Filtered Power Iteration

• Filtered power iteration is a numerical method that applies functional filters to matrix powers to extract desired spectral components and suppress unwanted modes.
• It employs polynomial, rational, and Chebyshev filtering techniques to address ill-posed linear inverse problems and improve convergence in large-scale eigenvalue computations.
• The approach extends to quantum and high-performance computing, offering robust regularization, preconditioning, and noise resilience in complex computational tasks.

Filtered power iteration refers to a broad class of numerical methods in which polynomial, rational, or operator-valued "filters" are systematically applied to matrix (or operator) powers within iterative algorithms to extract selected spectral components, suppress unwanted eigenmodes, stabilize convergence, or regularize ill-posed problems. The "filter" is typically a function designed to amplify contributions from specific regions of the spectrum, while suppressing the impact of noise, numerical error, or ill-conditioned components. Such methods have found central application in solving large-scale eigenproblems, linear inverse problems, and related numerical linear algebraic tasks, including in quantum and high-performance computing contexts.

1. Mathematical Foundations and Filter Representations

Filtered power iteration generalizes classical power or subspace iteration—iterative methods where repeated application of a matrix operator $A$ to a vector amplifies dominant eigenspace components—by composing each power with an explicit functional transformation ("filter") of $A$. Formally, the basic filtered power iteration at step $k$ applies a polynomial or rational operator $p_k(A)$ to an initial seed $x_0$:

$x_{k} = p_k(A) x_0$

The filter polynomial $p_k(\lambda)$ is constructed so that $|p_k(\lambda)|$ is maximized (or nearly constant) on the region of interest (e.g., containing the desired eigenvalues), and is small elsewhere. For linear inverse problems with $A$ severely ill-conditioned, spectral filtering explicitly suppresses noise-exacerbating components:

$$x_{\text{filt}} = \sum_j \phi_j \frac{u_j^{T} b}{\sigma_j} v_j$$

where $\phi_j$ are filter factors, and $(u_j, \sigma_j, v_j)$ are SVD components of $A$. The filter factors depend on the filter polynomial or rational function, and their structure determines regularization properties, noise amplification, and convergence behavior.

2. Filtered Power Iteration in Ill-Posed Linear Inverse Problems

For ill-posed problems such as $\min_x \|Ax - b\|$ with rapidly decaying singular values, truncated iteration (as in Landweber or Richardson methods) or Tikhonov-type regularization can be recast within the filtered iteration framework (Nagy et al., 12 Sep 2024).

Landweber Iteration as Filtering

Classic Landweber iteration:

$x^{(k)} = x^{(k-1)} + \zeta A^T (b - A x^{(k-1)})$

produces after $k$ steps

$$x^{(k)} = \sum_j \left[1 - (1-\zeta^2 \sigma_j^2)^k\right] \frac{u_j^T b}{\sigma_j} v_j$$

The associated filter factor $\phi_j^{(k)} = 1 - (1 - \zeta^2 \sigma_j^2)^k$ initially enhances large singular value contributions while suppressing noise from small singular value modes ("semi-convergence"). Preconditioned Landweber, with a Tikhonov-type preconditioner ($M^T M = A^T A + \alpha^2 I$), further accelerates and regularizes this process:

$$x^{(k+1)} = x^{(k)} + (M^T M)^{-1} A^T (b - A x^{(k)})$$

This yields a more favorable filter factor $\psi_j^{(k)}$ that approaches the Tikhonov regularized solution as $k \to \infty$.

Iterative Refinement as a Filtered Solution

Iterative refinement (IR) on the Tikhonov-regularized system fits into the filtered view as a recursively defined combination of preconditioned Landweber iterates. The IR filter factors $\phi_j^{(k)}$ satisfy a recurrence linking initial filtering through $\psi_j^{(k)}$ to a Tikhonov fixed point:

$$\phi_j^{(k)} = \psi_j^{(k)} - \frac{\alpha^2}{\sigma_{M,j}^2 + \alpha^2} \sum_{i=0}^{k-1} \left(1 - \frac{\sigma_{A,j}^2}{\sigma_{M,j}^2 + \alpha^2} \right)^i \phi_j^{(k-1-i)}$$

As $k\to\infty$, $$\phi_j^{(k)}\to \sigma_j^2 / (\sigma_j^2 + \alpha^2)$$. This formalizes the relationship of IR with filtered power iteration and explains its regularization and convergence properties, especially in mixed-precision contexts, where explicit filter analysis enables robust use of low-precision arithmetic without catastrophic loss of accuracy (Nagy et al., 12 Sep 2024).

3. Polynomial Filtering and Chebyshev Methods

Chebyshev (or more generally, polynomial) filtering is widely used to accelerate subspace iteration for computing extremal eigenvalues/eigenvectors in large symmetric or Hermitian problems (Kodali et al., 28 Mar 2025, Banerjee et al., 2016). Instead of repeated application of $A$, a high-degree polynomial filter $C_p(A)$—often built from Chebyshev recurrence—amplifies the desired spectral region while suppressing others:

$Y_{k+1} = C_p(A) Y_k$

The Chebyshev polynomial $T_m(x)$, mapped to the spectrum of $A$, provides close-to-optimal separation of the target interval. This approach is used in "Chebyshev filtered subspace iteration" (ChFSI), where efficient recursive evaluation and block orthonormalization are key. In variants such as R-ChFSI, the filter is applied to the residuals, enhancing robustness to inexact matrix-vector products and enabling the use of approximate inverses and low-precision arithmetic (Kodali et al., 28 Mar 2025).

Comparison Table: Standard vs. Residual-based CheFSI

Feature	ChFSI	R-ChFSI
Error propagation	Accumulates	Suppressed in residual
Matrix-vector precision	High required	Low/mixed allowed
Final residual	Limited by error	Matches machine

Chebyshev filtering is also deployed in discontinuous Galerkin DFT electronic structure calculations, where its block-sparse implementation and spectrum-adapted filtering enable routine large-scale simulations (Banerjee et al., 2016).

4. Rational Filters, Contour Integration, and FEAST-type Methods

Rational filtering, particularly via spectral projectors constructed from contour integrals, underpins methods such as FEAST and its filtered-power variants (Yeung et al., 2019). The core idea is to represent the desired spectral projector by:

$$P_{\Gamma} = \frac{1}{2\pi i} \oint_{\Gamma} (zI - A)^{-1} dz$$

where $\Gamma$ encircles the eigenvalues of interest. Rational approximations to $P_{\Gamma}$ serve as filters, and their numerical realization reduces to solving shifted linear systems.

Enhanced algorithms combine these rational projectors with power or Krylov subspace iteration (“filtered power subspace iteration” or F$_2$P):

• Double filtering with two contour integrals ($\Gamma_L, \Gamma_R$) enables relaxed subspace dimension requirements and alleviates conditioning problems when searching narrow spectral intervals.
• Power/subspace iteration within the filtered subspace amplifies targeted components even for small subspace sizes.

Empirical results confirm that this methodology is robust to subspace undersizing, ill-conditioning, and produces superior eigenpair approximations in large sparse matrix applications (Yeung et al., 2019).

5. Filtered Power Iteration in Modern and Quantum Algorithms

Filtered power iteration principles extend naturally into quantum algorithms via generalized quantum signal processing (GQSP) (Khinevich et al., 15 Jul 2025). In this framework, GQSP constructs arbitrary complex polynomial (and thus, filtered) transforms of a block-encoded Hamiltonian on a quantum computer. Classic power, Lanczos (Krylov), inverse iteration, and folded spectrum methods are all unified as specific filtered operations:

• Quantum Chebyshev/power iteration: $H^n$ filtering via GQSP.
• Quantum inverse iteration: Polynomial or rational approximations to $(H-\epsilon I)^{-n}$.
• Quantum folded spectrum filters: Approximations to $(I - c(H-\epsilon I)^2)^n$.

GQSP quantum circuits encode these polynomials efficiently in gate count and ancilla requirements. Resource analysis demonstrates that quantum filtered power iteration avoids the overhead of Trotter decomposition and achieves high flexibility, scalability, and convergence speed in molecular Hamiltonian benchmarks.

6. Asymptotic and Diagrammatic Analysis

Fourier diagrammatic analysis and state evolution theory (Jones et al., 11 Apr 2024) provide formal justification and deeper understanding of filtered power iteration and related message-passing algorithms. In high-dimensional random matrix settings, diagrammatic expansions show that only treelike structures contribute asymptotically; cycle-containing diagrams become negligible as $n \to \infty$. This underpins the validity of tree-based "cavity" approaches, explains the independence structure in large systems, and justifies widely-applied heuristics in approximate message passing and belief propagation.

A summary of diagrammatic contributions:

Diagram Shape	Asymptotic Contribution
Tree	Independent Gaussian
Forest	Hermite product of Gaussians
Cycle (not tree)	Negligible

In filtered power iteration variants, the tree approximation persists for much longer iteration regimes than in unfiltered power iteration, which is crucial for understanding long-run stability and accuracy in large-scale computational settings.

7. Implications and Practical Guidance

Filtered power iteration methods provide mathematically rigorous, controllable, and efficient techniques for:

• Regularizing ill-posed and inverse problems, especially when noise or low-precision arithmetic is present.
• Extracting selected eigenspaces in large matrices where direct methods are infeasible, and when robustness to subspace undersizing or spectral crowding is needed.
• Accelerating convergence in high-performance or quantum computing applications through polynomial/rational filtering frameworks.
• Leveraging analysis tools from statistical physics to characterize performance and algorithmic behavior in large random settings.

Filter factor analysis enables transparent diagnostics, informs preconditioner and filter design, and supports empirical validation of convergence and regularization properties in both classical and emerging computational paradigms.