This paper introduces novel deterministic quadrature methods for pricing European options under the rough Bergomi (rBergomi) model, a challenging non-Markovian stochastic volatility model. The authors propose two hierarchical approaches: adaptive sparse grids quadrature (ASGQ) and quasi Monte Carlo (QMC), both combined with a Brownian bridge construction and Richardson extrapolation. The Brownian bridge reduces the effective dimension of the integration problem, while Richardson extrapolation reduces the bias from time discretization, allowing coarser time grids. The methods exploit the smoothness of the integrand obtained via conditioning on the fractional Brownian motion path, which yields a Black-Scholes-type formula inside the expectation. Numerical experiments across various parameter sets, including very low Hurst parameters (H=0.05), demonstrate that ASGQ and QMC achieve substantial computational gains over standard Monte Carlo, often by one to two orders of magnitude, for relative error tolerances around 0.1%. The paper also provides numerical evidence that the weak error for both the hybrid and exact simulation schemes converges at order one, justifying the use of Richardson extrapolation. This work is the first to apply deterministic quadrature to rough volatility models, opening a new research direction beyond Monte Carlo for pricing and calibration.