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My questions:

  1. What does discarding the second register mean for the standard approach of hidden subgroup algorithm?
  2. Why does discarding let the first register end up in a mixed state?

My understanding so far:

I am trying to understand the effect of discarding the second register in the standard approach of hidden subgroup algorithm using this paper as resource. I will be using $S_5$ as my group of interest.

It is said in the paper that,

enter image description here

I assume $f$ maps $g$ to it's irrep. So, if we take the case of $S_5$, assuming an automorphism group of a $5$-node graph is hidden inside $S_5$, $|G| = |S_5| = 120$. Both $|g\rangle$ and $|f(g)\rangle$ are $120$ element column vectors.

So, when it is said that the second register is discarded what it actually means? Let's take the case of trapped ion qubits. When we say we discard the second register, does it mean that we will not do any unitary transformation or measurement on them anymore but they will always be there? In the Wikipedia page of Simon's problem the second register is measured.

I also don't understand why discarding (whatever it means) will let the first register end up in a mixed state. Shouldn't it be in a pure quantum state?

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Discarding in this context can be carried out by performing a Partial Trace. The system you're interested in is that of the first register. If you want to know only its state, as opposed to the combined state of the two registers, then you take the density matrix of the whole system and trace out the second register. The resultant state will, in this case, be mixed. What you're seeing at work is more or less purification in reverse. The resultant mixed state is the reduced state of some other pure system, namely the system comprised of the two registers.

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  • $\begingroup$ I have started with a pure state $$\frac{1}{\sqrt{|G|}} \sum_{g \in G} |g, f(g)\rangle$$. Why or how would I take a partial trace on it? $\endgroup$ – Omar Shehab Dec 19 '15 at 14:30
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    $\begingroup$ Partial traces are covered in all the standard texts on Quantum computation and information. I'm not even going to attempt to explain them better than they are there. $\endgroup$ – Logan Mayfield Dec 19 '15 at 16:00
  • $\begingroup$ that's correct. Moreover it is also true that we can treat a pure state as a mixed state with probability 1. Are you suggesting that I would consider $\frac{1}{\sqrt{|G|}} \sum_{g \in G} |g, f(g)\rangle$ as a mixed state i.e. $\frac{1}{|G|} \sum_{g \in G} |g, f(g)\rangle \langle, f(g)I $ and take the partial trace on the second register? $\endgroup$ – Omar Shehab Dec 19 '15 at 16:07
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    $\begingroup$ yes. that's the idea. $\endgroup$ – Logan Mayfield Dec 19 '15 at 16:28
  • $\begingroup$ is that the only way? What could be other alternatives? $\endgroup$ – Omar Shehab Dec 19 '15 at 16:47

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