# Correct Formulation of N&C Exercise 4.11 and other textbooks misquoting

Inspired by the comments in this question How to approximate $Rx$, $Ry$ and $Rz$ gates?, there is the errata for question 4.11 pg 176 in N&C. 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, http://www.michaelnielsen.org/qcqi/errata/errata/errata.html, corrects this s.t.

$$U = e^{i\alpha}R_n(\beta_1)R_m(\gamma_1)R_n(\beta_2)R_m(\gamma_2)\dots$$.

However I found that other textbooks such as Kayes and Mosca (pg. 66, thm 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?

## 1 Answer

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.

• Thanks! That is what I thought! I just wanted to make sure I wasn't crazy seeing it in all these other reputable places! I was trying to expand out the corrected form to prove it using quaterions to prove the statement, but things got messy quickly. – Sam Palmer May 19 '20 at 15:11