by the procedure, while we give bounds on the increase in number of physical qubits and in the effect on distance and other code parameters, such as soundness (as a locally testable code) and “cosoundness” (defined later). Applications are discussed, including to codes from high-dimensional manifolds which have logarithmic weight stabilizers. Assuming a conjecture in geometry[11], this allows the construction of CSS stabilizer codes with generator weight O(1) and almost linear distance. Another application of the construction is to increasing the distance to X or Z errors, whichever is smaller, so that the two distances are equal.
The realization of quantum error correction protocols whose logical error rates are suppressed far below physical error rates relies on an intricate combination: the error-correcting code's efficiency, the syndrome extraction circuit's fault tolerance and overhead, the decoder's quality, and the device's constraints, such as physical qubit count and connectivity. This work makes two contributions towards error-corrected quantum devices. First, we introduce mirror codes, a simple yet flexible construction of LDPC stabilizer codes parameterized by a group
G and two subsets of
G whose total size bounds the check weight. These codes contain all abelian two-block group algebra codes, such as bivariate bicycle (BB) codes. At the same time, they are manifestly not CSS in general, thus deviating substantially from most prior constructions. Fixing a check weight of 6, we find
[[60,4,10]],[[36,6,6]],[[48,8,6]], and
[[85,8,9]] codes, all of which are not CSS; we also find several weight-7 codes with
kd>n.
Next, we construct syndrome extraction circuits that trade overhead for provable fault tolerance. These circuits use 1-2, 3, and 6 ancillae per check, and respectively are partially fault-tolerant (FT), provably FT on weight-6 CSS codes, and provably FT on \emph{all} weight-6 stabilizer codes. Using our constructions, we perform end-to-end quantum memory experiments on several representative mirror codes under circuit-level noise. We achieve an error pseudothreshold on the order of
0.2%, approximately matching that of the
[[144,12,12]] BB code under the same model. These findings position mirror codes as a versatile candidate for fault-tolerant quantum memory, especially on smaller-scale devices in the near term.
Quantum error correction codes are needed to protect quantum systems from interference. An important family of quantum error correction codes are stabilizer codes. Many stabilizer codes in the literature belong to a subclass of stabilizer codes, referred to as CSS codes since they are based on the so-called CSS formalism. CSS codes have received significant attention because the dual-containing condition for the CSS formalism is relatively straightforward to satisfy and also because it is easy to produce CSS codes from classical error correction codes. However, some stabilizer codes in the literature show superior error correction performance than existing CSS codes, and are based on binary low-density parity-check (LDPC) codes. Since it has been shown that CSS codes based on nonbinary LDPC codes have good performance over quantum qubit channels, in this paper we investigate stabilizer codes based on nonbinary LDPC codes. In particular, we propose constructions of stabilizer codes based on nonbinary quasi-cyclic low-density parity-check (QC-LDPC) codes for qubit channels. The simulation results show that our QC-LDPC stabilizer codes decoded by a nonbinary sum-product algorithm have better performance than their binary counterparts.