Optimal Decoder for the Error Correcting Parity Code

TL;DR

The paper proposes a two-step decoder for the parity code that leverages one-dimensional materialized symmetries to decompose decoding into a matching step followed by post-processing. It shows that optimal decoding is achievable asymptotically by reducing to independent repetition codes and demonstrates a code-capacity threshold of $50\%$ in the large-d limit, with high fault-tolerant thresholds under noisy measurements. The decoder remains scalable: the first step can be performed via MWPM or ISM with parallelism, while the second step enforces a 1-line constraint, effectively turning the problem into a set of repetition-code decodings. The approach yields practical runtimes, robust performance in both ideal and faulty-measurement regimes, and favorable scalability for planar implementations, making the parity code a promising candidate for demonstrating quantum advantage on biased-noise platforms.

Abstract

We present a two-step decoder for the parity code and evaluate its performance in code-capacity and faulty-measurement settings. For noiseless measurements, we find that the decoding problem can be reduced to a series of repetition codes while yielding near-optimal decoding for intermediate code sizes and achieving optimality in the limit of large codes. In the regime of unreliable measurements, the decoder demonstrates fault-tolerant thresholds above 5% at the cost of decoding a series of independent repetition codes in (1 + 1) dimensions. Such high thresholds, in conjunction with a practical decoder, efficient long-range logical gates, and suitability for planar implementation, position the parity architecture as a promising candidate for demonstrating quantum advantage on qubit platforms with strong noise bias.

Figures (16)

• Figure 1: (a) Distance-5 LHZ code. Qubits (circles) lie on the vertices of the square lattice. Plaquettes correspond to stabilizers measuring the parity of the surrounding qubits in the $Z$ basis. The code contains five logical qubits. Logical Pauli-$Z$ operators correspond to single-qubit Pauli-$Z$ operators applied to base qubits labelled 1 to 5. Base qubits are arranged along the dashed line, which we refer to as qubit line 0. Logical Pauli-$X$ operators are formed by products of $X$ operators applied to ${d=5}$ qubits along solid lines of the same color, which we refer to as qubit lines 1 to $d$. A syndrome (red plaquettes) triggered by qubit errors (red circles) forms a closed contour (solid black line) that encloses the flipped qubits. (b)--(f) Symmetries of the distance-5 LHZ code formed by stabilizers shown in purple and red. Each qubit is in the support of either two stabilizers of a given symmetry or none. Stabilizers triggered along each symmetry by the error of panel (a) are shown in red and connected by the shortest-path matching (black lines). Virtual stabilizers shown with pale purple plaquettes are appended to ends of each symmetry line in (b)--(f) and form a virtual symmetry (g). Combined together, pairings of panels (b)--(g) form a cluster of panel (a) enclosing the error. In panels (b)--(g), qubit lines are omitted to avoid cluttering.
• Figure 2: Two distance-3 codes that (a) share one common qubit and (b) have no common qubits.
• Figure 3: Ambiguous matching clusters. Clusters in both panels correspond to the same measured syndrome (bright red plaquettes) but different virtual stabilizers (pale red plaquettes). Since the two clusters have the same perimeter of length 10, the direct implementation of the MWPM decoder does not distinguish between such degenerate solutions. However, the correction of panel (a) has weight 3, while the solution of panel (b) has weight 4. Executing the post-processing step described in Ref. \ref{['sec:minimization-step']} allows to remove the ambiguity and choose the correct, i.e., the lower-weight recovery operation.
• Figure 4: Logical failure rate of the code, i.e., the probability that at least one of $d$ logical qubits is flipped, versus physical error rate of data qubits. The two-step decoder is used to resolve the syndrome. Three families of curves correspond to parity codes with distances $d=11$ (red), $d=31$ (green), and $d=51$ (blue). Circles, diamonds, and squares correspond to a two-step decoder that uses, respectively, ISM, random matching, and MWPM during the first step of decoding. In all cases, post-processing is applied. Triangles correspond to the probability of a logical error along at least one 1-line $L_1^i$, that is, to a lower-bound failure rate of the optimal decoder.
• Figure 5: Code failure rate versus physical error rate for different distances $d$ (and, hence, logical system sizes $d$). In both panels we used MWPM decoder during the matching step. Panels (a) and (b) correspond to decoder operation with and without post-processing, respectively. Dashed lines (black dots) show the intersection of curves corresponding to $d$ and $d+6$, which shifts towards larger physical error probability as $d$ increases.

Theorems & Definitions (6)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof