pennylane-toroidal-noise

A phenomenological dephasing parameterization for PennyLane, where the effective dephasing rate is modulated by the spectral gap of an n × n discrete-torus Laplacian.

This package is a thin wrapper over qml.PhaseDamping. It exposes:

• spectral_gap(n) — the smallest non-zero eigenvalue of the cycle-graph Laplacian, λ₁(n) = 2 − 2cos(2π/n).
• effective_gamma(gamma, grid_n, alpha) — a one-parameter saturation of gamma against λ₁, returning γ · λ₁ / (λ₁ + α).
• ToroidalDephasing(gamma, grid_n, alpha, wires) — a named channel whose Kraus operators are delegated to qml.PhaseDamping.compute_kraus_matrices with effective_gamma(...) as the rate.

Status

Phenomenological model, not a derivation from physical first principles. The spectral-gap-to-dephasing mapping γ_eff = γ · λ₁ / (λ₁ + α) is offered as a tunable parameterization for studying lattice-geometry-dependent dephasing in PennyLane simulations. alpha is a knob, not a physically calibrated coupling. Treat results as exploratory unless you have an independent physical justification for this mapping in your specific setting.

Installation

pip install pennylane-toroidal-noise

Or from source:

git clone https://github.com/Paraxiom/pennylane-toroidal-noise
cd pennylane-toroidal-noise
pip install -e .[dev]
pytest

Usage

Recommended pattern — explicit composition

The cleanest way to use this library is to compose the spectral-gap calculation with qml.PhaseDamping at the call site:

import pennylane as qml
from pennylane_toroidal_noise import effective_gamma

dev = qml.device("default.mixed", wires=1)

@qml.qnode(dev)
def circuit(gamma):
    qml.Hadamard(wires=0)
    qml.PhaseDamping(effective_gamma(gamma, grid_n=12, alpha=1.0), wires=0)
    return qml.expval(qml.PauliX(0))

This makes the parameterization visible to anyone reading the circuit and keeps the noise model standard (a PhaseDamping channel).

Alternative — named operator

If you prefer a named operator for circuit-diagram readability or for reusing grid_n/alpha across many call sites, use ToroidalDephasing:

import pennylane as qml
from pennylane_toroidal_noise import ToroidalDephasing

dev = qml.device("default.mixed", wires=1)

@qml.qnode(dev)
def circuit(gamma):
    qml.Hadamard(wires=0)
    ToroidalDephasing(gamma, grid_n=12, alpha=1.0, wires=0)
    return qml.expval(qml.PauliX(0))

ToroidalDephasing.compute_kraus_matrices delegates to qml.PhaseDamping.compute_kraus_matrices; decomposition() returns a single qml.PhaseDamping(effective_gamma(...)).

Math

The discrete n × n torus is the graph product C_n □ C_n of two cycle graphs. The eigenvalues of its Laplacian are sums of cycle-graph eigenvalues:

λ_{j,k}(n) = (2 − 2cos(2πj/n)) + (2 − 2cos(2πk/n)),  0 ≤ j, k < n

The smallest non-zero eigenvalue is attained at (1, 0) or (0, 1):

λ_1(n) = 2 − 2cos(2π/n)

This is also the spectral gap of the cycle graph C_n. The library uses this as the geometry parameter in the saturation:

γ_eff(γ, n, α) = γ · λ_1(n) / (λ_1(n) + α)

Limitations and caveats

• Phenomenological direction. The mapping γ_eff = γ · λ₁ / (λ₁ + α) reduces dephasing as λ₁ → 0. This direction does not correspond to any particular standard physical mechanism — typical "spectral-gap protection" arguments in many-body physics suppress noise as the gap increases, not as it shrinks. Users should not interpret results from this parameterization as a physical prediction without independent justification.
• No T₂ / hardware claim. This package does not derive or claim a hardware T₂-extension figure. Earlier framings that quoted specific microsecond-to-millisecond figures have been removed; if you need those numbers, they must come from a calibrated device model, not from this parameterization.
• Single-qubit channel only. The spectral-gap calculation references an n × n lattice geometry, but the channel itself acts on a single wire. There is no multi-qubit lattice noise correlation in this implementation.

Related

• A pure-Rust companion crate is published as toroidal-noise for non-PennyLane use cases.

License

MIT.