Quantum measurement-based feedback simulation of complex dynamics of mean-field $p$-spin models

We study a method for simulating the nonlinear dynamics of many-body spin systems based on measurement-based feedback. We focus on $p$-spin models describing an Ising-like model on a completely connected graph with $p$-body interactions. These models exhibit diverse critical phenomena. For $p=2$ this recovers the Lipkin-Meshkov-Glick (LMG) model, exhibiting a continuous second-order phase transition between paramagnetic and ferromagnetic phases. For $p>2$, the phase transition is a first order and discontinuous. Our protocol considers the collective spin of an ensemble on $N$ qubits, and approximates the dynamics by weakly measuring one projection of the collective spin, followed by unitary evolution conditioned on the measurement outcome [1]. We numerically explore a variety of dynamical properties of phase transitions for different values of $p$, including our ability to recover the mean-field dynamics, and aspects of spontaneous symmetry breaking induced by the measurement. We characterize the simulated behavior in terms of the number of particles $N$, and study how the dynamics approaches the classical limit. Finally, we propose a possible experimental implementation of our $p$-spin simulation using an atom-light interface. [1] Munoz-Arias et al., arXiv:1907.12606