Linear-2 Eigenvector Quantization

• Eigenvector quantization using the Linear-2 method approximates the ground-state eigenvector of real symmetric matrices with nonpositive off-diagonals by exploiting a linear relationship with the row-sum vector.
• It employs a one-parameter variational optimization to minimize the Rayleigh quotient, achieving errors below 10⁻³ and dramatically lowering computational complexity compared to full diagonalization.
• The method is validated on diverse systems—including RRSMs, Hubbard models, and transverse-field Ising models—demonstrating robust performance across matrix sizes and configurations.

Eigenvector quantization in the form of the Linear-2 method provides an explicit, computationally efficient approach to directly approximate the ground-state eigenvector of real symmetric matrices with non-positive off-diagonal entries—commonly referred to as real random symmetric matrices (RRSM) of this specific type. The procedure leverages an observed linear relationship between the ground-state eigenvector and the row-sums of the matrix, allowing ground-state properties to be captured without full diagonalization. This approach is particularly significant for large-scale quantum and statistical systems modeled in condensed matter physics and related fields (Pan et al., 2019).

1. Class of Matrices and Assumptions

The Linear-2 method applies to $N \times N$ real symmetric matrices $H$ where all off-diagonal elements satisfy $H_{ij} \leq 0$ for $i \neq j$, with diagonal entries $H_{ii}$ arbitrary. The target is the unique ground-state eigenvector $G = (g_1, \ldots, g_N)^T$ associated with the smallest eigenvalue; uniqueness is guaranteed by the Perron–Frobenius theorem under these conditions. No further structure is required: the method applies to both dense and sparse matrices, with random (uniform or Gaussian) or model-derived entries, including systems such as the Hubbard and transverse-field Ising models (Pan et al., 2019).

2. Linear Scaling Law and Ansatz

Define the row-sum vector $S$ (sometimes denoted SME) by

$S_i = \sum_{j=1}^N H_{ij}$

for $i=1, \ldots, N$. Empirically, after normalizing $S$ and the ground-state eigenvector $G$ to unit Euclidean norm ($\|G\|_2 = \|S\|_2 = 1$), a robust linear relationship emerges: $g_i \simeq -S_i.$ For finite $N$, the Linear-2 ansatz introduces a one-parameter affine shift: $$v_i(c) = S_i + c, \quad g(c) = \frac{v(c)}{\|v(c)\|_2}$$ or equivalently,

$g_i = \alpha + \beta S_i$

with $\beta \approx 1$ and $\alpha$ (related to $c$) chosen via variational minimization with respect to the Rayleigh quotient. This ansatz maintains normalization and aligns with the optimal energy estimate (Pan et al., 2019).

3. Justification and Empirical Scaling

Justification for the scaling law arises from the Perron–Frobenius theorem, which ensures all ground-state components $g_i \geq 0$ and guarantees uniqueness. A mean-field-type argument relates minimizing $\langle G|H|G\rangle$ with $g_i \propto S_i$ under the described matrix conditions. Numerical tests on $\mathcal{O}(10^4)$ random RRSMs with $N$ ranging from $10^2$ up to $10^4$—including dense and sparse scenarios with uniform or Gaussian entries—demonstrate that $g_i$ versus $S_i$ shows an almost perfect linearity, slope $-1$, and zero intercept.

The root-mean-square (RMS) deviation

$$\text{rms} = \sqrt{\frac{1}{N}\sum_i (g_i + S_i)^2}$$

drops below $10^{-3}$ at $N \approx 10^3$ and decreases further for larger matrices. The linear correlation persists even with selective large-row rescalings. For diagonally-dominated or banded matrices, the strict $g_i \propto -S_i$ law is dampened but a positive correlation is retained (Pan et al., 2019).

4. Variational Algorithm for Eigenvector Quantization

The Linear-2 method is operationalized via a one-parameter variational optimization, minimizing the Rayleigh quotient with respect to $c$ in the ansatz $g_i = S_i + c$. The procedure is as follows:

• Inputs: $H$ ($N \times N$ real symmetric, $H_{ij} \leq 0$ for $i \neq j$).
• Precomputation:
  • $e \gets (1,1,\ldots,1)^T \in \mathbb{R}^N$
  • $S \gets H e$ (row sums)
  • $D \gets S^T S$
  • $E_1 \gets S^T e$
  • $F \gets e^T e = N$
  • $A \gets S^T(H S)$
  • $B \gets S^T(H e)$
  • $C \gets e^T(H e)$
• Rayleigh quotient:
  • $\text{numerator}(c) = A + 2cB + c^2C$
  • $\text{denominator}(c) = D + 2cE_1 + c^2F$
  • $$E(c) = \frac{\text{numerator}(c)}{\text{denominator}(c)}$$
• Minimization: Iteratively update $c$ (e.g., using Newton's method), stopping when $|\Delta c| < 10^{-8}$, to minimize $E(c)$.
• Final eigenvector: $v_i \gets S_i + c_0$, $g = v / \|v\|_2$, and $E_{\min} \approx g^T H g$.

For typical RRSMs as $N \gg 1$, $c_0 \to 0$ and the relationship $g \approx -S / \|S\|$ holds. The dominant computational cost is forming $S = H e$, an $\mathcal{O}(N^2)$ operation for dense matrices, reducing to $\mathcal{O}(\|H\|_0)$ for sparse inputs. The single-parameter optimization adds negligible cost (Pan et al., 2019).

5. Computational Cost and Accuracy

The Linear-2 method requires:

• $\mathcal{O}(N^2)$ operations for dense $H$ to compute $S$ and a few quadratic forms.
• $\mathcal{O}(\text{nnz})$ operations for sparse $H$ with nnz nonzero entries.
• Memory usage $\mathcal{O}(N^2)$ (dense) or $\mathcal{O}(\text{nnz})$ (sparse).
• Empirical ground-state energy errors $<10^{-3}$ for random $N \geq 100$, with errors diminishing as $N$ increases.

Compared to full diagonalization ($\mathcal{O}(N^3)$), the Linear-2 method offers significant computational savings, especially for large $N$ (Pan et al., 2019).

6. Representative Applications and Performance

The efficacy and generality of the Linear-2 method are demonstrated on several classes of matrices:

Matrix Type	N or Dimension	Observations
Random RRSM (uniform/Gaussian)	$N=100$–$10\,000$	$g_i$ vs $S_i$ linear (slope $-1$, intercept $\approx 0$); RMS $<10^{-3}$ for $N\gtrsim 10^3$
1D 4-site half-filled Hubbard model	$36 \times 36$	$U/t = 0,1$, all off-diagonal $\leq 0$; $c_0 \approx 0.00954$ ($U=0$), $-0.0137$ ($U=1$); $|\Delta E| \approx 0.002$ ($U=0$), $0.0077$ ($U=1$)
1D transverse-field Ising chain	$L=4$ to $14$ (dim $2^L$)	$g_i$ vs $S_i$ linear; $$|E_{\text{scaling}} - E_{\text{exact}}|/|E_{\text{exact}}| < 10^{-4}$$ across all $L$

Empirical performance indicates that accuracy improves further with increasing $N$. The methodology also maintains a strong positive correlation for matrices outside the strict class if diagonal dominance or bandedness is present, with the linear law holding less precisely (Pan et al., 2019).

7. Summary and Scope

The Linear-2 eigenvector quantization method provides a scalable, accurate tool for approximating the ground-state eigenvector of real symmetric matrices with nonpositive off-diagonal entries, with immediate applications in quantum many-body physics and network theory. The technique leverages a universal linear scaling law, justifiable by mean-field reasoning and empirically validated across both random and model Hamiltonians. Its computational advantages are most pronounced in large-scale or high-dimensional settings where traditional diagonalization is infeasible (Pan et al., 2019).