  We analyze the convergence of higher order Quasi-Monte Carlo (QMC)
quadratures of solution-functionals to countably-parametric, nonlinear operator
equations with distributed uncertain parameters taking values in a separable
Banach space $X$ admitting an unconditional Schauder basis.
  Such equations arise in numerical uncertainty quantification with random
field inputs. Unconditional bases of $X$ render the random inputs and the
solutions of the forward problem countably parametric, deterministic. We show
that these parametric solutions belong to a class of weighted Bochner spaces of
functions of countably many variables, with a particular structure of the QMC
quadrature weights: up to a (problem-dependent, and possibly large) finite
dimension, product weights can be used, and beyond this dimension, weighted
spaces with so-called SPOD weights recently introduced in [F.Y.~Kuo,
Ch.~Schwab, I.H.~Sloan, Quasi-Monte Carlo finite element methods for a class of
elliptic partial differential equations with random coefficients. SIAM J.
Numer. Anal., 50, 3351--3374, 2012.] can be used to describe the solution
regularity. The regularity results in the present paper extend those in [J.
Dick, F.Y.~Kuo, Q.T.~Le Gia, D.~Nuyens, Ch.~Schwab, Higher order QMC
(Petrov-)Galerkin discretization for parametric operator equations. SIAM J.
Numer. Anal., 52, 2676 -- 2702, 2014.] established for affine parametric,
linear operator families; they imply, in particular, efficient constructions of
(sequences of) QMC quadrature methods there, which are applicable to these
problem classes. We present a hybridized version of the fast
component-by-component (CBC for short) construction of a certain type of higher
order digital net.
