Proof on approximating adiabatic evolution by quantum circuit

Question

I am reading on how the adiabatic evolution can be approximated by a quantum circuit of size poly(nT) and I am trying to follow the derivation in the paper

W. van Dam, M. Mosca, and U. Vazirani, “How Powerful is Adiabatic Quantum Computation?,” Proceedings 2001 IEEE International Conference on Cluster Computing, pp. 279–287, 2001.

In section 4, page 4, it states that:

"The Campbell-Baker-Hausdorff theorem tells us how well we can approximate ‘parallel Hamiltonians’ by consecutive ones: $|||\exp(A+B) − \exp(A)\exp(B)||| \in O(|||AB|||)$."

The norm I believe is just the operator induced norm. I am familiar with the BCH formula but could not see the above relation directly coming out from the formula. So how is this relation derived?

I tried looking into the reference which is "matrix analysis" by Rajendra Bhatia but didn't get any success.

Answer 1

The Baker-Campbell-Hausdorff formula says that you can expand $$ \log(e^Ae^B)=A+B+[A,B]/2+\ldots=M $$ where higher order terms have 3 or more uses of $A$ and $B$. Now, let's say that $A$ and $B$ are anti-Hermitian so that $e^A$, and similar terms, are unitary. We have $$ \|\exp(A+B)-\exp(A)\exp(B)\|=\|e^{A+B}\left(\mathbb{I}-e^Me^{-(A+B)}\right)\|. $$ The matrix norm is invariant under the action of unitaries, so this is the same as $$ \|\mathbb{I}-e^Me^{-(A+B)}\|. $$ Now, you might apply the BCH formula again to get $$ \|\mathbb{I}-e^{M-A-B-[M,A+B]/2+\ldots}\|, $$ the point being that the leading order $A+B$ stuff cancels from $M-A-B$ and the commutator, leaving terms like $[A,B]$. If both $A$ and $B$ are small ($O(\epsilon)$), then higher order terms have vanishing relevance, so we have $$ \|\mathbb{I}-e^{[A,B]/2+O(\epsilon^3)}\|, $$ and if we do an expansion on that, we get $$ \|\mathbb{I}-(\mathbb{I}+[A,B]/2+O(\epsilon^3))\|=\|[A,B]\|/2+O(\epsilon^3)=O(\|AB\|). $$