Tesseract: A Search-Based Decoder for Quantum Error Correction

TL;DR

The paper presents Tesseract, a search-based decoder for quantum LDPC codes that casts decoding as a shortest-path problem on the power-set graph of potential errors, leveraging pruning and an admissible A* heuristic. It demonstrates near-optimal accuracy with substantial speedups over integer-programming decoders across multiple codes and protocols, including surface codes, color codes, and bivariate bicycle codes, under circuit-level noise. The work highlights significant practical benefits in decoding efficiency and protocol benchmarking, showing large qubit-efficiency gains for certain codes such as the [[144,12,12]] bicycle code. It also situates the approach among contemporary work (e.g., DTD) and provides an open-source C++ implementation for broader use.

Abstract

Tesseract is a Most-Likely Error decoder designed for low-density-parity-check quantum error-correcting codes. Tesseract conducts a search through a graph on the set of all subsets of errors to find the lowest cost subset of errors consistent with the input syndrome. Although this graph is exponentially large, the search can be made efficient in practice for random errors using $A^*$ search technique along with a few pruning heuristics. We show through benchmark circuits for surface, color, and bivariate-bicycle codes that Tesseract is significantly faster than integer programming-based decoders while retaining comparable accuracy at moderate physical error rates. We also find that Tesseract can decode transversal CNOT protocols for surface codes on neutral atom quantum computers. Finally, we compare surface code and bivariate bicycle code circuits, finding that the [[144,12,12]] bivariate bicycle code is $14\times$ to $19\times$ more efficient than surface codes using our most-likely error decoding, whereas using correlated matching and BP+OSD decoders would have implied only a $10\times$ improvement. Assuming instead that long-range couplers are $10\times$ noisier, the improvement drops to around $4\times$ using Tesseract or $2\times$ using correlated matching and BP+OSD.

Figures (8)

• Figure 1: The graph $G$ on all error sets, if $|\mathscr{E}| = 3$ and the predicate $P(F, F')$ is always $1$. Weights of the edges are not shown.
• Figure 2: The Tesseract decoder achieves very similar accuracy to the Integer Program decoder while approximately 5$\times$ faster in runtime. Moreover, both Tesseract and the Integer Program decoders are order(s) of magnitude more accurate than fast algorithmic decoders on color codes (comparing with chromobiusgidney2023new) and bicycle codes (comparing with BP+OSD panteleev2021degenerate via the ldpc module ldpc). A square-root scaling is used for the number of physical qubits axis. Note that not all combinations of decoder, protocol, and physical error rate have an observed logical error, and these points are omitted from the upper row of plots. For example, the $d=13$ superdense color code circuit at $p=0.0001$ decoded with the Integer Program decoder had no observed logical errors within the scope of our simulations.
• Figure 3: We compare the performance of the [[144,12,12]] bivariate bicycle (BB) code from Ref. bravyi2024high with surface codes. The circuit for the [[144,12,12]] code uses 288 qubits including ancillas, and has circuit distance 10 (hence is referred to as $[[288,12,10]]$ in the figure). We compare with the performance of 12 copies of surface codes. In (a) we use correlated sparse blossom to decode the surface codes and BPOSD to decode the BB code, whereas in (b) we use our Tesseract decoder to decode both. We use an SI1000 noise model Gidney2022benchmarkingplanar for all surface code circuits and the BB code noise models are given in the legend. The "NLR5" and "NLR10" noise models use $5\times$ and $10\times$ higher noise strengths for couplers that are long-range on the toric layout.
• Figure 4: Comparison of the DTD and MaxSAT decoder timing data from ott2025decision with our Integer Program and Tesseract decoder implementation. All of the above decoders are exact -- in particular, none of Tesseract's beam cutoffs were used -- guaranteeing that the most likely error is returned every time. It is worth noting that in practice, judicious use of the cutoffs such as Tesseract's beam parameter can make both Tesseract and the DTD decoder significantly faster without comprimising much accuracy.
• Figure 5: Comparison of uncorrelated vs. correlated decoding using BP+OSD for the [[72,12,6]] bivariate bicycle code, using the same circuit and uniform circuit-level depolarizing noise model as given in Ref. bravyi2024high. We perform a 6-round $X$ memory experiment. For uncorrelated BP (labeled "X detectors"), we decode a stim circuit with only the $X$-type detectors annotated, whereas for correlated BP+OSD we annotate all detectors ($X$-type and $Z$-type).

Theorems & Definitions (2)

• Theorem 3.1
• proof