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Sunday, January 11, 2004

Complexity Class of the Week: PP (by guest blogger Scott Aaronson)

Yeah, I know: PP has already been this weblog's complexity class of the week. But once you've seen how to define PP using super-powerful variants of quantum mechanics, you might never look at Probabilistic Polynomial-Time the same way again! (Then again, you might.)

Let's define PostBQP (or BQP with postselection) as the class of languages L for which there exists a uniform family of polynomial-size quantum circuits such that

  • For all inputs x, the circuit's first qubit has a nonzero probability of being measured '1' at the end of the computation.
  • If x is in L, the second qubit will be measured '1' with probability at least 2/3, conditioned on the first qubit being measured '1'.
  • If x is not in L, the second qubit will be measured '1' with probability at most 1/3, conditioned on the first qubit being measured '1'.
In physics, "postselection" means you throw away all runs of an experiment for which a measurement of some quantity X doesn't yield a desired outcome. (Hopefully, X isn't itself what you're trying to measure - otherwise it's not postselection, it's fraud!) But you can also think of PostBQP as the quantum analogue of the classical complexity class BPPpath (another previous CCW).

Clearly PostBQP sits between BQP and PP. I became interested in PostBQP when I realized that the containment BQP/qpoly in EXP/poly (discussed earlier in this weblog) can be improved to BQP/qpoly in PostBQP/poly.

Exercise 1. Imagine that the gates of a quantum computer only needed to be invertible - not unitary. Since states might no longer be normalized, let's define the probability of measuring a basis state x with amplitude αx to be |αx|2 divided by the sum over all basis states y of |αy|2. Show that we could decide exactly the languages in PostBQP.

Exercise 2. Now imagine that the gates are unitary, but the probability of measuring a basis state x equals |αx|p divided by the sum over all basis states y of |αy|p, where p is a nonnegative real number not equal to 2. Show that we could decide all languages in PostBQP.

I was getting more and more excited about the fundamental new complexity class I'd discovered. Alas:

Exercise 3. Show that PostBQP equals PP.

The moral is that when you make a quantum class too powerful, it turns into a classical class! (Another example of this is NQP = coC=P.)

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