Multiphoton quantum simulation of the generalized Hopfield memory model

TL;DR

The present work shows that combining a system composed of Nph indistinguishable photons in superposition over M field modes, a controlled array of M binary phase-shifters, and a linear-optical interferometer, yields output photon statistics described by means of a p-body Hopfield Hamiltonian of M Ising-like neurons, with p = 2Nph.

Abstract

In the present work, we introduce, develop, and investigate a connection between multiphoton quantum interference, a core element of emerging photonic quantum technologies, and Hopfieldlike Hamiltonians of classical neural networks, the paradigmatic models for associative memory and machine learning in systems of artificial intelligence. Specifically, we show that combining a system composed of Nph indistinguishable photons in superposition over M field modes, a controlled array of M binary phase-shifters, and a linear-optical interferometer, yields output photon statistics described by means of a p-body Hopfield Hamiltonian of M Ising-like neurons +-1, with p = 2Nph. We investigate in detail the generalized 4-body Hopfield model obtained through this procedure and show that it realizes a transition from a memory retrieval to a memory black-out regime, i.e. a spin-glass phase, as the amount of stored memory increases. The mapping enables novel routes to the realization and investigation of disordered and complex classical systems via efficient photonic quantum simulators, as well as the description of aspects of structured photonic systems in terms of classical spin Hamiltonians.

Figures (6)

• Figure 1: a) Standard description of linear optical transformation from an input configuration $\vec{c}$ to an output configuration $\vec{k}$ through a linear scattering matrix $U$. An example here is depicted for ${N_{\rm ph}}=3$ photons and $M=5$ modes. b) Schematic of the mapping of an $p$HM to a photonic system composed of an input state $\ket{\psi}$, a set of $M$ phase-shifters which can have binary values $\phi_i \in \{0, \pi \}$ and map to the $M$ spin states $\sigma_i = \text{exp}(i \phi_i)=\pm 1$, and a scattering matrix $S$. c) Full schematic including the initial Discrete Fourier Transform $U^{\text{DFT}}$ for generating the near-uniform input state $\ket{\psi_\text{DFT}}$. d) Simplified scheme with the set of output states $\Lambda_{\text{FB}}$ given by fully-bunched configurations with all photons exiting in the same output mode. Panel e) reports the "on-chip" experimental scheme, correspondent to c).
• Figure 2: Panels a) and b) report $\mathcal{F(\tau)}$ for various temperatures (see legend), for values of the storage size ration $\alpha=0.0004$ (retrieval regime) and $\alpha=0.02$ (spin-glass phase), respectively. Panel c) compares the Metropolis standard deviation $\sigma_T=\sigma_{\rm Metropolis}$ with the experimental standard deviation $\sigma_{Exp}$ resulting from a limited number of photon pairs per iteration step. $\sigma_{Exp}$ for various "Experimental time per iteration" (ETI) is reported, assuming a source of photon pairs capable of generating 20 million photon pairs per second.
• Figure 3: Phase Diagram of the $4$-Hopfield model realized with a quantum interferometer of $2$-photons on $M=50$ modes. The insets report $P(q)$ and $P(m)$ for the paramagnetic phase (a, b, $\alpha=0.0032$ ; $T= 0.55$), retrieval phase (c, d $\alpha=0.0004$ ; $T= 0.1$), and spin glass phase (e, f $\alpha=0.0032$ ; $T= 0.15$).
• Figure 4: Sketch of the experimental setup. SMF: single mode fiber; DMD : Digital Micromirror Device; S: Sample; OBJ: Objective; $i$ and $j$ represent the populated mode index.
• Figure 5: Values of the coefficient $C_\alpha\simeq t_{\rm MCU}/|\Lambda |$ of the scaling with $\alpha M^{N_{\rm ph}}$ of the computational cost of Monte Carlo Update in the digital simulation of the dynamics.