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

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.