Population dynamics theory for noisy random circuits: out-of-time ordered correlators and circuit fidelity

Random circuits were used to realize a quantum computation challenging the capacity of classical computers, Nature, 574, 505, (2019). Random circuits are also subsequently used as a metrology tool in NISQ devices. Both these applications rely on convergence to the universal unitary random matrix statistics and the associated Porter-Thomas distribution. However, depth of practically implemented circuits is often linear in system size and therefore convergence to Porter-Thomas needs to be analyzed carefully especially in presence of noise. In this work we address such intermediate depth dynamics theoretically. We developed a mapping of quantum dynamics in random unitary circuits to the classical random process akin to the population dynamics in biology. This population dynamics is defined by classical rules of propagation, creation and annihilation of "charges", invariants under rotations in the space of operators. These rules are derived from the entangling two qubit gate of the random circuit. Population dynamics predicts the dynamics at all time scales between an initial product state and the Porter-Thomas limit. It also accounts for the effect of noise and errors in the hardware. We predict dynamics of out-of-time order correlators (OTOCs), Loschmidt echo (LE) and linear cross entropy (lXEB). Theoretical predictions for OTOCs demonstrate good agreement with 53-qubit experiments.