Statistical Validation of the Formulon Physics Legendre Encoder: Shot-Noise Scaling, Decoder Efficiency and Deterministic Accuracy on a PennyLane Simulator

Sri Kaleeswarar S

Published Oct 29, 2026• 0 citations

Abstract

WHAT THIS PAPER IS ABOUT Imagine a quantum computer wants to understand a number, say 0.35. It cannot just read it. So the number is first converted into a special pattern called Legendre features. These come from Legendre polynomials, a set of smooth curves in maths that act like a fingerprint of the number. WHAT WAS TESTED Formulon Physics is a Python library that does this conversion and can also reverse it, getting the original number back. I wanted to check two things: is it correct, and how accurate is it? HOW I TESTED IT 1. Correctness check: I compared Formulon's answers with SciPy, a trusted maths library. Over 1,312 values, the biggest difference was about one part in a quadrillion, which is just computer rounding. 2. Accuracy check: A quantum computer gives answers by measuring many times. Each measurement is called a shot. I ran 4,100 tests using 100, 500, 1,000 and 5,000 shots, on a simulator in Google Colab. WHAT I FOUND - More shots means a smaller error. It follows a simple rule: 4 times more shots cuts the error to half. This is just like a coin toss. Flip it 10 times and the result is rough; flip it 1,000 times and you get close to 50-50. - The error matched the maths prediction. It was within about 5% of what theory says, so nothing strange is happening inside the tool. - Higher degree recovers the number better. Using more features (degree 12 instead of 2) brought back the original number much more accurately. - The decoder is nearly perfect. It gets almost the best accuracy that is mathematically possible. LIMITS This was a simulator, not a real noisy quantum machine. I ran only 5 repeats per setting. The code also reused some random seeds. I also did not claim any quantum advantage. WHY IT MATTERS Anyone using Formulon now knows how many shots they need for the accuracy they want. For example, about 516 shots give an error of 0.01 with degree 8.

DOI: 10.5281/zenodo.23753749

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Statistical Validation of the Formulon Physics Legendre Encoder: Shot-Noise Scaling, Decoder Efficiency and Deterministic Accuracy on a PennyLane Simulator

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Publisher

Abstract

DOI: 10.5281/zenodo.23753749

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arXiv:2002.05180IEEE Transactions on Information TheoryOct 16, 202130 citations

We first provide a simple proof showing that any single-shot code is self-correcting. We then show that any

[[n,k]]

CSS quantum low-density parity-check (QLDPC) code can be transformed into an

[[nm,k]]

CSS QLDPC code with single-shot (SS) quantum error correction and an efficient local decoder which runs in

O(\log m)

parallel time. Below a constant threshold

p_{\rm RG}>0

for joint local stochastic physical and measurement noise, the logical failure probability over

T

correction rounds is bounded by \begin{equation*} O(T n)\exp[-Ω(m^α)] \end{equation*} for a constant

α>0

; we say that the \textit{threshold} is

p_{\rm RG}

and \textit{error suppression} is

Ω(m^α)

. Moreover, if the input code is equipped with a single-shot decoder with threshold

p_c>0

and error suppression

Ω(n^β)

, then the output code \textit{enhances} the error suppression to

Ω(m^α n^β)

, while maintaining the decoder-independent threshold

p_{\rm RG}

, and thus \textit{enhanced} if

p_c < p_{\rm RG}

\Omega (r d^{2})

parity check measurements for a distance-

d

code defined by

r

parity checks. For some specific highly structured quantum codes, Bombin has shown that single-shot fault-tolerant quantum error correction is possible using only

r

measurements. In this work, we consider a phenomenological noise model for parity check measurements assuming that each bit of a codeword and the measurement outcome suffer from independent bit flips with some error rate

p

. For this model, we demonstrate that fault-tolerant quantum error correction can be achieved using

O(d \log (d))

measurements for any code with distance

d \geq \Omega (n^\alpha)

for some constant

\alpha > 0

. Moreover, we prove the existence of a sub-single-shot fault-tolerant quantum error correction scheme using fewer than

r

measurements. In some cases, the number of parity check measurements required for fault-tolerant quantum error correction is exponentially smaller than the number of parity checks defining the code. The short measurement sequences constructed generally have high weight and our phenomenological noise model is not realistic in this regime. Our error correction strategy could find applications to small codes and LDPC codes if one can manage to keep the weight of the measured parity checks low.