Brownian Random Circuit Dynamics for Efficient Generation of Random Dicke States with applications to Axis-Agnostic Heisenberg-scaling Metrology

Quantum states chosen at random from the symmetric Dicke subspace are useful for precision metrology, as they feature Heisenberg-scaling sensitivity for external fields applied along arbitrary, a priori unknown axes. We propose and analyze a Brownian random circuit model to efficiently generate these random Dicke states. Using the Choi–Jamiołkowski Isomorphism, we map the dynamics onto the thermodynamics of an effective Hamiltonian acting on 2k replicas and study the approach to randomness by computing the spectrum of this effective Hamiltonian using mean field theory and exact diagonalization. The many-body energy gap dictates the timescale required to approach randomness. We present evidence showing that this energy gap is constant as a function of spin size, suggesting that constant-depth circuits are sufficient to generate metrologically-useful random Dicke states in the limit of large spin and in the presence of noise and dissipation.