Useful quantum computers will require fault tolerant logical operations. Researchers are actively exploring many different quantum error correction (QEC) codes to enable this, improving the Logical Error Rates (LER) of Quantum Processing Units (QPUs). While it is well understood how to run logical operations with surface codes (which belong to the topological code family) via lattice surgery, they are qubit efficient for memory but suboptimal relative to other topological codes for performing fault-tolerant logical computation. 

Quantum low-density parity-check (QLDPC) codes are another type of QEC scheme that require the least physical qubits for memory due to their constant rates. However, it is still unclear how both Clifford and non-Clifford gates can be performed efficiently at the logical level with QLDPC codes. 

Color Codes are another type of topological code that are well understood. Although they require more physical qubits for memory than surface codes to achieve the same target logical failure rate, they can perform logical gates more efficiently than surface codes due to their ability to do all Clifford gates transversally and the symmetry of their representation resulting in simpler lattice surgery operations. With fast and accurate decoders, it is likely that parameter regimes exist where logic with color codes could be more efficient than logic with surface codes. 

Given that decoding color codes is much harder than decoding surface codes, color codes have historically been shelved due to a lack of fast and accurate decoders, which enable users to leverage these codes in real time. 

NVIDIA Ising Decoder ColorCode 1 Fast is designed to accelerate and improve the LER of color code decoders, enabling more than 347.7x better LER, and 7.3x faster runtime compared with the state of the art color code decoder Chromobius for d=31 and physical error rate of 0.3%. Such results bring color codes back into focus for potential use in building and operating truly useful quantum computers.

The following video demonstrates how to train the Ising Decoder ColorCode 1:

Ising Decoding

Using the Ising Decoding training pipeline, QPU builders, operators, and decoder developers can train small 3D Convolutional Neural Network (CNN) AI-based pre-decoders designed for triangular Color Codes. Real-time operations that scale in both space and time help improve latency and LERs. These pre-decoders accelerate and improve decoder accuracy by handling a large quantity of localized error syndromes. They can also scale to arbitrary code distances, helping teams deploy quantum error correction decoders that will scale with their QPUs all the way to lattice surgery. 

Further, since they predict full space-time corrections and are local (and thus independent of the input size and geometry), they can be used to perform lattice surgery in a parallel space-time blockwise decoding architecture, which is essential for decoding errors in real time during the execution of a quantum algorithm. 

Users need only define their noise model, the triangular color code distance, and model depth (deeper is more accurate). The training architecture then uses the NVIDIA cuStabilizer library within NVIDIA cuQuantum and NVIDIA PyTorch to generate synthetic training data and train a 3D CNN that optimizes decoding performance for the task. Users can also opt for more or fewer layers, which creates a runtime and accuracy trade-off. The best CNN model depends on the code distance, physical error rate, global decoder effectiveness, and round-trip latency budget.

This can be evaluated in roughly 20 lines of python code outlined below. 

1. Build a distance-5 color-code memory experiment: superdense syndrome extraction, Si1000 circuit-level noise at p = 0.3%

 circuit = build_color_memory_circuit(
      distance=5,
      n_rounds=5,
      basis="X",
      p_error=1e-3,
      noise_model_family="si1000",
      noise_instruction_semantics="reference",
  ).stim_circuit

 2. Compile a Chromobius decoder from the circuit’s detector error model

 dem = circuit.detector_error_model(
      approximate_disjoint_errors=True,
      ignore_decomposition_failures=True,
  )
  decoder = chromobius.compile_decoder_for_dem(dem)

3. Sample noisy syndromes and decode

 dets, obs = circuit.compile_detector_sampler().sample(
      100_000, separate_observables=True
  )
  packed = np.packbits(dets.astype(np.uint8), axis=1, bitorder="little")
  predictions = decoder.predict_obs_flips_from_dets_bit_packed(packed)
  pred = np.unpackbits(predictions, axis=1, bitorder="little")[:, : circuit.num_observables]

4. Logical error rate

ler = np.mean(np.any(pred != obs, axis=1))
  print(f"logical error rate: {ler:.2e}  ({ler / 5:.2e} per round)")

Running this prints an LER around 3.6e-03 (7.3e-04 per round). From here you can sweep distances and error rates to trace out the threshold curve code/scripts/color_code_threshold_chromobius.py does the full sweep in one command, or drop the Ising-Decoder-ColorCode-1 pre-decoder in front of Chromobius to sparsify syndromes before the final decode.

Accuracy or speed: Selecting the right model

Ising Decoder ColorCode 1 Fast has 17 layers, a receptive field of 13, and is trained on input volumes of size 13x13x19 (although arbitrary input volumes can be used during inference). This model has roughly 2,900,000 parameters. Due to its small size, it runs efficiently on a GPU but provides less improvement to the LER compared to a larger model.

Figure 1, below, shows the value Ising Decoder ColorCode 1 can add to a common open source color code decoder like Chromobius. It shows the ideal regime for deploying the pre-decoder model given some physical error rate and code distance. 

Learn more about NVIDIA Ising Decoder ColorCode model architecture and explore a wide range of analysis and results. 

Start building with NVIDIA Ising open resources

The NVIDIA Ising model family is fully open. Weights, training architectures, data, benchmarks, and recipes are provided to enable others to modify, deploy, train, and fine-tune their own models and variants for their specific QPUs. 

Training Recipes

NVIDIA is releasing a complete training pipeline for Ising Decoder ColorCode 1 Fast that enables users to generate synthetic data with the NVIDIA cuQuantum library, NVIDIA cuStabilizer on the fly, while training with PyTorch. This architecture enables developers to produce decoder models tailored to their specific QPU noise characteristics. 

Get started

NVIDIA Ising is available with the following resources for getting started: 

• Ising Decoding training architecture and cookbook GitHub under Apache 2.0. 
• Read the Color Code Model Architecture Paper.