Quantum Flywheel four-page prototype

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Exact normalized partition benchmark and algorithm comparisons with source code and saved instances.

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{\sffamily\bfseries\color{ink}\fontsize{22}{25}\selectfont Exact Normalized Minimum-Information Partitions\par} \vspace{3pt} {\sffamily\fontsize{12}{15}\selectfont Classical and simulated-quantum search in recurrent neural networks\par} \vspace{5pt} {\bfseries Joseph W. Anady}\quad ThatAIGuy Research

Abstract

We executed a reproducible benchmark on 60 synthetic stochastic recurrent networks with 8--14 nodes, exhaustively evaluating 163,140 bipartitions. The target is entropy-normalized stochastic interaction, not an unspecified consciousness score. Exact-MIP recovery was 18.3% for direct Queyranne search, 85.8% for simulated annealing, 68.3% for depth-1 simulated QAOA, and 76.2% for depth-2 simulated QAOA; matched 256-draw uniform sampling reached 35.7%. Annealing was the strongest tested heuristic. QAOA enriched optimal-partition sampling but used an exponentially constructed diagonal cost table, so these results do not establish quantum advantage. The completed output is an auditable reference dataset and search pipeline for a subsequent hardware-oriented study.

1. Research question and scope

Can classical and variational quantum search recover the normalized minimum-information partition (MIP) when an exhaustive reference is available? This operationalizes the information-integration question motivating the author's broader manuscript [6], without reusing its historical scores as new evidence. The prototype evaluates synthetic linear stochastic recurrent neural-network models; it does not train language models, analyze biological recordings, or establish subjective experience. Stochastic interaction is a specific observational information-loss measure [1], not the full IIT 3.0/4.0 formalism.

2. Generated network instances

For $n\in\{8,10,12,14\}$, we generated five independent instances in each of three families: dense, two-module, and sparse-with-ring-backbone (15 per size; 60 total). Initial weights are independent $\mathcal N(0,1/n)$. Modular cross-block weights are multiplied by 0.15. Sparse masks retain entries with probability 0.2, plus the diagonal and two ring neighbors. Nodes are randomly permuted; each weight matrix is rescaled to spectral radius 0.9. With independent unit-covariance innovations,

We solve for the stationary population covariance analytically. Thus the experiment tests partition search without finite-sample covariance-estimation error. Root seed: 20261012; all matrices and seed derivations are retained.

3. Exact normalized objective

For a nonempty proper subset $S$ and complement $\bar S$, define [1,2]

The optimized quantity $J$ is dimensionless; the unnormalized $\Phi_{\rm SI}(S^*)$ is separately saved in nats. All logarithms are natural. Differential-entropy normalization depends on variable units: the innovation scale is fixed, not claimed invariant to rescaling. "Exact" means exhaustive global search using float64 arithmetic and the tolerance stated on page 2, not symbolic exact arithmetic.

4. Computation and comparison protocol

Gaussian evaluation and exhaustive reference

Let $G=W\Sigma$ and $R_S=\Sigma_{SS}-G_{SS}\Sigma_{SS}^{-1}G_{SS}^{\top}$. The implementation uses Cholesky solves and log-determinants, not explicit matrix inversion:

The full-system innovation determinant is one. We pin node 1 to $S$ to remove complement duplicates, leaving $2^{n-1}-1$ valid bipartitions: 127, 511, 2,047 and 8,191 per instance. Enumeration records every $J$, numerator, denominator and optimal partition. Recovery requires $J_{\rm returned}\le J^*+10^{-10}+10^{-8}|J^*|$. No tied optima occurred at this tolerance.

Queyranne: direct normalized-objective search

We use pendant-pair contraction with a deterministic starting order and cache objective values within each search. The minimum candidate is returned without an added local-search step. Queyranne has a guarantee for symmetric submodular functions [3]; that guarantee is not assumed for normalized stochastic interaction. Consequently, its outputs are judged against enumeration, not treated as reference answers. Each run has a fresh on-demand oracle and cannot read the exhaustive table.

Simulated annealing: independently repeated searches

Each of ten independent runs per network uses four random restarts and 512 single-bit proposals per restart. Metropolis acceptance is $\min\{1,e^{-\Delta J/T}\}$. Temperature decreases geometrically from $T_0$ to $10^{-3}T_0$, with $T_0=\max\{\operatorname{SD}(32\text{ random pilot scores}),0.001\}$. All-in-one proposals are rejected and consume a proposal step. The best visited valid partition, including pilot points, is retained. Each run uses a fresh cached oracle; it receives neither the optimum nor the reference table. Maximum budget: 2,048 proposals plus 36 pilot/initial evaluations.

QAOA: exact statevector simulation, finite-shot output

With $m=n-1$ qubits (7--13 here), the circuit alternates diagonal cost phases and transverse-field mixing [4]:

Valid diagonal entries are $J(S)/\max J$; the single invalid all-in-one bitstring has cost 2. Three COBYLA starts minimize the exact penalized expectation, with 160 evaluations per start at $p=1$ and 240 at $p=2$. Selection never uses optimal-state probability. All $p=2$ starts exhausted this budget; convergence is not asserted. After training, ten independent 256-shot batches are sampled, retaining the lowest-cost valid observed partition. Invalid shots are counted, not replaced. Uniform search uses ten batches of 256 draws from valid partitions.

\callout{Resource accounting. This is a dense-diagonal QAOA simulator, not an efficient QUBO encoding or a transpiled hardware circuit. Building its full cost table already performs exhaustive search. QAOA also receives noiseless expectation values during training. The comparison measures output quality at stated, unequal budgets; it does not demonstrate a speedup.}

Implementation checks

Before the main run, Queyranne matched enumeration on 70 weighted-cut and 35 Gaussian-mutual-information controls. QAOA agreed with independent dense matrix exponentials to $4.66\times10^{-16}$ maximum amplitude error. Across the 60 networks, the covariance-equation relative residual was at most $4.54\times10^{-15}$; independent joint-versus-conditional log-determinant checks differed by at most $7.11\times10^{-15}$. Independent uncoupled nodes gave zero loss. Saved trial outcomes were subsequently audited against the saved tables and shots.

5. Measured results

Primary endpoint: exact normalized-MIP recovery. Queyranne has one deterministic attempt per network. Other methods have ten attempts per network, giving 600 attempts each. For QAOA these are ten readout batches from one trained circuit per depth and network, not ten independent trainings. Statistical resampling therefore uses networks, not shots or batches, as the unit.

\begin{center} \renewcommand{\arraystretch}{1.18} \begin{tabular}{lrrrr} \toprule Method & Recovered & Rate & 95% interval & Mean gap\,%\\ \midrule Queyranne, normalized & 11/60 & 18.3% & 10.0--26.7% & 154.83\\ Simulated annealing & 515/600 & 85.8% & 81.5--89.8% & 2.37\\ Simulated QAOA, $p=1$ & 410/600 & 68.3% & 64.8--71.7% & 9.62\\ Simulated QAOA, $p=2$ & 457/600 & 76.2% & 72.3--79.8% & 3.63\\ Uniform, 256 draws & 214/600 & 35.7% & 33.3--38.2% & 86.09\\ \bottomrule \end{tabular} \end{center} \smallnote{Table 1. Gap is $100(J_{\rm returned}-J^*)/J^*$, averaged over all attempts, including failures. Intervals use 10,000 stratified cluster-bootstrap resamples: five networks resampled within each size $\times$ family cell. They describe this synthetic ensemble and fixed protocol, not universal algorithm performance.}

Recovery by network size

\begin{center} \renewcommand{\arraystretch}{1.16} \begin{tabular}{rrrrrrr} \toprule Nodes & Cuts each & Queyranne & Annealing & QAOA 1 & QAOA 2 & Uniform\\ \midrule 8 & 127 & 33.3% & 99.3% & 100.0% & 98.7% & 87.3%\\ 10 & 511 & 20.0% & 91.3% & 86.7% & 84.0% & 42.7%\\ 12 & 2,047 & 6.7% & 82.7% & 59.3% & 74.0% & 10.0%\\ 14 & 8,191 & 13.3% & 70.0% & 27.3% & 48.0% & 2.7%\\ \bottomrule \end{tabular} \end{center} \smallnote{Table 2. Each row contains 15 networks; denominators are 15 for Queyranne and 150 for each repeated method. Increasing QAOA depth helped overall but did not monotonically improve every size group under finite optimization and sampling budgets.}

Measured wall time, with preprocessing visible

\begin{center} \renewcommand{\arraystretch}{1.16} \begin{tabular}{rrrrr} \toprule Nodes & Exact table (ms) & Queyranne (ms) & Annealing (ms) & QAOA 2: table + train (ms)\\ \midrule 8 & 8.32 & 2.94 & 11.50 & 513.46\\ 10 & 32.74 & 4.93 & 20.16 & 547.96\\ 12 & 131.92 & 7.53 & 27.50 & 744.40\\ 14 & 561.38 & 11.64 & 35.00 & 1,528.62\\ \bottomrule \end{tabular} \end{center} \smallnote{Table 3. Medians across instances (annealing: all runs within each size). QAOA includes one full table and all three optimizer starts; readout adds about 0.1 ms per batch. Shared covariance setup and JIT warm-up are excluded. Uniform/readout table-lookup times are not presented as comparable search runtimes.}

Execution and an inspectable example

The 60 instance computations totaled 87.24 seconds, excluding initial validation/JIT warm-up. Execution was serial with one BLAS thread on a Linux x86-64 environment reporting an AMD EPYC 9V74 CPU; Python 3.13.5, NumPy 2.3.5, SciPy 1.17.0 and Numba 0.65.1. Timings are environment-specific.

For the first fixed instance, n08_dense_r00, the exact partition is $\{1,2,3,7\}\mid\{4,5,6,8\}$: $J^*=0.2578092851$, $\Phi_{\rm SI}=2.1867917397$ nats and $K=8.4822070674$ nats. Queyranne instead returns $\{1,2,3,6\}\mid\{4,5,7,8\}$ with $J=0.2788420313$. Every other instance, including unsuccessful searches, is retained in the companion data.

6. Interpretation and limits

Normalization materially changes this benchmark. A separately labeled post-run diagnostic found that Queyranne recovered all 60 unnormalized stochastic-interaction minima, versus 11 normalized minima. Normalization changed the exact minimizing partition in 37/60 networks. An exhaustive post-run check of 6,804,480 elementary submodularity inequalities found violations in all 60 normalized objective tables. Thus the direct normalized search cannot inherit an exactness guarantee merely from success on a related objective. These post-run diagnostics leave the primary results unchanged.

Quantum sampling enrichment is not quantum advantage. Depth-2 QAOA exceeded uniform recovery by 40.5 percentage points (paired bootstrap interval 36.7--44.3), but annealing exceeded it by 9.7 points (5.5--13.7). The median optimal-state probability was 7.93 times the valid-uniform probability, with some instances below uniform. The dense cost table, classical optimizer, exact expectations and statevector simulation are substantial resources. There was no QPU execution, noise model, efficient cost-oracle synthesis, or BlueQubit cloud job in this prototype.

Scientific boundary. The result concerns this specified Gaussian, entropy-normalized bipartition problem. It does not establish a general hardness theorem, an efficient hardware implementation, performance on trained nonlinear networks, a measurement of full IIT $\Phi$, or evidence of machine consciousness. Sampling enrichment is not a substitute for full resource accounting.

7. Proposed three-month extension

The completed prototype supplies the exact-answer testbed for a proposed quantum-optimization study, aligned with Quantum Flywheel's three-month research structure [5]. Compute should be used to test a genuinely implementable encoding, not simply to enlarge the current exponential lookup table.

Month 1 — encoding and controls. Develop sparse Ising/QUBO surrogates or an explicitly costed reversible numerical oracle. Measure distortion against exact normalized tables and retain the true objective for scoring. Freeze held-out seeds; add normalization and parameter-budget controls.

Month 2 — scaling and robustness. Extend exact references where feasible, including 16--20-node instances. Compare deeper QAOA, annealing and stronger classical methods at matched budgets. Measure optimizer variability, noise, state-preparation and phase-oracle costs; heuristic agreement is not an exact reference.

Month 3 — hardware-ready evidence. For viable encodings, compile small circuits, report two-qubit gate counts and depth, and run matched simulator/QPU tests subject to granted access. Deliver the benchmark release and preprint, including negative findings. Advance only on full-pipeline cost and held-out accuracy.

8. Reproducibility and sources

AI assistance covered implementation, experiment orchestration, numerical checks and report assembly. The companion bundle contains the frozen protocol, source code, all 60 weight/covariance matrices, every objective table, optimized angles, optimizer traces, sampled bitstrings, 2,400 trial records, and validation artifacts. The main implementation and protocol were hashed before execution. Run python code/benchmark.py in a fresh result directory, then python code/analyze.py; existing results are preserved. No manuscript benchmark numbers were imported into this experiment.

\vspace{1pt} {\fontsize{8.5}{10.3}\selectfont\setlength{\parskip}{2pt} [1] J. Kitazono, R. Kanai and M. Oizumi. Efficient Algorithms for Searching the Minimum Information Partition in Integrated Information Theory. Entropy 20, 173 (2018). DOI: 10.3390/e20030173.\par [2] A. B. Barrett and A. K. Seth. Practical Measures of Integrated Information for Time-Series Data. PLoS Comput. Biol. 7, e1001052 (2011). DOI: 10.1371/journal.pcbi.1001052.\par [3] S. Hidaka and M. Oizumi. Fast and exact search for the partition with minimal information loss. PLoS ONE 13, e0201126 (2018). DOI: 10.1371/journal.pone.0201126.\par [4] E. Farhi, J. Goldstone and S. Gutmann. A Quantum Approximate Optimization Algorithm. arXiv:1411.4028 (2014).\par [5] BlueQubit. Quantum Flywheel grant-program announcement. Official program release, 14 September 2026.\par [6] J. W. Anady. Measurable Artificial Consciousness Through Fibonacci Dynamics: A Mathematical Framework with Empirical Validation. Author-supplied 17-page manuscript; background motivation only.\par }