Quantum speedups need structure
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We prove the following conjecture, raised by Aaronson and Ambainis in 2008: Let f:{-1,1}^n → [-1,1] be a multilinear polynomial of degree d. Then there exists a variable x_i whose influence on f is at least poly(Var(f)/d). As was shown by Aaronson and Ambainis, this result implies the following well-known conjecture on the power of quantum computing, dating back to 1999: Let Q be a quantum algorithm that makes T queries to a Boolean input and let ϵ,δ > 0. Then there exists a deterministic classical algorithm that makes poly(T,1/ϵ,1/δ) queries to the input and that approximates Q's acceptance probability to within an additive error ϵ on a 1-δ fraction of inputs. In other words, any quantum algorithm can be simulated on most inputs by a classical algorithm which is only polynomially slower, in terms of query complexity.
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