Upper and Lower Bounds on the Computational Complexity of Polar Encoding and Decoding
Bibliographic record
Abstract
It is shown that all polar encoding schemes using a standard encoding matrix with rate R>1/2 and block length N have energy within the Thompson circuit model that scales at least as E ≥ Ω (N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3/2</sub> ). This lower bound is achievable up to polylogarithmic factors using a mesh network topology defined by Thompson and the encoding algorithm defined by Arıkan. A general class of circuits that compute successive cancellation decoding adapted from Arıkan's butterfly network algorithm is defined. It is shown that such decoders implemented on a rectangle grid for codes of rate R > 2/3 must take energy E ≥ Ω (N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3/2</sup> ). The energy of a Mead memory architecture and a mesh network memory architecture are analyzed and it is shown that a processor architecture using these memory elements can reach the decoding energy lower bounds to within a polylogarithmic factor. Similar scaling rules are derived for polar list decoders and belief propagation decoders. Capacity approaching sequences of energy optimal polar encoders and decoders, as a function of reciprocal gap to capacity χ = (1- R/C) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> (where R is rate C and is channel capacity), have energy that scales as Ω (χ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5.3685</sup> ) ≤ E ≤ O (χ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">7.071</sup> log <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> (χ)). Known results in constant depth circuit complexity theory imply that no polynomial size classical circuits can compute polar encoding, but this is possible in quantum circuits that include a constant depth quantum fan-out gate.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".