From Channels to Fields: A Spatiotemporal Formalism for Noise in Quantum Circuits (QNaF)

We introduce QNaF, a spatiotemporal noise formalism for quantum circuits. We also provide a GPU-accelerated, PyTorch-based differentiable quantum simulator implementing this model, enabling gradient-based optimization of quantum systems. The implementation is actively used with ~200+ monthly downloads via PyPI. Noise in quantum circuits is not a static decoration applied uniformly to gates — it is a dynamical processshaped by the causal history of the computation. We introduce the Quantum Noise as Field (QNaF) formalism,which models decoherence as a latent field Φ(x, t) ∈ R6 evolving causally over the spacetime graph of a quantum circuit.The field decomposes into six physically motivated channels — Memory, Spatial Diffusion, Disturbance Propagation,Stochastic Kicks, Nonlinear Saturation, and Nonlocal Bleed — each encoding a distinct mechanism by which priorcircuit events influence future error probabilities. A trainable DecoherenceProjectionMatrix (DPM) projects thefield onto per-node Pauli error probabilities, serialisable as a compact hardware noise fingerprint. We demonstratethat without any task-specific calibration, the same DPM gives rise to qualitatively different field organisationsfor topologically distinct circuits: Memory-dominant (60.2%) for quantum teleportation and Disturbance-dominant(57.2%) for the three-qubit quantum Fourier transform — a circuit-adaptive reorganisation that emerges purely fromthe field dynamics. Quantitative benchmarks against IBM Aer show TVD < 0.03 on Bell, GHZ, and QFT circuits,with QNaF fidelity meeting or exceeding Aer’s on all structured circuits. The Pauli error hierarchy (Z ≈ 64–66%,X ≈ 22%, Y ≈ 13%) is stable across circuits, providing cross-circuit evidence that the DPM encodes a hardware priorrather than circuit-specific artefacts. We situate the formalism within the Lindblad framework — showing Lindbladdynamics recover as a fixed point in the Markovian limit — and draw a structural analogy to Ginzburg–Landau fieldtheory, in which the tanh-bounded field plays the role of an order parameter relaxing toward a temperature-dependentequilibrium. Limitations and open questions, including microscopic derivation, sparse scaling, and correlated errorextensions, are discussed.