Multiphoton quantum simulation of the generalized Hopfield memory model

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.

💡 Research Summary

The paper establishes a concrete mapping between multiphoton quantum interference in linear‑optical networks and generalized p‑body Hopfield Hamiltonians, thereby providing a photonic platform for simulating complex associative‑memory models. The authors consider Nₚ indistinguishable photons distributed over M optical modes. Each mode is equipped with a binary phase shifter that can impose a phase of 0 or π, which is identified with an Ising spin σᵢ = ±1. After the phase encoding, the photons propagate through a universal linear interferometer described by a scattering matrix S (or a unitary U). The transition amplitude from an input Fock configuration |c⟩ to an output configuration |k⟩ is given by a permanent of a sub‑matrix of S, and the detection probability Pr(Λ|σ) for a chosen subset Λ of output patterns can be written as a sum over products of the spin variables. This probability is directly proportional to the Boltzmann weight of a p‑body Hopfield Hamiltonian with p = 2Nₚ, i.e. H