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Let $A$ and $B$ be finite-dimensional Hermitian matrices, and let $v$ $:=$ $max${$||A||,||B||$}.

Show that $||(e^{-iAt/m}e^{-iBt/m})^m -e^{-i(A+B)t}||\leq\epsilon$ provided $m = \Omega(v^2t^2/\epsilon)$.

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    Is that supposed to be a 2-norm?2010-11-26
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    It is generally considered impolite to phrase your questions in the imperative voice. If this is homework, you should also indicate what you've done so far and where you've gotten stuck. It is not our job to do your homework for you every time you tell us to.2010-11-26

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HINT: Consider the product of power series $\exp(A/m)\exp(B/m) = I + \frac{A}{m} + \frac{B}{m} + O\left(\frac{1}{m^2}\right)$. Then use this to estimate the logarithm for sufficiently large $m$, then exponentiate and raise to the $m$-th power.

This is from the proof in section 2.4 of "Lie Groups, Lie Algebras, and Representations" by Brian C. Hall.

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As long as the limit definition of exponential function can be extended to operators, $$\lim_{N \rightarrow \infty }(1+\frac{\alpha}{N})^N=e^{\alpha}$$

$$ \alpha \in \mathbb{R} $$ $$ \alpha \in \operatorname{ \{Matrix,Operators, etc. ?\} } $$

then we can use it similarly

$$ \lim_{N \rightarrow \infty} \bigg( 1+ \frac{-it (A+B)}{N}+o(\frac{1}{N^2})\bigg)^N =e^{-it(A+B)} $$

Trotter formula, widely used in Path Integral, Quantum Monte-Carlo methods,