Crate toroidal_noise

Expand description

§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.

A PennyLane-compatible Python sibling is published as pennylane-toroidal-noise, which delegates to qml.PhaseDamping.

§License

MIT.

Structs§

KrausOperator: A Kraus operator: a 2×2 complex matrix K such that the channel maps ρ → Σᵢ Kᵢ ρ Kᵢ†.

Constants§

HADAMARD: Hadamard gate.
IDENTITY: Identity matrix.
RHO_ZERO: |0⟩⟨0| density matrix.
SIGMA_X: Pauli X gate.

Functions§

amplitude_damping: Single-qubit amplitude damping channel.
apply_channel: Apply a channel (list of Kraus operators) to a 2×2 density matrix: ρ → Σᵢ Kᵢ ρ Kᵢ†
apply_unitary: Apply a unitary gate to a density matrix: ρ → U ρ U†
dephasing: Single-qubit phase damping channel.
depolarizing: Single-qubit symmetric depolarizing channel.
effective_gamma: Phenomenological effective dephasing rate from a toroidal lattice spectral gap.
is_trace_preserving: Check if Kraus operators satisfy trace preservation: Σᵢ Kᵢ† Kᵢ = I.
spectral_gap: Spectral gap of the cycle graph Cₙ Laplacian.
toroidal_dephasing: Single-qubit dephasing parameterized by a toroidal lattice spectral gap.
trace: Trace of a 2×2 matrix.

Type Aliases§

Matrix2x2: A 2×2 complex matrix stored in row-major order.