# Phase-only quantum generative flow

- ID: 230
- Canonical URL: https://synthcore.org/idea/230/phase-only-quantum-generative-flow
- API JSON: https://synthcore.org/api/idea/230.json
- API Markdown: https://synthcore.org/api/idea/230.md
- Verification status: mechanism_failed
- Source: [arXiv:2608.27216](https://arxiv.org/abs/2608.27216)
- Category: sampling
- Solves: sample-efficiency, stability, accuracy
- ML areas: diffusion-sampling, vae, training-dynamics
- Math tags: pde, dynamical-systems, geometry, probability, optimization
- Ratings: usefulness 6/10; difficulty 7/10; novelty 6/10

## Idea description

Replace a time-dependent neural velocity field with a neural initial phase whose evolution is determined by the Madelung equations. Particles are sampled once from a reference density and then moved deterministically along the characteristic velocity field, while the quantum potential supplies a density-dependent smoothing and curvature correction.

## Mathematical statement

The paper starts from the free-particle Schrödinger equation $\mathrm{i}\hbar\partial_t\Psi=-(\hbar^2/(2m))\Delta\Psi$, where $\Psi(x,t)\in\mathbb{C}$ is the wave function, $\hbar>0$ is the reduced Planck constant, $m>0$ is the mass, $x\in\mathbb{R}^d$, and $t\in[0,T]$. Write $\Psi=\sqrt{\rho}\exp(\mathrm{i}S/\hbar)$, with density $\rho(x,t)=|\Psi(x,t)|^2$ and phase $S(x,t)\in\mathbb{R}$. The Madelung system is the continuity equation $\partial_t\rho+\nabla\cdot(\rho v)=0$ and Hamilton-Jacobi equation $\partial_tS+\frac{\|\nabla S\|^2}{2m}+Q[\rho]=0$, with velocity $v=\nabla S/m$ and Bohm quantum potential $Q[\rho]=-(\hbar^2/(2m))\Delta\sqrt{\rho}/\sqrt{\rho}$. If $\rho$ stays positive and the characteristic ODE $\dot X_t=v(X_t,t)$ has a unique global flow, then $X_t\sim\rho(\cdot,t)$ whenever $X_0\sim\rho(\cdot,0)$. The quantum term is related to the variational derivative of Fisher information $I(\rho)=\int\|\nabla\log\rho(x)\|^2\rho(x)\,dx$. The ML adaptation parameterizes only $S_\theta(x,0)$ and optimizes the terminal law $\rho_\theta(\cdot,T)$ against a target distribution.

## Key formulas

- $$\mathrm{i}\hbar\,\partial_t\Psi=-\frac{\hbar^2}{2m}\Delta\Psi,\qquad \Psi(x,0)=\Psi_0(x),\qquad \|\Psi_0\|_{L^2}=1.$$
- $$\Psi=\sqrt{\rho}\,e^{\mathrm{i}S/\hbar},\qquad \partial_t\rho+\nabla\!\cdot\left(\rho\frac{\nabla S}{m}\right)=0,\qquad \partial_tS+\frac{\|\nabla S\|^2}{2m}+Q[\rho]=0,$$
- $$Q[\rho](x,t)=-\frac{\hbar^2}{2m}\frac{\Delta\sqrt{\rho(x,t)}}{\sqrt{\rho(x,t)}},\qquad \dot X_t=\frac{1}{m}\nabla S(X_t,t),\quad X_0\sim\rho_0\Longrightarrow X_t\sim\rho_t.$$
- $$\theta^*=\arg\min_\theta\left[\mathcal{D}\!\left(\{X_\theta^{(i)}(T)\}_{i=1}^N,p_{\mathrm{target}}\right)+\lambda\int_0^T\!\int\rho_\theta\|\nabla\log\rho_\theta\|^2\,dx\,dt\right],\quad S_\theta(x,0)=s_\theta(x).$$

## Implementation notes

(1) Integration point: implement a deterministic generative sampler replacing a diffusion reverse process or continuous normalizing flow. Start with a positive Eulerian density field $\rho_t$ on a 1D or 2D grid; represent the initial phase as an MLP $s_\theta(x)$ and set $S_\theta(x,0)=s_\theta(x)$. Initialize $\rho_0$ as a known Gaussian latent density. (2) Pseudocode: sample grid density $\rho_0$ and particles $X_0^{(i)}\sim\rho_0$; set $S\leftarrow s_\theta$; for time steps $k=0,\ldots,K-1$, compute $v_k=\nabla S_k/m$, update particles with RK4, $X_{k+1}=X_k+\operatorname{RK4}(v_k,X_k,\Delta t)$, update the density by a conservative finite-volume continuity solver, compute $Q_k=-(\hbar^2/(2m))\Delta\sqrt{\rho_k}/(\sqrt{\rho_k}+\epsilon)$, and evolve $S_{k+1}=S_k-\Delta t(\|\nabla S_k\|^2/(2m)+Q_k)$. Backpropagate through the solver using a terminal Sinkhorn, MMD, or sliced-Wasserstein loss between $X_K$ and target samples. (3) Computed from the mathematics: the velocity $\nabla S/m$, quantum potential $Q$, continuity update, and deterministic pushforward property. Estimated empirically: target discrepancy, positivity violation, mass conservation, and the Fisher-information coefficient $\lambda$; use $\epsilon=10^{-6}$ in $Q$ and optionally parameterize $\rho=\operatorname{softplus}(r)+\epsilon$. (4) First experiment: train on a 2D standard Gaussian source to an asymmetric bimodal Gaussian-mixture target, using a 3-layer MLP phase, $K=64$ steps, and compare with a neural ODE flow-matching model having the same parameter count and function evaluations. Measure terminal sliced-Wasserstein distance and MMD, mode coverage, trajectory norm, Jacobian collapse, and wall-clock sampling cost. Success is lower mode dropping at equal solver steps or a smoother loss curve with no exploding trajectories. Also verify that empirical particle histograms match the PDE density and that total mass remains within $10^{-4}$ of one.

## Verification

- Status: mechanism_failed
- Mechanism evidence: yes
- Mechanism confirmed: no
- Verdict: Built a self-contained phase-only quantum-flow MVP with spectral Schrödinger evolution, Madelung residual verification, deterministic particle transport, and a flow-matching baseline. The core equations were verified to machine precision, with continuity residual 1.11e-14 and Hamilton-Jacobi residual 8.78e-14, but the learned phase sampler did not show the promised generative advantage: its terminal W1 and MMD were substantially worse than the baseline and it was slower. The particle-density L1 discrepancy was about 0.195 in the small finite-particle check.

### Mechanism check

- Verdict: Built a self-contained phase-only quantum-flow MVP with spectral Schrödinger evolution, Madelung residual verification, deterministic particle transport, and a flow-matching baseline. The core equations were verified to machine precision, with continuity residual 1.11e-14 and Hamilton-Jacobi residual 8.78e-14, but the learned phase sampler did not show the promised generative advantage: its terminal W1 and MMD were substantially worse than the baseline and it was slower. The particle-density L1 discrepancy was about 0.195 in the small finite-particle check.
- Confidence: 9/10
- Limitations: Only a 1D toy problem was tested. The phase model was optimized against grid density with a small fixed-step spectral solver rather than end-to-end particle Sinkhorn training. No 2D experiment, equal-FLOP tuning study, positivity stress test, long-horizon stability study, or larger-scale comparison was performed.

## Artifacts

- [phase_quantum_mvp.py](https://synthcore.org/code/1207/phase_quantum_mvp.py)
- [report.md](https://synthcore.org/code/1207/report.md)
- [Download all files as ZIP](https://synthcore.org/download/1207)

## Disclaimer

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