We study algorithms for approximation of the mild solution

of stochastic heat equations on the spatial domain ]0,1[^d.

The error of an algorithm is defined in L_2-sense.

We derive lower bounds for the error of every algorithm

that uses a total of N evaluations of one-dimensional components

of the driving Wiener process W. For equations with additive

noise we derive matching upper bounds and we construct

asymptotically optimal algorithms. The error bounds depend on

N and d, and on the decay of eigenvalues of the covariance of W

in the case of nuclear noise. In the latter case the use of

non-uniform time discretizations is crucial.