Incorrect Bounds on Trotter Error

[kapawlak]

The 'loose' and possibly 'tight' bounds calculated by openfermion.circuits.error_bound are incorrect. Evidence:

• The 'loose' bound is numerically smaller than the 'tight' bound for simple 2D Ising models, which can be easily verified. See image below.
• For a two-summand Hamiltonian $H=A+B$, the loose bound is essentially set to $\mathcal E \le ||[A,[B,A] ||$. The correct bound is $\mathcal E_{2} \le \frac{1}{12} (||[A,[B,A] || + \frac{1}{2}||[B,[B,A]|| )$. In the case of the 2D Ising model, the second term is combinatorially large in the number of lattice edges and quickly overwhelms the first term, resulting in a bound much larger than the provided loose bound.
• For my specific problem, I derived a formula for the value of the second order error by hand. The 'loose' bound is always equal to the term |[A[B,A]]|, which is incorrect as mentioned. The 'tight' bound seems to overshoot my analytical expression, which is acceptable, but indicates that there might be some room for simple improvements here.

Tight on time, I'm going to commit a faux-paux here and post a screenshot rather than minimal code, and I'll come back and add minimal code to reproduce after my sprint if anyone needs it :).