  The Gottesman-Knill theorem states that a Clifford circuit acting on
stabilizer states can be simulated efficiently on a classical computer.
Recently, this result has been generalized to cover inputs that are close to a
coherent superposition of logarithmically many stabilizer states. The runtime
of the classical simulation is governed by the stabilizer extent, which roughly
measures how many stabilizer states are needed to approximate the state. An
important open problem is to decide whether the extent is multiplicative under
tensor products. An affirmative answer would yield an efficient algorithm for
computing the extent of product inputs, while a negative result implies the
existence of more efficient classical algorithms for simulating largescale
quantum circuits. Here, we answer this question in the negative. Our result
follows from very general properties of the set of stabilizer states, such as
having a size that scales subexponentially in the dimension, and can thus be
readily adapted to similar constructions for other resource theories.
