  We study the influence of conservation laws on entanglement growth. Focusing
on systems with U(1) symmetry, i.e., conservation of charge or magnetization,
that exhibits diffusive dynamics, we theoretically predict the growth of
entanglement, as quantified by the Renyi entropy, in lattice systems in any
spatial dimension d and for any local Hilbert space dimension q (qudits). We
find that the growth depends both on d and q, and is in generic case first
linear in time, similarly as for generic systems without any conservation laws.
Exception to this rule are chains of 2-level systems where the dependence is a
square-root of time at all times. Predictions are numerically verified by
simulations of diffusive Clifford circuits with upto 10^5 qubits. Such
efficiently simulable circuits should be a useful tool for other many-body
problems.
