Spatially Correlated Flux-Noise Model

• Spatially correlated flux noise is defined by nonzero covariance between noise at different spatial sites, necessitating full covariance matrix modeling.
• It originates from convolution of white noise with instrument response functions and spin interactions, impacting device performance in astronomy and quantum technology.
• Accurate simulation using methods like Cholesky decomposition and FFT techniques is essential for precise uncertainty quantification and error mitigation.

A spatially correlated flux-noise model characterizes the statistical properties, origins, and computational methodologies associated with noise that exhibits nontrivial spatial correlations—a critical consideration in a diverse range of fields including astronomical interferometry, superconducting devices, and quantum information processors. Unlike uncorrelated (white) noise models, spatially correlated flux noise requires explicit consideration of inter-pixel or inter-site covariance, substantially impacting uncertainty quantification, detection thresholds, and device performance.

1. Definition and Theoretical Foundations

Spatially correlated flux noise is defined by a nonzero covariance between the noise contributions at distinct spatial locations. Let $N(x)$ denote the flux noise at position $x$. The spatial structure of the noise is fully characterized by its autocorrelation function (ACF), $\xi(\Delta x) = \langle N(x+\Delta x)N(x)\rangle$, with $\xi(0)$ equal to the per-pixel noise variance. For spatially uncorrelated noise, $\xi(\Delta x)=0$ for $\Delta x\ne 0$, whereas in real interferometric images and quantum devices, $\xi(\Delta x)$ has significant nonzero support over finite ranges determined by instrument response, physical coupling, or environmental fluctuations.

In the Gaussian noise regime, the noise statistics are fully specified by the covariance matrix $$C_{mn} \equiv \langle N_m N_n \rangle = \xi(x_m-x_n)$$, where $m$, $n$ index pixel or site locations. The noise spectral properties are commonly described by the Fourier transform of the ACF, yielding the spatial power spectrum relevant for simulating and analyzing noise realisations and their statistical effects (Tsukui et al., 2023, Tsukui et al., 2022).

2. Origins and Physical Mechanisms

In radio interferometric imaging (e.g., ALMA, NOEMA), spatial correlation in noise arises predominantly from convolution of white (uncorrelated) visibility noise with the synthesized beam (dirty beam). This convolution induces spatial correlations on the scale of the beam and its side lobes, directly shaping the effective ACF. For a beam $b(x)$ and ideal image-plane noise $N_\text{ideal}(x)$, the observed noise is $N(x) = N_\text{ideal} * b(x)$, thus $\xi(\Delta x) = \sigma_N^2 [b*b](\Delta x)$, where $*$ denotes convolution and $\sigma_N^2$ is the underlying white-noise variance (Tsukui et al., 2022).

In superconducting circuits (e.g., SQUIDs, qubits), flux noise is attributed to ensembles of localized spins residing at material interfaces or surfaces, whose fluctuations are correlated via direct exchange interactions, conduction electron-mediated couplings, or environmental common modes. The correlation decay length is set by the physics of these couplings, for instance, via long-range ferromagnetic or RKKY-like interactions. In this context, spatially correlated $1/f^\alpha$ flux noise is observed, often dominating device performance limits (LaForest et al., 2015, De, 2017).

Spatial correlation also mediates the emergence of Ising-type coherent couplings and correlated dephasing in multi-qubit quantum processors, where a fluctuating magnetic environment supplies the noise source with a characteristic spatial decay length $\xi$ (Zou et al., 2023).

3. Mathematical Framework and Simulation Techniques

The estimation and accurate treatment of spatially correlated flux noise require construction and usage of the full covariance matrix $C$ as defined by measured or modeled ACFs. For any spatially integrated quantity (e.g., aperture flux, model fit parameters), the variance is given by

$$\mathrm{Var}(F) = \sum_{m \in S} \sum_{n \in S} C_{mn}$$

where $S$ is the subset of relevant pixels or measurement locations. Unlike the uncorrelated limit, off-diagonal terms in $C$ substantially increase the integrated uncertainty, particularly for large apertures or extended measurements (Tsukui et al., 2023, Tsukui et al., 2022).

Monte Carlo simulations of spatially correlated noise utilize either:

• Cholesky Decomposition: Given $C$, draw a vector of IID standard normal random variables $z$ and compute $n = L z$ with $L$ the Cholesky (lower-triangular) factor of $C$; $n$ represents a noise realization with the prescribed covariance.
• Spectral (FFT) Methods: Generate Fourier space Gaussian fields with prescribed power spectrum (the Fourier transform of $\xi$), ensuring Hermitian symmetry for real-valued outputs, and perform an inverse FFT to recover real-space noise fields.

Both methods yield noise realisations strictly consistent with the target spatial correlation structure and underpin uncertainty propagation, model fitting, and significance estimation tasks (Tsukui et al., 2023, Tsukui et al., 2022).

4. Impact on Uncertainty Quantification and Model Fitting

Ignoring spatial correlation leads to systematic underestimation of statistical uncertainties in integrated flux measurements, source detection, or model parameter inference, with errors exceeding 50% once the spatial summation domain includes several beam areas or exceeds the correlation length. For $\chi^2$ model fitting, the standard goodness-of-fit statistic generalizes to

$\chi^2(\theta) = r^T C^{-1} r$

with $r$ the residual between data and model, and $C$ the covariance. In the presence of correlated noise, the effective degrees of freedom shrink, resulting in broader confidence intervals and reduced detection significance (Tsukui et al., 2023).

Proper incorporation of spatial correlation is essential in astrophysical applications such as ALMA and NOEMA photometry, where correlated side lobes and beam structures cannot be neglected, and in quantum devices where spatial noise correlations control decoherence and error rates in multi-qubit operations (Zou et al., 2023, LaForest et al., 2015).

5. Origins and Effects in Superconducting and Quantum Devices

The flux-noise model in superconducting circuits directly incorporates the vectorial coupling $g(r)$ of spins to device geometry. The spatially correlated noise power,

$$P = \langle [\delta \Phi]^2 \rangle = \sum_{i,j} g(r_i)\cdot g(r_j) \left[ \frac{1}{2}\langle\{s_i,s_j\}\rangle - \langle s_i\rangle\langle s_j\rangle \right],$$

is controlled not only by single-spin densities but by their spatial correlations. The correlation length $\xi$ associated with phase transitions in the spin bath alters the scaling of noise power with device geometry, allowing for suppression strategies via engineered spin order (e.g., poloidal alignment, inter-surface ferromagnetic correlations) (LaForest et al., 2015).

Random telegraph-like macro-spin switching in defect clusters, governed by superparamagnetic transitions and Glauber/Ising dynamics, provides a natural origin for observed $1/f^\alpha$ noise spectra in SQUID flux noise, with exponent $\alpha\sim0.8$ emerging from a broad range of fluctuator lifetimes (De, 2017).

6. Quantum Implications: Entanglement Generation and Decoherence

In quantum information processors, spatially correlated flux noise couples qubits via both classical and quantum channels. The noise cross-spectral density $$S_{ij}(\omega) = A_{ij}/|\omega| \exp(-|r_i - r_j|/\xi)$$ quantifies both Markovian and non-Markovian effects. In the deep quantum limit, cross correlations induce long-range Ising coupling $J^z_{ij}$ and can generate entanglement among nonlocally separated qubits. Contrary to purely classical correlated noise, which only produces correlated dephasing, quantum correlations in the noise lead to both relaxation and coherent interaction terms, including Dzyaloshinskii-Moriya and exchange interactions under transverse drives (Zou et al., 2023).

Time-dependent dephasing rates, given by

$\Gamma^z_{ij}(t) = \int_0^t ds\, \gamma^z_{ij}(s),$

exhibit non-monotonic behavior characteristic of non-Markovian memory, resulting in partial restoration of lost entanglement and nontrivial temporal dynamics in metrics such as concurrence. Effective error mitigation and protocol design in such systems require quantitative incorporation of the measured or modeled $S_{ij}(\omega)$ and spatial correlation length $\xi$.

7. Practical Implementation and Recommendations

Robust spatially correlated flux-noise modeling mandates empirically driven estimation of the noise ACF from emission-free, instrumentally representative regions. Coverage should extend over multiple beam widths or correlation lengths, with explicit masking of signal-bearing regions. The measured ACF, once normalized to the per-pixel variance, serves as the foundation for covariance matrix construction, generation of correlated noise cubes for error propagation, and correct evaluation of model uncertainties.

In astronomical applications, direct ACF-based uncertainty computation and Monte Carlo validation (e.g., via the ESSENCE tool) are recommended; block-averaging or “per-beam” scalings are inadequate and may lead to overstated confidence. For spectral cubes, the ACF should be measured separately per channel to account for any frequency dependence in noise correlation. In quantum and superconducting systems, spatial engineering and characterization of the noise environment (e.g., control of cluster correlations or spin texture alignment) can achieve tangible suppression of flux noise and enhanced control over quantum coherence (Tsukui et al., 2023, LaForest et al., 2015, De, 2017).

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The spatially correlated flux-noise model is thus a multiscale, physically motivated statistical framework critical for rigorous uncertainty quantification, significance estimation, and device optimization in both astronomical and quantum technological contexts (Tsukui et al., 2023, Tsukui et al., 2022, Zou et al., 2023, LaForest et al., 2015, De, 2017).