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I'm trying to do something similar to this question, where I want to partially measure the system before getting the output. In particular, say I have 4 qubits. I want to measure two of them, and then get the state vector associated with the other two. I know that I can do this the following way:

  1. Measure the 2 qubits.
  2. Perform a partial trace on those two qubits (so that my output vector only has $2^2$ components instead of $2^4$).
  3. Call the statevector function to get the state on my other 2 qubits.

The problem is that I need to actually get the statevector in Step 2 before performing the partial trace. This means the system has to produce the statevector, taking up a lot of memory and limiting the number of qubits I can scale to.

My question is: Is there a way to collapse the wavefunction and then get the statevector out on only the remaining qubits without having the system calculate the whole statevector beforehand?

I don't know if this is a reasonable question, but I was wondering if it could be done.

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  • $\begingroup$ Saying "Measure the 2 qubits" without specifying basis IMO means single-qubit measurements in computational basis. Is it what you mean? $\endgroup$
    – kludg
    Apr 9 at 2:12
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Short answer, 'no'.

Long answer: I think a lot of mystery about quantum mechanics has the possibility to disappear when we do simulations, like the ones you are suggesting.

For my answer, I will assume that whatever you are going to do with your 4 (or larger) qubit system, it will be at least 'entangled', in the sense that the final state is not a product state (not interesting, in simple words).

The fundamental limitation of using a quantum mechanical description of reality is that one has to use all 2^4 complex numbers to represent the state.

Now, there are multiple ways to answer your question. One would be to call for a no-go sort of argument like this. If what you suggest was possible, we would not require a 2^4 complex numbers representation, which implies that we have an equivalent description of reality that does not require quantum mechanics. But since quantum mechanics is the ultimate theory of the univer..., I would say no-go to this argument.

The other way to answer your question is to ask: Can you represent (and scale it up) all the information stored in 2^N complex numbers, in <2^N complex numbers? I would guess, in general, 'no'.

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  • $\begingroup$ Thanks for the response @quantumdip. I guess what I was thinking about more is the fact that, if I have a quantum state like $|a\rangle |b\rangle |c\rangle$ and I measure the third qubit in the computational basis, I will get a quantum state like $|a\rangle |b\rangle |0/1\rangle$. I could then just look at the first two qubits, which gives me $|a\rangle |b\rangle$. This only requires $2^2$ complex numbers to specify. This was the sort of thing I had in mind. I know you can do things with matrix product states that require less parameters for some states, so that could be an option. $\endgroup$
    – Germ
    Apr 8 at 20:50
  • $\begingroup$ Yes, if you are using a product state as you mention, you can in principle do what you are suggesting. However, perhaps it is imminent to ask: what is your motivation or end goal with this undertaking? Because product states descriptions are within the realm of 'classical' and may not require quantum mechanics at all. $\endgroup$
    – quantumdip
    Apr 9 at 10:33
  • $\begingroup$ My problem does indeed require entangled states, so I do need the machinery of quantum theory to do this. Fortunately, I was able to implement this with tensor networks, so it seems like my problem isn't one anymore. $\endgroup$
    – Germ
    Apr 14 at 18:10

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