Summary: Quantum Computing, Postselection, and Probabilistic
Institute for Advanced Study, Princeton, NJ (USA)
I study the class of problems efficiently solvable by a quantum computer, given the ability to "postselect"
on the outcomes of measurements. I prove that this class coincides with a classical complexity class called
PP, or Probabilistic Polynomial-Time. Using this result, I show that several simple changes to the axioms
of quantum mechanics would let us solve PP-complete problems efficiently. The result also implies, as an
easy corollary, a celebrated theorem of Beigel, Reingold, and Spielman that PP is closed under intersection,
as well as a generalization of that theorem due to Fortnow and Reingold. This illustrates that quantum
computing can yield new and simpler proofs of major results about classical computation.
Keywords: quantum computing, computational complexity, postselection.
Postselection is the power of discarding all runs of a computation in which a given event does not occur. To
illustrate, suppose we are given a Boolean formula in a large number of variables, and we wish to find a setting
of the variables that makes the formula true. Provided such a setting exists, this problem is easy to solve
using postselection: we simply set the variables randomly, then postselect on the formula being true.
This paper studies the power of postselection in a quantum computing context. I define a new complexity