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.