Attractor-Keyed Memory

Abstract

Physical selectors (lasers choosing a mode, Ising machines settling on a ground state, condensates occupying a spin state) produce high-dimensional signatures at the moment of decision: full field amplitudes, multimode interference patterns, or scattering responses. These signatures are richer than the winner's index, yet they are routinely discarded. We show that when the signatures are repeatable across trials (stereotyped) and linearly independent across routes, a single linear decoder compiled from calibration data maps them to arbitrary payloads, merging selection and memory access into one event and eliminating the fetch that dominates latency and energy in sparse routing architectures. The construction requires one SVD of measured device responses, which certifies capability and bounds worst-case error for any downstream payload before the task is chosen. Runtime error separates into two independently diagnosable channels, decoding fidelity (controlled by dictionary conditioning $σ_{\min}(Φ)$) and routing reliability (controlled by the margin-to-noise ratio $Δ/T_{\mathrm{eff}}$), each with a distinct physical origin and targeted remedy. We derive the full error decomposition, give Ising-machine selector constructions, and validate the predicted scalings on synthetic speckle-signature simulations across three measurement modalities. No hardware demonstration exists; we provide a falsifiable four-step experimental protocol specifying what a first experiment must measure. Whether real device signatures satisfy stereotypy is the central open question.

Figures (2)

• Figure 1: Attractor-keyed memory versus reservoir computing. Both use a linear readout from a physical state, but differ structurally: reservoirs operate on a continuous driven trajectory, AKM on a discrete set of attractor signatures; reservoir states are input-sensitive (separability), AKM signatures are stereotyped (same pattern each time route $k$ wins); the reservoir decoder is trained on input--output pairs, the AKM decoder is compiled from calibration data via $W = Y\Phi^{+}$; reservoir analyses fold routing-like failures into an overall readout error budget, AKM decomposes into two separately diagnosable channels (decoding fidelity $\sigma_{\min}(\Phi)$, routing reliability $\Delta/T_{\mathrm{eff}}$); reservoir analysis yields a capacity bound, AKM yields a predeployment certification protocol. In AKM the payload $y_k$ need not resemble the attractor state; the attractor is the key, not the stored value.
• Figure 2: Photonic implementation and numerical validation of attractor-keyed memory.(a) Schematic photonic pipeline. A spatial light modulator (SLM) encodes the input $x$; a Fourier mask implements $B_{\mathrm{wide}}$, producing candidate scores. A block-diagonal routing map $R_{\mathrm{route}}$ compresses scores into $K$ competing routes. A spatial photonic Ising machine (SPIM) Pierangeli2019PhotonicIsing_PRL selects the winner $\hat{k}_b$ with margin $\Delta$. The selected mode's field propagates through a scattering medium (diffuser or multimode fiber); a detector array records the speckle signature $\tilde{\phi}_{\hat{k}} \in \mathbb{R}^{M_{\mathrm{sig}}}$. A pre-compiled decoder $W^{(b)}$ maps each signature to output; summing over $B$ blocks yields $y$. (b)--(f) Error decomposition on a controlled speckle-signature model (Monte Carlo, not experiment; $K\!=\!64$, $M_{\mathrm{sig}}\!=\!128$, $D\!=\!16$; $R$: calibration trials per route; $\sigma_{\mathrm{read}}$: readout noise s.d.). (b) Relative reconstruction error $\|W_{\hat{\Phi}}\,\Phi - Y\|_F/\|Y\|_F$ versus calibration trials $R$, where $W_{\hat{\Phi}} = Y\hat{\Phi}^{+}$ is the decoder compiled from the noisy estimate $\hat{\Phi}$ and $\Phi$ is the true mean-signature dictionary; four dictionary types (amplitude $|z|$, intensity $|z|^2$, heterodyne $[\Re\,z;\Im\,z]$, orthogonal); $\sigma_{\min}(\Phi)$ controls convergence rate. (c)--(f) All four panels use the amplitude-speckle dictionary. Solid lines: analytic RMS prediction $\sqrt{\mathbb{E}\|e\|_2^2}$ from the two-channel second-moment decomposition (Supp. Mat., Remark 3, Eq. (S17); see text for modeling assumptions); filled circles: Monte Carlo RMS ($10{,}000$ trials); faint circles: Monte Carlo mean $\mathbb{E}[\|e\|_2]$, which lies ${\approx}\,2.5\%$ below the RMS by Jensen's inequality (Supp. Mat., Remark 1); dashed: expected worst-case bounds obtained by multiplying per-trial bounds (Eq. \ref{['eq:driftbound']} for decoding; $d_Y + \|Y\|_2\,\bar{\delta}/\sigma_{\min}(\Phi)$ for a single misrouted trial) by appropriate probabilities, with $P_{\mathrm{mis}}$ from Eq. \ref{['eq:misbound']}. (c) Routing-dominated regime (varying $T_{\mathrm{eff}}$ at fixed $\sigma_{\mathrm{read}}\!=\!0.03$). (d) Decoding-dominated regime (varying $\sigma_{\mathrm{read}}$ at fixed $T_{\mathrm{eff}}\!=\!0.05$). The two error channels are separable and each matches its predicted scaling. (e) Decoding error under stereotypy violation $\varepsilon_{\mathrm{stereo}}$ (correct routing enforced). (f) Full pipeline at a realistic operating point; the crossover at $T_{\mathrm{eff}}\!\approx\!0.71$ separates the decoding-limited from the routing-limited regime. (g), (h) Dictionary conditioning $\sigma_{\min}(\Phi)$ versus table size $K$ at fixed real aspect ratio $M_{\mathrm{real}}/K$, where $M_{\mathrm{real}}$ is the real signature dimensionality ($M_{\mathrm{sig}}$ for real-valued modalities, $2M_c$ for heterodyne with $M_c$ complex modes). Three measurement modalities, 30 random seeds per point; shaded bands: $\pm 1$ s.d. (g)$M_{\mathrm{real}}/K\!=\!2$. (h)$M_{\mathrm{real}}/K\!=\!4$. From $K\!=\!64$ onward, $\sigma_{\min}$ tracks the Bai--Yin prediction (dashed) with modest finite-size corrections at small $K$. At $K\!=\!1024$ the heterodyne value lies within $1\%$ of the asymptote at both aspect ratios. See Supp. Mat. SM for full panel specifications.