Post-selection and complexity theory

Question

I read about post-selection and didn't understand the meaning behind this thing. I didn't understand the Wikipedia article well, so what is a simple but understandable explanation of post-selection and how to use it in complexity?

Answer 1

I went ahead and looked to see if there was a good resource that I could provide, but beyond some resources that Aaronson wrote, I could not find any good additional supplements. You should check out what he says and see if that helps. However, I'll do my best to provide a very quick and potentially hand wavy interpretation.

Basic Quantum Computing: (If Needed)

Consider a quantum computer with some number of quantum bits. Before we measure quantum bits, they are in a superposition of many possible output states. When measured, they collapse to a single state that is sampled from a probability distribution of states. In other words, during measurement, some state is "chosen" randomly from all possible states, with some more likely than others to be "chosen".

A quantum computation takes these unmeasured qbits and methodically applies quantum logic gates. Doing this influences that superposition (the probability distribution of final states). When we are done applying quantum gates, we perform a measurement on the entangled qbits (final quantum state). This gives us a result we can use, but all the underlying quantum information is lost, preventing us from measuring again. The quantum computation would have to be repeated from scratch in order to perform another measurement. To account for errors, quantum computations are usually performed multiple times to build up additional confidence in the final result.

Postselection in a Computation:

When computing a decision problem, let us just designate a single quantum bit, $q_1$, as our "answer" where $\lvert1\rangle$ and $\lvert0\rangle$ correspond to accept and reject.

Postselection introduces conditional probability into the mix. We say we want to postselect a quantum bit, $q_1$,for a specified outcome of a second bit, $q_2$. To simplify matters, let us assume that this is the last thing that we do before measuring. Without postselecting, our machine would accept with $\mathbb{P}\left[q_1=\lvert1\rangle\right]$. If we use post selection as described above, we are telling $q_1$ to only be in a superposition of states that work given that $q_2=\lvert1\rangle$. The probability that the computation accepts now becomes $\mathbb{P}\left[q_1=\lvert1\rangle\mid q_2=\lvert1\rangle \right]$. This essentially filters out all possible states that $q_1$ could have been in if $q_2 = \lvert0\rangle$.

If $q_2$ happened to be defined as an indicator of a "good" computation in our experiment, then, by using postselection, we would have thrown out all the "bad" computations.

Note: $\mathbb{P}\left[q_2=\lvert1\rangle\right]$ must be $>0$ or else the conditional probability measure is not well defined.

So in short, postselection gives us the ability to force quantum bits to think that another quantum bit is in a specific state we choose. This lets us filter out a lot of results that we consider to be useless to us when we measure our quantum state. As demonstrated in Aaronson's proof of $\text{PostBQP = PP}$, the ability to postselect provides significant computational advantage over vanilla models of quantum computation.