 
Summary: Quantum Lower Bound for Recursive Fourier Sampling
Scott Aaronson
Institute for Advanced Study, Princeton
aaronson@ias.edu
Abstract
One of the earliest quantum algorithms was discovered by Bernstein and Vazirani, for a problem called
Recursive Fourier Sampling. This paper shows that the BernsteinVazirani algorithm is not far from
optimal. The moral is that the need to ``uncompute'' garbage can impose a fundamental limit on e#cient
quantum computation. The proof introduces a new parameter of Boolean functions called the ``nonparity
coe#cient,'' which might be of independent interest.
Like a classical algorithm, a quantum algorithm can solve problems recursively by calling itself as a sub
routine. When this is done, though, the algorithm typically needs to call itself twice for each subproblem to
be solved. The second call's purpose is to uncompute `garbage' left over by the first call, and thereby enable
interference between di#erent branches of the computation. Of course, a factor of 2 increase in running time
hardly seems like a big deal, when set against the speedups promised by quantum computing. The problem
is that these factors of 2 multiply, with each level of recursion producing an additional factor. Thus, one
might wonder whether the uncomputing step is really necessary, or whether a cleverly designed algorithm
might avoid it. This paper gives the first nontrivial example in which recursive uncomputation is provably
necessary.
The example concerns a longneglected problem called Recursive Fourier Sampling (henceforth RFS),
