Decision-tree decoders for general quantum LDPC codes

TL;DR

This work proposes Decision Tree Decoders (DTDs) as a general framework for decoding quantum LDPC codes, relying only on the sparsity of the check matrix $H$ to enable broad applicability. It introduces two explicit decoders: a provable Height-bound DTD that guarantees a minimum-weight correction (often with fast median-case runtime for practical codes) and a heuristic Belief-Propagation–guided DTD (BP-DTD) that achieves higher accuracy and faster empirical performance under circuit noise. The study provides rigorous bounds, complexity considerations, and extensive numerical results for color codes and bivariate bicycle codes, demonstrating near-minimal search costs in median cases and practical performance advantages over existing BP-OSD approaches in realistic regimes. The work highlights potential uses in ensemble decoding, distance estimation, and hardware-friendly implementations, while outlining open questions about achieving provable, efficient decoding for all general qLDPC codes. Overall, the DT D framework advances provable and practical decoding for broad qLDPC code families and fault-tolerant quantum circuits.

Abstract

We introduce Decision Tree Decoders (DTDs), which rely only on the sparsity of the binary check matrix, making them broadly applicable for decoding any quantum low-density parity-check (qLDPC) code and fault-tolerant quantum circuits. DTDs construct corrections incrementally by adding faults one-by-one, forming a path through a Decision Tree (DT). Each DTD algorithm is defined by its strategy for exploring the tree, with well-designed algorithms typically needing to explore only a small portion before finding a correction. We propose two explicit DTD algorithms that can be applied to any qLDPC code: (1) A provable decoder: Guaranteed to find a minimum-weight correction. While it can be slow in the worst case, numerical results show surprisingly fast median-case runtime, exploring only $w$ DT nodes to find a correction for weight-$w$ errors in notable qLDPC codes, such as bivariate bicycle and color codes. This decoder may be useful for ensemble decoding and determining provable code distances, and can be adapted to compute all minimum-weight logical operators of a code. (2) A heuristic decoder: Achieves higher accuracy and faster performance than BP-OSD on the gross code with circuit noise in realistic parameter regimes.

Figures (14)

• Figure 1: (Left) An illustration of the first three levels of the decision tree (DT) for a distance-5 color code. Each DT node (blue triangle) represents a partial correction, with the root DT node (red outline) at the center. DT nodes are labeled with the code’s Tanner graph, where black circles are data vertices, white squares are check vertices, red circles indicate partial corrections, and yellow stars mark syndrome vertices. A syndrome vertex is selected from each DT node, and its neighboring fault vertices are added to form child nodes. DT nodes with trivial syndromes (green outline) represent valid corrections. (Right) Starting at the root node, DT decoders iteratively explore by assigning a cost to each child of the lowest-cost node in the tree. For example, using the syndrome height $h(\sigma)$, the minimum weight of any error with syndrome $\sigma$, as the cost function yields a minimum-weight correction in $w$ steps for weight-$w$ errors.
• Figure 2: Provable decoder. For several distance-$d$ color codes (CC) and bivariate bicycle codes (BB), for each $w \leq \frac{d-1}{2}$ we randomly sample all weight-$w$$X$-type faults. We report the median number of decision tree nodes explored by two decoding algorithms: height-bound DTD and naive breadth-first DTD. As expected, breadth-first DTD explores exponentially many nodes in $w$ (with dashed, fit parameter $b_\text{CC}=3.825$ and $b_\text{BB}=4.902$), while height-bound DTD, perhaps surprisingly, explores only $w$ nodes.
• Figure 3: Heuristic decoder. Cutoff-time performance curves for BP-OSD and BP-DTD for the gross code under circuit noise of strength $p = 10^{-3}$. Failure probability arises from two sources: exceeding the cutoff time $T$ or decoding within $T$ but producing an incorrect correction. At early and late times, both decoders perform similarly—either relying on a BP pre-decoder or having enough time to terminate, with BP-DTD showing slightly lower logical error rates in the late regime. In the intermediate regime, BP-DTD terminates more often than BP-OSD, leading to significantly lower logical error rates.
• Figure 4: (a) Three instances of the color code family with distances 3, 5 and 7 (larger distance codes are defined by extending the pattern). Qubits are at vertices, with an $X$- and a $Z$-type stabilizer generator supported on the qubits of each face. (b) The gross code with qubits at vertices on this tiling of the torus. There is a weight-6 $X$-type ($Z$-type) stabilizer generator supported on the four qubits on each red (green) square face and on two additional qubits located at fixed translated vectors relative to the face, indicated by red (green) arrows for one highlighted face.
• Figure 5: (a) The logical action of any stabilizer channel from $k$ to $k'$ logical qubits is to measure a set of $l$ independent commuting logical Paulis of the input code, and to apply an isometry from the remaining $k-l$ to $k'$ logical qubits of the output code. This is fully specified by $k$-dimensional and $k'$-dimensional Clifford unitaries $C$ and $B$, and the number $l = 0, 1, 2,\dots k$. (b) We use orange and blue disks to indicate logical $X$- and $Z$-type faults in this logical circuit that can have non-trivial action (note that a $Z$ error after a $|0\rangle$ preparation is trivial). We can specify all logical failures by the bitstrings $a\in\mathbb{F}_2^{l}$, $b\in\mathbb{F}_2^{k'-k+l}$, $c\in\mathbb{F}_2^{k-l}$ and $d\in\mathbb{F}_2^{k-l}$. Each of the $k+k'$ rows of $A$ corresponds to a bit in one of these bitstrings.

Theorems & Definitions (9)

• Definition 1: Faults and decoding
• Lemma 1: Breadth-first DTD: min-weight
• proof
• Lemma 2: Height-oracle DTD: min-weight, min explored nodes
• proof
• Lemma 3: Height-bound DTD: min-weight
• proof
• Lemma 4
• proof