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Suppose I have three qubits that of course represent $2^3=8$ states. I want to put these qubits so that they are in the particular superposition say...

$$a|000\rangle + b|001\rangle + c|010\rangle + d|011\rangle + e|100\rangle + f|101\rangle + g|110\rangle + h|111\rangle\,.$$

Here's the question: do I perform a 2d rotation on each qubit (each having its own $2\times2$ rotation matrix) for a total of three operations, or do I perform a single 8d rotation using an $8\times8$ rotation matrix?

Follow up question: if the answer is 'make an $8\times8$ matrix', doesn't this at some point boil down to just applying a rotation to each qubit?

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You don't need a full 8x8 three-qubit operation, but you will need a two-qubit operation of some kind. If you only rotate each qubit independently, you'll find you can only make un-entangled states like:

$(a_1 \left| 0 \right\rangle + b_1 \left| 1 \right\rangle) \otimes (a_2 \left| 0 \right\rangle + b_2 \left| 1 \right\rangle) \otimes (a_3 \left| 0 \right\rangle + b_3 \left| 1 \right\rangle)$

$= a_1 a_2 a_3 |000\rangle + a_1 a_2 b_3 |001\rangle + a_1 b_2 a_3 |010\rangle + a_1 b_2 b_3 |011\rangle + b_1 a_2 a_3 |100\rangle + b_1 a_2 b_3 |101\rangle + b_1 b_2 a_3 |110\rangle + b_1 b_2 b_3 |111\rangle$

But as soon as you have a non-trivial two-qubit operation, like a controlled-NOT, to go with your single qubit gates, you can approximate any larger operation or state arbitrarily well. (This is different from classical reversible computing, where you need a doubly-controlled-not before you get universality.)

The basic idea of how it's done is:

  • Break a large operation into lots of single-qubit operations, but allowing each to have lots of controls.
  • Use the fact that all single-qubit operations have a square root to knock controls off of multi-controlled operations until you're left with singly-controlled operations.
  • Replace controlled-whatevers with controlled-NOTs by finding $A$, $B$, $C$, $\theta$ such that $ABC = I$ but $AXBXC e^{i \theta} = U$ for each single-qubit operation $U$.

The details of these kinds of constructions are covered in textbooks.

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  • $\begingroup$ Thanks so much - is there a textbook(or site) in particular that you could recommend? $\endgroup$ – C Shreve Dec 19 '15 at 22:54
  • $\begingroup$ @CShreve I used Nielsen and Chuang. It's the de-facto standard textbook for the field. $\endgroup$ – Craig Gidney Dec 19 '15 at 23:40

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