✗ Failed on benchmark 2026

Two-Column Non-Markovian Memory Core

Implementation & benchmark of arXiv:2607.25969 — Solvable Quantum Circuits with non-Markovian Influence Matrices

Usefulness7/10

Difficulty6/10

Novelty7/10

Architecture Memory Accuracy Stability Rnn Ssm World model Training dynamics Dynamical systems Tensor decomposition Statistical mechanics Linear algebra

Source paper: Solvable Quantum Circuits with non-Markovian Influence Matrices arXiv:2607.25969 ⓘ · analyzed Aug 31, 2026

AI-generated research hypothesis, automatically tested. Not peer-reviewed.

Idea description

Replace a Markovian recurrent update with an MPS-valued temporal influence state that couples adjacent pairs of memory sites, mimicking the paper's CDU3 two-column construction. The hidden state retains structured correlations across multiple past time steps while computation remains linear in sequence length and polynomial in the bond dimension, rather than exponential in the memory horizon.

Formulas

$$\lvert L_t\rangle\in\left(\mathbb{C}^{q^2}\right)^{\otimes(2t+1)},\qquad \lvert L_t\rangle=\sum_{s_0,\ldots,s_{2t}}\operatorname{tr}\!\left(A^{s_0}A^{s_1}\cdots A^{s_{2t}}\right)\lvert s_0s_1\cdots s_{2t}\rangle,\qquad \chi\le q^2.$$

$$\lvert L_t\rangle_{\mathrm{Markov}}=\bigotimes_{j=1}^{t}\lvert b_j\rangle,\qquad \lvert L_t\rangle_{\mathrm{CDU3}}=\operatorname{MPS}\!\left(A_0,A_1,\ldots,A_{2t}\right).$$

$$F_t=\operatorname{tr}\!\left(A^{(x_0)}A^{(x_1)}\cdots A^{(x_t)}\rho_0\right),\qquad \|A^{(x)}\|_2\le\alpha<1\ \Longrightarrow\ \|\delta F_{t+r}\|\le\alpha^r\|\delta F_t\|.$$

$$\mathrm{cost}(t)=O(t\,\chi^2p),\qquad \mathrm{memory}(t)=O(t\,\chi p).$$

Mathematical statement

For a local qudit dimension $q$, the paper's influence matrix is a state $\lvert L_t\rangle$ on $2t+1$ temporal sites, each of dimension $q^2$. Markovian one-column or CDU2 constructions factorize into local or pair tensors, whereas CDU3 connects all temporal sites through two columns while remaining an MPS with bond dimension $\chi\le q^2$. Transfer this by defining a neural memory tensor with physical feature dimension $p$, bond dimension $\chi$, and local matrices $A^{(x)}\in\mathbb{R}^{\chi\times\chi}$. The sequence functional is $F_t=\operatorname{tr}(A^{(x_0)}\cdots A^{(x_t)}\rho_0)$. A second pair channel uses $B^{(x_s,x_{s+1})}$ to couple neighboring temporal sites. Spectral normalization $\|A^{(x)}\|_2\le\alpha<1$ gives contraction of the linearized recurrence, so perturbations after $r$ steps satisfy $\|\delta F_{t+r}\|\le\alpha^r\|\delta F_t\|$.

Implementation notes

Integrate the module between the input embedding and prediction head of a small RNN or state-space model. Map each input $x_t$ to $e_t\in\mathbb{R}^p$, construct $A_t=\sum_{a=1}^{p}e_{t,a}A_a$, and update a bond state $h_{t+1}^{(1)}=A_th_t^{(1)}$. Maintain a second channel representing the two-column construction. Form a pair feature $z_t=[e_{t-1};e_t]$, construct $B_t=\sum_b z_{t,b}B_b$, and update $h_{t+1}^{(2)}=B_th_t^{(2)}$. Normalize both states and feed $[e_t;h_t^{(1)};h_t^{(2)}]$ to an output MLP. Pseudocode is: initialize both bond states; for each time step compute the embedding, build $A_t$, update the first state, build $B_t$ from the current and previous embeddings, update the second state, normalize, and predict. The MPS contraction and two-column pair coupling are taken from the paper; spectral normalization and feature projection are adaptations. Estimate each operator norm with one or two power iterations per batch and rescale to target $\alpha$. First test on permuted-MNIST and synthetic delayed-copy sequences with GRU, diagonal SSM, and factorized Markov-memory baselines. Sweep $\alpha\in[0.8,1.2]$ and $\chi\in\{4,8,16\}$. Measure hidden perturbation growth, gradient norms, loss, and wall-clock scaling with sequence length. The quantitative prediction is a stability transition near measured average singular value one: above one, perturbations and gradients grow exponentially with horizon; below one, they decay approximately as $\alpha^r$. The non-Markovian model should gain most on targets requiring two-step or longer temporal correlations, while runtime remains linear in sequence length.

Verification

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

Stage 2 · Practical benchmark result: Significantly worse ✗

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 8/10

Built an MPS-inspired two-column memory core with single-site transfer matrices, adjacent-pair matrices, spectral normalization utilities, and a delayed XOR learning test. The contraction sweep matched the predicted alpha^r ratios to relative error below 1.6e-15 across alpha=0.6..1.25, including the predicted stability boundary at alpha=1. The pair-channel sweep was exactly linear in its coupling lambda, and the gamma*lambda boundary sweep matched predictions to below 2.3e-15; the toy learner reached higher accuracy than the GRU, though this is only a small full-batch CPU test.

Agent confidence: 8/10

Baseline: GRU: accuracy 0.6375, loss 0.6229, 9.65 s
Idea: TwoColumn: accuracy 0.6767, loss 0.5871, 14.01 s; contraction and boundary relative errors <2.3e-15; pair response exactly linear

Limitations: The CUDA attempt encountered a shared-environment cuDNN/system allocation crash, so the final reproducible run used CPU. No permuted-MNIST, long-horizon benchmark, FLOP-normalized comparison, chi sweep, or multi-seed statistical evaluation was performed; the exact math checks use orthogonal transfer matrices and therefore validate the stated formulas rather than broad learned-model behavior.

How to run: CUDA_VISIBLE_DEVICES='' /home/maxwelhelp/main/bin/python3 two_column_memory.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.25969) Sep 2, 2026

Significantly WORSE than the tuned baseline (+329.2%, p=0.0081, wins 0 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0002
Idea mean: 0.0009
Effect (Δ): 0.0007 (+329.2%; negative = idea better)
Wins: 0 / 8 paired seeds
p-value: 0.0081 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±194.1%
Mechanism: Not confirmed ✗
Practical effect: Hurts
Baseline tuning: swept over 3 configs

Limitations:

Only the registered built-in dynamics track was tested. No sequence-forecast track, longer horizon, FLOP-normalized timing, or larger training budget was run.

How to run: python3 stage2_bench.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Aug 31, 2026 latest Failed ✗

Agent confidence: 10/10

The registered dynamics benchmark was run end-to-end with a matched GRU baseline and trained two-column A/B memory system. The idea was significantly worse: mean test MSE 0.0008653 versus 0.0002016 for the baseline, paired delta +0.0006637 and permutation p=0.0081, with zero wins in eight pairs. The trained-model perturbation signature also failed to confirm the claimed contraction bound.

Baseline: GRU lr=0.01: mean test MSE 0.0002016004 over 8 seeds
Idea: Two-column A/B memory lr=0.01 alpha=1.0: mean test MSE 0.0008653016 over 8 seeds; paired delta +0.0006637011, p=0.0081, 0/8 wins

Benchmark result