Tesseract is a Most Likely Error decoder designed for Low Density Parity Check (LDPC) quantum error-correcting codes. It applies pruning heuristics and manifold orientation techniques during a search over the error subsets to identify the most likely error configuration consistent with the observed syndrome. Tesseract achieves significant speed improvements over traditional integer programming-based decoders while maintaining comparable accuracy at moderate physical error rates.

We tested the Tesseract decoder for:

• Surface codes
• Color codes
• Bivariate-bicycle codes
• Transversal CNOT protocols for surface codes

Features

• A* search: deploys A* search while running a Dijkstra algorithm with early stop for high performance.
• Stim and DEM Support: processes Stim circuit files and Detector Error Model (DEM) files with arbitrary error models. Zero-probability error instructions are automatically removed when a DEM is loaded.
• Parallel Decoding: uses multithreading to accelerate the decoding process, making it suitable for large-scale simulations.
• Efficient Beam Search: implements a beam search algorithm to minimize decoding cost and enhance efficiency.
• Sampling and Shot Range Processing: supports sampling shots from circuits. When a detection error model is provided without an accompanying circuit, Tesseract requires detection events from files using --in. The decoder can also process specific shot ranges for flexible experiment setups.
• Detailed Statistics: provides comprehensive statistics output, including shot counts, error counts, and processing times.
• Heuristics: includes flexible heuristic options: --beam, --det-penalty, --beam-climbing, --no-revisit-dets, --at-most-two-errors-per-detector, and --pqlimit to improve performance while maintaining a low logical error rate. To learn more about these options, use ./bazel-bin/src/tesseract --help.
• Visualization tool: open the viz directory in your browser to view decoding results. See viz/README.md for instructions on generating the visualization JSON.

Installation

Tesseract relies on the following external libraries:

• argparse: For command-line argument parsing.
• nlohmann/json: For JSON handling (used for statistics output).
• Stim: For quantum circuit simulation and error model handling.

Build Instructions

Tesseract uses Bazel as its build system. To build the decoder:

bazel build src:all

Running Tests

Unit tests are executed with Bazel. Run the quick test suite using:

bazel test //src:all

By default the tests use reduced parameters and finish in under 30 seconds. To run a more exhaustive suite with additional shots and larger distances, set:

TESSERACT_LONG_TESTS=1 bazel test //src:all

Usage

The file tesseract_main.cc provides the main entry point for Tesseract Decoder. It can decode error events from Stim circuits, DEM files, and pre-existing detection event files.

Basic Usage:

./tesseract --circuit CIRCUIT_FILE.stim --sample-num-shots N --print-stats

To decode pre-generated detection events, provide the input file using --in SHOTS_FILE --in-format FORMAT.

Example with Advanced Options:

./tesseract \
        --pqlimit 1000000 \
        --no-revisit-dets \
        --det-order-seed 232852747 \
        --num-det-orders 24 \
        --circuit circuit_file.stim \
        --sample-seed 232856747 \
        --sample-num-shots 10000 \
        --threads 32 \
        --print-stats \
        --beam 23 --beam-climbing \
        --shot-range-begin 582 \
        --shot-range-end 583

Example Usage

Sampling Shots from a Circuit:

./tesseract --circuit surface_code.stim --sample-num-shots 1000 --out predictions.01 --out-format 01

Using a Detection Event File:

./tesseract --in events.01 --in-format 01 --dem surface_code.dem --out decoded.txt

Using a Detection Event File and Observable Flips:

./tesseract --in events.01 --in-format 01 --obs_in obs.01 --obs-in-format 01 --dem surface_code.dem --out decoded.txt

Tesseract can combine generated detector-order methods with explicit detector traversal orders from one or more JSON files. Sources are concatenated in command-line order. For example, this uses three BFS orders, every order in the file, and then three coordinate orders:

./tesseract \
    --num-det-orders 3 \
    --det-order-bfs \
    --detector-orders orders.json \
    --det-order-coordinate \
    ...

Each selected generated method contributes --num-det-orders orders and uses --det-order-seed independently as its base seed. Within a method, order k uses seed + k with an independent random-number stream; this differs from the older shared-stream ensemble generator. --detector-orders FILE may be repeated, and each occurrence contributes every order in that file. The JSON format is the same list-of-lists form as Python's TesseractConfig.det_orders:

[[0, 2, 1, 3], [3, 1, 2, 0]]

Every inner list must be a complete permutation of the DEM detector IDs. With no detector-order source option, Tesseract uses the existing generated Index source. A file-only invocation remains file-only; use --det-order-index explicitly to combine file orders with generated Index orders.

Tesseract supports reading and writing from all of Stim's standard output formats.

Decoding with GARI

Generate a GARI matrix DEM from a source circuit:

python src/py/_tesseract_py_util/gari.py \
    --circuit circuit_file.stim \
    --prior xor \
    --out-dir gari_output

This writes gari_output/circuit_file_gari_xor.dem. Its physical detector rows preserve the source detector IDs, and its added virtual rows form a suffix. GARI-aware detector orders can be written in the generic CLI format when reordering is wanted:

import json
import stim
from tesseract_decoder import demutil

circuit = stim.Circuit.from_file("circuit_file.stim")
gari_dem = stim.DetectorErrorModel.from_file("gari_output/circuit_file_gari_xor.dem")
orders = demutil.gari.build_detector_orders(circuit, gari_dem, num_det_orders=5)
with open("gari_orders.json", "w") as f:
    json.dump(orders, f)

Sample from the source circuit and decode with the generated DEM:

./bazel-bin/src/tesseract \
    --circuit circuit_file.stim \
    --dem gari_output/circuit_file_gari_xor.dem \
    --detector-orders gari_orders.json \
    --sample-num-shots 100 \
    --sample-seed 1234 \
    --threads 1 \
    --pqlimit 1000000 \
    --beam 5 \
    --beam-climbing \
    --no-revisit-dets \
    --print-stats \
    --stats-out gari-stats.json

The GARI matrix DEM is a decoding representation and must not be sampled. When a circuit and a larger DEM are supplied together, both command-line decoders read or sample the circuit's detector prefix and leave the DEM's virtual suffix zero. Their observable counts must agree. When reading a source-width event file, pass both --circuit and --dem; with --dem alone the input width is the full DEM width.

For matrix analysis, gari.py --row-order block writes a _block.dem file in the internal physical-X, physical-Z, virtual-Z, virtual-X row order. This research form does not accept source syndromes as a direct prefix. See GARI transformed matrices for supported circuit conventions and Python API details.

Performance Optimization

Here are some tips for improving performance:

• Parallelism over shots: increase --threads to leverage multicore processors for faster decoding.
• Beam Search: use --beam to control the trade-off between accuracy and speed. Smaller beam sizes result in faster decoding but potentially lower accuracy.
• Beam Climbing: enable --beam-climbing for enhanced cost-based decoding.
• At most two errors per detector: enable --at-most-two-errors-per-detector to improve performance.
• Priority Queue limit: use --pqlimit to limit the size of the priority queue.
• Error sparsification: enable --sparsify-errors to always keep low-degree errors while selectively reactivating high-degree errors per shot. This can improve runtime on DEMs with many high-degree errors, at the cost of a tunable accuracy/speed tradeoff.

Example with error sparsification:

./tesseract \
    --circuit circuit_file.stim \
    --sample-num-shots 10000 \
    --beam 20 \
    --beam-climbing \
    --sparsify-errors \
    --sparsify-base-degree 3 \
    --print-stats

--sparsify-base-degree K is required when --sparsify-errors is enabled. Errors touching at most K detectors are always active. Errors above K are optional and are ranked per shot by overlap with the fired detectors. In the surface code (or other 'mostly graphlike' codes) try K = 2. In the color code or bivariate bicycle codes, try K = 3. In general, it is recommended to set K to the number of activated detectors created by a single data qubit error in the bulk, restricting to X or Z errors only for CSS codes.

--sparsify-reactivate-limit M caps the number of optional high-degree errors reactivated per shot. If omitted, Tesseract uses round((4.5^(K - 2) / 3) * num_detectors).

--sparsify-max-degree D optionally excludes optional errors above degree D. If omitted, optional errors are not capped by degree.

Output Formats

• Observable flips output: predictions of logical errors.
• DEM usage frequency output: if --dem-out is specified, outputs estimated error frequencies.
• Statistics output: includes number of shots, errors, low confidence shots, and processing time.

Multi-Pass Graph Shattering

Multi-pass graph shattering partitions a correlated detector error model into two detector components and decodes the smaller component models separately. With two passes, predictions from the first pass update error priors used during the final pass. The current implementation requires exactly two components and accepts one or two passes.

Two-pass reweighting is a correlated-matching-style heuristic. A component symptom's XOR marginal includes every mechanism that produces it, including one-sided mechanisms, while paired evidence includes only mechanisms shared with the other component symptom. Their ratio is therefore not, in general, an exact joint or conditional probability. Reweighted probabilities are capped at 0.499 so every retained error continues to have positive decoding cost. Because reweighting is defined on aggregate component symptoms, two-pass decoding requires merge_errors=true. Unmerged mechanisms remain supported with one-pass decoding.

Detector classification

The standalone CLI deliberately accepts one canonical convention only: every detector instruction must have a JSON tag with a top-level "measure_basis" whose value is exactly "X" or "Z":

detector[{"measure_basis":"X"}](0, 0, 0) D0
detector[{"measure_basis":"Z"}](1, 0, 0) D1

The CLI does not infer bases from legacy metadata or coordinates. Tagged DETECTOR instructions in a .stim circuit retain their tags during circuit-to-DEM conversion, so a canonically tagged circuit can be passed directly. Otherwise, normalize its DEM in Python first. Automatic normalization covers only the named automatic metadata and Chromobius-coordinate conventions. For a Stim-generated surface-code circuit, select the explicit parity adapter as shown below. The helper preserves coordinates, instruction order, repeats, shifts, errors, tags, and unrelated JSON metadata:

from pathlib import Path

import stim
import tesseract_decoder

circuit = stim.Circuit(Path("circuit.stim").read_text())
dem = circuit.detector_error_model()
canonical_dem = tesseract_decoder.demutil.annotate_detector_bases(
    dem,
    detector_basis_classifier=(
        tesseract_decoder.demutil.stim_surface_code_detector_basis_classifier
    ),
)
Path("canonical.dem").write_text(str(canonical_dem))

annotate_detector_bases(dem) writes top-level measure_basis, the authoritative field in both Python and the CLI. It rejects an invalid or conflicting existing top-level measure_basis, but preserves lower-priority fields such as basis and md without requiring agreement. These fields may describe something other than the decoding component. Non-JSON detector tags are rejected rather than overwritten. To migrate a DEM using the previous CLI convention, top-level basis, pass it through this helper before using it with the CLI.

Python and Sinter use the shared automatic classifier by default, including its supported legacy metadata and Chromobius-coordinate adapters. Multi-pass decoding still requires every detector to be classified and the result to contain exactly two components.

CLI options

• --multipass: Enables multi-pass graph shattering.
• --num-passes, --num_passes: Selects one or two passes (default: 2). One pass performs no inter-pass prior update; two passes perform one round of prior propagation. Other values are rejected.
• --multipass-strategy, --multipass_strategy: Selects causal (default), which derives the pass schedule from component dependencies, or experimental static, which schedules both components in every pass.
• --print-multipass-plan: Prints the monolithic and component model statistics, dependencies, and pass schedule to standard error. It requires --multipass; these statistics are calculated only when this flag is present.

--dem-out is not supported with --multipass.

CLI example

When the circuit is not already canonically tagged, sample from the circuit while decoding against the normalized DEM:

./bazel-bin/src/tesseract \
    --circuit circuit.stim \
    --dem canonical.dem \
    --sample-num-shots 1000 \
    --multipass \
    --num-passes 2 \
    --multipass-strategy causal \
    --pqlimit 1000000 \
    --beam 20 \
    --beam-climbing \
    --no-revisit-dets \
    --num-det-orders 21 \
    --print-multipass-plan \
    --print-stats

Python and Sinter

The ordinary Sinter workflows need no normalization or custom callback:

from multi_pass_sinter_decoders import MultiPassSinterDecoder, get_sinter_decoders

decoder = MultiPassSinterDecoder()
custom_decoders = get_sinter_decoders()

Pass detector_basis_classifier=... to use another X/Z convention. Standard TesseractSinterDecoder configuration keywords can be supplied directly to MultiPassSinterDecoder; unknown keywords are rejected.

Python Interface

Full Python wrapper documentation

This repository contains the C++ implementation of the Tesseract quantum error correction decoder, along with a Python wrapper. The Python wrapper/interface exposes the decoding algorithms and helper utilities, allowing Python users to leverage this high-performance decoding algorithm.

For installation:

pip install tesseract-decoder

The following example demonstrates how to create and use the Tesseract decoder using the Python interface.

from tesseract_decoder import tesseract
import stim
import numpy as np


# 1. Define a detector error model (DEM)
dem = stim.DetectorErrorModel("""
    error(0.1) D0 D1 L0
    error(0.2) D1 D2 L1
    detector(0, 0, 0) D0
    detector(1, 0, 0) D1
    detector(2, 0, 0) D2
""")

# 2. Create the decoder configuration
config = tesseract.TesseractConfig(dem=dem, det_beam=50)

# To enable sparse activation for high-degree DEMs:
config = tesseract.TesseractConfig(
    dem=dem,
    det_beam=50,
    sparsify_errors=True,
    sparsify_base_degree=3,
    sparsify_reactivate_limit=-1,  # Use the built-in heuristic, clamped to error count.
)

# 3. Create a decoder instance
decoder = config.compile_decoder()
print(
    "Resolved sparsify reactivation limit:",
    decoder.config.sparsify_reactivate_limit,
)

# 4. Simulate detector outcomes
syndrome = np.array([0, 1, 1], dtype=bool)

# 5a. Decode to observables
flipped_observables = decoder.decode(syndrome)
print(f"Flipped observables: {flipped_observables}")

# 5b. Alternatively, decode to errors
decoder.decode_to_errors(syndrome)
predicted_errors = decoder.predicted_errors_buffer
# Indices of predicted errors
print(f"Predicted errors indices: {predicted_errors}")
# Print properties of predicted errors
for i in predicted_errors:
    print(f"    {i}: {decoder.errors[i]}")

Using Tesseract with Sinter

Tesseract can be easily integrated into Sinter workflows. Sinter is a tool for running and organizing quantum error correction simulations.

Here's an example of how to use Tesseract as a decoder for multiple Sinter tasks:

import stim
import sinter
from tesseract_decoder import make_tesseract_sinter_decoders_dict, TesseractSinterDecoder
import tesseract_decoder

if __name__ == "__main__":  
    # Define a list of Sinter task(s) with different circuits/decoders.
    tasks = []
    # Depolarizing noise probability.
    p = 0.005
    # These are the sensible defaults given by make_tesseract_sinter_decoders_dict().
    # Note that `tesseract-short-beam` and `tesseract-long-beam` are the two sets of parameters used in the [Tesseract paper](https://arxiv.org/pdf/2503.10988).
    decoders = [
        'tesseract',
        'tesseract-long-beam',
        'tesseract-short-beam',
        'tesseract-long-beam-sparsify-color-code-like',
        'tesseract-short-beam-sparsify-surface-code-like',
    ]
    decoder_dict = make_tesseract_sinter_decoders_dict()
    # You can also make your own custom Tesseract Decoder to be used with Sinter.
    decoders.append('custom-tesseract-decoder')
    decoder_dict['custom-tesseract-decoder'] = TesseractSinterDecoder(
        det_beam=10,
        beam_climbing=True,
        no_revisit_dets=True,
        merge_errors=True,
        pqlimit=1_000,
        num_det_orders=5,
        det_order_method=tesseract_decoder.utils.DetectorOrderMethod.Index,
        seed=2384753,
        sparsify_errors=True,
        sparsify_base_degree=3,
        sparsify_reactivate_limit=-1,
    )

    for distance in [3, 5, 7]:
        for decoder in decoders:
            circuit = stim.Circuit.generated(
                "surface_code:rotated_memory_x",
                distance=distance,
                rounds=3,
                after_clifford_depolarization=p
            )
            tasks.append(sinter.Task(
                circuit=circuit,
                decoder=decoder,
                json_metadata={"d": distance, "decoder": decoder},
            ))

    # Collect decoding outcomes per task from Sinter.
    results = sinter.collect(
        num_workers=8,
        tasks=tasks,
        max_shots=10_000,
        decoders=decoders,
        custom_decoders=decoder_dict,
        print_progress=True,
    )

    # Print samples as CSV data.
    print(sinter.CSV_HEADER)
    for sample in results:
        print(sample.to_csv_line())

You should get something like:

    shots,    errors,  discards, seconds,decoder,strong_id,json_metadata,custom_counts  
    10000,        42,         0,   0.071,tesseract,1b3fce6286e438f38c00c8f6a5005947373515ab08e6446a7dd9ecdbef12d4cc,"{""d"":3,""decoder"":""tesseract""}",  
    10000,        49,         0,   0.546,custom-tesseract-decoder,7b082bec7541be858e239d7828a432e329cd448356bbdf051b8b8aa76c86625a,"{""d"":3,""decoder"":""custom-tesseract-decoder""}", 
    10000,        13,         0,    7.64,tesseract-long-beam,217a3542f56319924576658a6da7081ea2833f5167cf6d77fbc7071548e386a9,"{""d"":5,""decoder"":""tesseract-long-beam""}",  
    10000,        14,         0,    4.12,tesseract-long-beam-sparsify-color-code-like,14fa5f9f08381d760f6c1f59805b75f2c70cfb83e50d9f1f40d92820a20eeb13,"{""d"":5,""decoder"":""tesseract-long-beam-sparsify-color-code-like""}",
    10000,        42,         0,   0.743,tesseract-short-beam,cf4a4b0ce0e4c7beec1171f58eddffe403ed7359db5016fca2e16174ea577057,"{""d"":3,""decoder"":""tesseract-short-beam""}",  
    10000,        34,         0,   0.924,tesseract-long-beam,8cfa0f2e4061629e13bc98fe213285dc00eb90f21bba36e08c76bcdf213a1c09,"{""d"":3,""decoder"":""tesseract-long-beam""}",  
    10000,        35,         0,   0.681,tesseract-long-beam-sparsify-color-code-like,f41bdb1bde3f5cf4893a9a9e33fc7d4c47d742f22b13dfec9195347e780119bc,"{""d"":3,""decoder"":""tesseract-long-beam-sparsify-color-code-like""}",
    10000,        10,         0,   0.439,tesseract,8274ea5ffec15d6e71faed5ee1057cdd7e497cbaee4c6109784f8a74669d7f96,"{""d"":5,""decoder"":""tesseract""}",  
    10000,         8,         0,    3.93,custom-tesseract-decoder,8e4f5ab5dde00fec74127eea39ea52d5a98ae6ccfc277b5d9be450f78acc1c45,"{""d"":5,""decoder"":""custom-tesseract-decoder""}",  
    10000,        10,         0,    5.74,tesseract-short-beam,bf696535d62a25720c3a0c624ec5624002efe3f6cb0468963eee702efb48abc1,"{""d"":5,""decoder"":""tesseract-short-beam""}",  
    10000,         5,         0,    1.27,tesseract,3f94c61f1503844df6cf0d200b74ac01bfbc5e29e70cedbfc2faad67047e7887,"{""d"":7,""decoder"":""tesseract""}",  
    10000,         4,         0,    25.0,tesseract-long-beam,4d510f0acf511e24a833a93c956b683346696d8086866fadc73063fb09014c23,"{""d"":7,""decoder"":""tesseract-long-beam""}",  
    10000,         4,         0,    14.8,tesseract-long-beam-sparsify-color-code-like,80868acc6e43c62cb73b242b66ae27d3ea08fe970ea879db5a8425c2454fc8a1,"{""d"":7,""decoder"":""tesseract-long-beam-sparsify-color-code-like""}",
    10000,         1,         0,    18.6,tesseract-short-beam,75782ce4593022fcedad4c73104711f05c9c635db92869531f78da336945b121,"{""d"":7,""decoder"":""tesseract-short-beam""}",  
    10000,         4,         0,    11.6,custom-tesseract-decoder,48f256a28fff47c58af7bffdf98fdee1d41a721751ee965c5d3c5712ac795dc8,"{""d"":7,""decoder"":""custom-tesseract-decoder""}",  

This example runs simulations for a repetition code with different distances [3, 5, 7] with different Tesseract default decoders.

Sinter can also be used at the command line. Here is an example of this using Tesseract:

sinter collect \
    --circuits "example_circuit.stim" \
    --decoders tesseract \
    --custom_decoders_module_function "tesseract_decoder:make_tesseract_sinter_decoders_dict" \
    --max_shots 100_000 \
    --max_errors 100
    --processes auto \
    --save_resume_filepath "stats.csv" \

Sinter efficiently manages the execution of these tasks, and Tesseract is used for decoding. For more usage examples, see the tests in src/py/tesseract_sinter_compat_test.py.

Good Starting Points for Tesseract Configurations:

The Tesseract paper recommends two setups for starting your exploration with Tesseract:

(1) Long-beam setup:

tesseract_config = tesseract.TesseractConfig(
    dem=dem,
    pqlimit=1_000_000,
    det_beam=20,
    beam_climbing=True,
    det_orders=tesseract_decoder.utils.build_det_orders(
        dem=dem,
        num_det_orders=21,
        method=tesseract_decoder.utils.DetectorOrderMethod.Index,
    ),
    no_revisit_dets=True,
)

(2) Short-beam setup:

tesseract_config = tesseract.TesseractConfig(
    dem=dem,
    pqlimit=200_000,
    det_beam=15,
    beam_climbing=True,
    det_orders=tesseract_decoder.utils.build_det_orders(
        dem=dem,
        num_det_orders=16,
        method=tesseract_decoder.utils.DetectorOrderMethod.Index,
    ),
    no_revisit_dets=True,
)

DetIndex is the default detector ordering. You can also pass DetBFS or DetCoordinate explicitly. Detector orders are complete detector-ID permutations in traversal order: order[position] = detector_id. These values balance decoding speed and accuracy across the benchmarks reported in the paper and can be adjusted for specific use cases.

The Sinter decoder dictionary also provides sparsified variants: tesseract-long-beam-sparsify-color-code-like, tesseract-long-beam-sparsify-surface-code-like, tesseract-short-beam-sparsify-color-code-like, and tesseract-short-beam-sparsify-surface-code-like.

As a quick rule of thumb, use the non-sparsified decoders as the safest baseline. Use the surface-code-like variants for surface-code-like or mostly graphlike DEMs, and use the color-code-like variants for color-code, bivariate-bicycle-code, or other DEMs where a typical bulk data error activates about three detectors. Within either family, prefer the long-beam variants when accuracy matters more and the short-beam variants when runtime matters more. See the Performance Optimization section for the full sparsification details.

Equivalent Python configs can enable sparsification with sparsify_errors=True, sparsify_base_degree=2 or 3, and sparsify_reactivate_limit=-1 to use the built-in heuristic clamped to the compiled error count.

Help

• Do you have a feature request or want to report a bug? Open an issue on GitHub to report it!
• Do you have a code contribution? Read our contribution guidelines, then open a pull request!

We are committed to providing a friendly, safe, and welcoming environment for all. Please read and respect our Code of Conduct.

Citation

When publishing articles or otherwise writing about Tesseract Decoder, please cite the following:

@misc{beni2025tesseractdecoder,
    title={Tesseract: A Search-Based Decoder for Quantum Error Correction},
    author = {Aghababaie Beni, Laleh and Higgott, Oscar and Shutty, Noah},
    year={2025},
    eprint={2503.10988},
    archivePrefix={arXiv},
    primaryClass={quant-ph},
    doi = {10.48550/arXiv.2503.10988},
    url={https://arxiv.org/abs/2503.10988},
}

Hacking on the Python module locally

To install your own build of the Tesseract Python module locally so that you can easily modify and hack on it, use something like the following:

bazel build --define ABI_TAG="cp313" --define TARGET_VERSION="cp3.13" --define VERSION="v0.0.0dev"  :tesseract_decoder_wheel
pip uninstall --y tesseract_decoder
pip install bazel-bin/tesseract_decoder-0.0.0.dev0-py3.12.9-none-manylinux_2_17_x86_64.manylinux2014_x86_64.whl
python testscript.py

Contact

For any questions or concerns not addressed here, please email tesseract-decoder-dev@google.com.

Disclaimer

Tesseract Decoder is not an officially supported Google product. This project is not eligible for the Google Open Source Software Vulnerability Rewards Program.

Copyright 2025 Google LLC.