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I want to consider a classical computer without any artificial restrictions, which has a single quantum coprocessor with a fixed finite number of qubits. It can perform the following three types of operations:

  1. The quantum coprocessor is able to initialize (all) its qubits to a well defined initial quantum state.
  2. The quantum coprocessor has a fixed number of gates it can apply to its quantum state, among them a (small) generating set for all permutations, and a universal set of quantum gates.
  3. The quantum coprocessor is able to perform "some" quantum measurements, whose (finite number of different) classical results can be read by the classical computer.

The second type of operation looks a bit questionable to me. In practice, the classical computer would probably configure an arrangement of quantum gates first, before it would start a quantum computation on the coprocessor. But does it violate any physical principles, to be able to reliably apply (classically controlled) an arbitrary long sequence of quantum gates? I'm also a bit unsure about the first operation (even so it looks benign), because it erases any previous content of the qubits (thereby violating the no-deletion theorem). The third type of operation on the other hand should definitively be physically realizable, and might even help to avoid the need for the first operation, because the quantum state of the coprocessor could theoretically claimed to be known after a measurement. (But this interpretation of the third operation would also mean that the second and third type of operation together are able to emulate the first operation, hence my doubts about violating physical principles remain.)

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But does it violate any physical principles, to be able to reliably apply (classically controlled) an arbitrary long sequence of quantum gates?

No, it does not. However you must be prepared to require the arbitrarily long sequence of gates, as when you only have a finite set of gates and an uncountable set of possible unitary transformations of qubits, you must commit yourself to approximation. See this note on Wikipedia for some example "universal quantum gates".

I'm also a bit unsure about the first operation (even so it looks benign), because it erases any previous content of the qubits (thereby violating the no-deletion theorem).

Indeed, thermalizing the qubits with some bath so that you can take them down to a common starting state $|00\dots 0\rangle$ is going to form a barrier between two quantum descriptions of the coprocessor: one before, and one after this "reset" operation. However you can get around this theoretical limitation by simply taking the previous quantum computation and arbitrarily adding qubits, so that you cease to talk about the "deleted" ones. If you really want to enforce this "deletion" operation, insert an explicit measurement of those qubits so that you don't forget that they're "erased" from the quantum perspective.

The third type of operation on the other hand should definitively be physically realizable.

Yes, but there is also a theoretical trick you may wish to play with these things: a well-known property of quantum circuits (I think it's even in Nielsen and Chuang) is that if you have a measurement in the middle of your operations, you can always analyze the system by delaying that measurement to the end of the computation. (If you have to evolve this measurement past any logic, I think one just substitutes the equivalent quantum gates while doing this.)

So even if you have these interactive measurements you may simply want to model them as all happening at the end of the computation.

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I want to consider a classical computer without any artificial restrictions,

Then you might have failed already becsuse there are relevant things you didn't specify. The problem is whether you wanted to allow and/or prevent superpositions of different initializations in step one.

  1. The quantum coprocessor is able to initialize (all) its qubits to a well defined initial quantum state.

Yes and no. If, for instsnce, your quibits are the spin of a spin 1/2 particle. You can put each individual qubit into an individual state by measuring the spin along an axis (and then if you get the "wrong" result you can flip it). However there is no way to assign a phase to a single particle, more on that later. But for now, this process leaves each qubit in a general state except each has worse than an unknown phase. The whole collection of all the states is what has the phase and that is unknown.

Since this is unavoidable you could argue that no one expected you to do this. However you haven't been completely general. You could have started with your registers having every possible arrangement of qubits with possible arrangement of quibits assigned an overall magnitude and phase. And that only the superposition lacks an overall magnitude and phase. It's like you can pick a point on the unit sphere in $\mathbb C^{2^n}$ and assign those complex numbers to every possible arrangement of qubits to the individual states.

This is called entanglement. And by assigning a well defined state to every individual qubit you have excluded that possibility and hence potentially limited your computer.

  1. The quantum coprocessor has a fixed number of gates it can apply to its quantum state, among them a (small) generating set for all permutations, and a universal set of quantum gates.

Again, if you have a classical controller that uses the generating set to get the arbitrarily permutations then it might not handle the superpositions of different initial qubits properly. It is hard to tell since you didn't specify it. If the quantum computer is good st holding qubits (which is in my understanding one of the biggest problems) then the classical computer should be able to set up a generating permutation between two holding regions and transfer them over from one to the other.

By permutation I assumed you meant shuffling the states from the different registers. So e.g. swap registers 1 and 2, swap registers 1 and 3 etc.

As for the other gates. Keep in mind that the full possibilities of measurements should be able to detect whether it is in the exact superposition of each possible arrangement of states to qubits. There are measurements associated with every possible state, and there are a continuum of possibilities. I fail to see you getting them all, or not limiting yourself from what is physically possible.

But you could have the same set other people have. Which might get exponentially or factorially larger as the number of qubits increases.

  1. The quantum coprocessor is able to perform "some" quantum measurements, whose (finite number of different) classical results can be read by the classical computer.

This is so vague it must be possible. But a measurement is just another interaction but one that couples results to an environment so that the superposition from the entanglement of the result with the environment becomes unexploitable. Again, you should have been able to read and measure much more than the state of each qubit.

Now if the limited tools you want are the tools your algorithm assume you have then you should be fine.

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  • $\begingroup$ For 1., I don't want to initialize the qubits independently, but initialize them all at once, for example to $|00\dots 0\rangle$. The global phase is a red herring, but note that for $n$-qubits, the unit sphere is not in $\mathbb C^n$, but in $\mathbb C^{2^n}$. Apart from that, I interpret your answer to say "the quantum coprocessor does not violate any physical principles, but its abilities seem to be quite limited compared to what is physically possible." $\endgroup$ – Thomas Klimpel Sep 6 '15 at 15:50

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