toroidal-noise 0.2.0

Single-qubit quantum noise channels (dephasing, amplitude damping, depolarizing) with a phenomenological spectral-gap dephasing parameterization on discrete tori.
Documentation

toroidal-noise

A small Rust crate for single-qubit quantum noise channels and 2×2 density-matrix simulation, including a phenomenological dephasing parameterization driven by the spectral gap of an n × n discrete-torus Laplacian.

Channels

Function Parameters Kraus operators
dephasing(gamma) gamma ∈ [0, 1] 2
amplitude_damping(gamma) gamma ∈ [0, 1] 2
depolarizing(p) p ∈ [0, 1] 4 (I, X, Y, Z)
toroidal_dephasing(gamma, n, alpha) gamma ∈ [0, 1], n >= 2, alpha > 0 2

Plus utility functions:

  • spectral_gap(n) — λ₁(n) = 2 − 2cos(2π/n), the smallest non-zero eigenvalue of the cycle-graph Laplacian (also the spectral gap of the n × n torus).
  • effective_gamma(gamma, grid_n, alpha) — the phenomenological mapping γ · λ₁ / (λ₁ + α).

Status

toroidal_dephasing is a phenomenological model, not a derivation from physical first principles. It is a thin wrapper that delegates to dephasing(effective_gamma(γ, n, α)). The mapping is offered as a tunable parameterization for studying lattice-geometry-dependent dephasing in simulations; alpha is a knob, not a physically calibrated coupling. Treat results as exploratory unless you have an independent physical justification.

The other channels (dephasing, amplitude_damping, depolarizing) are standard textbook noise channels and have no such caveat.

Usage

use toroidal_noise::{
    apply_channel, apply_unitary, dephasing, effective_gamma,
    toroidal_dephasing, HADAMARD, RHO_ZERO,
};

// Start with |+⟩
let rho = apply_unitary(&RHO_ZERO, &HADAMARD);

// Two equivalent ways to apply toroidal-parameterized dephasing:

// 1. Named wrapper
let rho_a = apply_channel(&rho, &toroidal_dephasing(0.5, 12, 1.0));

// 2. Explicit composition (preferred when readers should see the parameterization)
let rho_b = apply_channel(&rho, &dephasing(effective_gamma(0.5, 12, 1.0)));

// Off-diagonal coherence comparable in both
assert!((rho_a[0][1].re - rho_b[0][1].re).abs() < 1e-12);

Math

The discrete n × n torus is the graph product Cₙ □ Cₙ of two cycle graphs. Eigenvalues of its Laplacian are pairwise 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, attained at (1, 0) or (0, 1), is

λ₁(n) = 2 − 2cos(2π/n)

The library uses this in the saturation:

γ_eff(γ, n, α) = γ · λ₁(n) / (λ₁(n) + α)
grid_n λ₁(n) γ_eff/γ (α=1)
4 2.000 0.667
6 1.000 0.500
8 0.586 0.369
12 0.268 0.211
32 0.0383 0.0369
64 0.00964 0.00955

Limitations and caveats

  • Phenomenological direction. 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. Do not interpret results from this parameterization as a physical prediction without independent justification.
  • No T₂ / hardware claim. This crate does not derive or claim any hardware T₂-extension figure. If you need such numbers, they must come from a calibrated device model, not from this parameterization.
  • Single-qubit channels only. The spectral-gap calculation references an n × n lattice geometry, but every channel here acts on a single qubit. There is no multi-qubit lattice noise correlation.

Related

License

MIT.