From celestial mechanics to quantum theory of atoms and molecules, perturbation theory has played a central role in natural sciences. Particularly in quantum mechanics, the amount of information needed for specifying the state of a many-body system commonly scales exponentially as the system size. This poses a fundamental difficulty in using perturbation theory at arbitrary order. As one computes the terms in the perturbation series at increasingly higher orders, it is often important to determine whether the series converges and if so, what is an accurate estimation of the total error that comes from the next order of perturbation up to infinity. Here we present a set of efficient algorithms that compute tight upper bounds to perturbation terms at arbitrary order. We argue that these tight bounds often take the form of symmetric polynomials on the parameter of the quantum system. We then use cellular automata as our basic model of computation to compute the symmetric polynomials that account for all of the virtual transitions at any given order. At any fixed order, the computational cost of our algorithm scales polynomially as a function of the system size. We present a non-trivial example which shows that our error estimation is nearly tight with respect to exact calculation.
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