I found that parameterized quantum circuits always produce outputs inside the Mandelbrot set — even across 130 inputs and 100 random seeds
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I’ve been studying the geometry of parameterized quantum circuit (PQC) outputs, and I stumbled onto something unexpected. If you take adjacent qubit probabilities from a PQC, center them around 0.5, and treat them as a complex number c, then iterate the Mandelbrot map z\rightarrow z^2+c, the orbit never escapes. Across: 130 inputs 35 mathematical families 6‑qubit, 6‑layer PQC 4096–8192 shots per input …every single output landed inside the Mandelbrot set. I stress‑tested this across 2–20 qubits, 1–50 layers, and 100 random parameter seeds. The result held 97.5% of the time. The only failures were circuits with just one entanglement layer — with 2+ layers, boundedness was universal. The mechanism seems to be entanglement: CNOT cascades prevent adjacent qubits from simultaneously reaching extreme probabilities, which keeps |c|\leq 0.606, well inside the Mandelbrot cardioid. This doesn’t encode input identity (p = 0.83), so it’s not a classifier. But it is a geometric constraint on PQC output space that I haven’t seen reported before. Full paper (Zenodo): https://zenodo.org/records/19367794 submitted by /u/Clean-Swordfish-5977 [link] [comments]