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Inspired by the comments in this question, there is the errata for question 4.11 pg 176 in Nielsen and Chuang. The original form states that for any non parallel $m$ and $n$, then for an arbitrary $U$:

$$U = e^{i\alpha}R_n(\beta)R_m(\gamma)R_n(\delta)$$ for appropriate $\alpha,\beta, \gamma, \delta$. The errata of Nielsen and Chuang corrects this such that $$U = e^{i\alpha}R_n(\beta_1)R_m(\gamma_1)R_n(\beta_2)R_m(\gamma_2)\dots$$ However I found that other textbooks such as "An Introduction to Quantum Computing" by Kaye, Laflamme, and Mosca (alt link) (p.66, Theorem 4.2.2), and various online material still quotes the original form of the theorem. As such I am wondering is the errata correct, and is just that all the other material has 'incorrectly' quoted the result from N&C?

Frederik vom Ende
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Sam Palmer
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1 Answers1

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The errata is correct. I had a project student who erroneously took one of these mis-quotes and she spent ages working with it, realising it didn't make sense, and subsequently proving that the stated formula was incorrect, only later to find the N&C erratum. As you say, it has propagated far and wide!

If you want some insight about the problem, imagine $n$ and $m$ are two axes that are almost parallel (visualise this on the Bloch sphere). Now imagine I start with a state that is aligned with the $n$ axis. With a sequence $n-m-n$, so the claim goes, I should be able to produce any state on the surface of the Bloch sphere. But the first $n$ does nothing because of what our initial state is. Then the $m$ only creates a rotation preserving the angle of the initial state with the axis, and so it's always close to where it stated. The same again with the final rotation about the $n$ axis.

DaftWullie
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