Gennaro Zanfardino - Università del Salento

Quantum simulation of associative memory and high-order spin-glass models 

We present a unified framework connecting multiphoton quantum interference, generalized Hopfield neural networks, spin-glasses, and quantum computational complexity.

First, we show that systems of indistinguishable photons propagating through programmable linear-optical interferometers naturally generate effective high-order classical spin Hamiltonians [1]. In these architectures, binary phase shifters act as Ising-like neurons, while multiphoton interference produces emergent p-body interactions. In particular, two-photon processes realize effective four-body Hopfield models, enabling the investigation of associative memory, retrieval dynamics, and spin-glass behavior on programmable photonic quantum processors. The resulting phase diagrams reveal transitions between memory-retrieval, spin-glass “black-out”, and paramagnetic regimes as the storage capacity and effective temperature are varied. We also discuss the experimental realization of these phenomena on integrated photonic platforms.

Motivated by these photonic implementations, we then introduce a broader quantum-information framework based on symmetric multiqubit Dicke states with Hamming weight p. We show that Dicke-state encodings combined with polynomial-size quantum circuits generate effective 2p-body spin Hamiltonians whose couplings emerge from quantum interference in fixed-Hamming-weight sectors. This provides an efficient quantum representation of correlated high-order spin-glass models without explicit enumeration of exponentially many interaction terms.

Within this setting, we define the Quantum Energy Evaluation Problem (QEEP), in which a classical spin configuration is encoded through diagonal phase operators and evaluated via quantum measurements after unitary evolution. We show that the threshold version of QEEP with a single-qubit observable is BQP-complete.

These results establish a bridge between photonic quantum simulators, associative neural-network models, disordered many-body systems, and quantum complexity theory, opening new directions for programmable quantum simulations of machine-learning-inspired Hamiltonians and complex spin systems.

References

[1] Zanfardino, Gennaro, et al. "Multiphoton quantum simulation of the generalized Hopfield memory model." Physical Review Letters 136.7 (2026): 070602.