Blockwise Parallel Decoding Methods

• Blockwise parallel decoding is a strategy that decomposes sequential decoding into overlapping blocks to exploit concurrency and improve throughput on modern hardware.
• It has been applied to convolutional and polar codes, quantum surface codes, and deep autoregressive models, delivering significant speedups with minimal accuracy loss.
• Practical implementations require careful tuning of block lengths, buffer sizes, and synchronization to balance computational efficiency and error-correction performance.

Blockwise parallel decoding refers to a family of algorithmic strategies—used in both classical and quantum error correction, as well as deep generative modeling—where the original sequential decoding problem is decomposed into overlapping or partitioned blocks that can be processed in parallel. The key objective is to exploit computational concurrency inherent to modern hardware architectures, such as GPUs or parallel CPUs, to greatly improve throughput and reduce wall-clock latency, while maintaining accuracy comparable to traditional serial algorithms. Blockwise parallel decoding has been applied to convolutional codes via Viterbi decoding, polar codes, quantum surface code error correction, and autoregressive sequence generation in neural network models. Central to the approach are trade-offs among buffer size, block length, parallel scheduling, and the potential for small, controlled performance losses depending on overlap and verification strategies (Peng et al., 2016, Li et al., 2013, Tan et al., 2022, Bombín et al., 2023, Stern et al., 2018, Kim et al., 14 Apr 2024).

1. Blockwise Parallel Decoding in Classical Error Correction

In classical channel coding, blockwise parallelization enables real-time high-throughput decoding for trellis-structured codes and polar codes, with minimal degradation of error-correcting performance.

Viterbi Decoding for Convolutional Codes:

The block-based parallel Viterbi decoder divides the trellis into overlapping “parallel blocks” (PBs). Each PB extends over a region [t–M, t+D+L], where D is the main decoding region, and M, L are the lengths of the pre- and post-buffers (truncation and traceback blocks, typically ≈5K for constraint length K). Two main parallel phases are deployed:

• Forward add–compare–select (ACS) is executed in parallel for all states, using group-based processing to exploit CUDA warp-level primitives and shared-memory layouts for maximum coalescing.
• Backward traceback is performed for each PB, extracting decoded bits for the central D region after traversing the buffer zones.

This design realizes 1.5× throughput gains (up to 1802 Mbps for a 64-state code on GTX980), primarily by multiplexing across PBs and state groups, using memory-alignment and packing optimizations to fully utilize the GPU (Peng et al., 2016).

Parallel Decoding of Polar Codes:

Blockwise parallel SC and SC-List decoders exploit the Kronecker product structure of the polar transform. A codeword of length N=2ⁿ is partitioned into M=2ᵐ blocks, each of length N/M, and decoded in parallel. For each block, correlations due to shared bits (as enforced by XOR-based constraints from the polar construction) are merged at synchronization points, while maintaining path metrics across list-decoder instances. This approach yields M× speedup without observable degradation in bit error rate (BER) or frame error rate (FER); global path pruning preserves optimality (Li et al., 2013).

2. Blockwise Decoding in Quantum Error Correction

Blockwise or windowed parallelization strategies have become critical for surface codes and lattice-surgery quantum architectures, where low-latency decoding is vital for real-time quantum error correction.

Parallel Sliding-Window (Sandwich) Decoding for Surface Codes:

The syndrome data stream is segmented into overlapping time windows of length w with core s and buffer b=(w–s)/2. Decoding proceeds in two pipeline stages:

1. Each window is decoded in parallel using an “inner decoder” (e.g., MWPM or union-find) with only the core layer contributing final corrections.
2. Seam decoders annihilate residual defects on overlap regions, again in parallel.

This “sandwich decoder” achieves nearly constant latency in experiment depth n and circuit-level thresholds indistinguishable from batch decoding: e.g., 0.68% (MWPM), 0.55% (UF), matching the non-parallel implementation. Optimal window and buffer size scales with code distance d, with constant-depth causal dependency and O(n/d) parallel efficiency (Tan et al., 2022).

Modular Edge-Vertex Decoding:

Quantum circuits are modeled as graphs with vertex-tasks (blocks) and edge-tasks (ports), each assigned commit and buffer regions. Buffer regions must satisfy a “buffering condition,” ensuring that undetected error clusters involving future (unseen) data require weight at least d (full code distance). Decoding proceeds in two phases: all edge-tasks run in parallel (using only local information plus buffer), then after boundary syndromes have been set, all vertex-tasks run in parallel. Empirical LER matches global monolithic decoding when buffer size b≥d, and buffer reduction below d results in exponential increase in error (Bombín et al., 2023).

3. Blockwise Parallel Decoding for Deep Autoregressive Models

Blockwise parallel decoding enables substantial acceleration of sequence generation in LLMs and vision transformers by decoupling token generation steps.

Mechanism:

At each decoding step, the model predicts B tokens in parallel using auxiliary proposal heads, then greedily verifies the longest prefix consistent with the sequential base model’s greedy predictions. Accepted tokens are appended, and the process iterates. If a single model implements B predictive heads (outputting p₁…p_B in one pass), efficiency is maximized. Variants use either strict (exact match) or approximate (top-K or ε-distance) verification (Stern et al., 2018).

Block Efficiency and Speedup:

Let E[𝑘̂] be the mean accepted block length; iteration (model call) count reduces from T to T/E[𝑘̂], with observed reduction factors up to 2× (no quality loss) and up to 7× (with relaxed acceptance). Experiments on translation and image super-resolution tasks demonstrate real wall-clock speedups up to 4× (e.g., B=8, 3.3× speedup, –1 BLEU on WMT14 En→De).

Improvements via Draft Refinement:

Recent work refines BPD via parallel rescoring methods:

• Neural LM Rescoring: A small neural LM interpolates its logits with each prediction head; tokens are re-ranked using the sum of logits from both models.
• n-gram FST Rescoring: The top-K token lattice from all heads is globally rescored using a trained n-gram LM, with n-best path extraction and batch verification.

These refinements realize +5–21% gains in block efficiency across datasets (e.g., NewsRoom: B_eff=1.08→1.31 with neural rescoring and n-gram FST), directly reducing total model calls and decoding latency (Kim et al., 14 Apr 2024).

4. Memory Layout, Scheduling, and GPU/Parallel Implementation

Efficient blockwise parallel decoding generally demands careful architecture-specific mapping:

• For Convolutional Codes: Coalesced access through 3D survivor-path arrays (SP) and bank-aligned shared-memory for path metrics. Group-based CUDA kernels map warps to state groups, suppressing memory conflicts and maximizing arithmetic density (Peng et al., 2016).
• For Polar Codes: Interleaved sub-blocks and incorporation of XOR constraints at combination points permit independent SC/SC-List extensions, while global path metric pruning governs survivor selection (Li et al., 2013).
• For Neural Models: Multi-head output layers can be realized via B·d_hidden projections, and draft lattices can be processed batch-wise for proposal and verification. Efficient implementations leverage FST frameworks for n-gram rescoring and light-weight secondary LMs for neural interpolation (Kim et al., 14 Apr 2024).
• For Quantum Decoders: Topologies (window/time or edge/vertex) are mapped onto processor arrays corresponding to code geometry, with explicit causal barriers and one- or two-round communication of boundary conditions (Tan et al., 2022, Bombín et al., 2023).

5. Trade-offs, Performance, and Scaling Laws

Across classical, quantum, and neural applications, performance is governed by block length, buffer size, and model architecture:

Domain	Buffer/Block Parameter	Scaling of Latency/Throughput	Error-Rate/Quality Degradation	Max Speedup
Viterbi (Convolutional)	D, M=L≈5K (K=7), D≈512	Throughput ∝ block count	≤0.1dB (8-bit quantization), no BER floor up to block overlap sweet spot	1.5× (PBVD, GTX980: 1802Mbps)
Polar Codes	M=2^m	Speedup ×M, complexity drops	BER/FER unchanged up to M=8	8× SC/SC-List
Quantum (Surface, Modular)	w≈3d, b≈d or b≥d	O(n/d) parallel chains, constant-depth	LER matches global for b≥d, sharp loss for b<d	Processor count, code geometry-bound
Neural BPD	B, h	Iteration depth T/E[k̂], wall-clock ∝ block efficiency	ΔBLEU≤0.8 (medium B, exact), up to –1.2 BLEU (large B, approx)	≈4× (Transformer, B=8)

Larger block/B/buffer sizes improve parallel efficiency and reduce causal dependencies, but raise per-block computation and marginal memory cost. Excessive block length or inadequate overlap can yield diminishing returns or, in quantum error correction, degrade logical performance below threshold.

6. Practical Guidelines and Applicability

To maintain target error rates or output fidelity:

• In Viterbi and polar decoding, select block/truncation/buffer sizes to maximize arithmetic density without crossing over lossless performance regime (Peng et al., 2016, Li et al., 2013).
• For quantum codes, ensure buffer (window or edge-vertex) at least matches code distance (b≥d), as smaller buffers risk catastrophic logical failures (Tan et al., 2022, Bombín et al., 2023).
• In deep autoregressive models, start with small block sizes (B=2–4) for zero degradation, scaling to B=6–10 with draft refinement and approximate verification for maximal speed (Stern et al., 2018, Kim et al., 14 Apr 2024).

Real-world deployments require hardware-aware thread/block scheduling, asynchronous I/O overlap, and dynamic autotuning (e.g., for shared-memory and register availability). For neural decoders, draft refinement via n-gram or neural rescoring attains additional speedups while incurring minimal latency overhead relative to main-model inference time (Kim et al., 14 Apr 2024).

Blockwise parallel decoding thus defines a general and effective paradigm for extracting maximal concurrency from sequential error correction and generation tasks, with minimal impact on accuracy when buffer and block parameters are chosen according to domain-specific scaling laws.