Correlated Long-Range Residual Mixer
Implementation & benchmark of arXiv:2609.04091 — Effective Hamiltonian description on monitored Majorana chains: correlated power-law hoppings and unconventional entanglement scaling
Source paper: Effective Hamiltonian description on monitored Majorana chains: correlated power-law hoppings and unconventional entanglement scaling arXiv:2609.04091 ⓘ · analyzed Sep 4, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Replace an iid local or randomly sparse residual mixer with a distance-decaying long-range operator whose edge magnitudes are correlated through a shared latent Gaussian field. The paper predicts that these correlations qualitatively change low-energy spectral scaling and increase multiscale information propagation relative to iid long-range weights. Apply the operator as a spectrally normalized residual block so that the benefit comes from correlated scale coverage rather than uncontrolled amplification.
Formulas
Mathematical statement
The paper studies effective Majorana Hamiltonians with random long-range hopping terms, schematically \(H=i\sum_{i<j}t_{ij}\gamma_i\gamma_j\), where \(\gamma_i\) are Majorana operators and hopping magnitudes \(|t_{ij}|\) decay algebraically with distance while possessing nontrivial correlations. Its finite-size signatures are \(\Delta(L)\propto L^{-z}\) with \(1<z<2\), and entanglement \(S(L)\propto[\ln L]^2\) for correlated hoppings, versus approximately \(S(L)\propto\ln L\) for uncorrelated hoppings. Transfer this by defining a neural mixer \(M\) with \(M_{ij}=\sigma_i\sigma_j\epsilon_{ij}(1+|i-j|)^{-\alpha}\), where \(\alpha>0\) is a distance exponent, \(\epsilon_{ij}\) is a zero-mean sign variable, and \(\sigma_i=\exp(g_i-\operatorname{Var}(g)/2)\) is a positive correlated amplitude field. Generate \(g\) with Fourier power spectrum \(P_g(k)\propto(k+k_0)^{-\beta}\), where \(\beta\) controls correlation strength. Use \(x^{+}=x+\eta M\phi(x)\), with \(\eta\|M\|_2\le c<1\). The transferred predictions are an anomalous mixer spectral-gap exponent \(1<z<2\) and a multiscale participation entropy closer to \([\ln L]^2\) than \(\ln L\).
Implementation notes
1. Integration point: implement this as a drop-in replacement for the token-mixing matrix in a residual Transformer, MLP-Mixer, or linear state-space block. For sequence length L, construct distances d(i,j)=|i-j|, sample a low-frequency latent vector g by drawing white noise in Fourier space and multiplying mode k by (k+k0)^(-beta/2), then apply an inverse FFT. Set sigma_i=exp(g_i-0.5*Var(g)), sample a fixed random sign matrix epsilon, multiply by (1+d)^(-alpha), and normalize the operator. 2. Pseudocode: g=ifft(noise*((abs(k)+k0)**(-beta/2))).real; sigma=exp(g-0.5*var(g)); M=epsilon*(sigma[:,None]*sigma[None,:])/(1+dist)**alpha; M=M/spectral_norm(M); x=x+eta*M@gelu(W@x). Estimate the norm using 3-10 power iterations and set eta=c/(norm+1e-8), initially c=0.5. 3. Computed from the paper: use algebraic distance decay and compare correlated against iid amplitudes. Estimated empirically: spectral gap, singular-value density, gradient norms, and a propagation-entropy proxy. Define the gap as 1-s2/s1, where s1 and s2 are the two largest singular values, or use the smallest nonzero eigenvalue of M^T M. 4. First cheap experiment: train 4-layer residual token mixers on CIFAR-10 patches or a character-level next-token task, comparing local mixing, iid long-range weights, and correlated long-range weights at matched parameter count and spectral norm. Sweep L in {64,128,256,512} and beta in {0,0.5,1,2}. The falsifiable prediction is that correlated models have a fitted finite-size exponent z between 1 and 2, while iid controls do not show the same anomalous scaling; their participation entropy should fit a*(log L)^2+b with lower residual error than a*log L+b for beta at least 1. Training should remain stable when eta*||M||2 <= 0.5, while removing normalization should cause exploding residual Jacobians near eta*||M||2 approximately 1.
Verification
Stage 1 · Toy mechanism gate: Failed ✗
Stage 2 · Mechanism transferred to benchmark: Not tested
Stage 2 · Practical benchmark result: Not run
Methodology: Toy-system gate first; the benchmark stage runs only after a pass. How verification works
Stage 1 — Mechanism check agent confidence 9/10
Built a readable correlated long-range residual mixer with Fourier-generated latent amplitudes, algebraic distance decay, random signs, spectral normalization, and propagation diagnostics. The math check confirmed ||M||2=1 after normalization and the residual amplification matched the bound 1+c; removing normalization caused catastrophic growth. However, the correlated mixer did not clearly outperform iid: fitted gap exponents were z=1.884 versus 1.818, while both controls had 1<z<2, and the squared-log entropy fit was better for both but had higher residual error for correlated (0.0237) than iid (0.0138).
- Agent confidence
- 9/10
- Baseline
- iid long-range: z=1.818, R2=0.949, entropy RSS(log L)=0.04321, RSS((log L)^2)=0.01378; unnormalized c=0.5 final norm=2.70e26
- Idea
- correlated long-range beta=1.5: z=1.884, R2=0.963, entropy RSS(log L)=0.04546, RSS((log L)^2)=0.02367; normalized ||M||2=1.000, c=0.5 residual amplification=1.500
Limitations: This is a small dense-matrix toy study, not a trained Transformer/MLP-Mixer task. The propagation entropy is only a proxy for entanglement, system sizes and realization counts are limited, and no GPU or matched-FLOP training benchmark was performed.
How to run: python3 experiment.py
Verdict computed by deterministic test code from paired-seed statistics — not by the language model.
Artifacts
- experiment.py 5.1 KB View Raw
- report.md 1.4 KB View