 
Summary: Are Quantum States Exponentially Long Vectors?
Scott Aaronson
I'm grateful to Oded Goldreich for inviting me to the 2005 Oberwolfach Meeting on Complexity Theory.
In this extended abstract, which is based on a talk that I gave there, I demonstrate that gratitude by
explaining why Goldreich's views about quantum computing are wrong.
Why should anyone care? Because in my opinion, Goldreich, along with Leonid Levin [10] and other
"extreme" quantum computing skeptics, deserves credit for focusing attention on the key issues, the ones
that ought to motivate quantum computing research in the first place. Personally, I have never lain awake
at night yearning for the factors of a 1024bit RSA integer, let alone the class group of a number field. The
real reason to study quantum computing is not to learn other people's secrets, but to unravel the ultimate
Secret of Secrets: is our universe a polynomial or an exponential place?
Last year Goldreich [7] came down firmly on the "polynomial" side, in a short essay expressing his belief
that quantum computing is impossible not only in practice but also in principle:
As far as I am concern[ed], the QC model consists of exponentiallylong vectors (possible
configurations) and some "uniform" (or "simple") operations (computation steps) on such vectors
. . . The key point is that the associated complexity measure postulates that each such operation
can be effected at unit cost (or unit time). My main concern is with this postulate. My own
intuition is that the cost of such an operation or of maintaining such vectors should be linearly
related to the amount of "nondegeneracy" of these vectors, where the "nondegeneracy" may
vary from a constant to linear in the length of the vector (depending on the vector). Needless
