openfermion.circuits.error_bound

Numerically upper bound the error in the ground state energy for the second order Trotter-Suzuki expansion.

openfermion.circuits.error_bound(
    terms, tight=False
)

terms a list of single-term QubitOperators in the Hamiltonian to be simulated.
tight whether to use the triangle inequality to give a loose upper bound on the error (default) or to calculate the norm of the error operator.

A float upper bound on norm of error in the ground state energy.

Notes: follows Poulin et al.'s work in "The Trotter Step Size Required for Accurate Quantum Simulation of Quantum Chemistry". For a tighter bound, the 1-norm of Equation 9 is used with the error operator from error_operator:

   $$
   \Delta E_{\text{tight} } = \frac{1}{12} \left\|
   \sum_{\beta} \sum_{\alpha \leq \beta} \sum_{\alpha' < \beta}
   \left[ H_\alpha \left(1 - \frac{\delta_{\alpha\beta} }{2}\right),
   [H_{\beta}, H_{\alpha'}] \right] \right\| \Delta t^2
   $$


   For the loose bound, Equation 16 is used to bound term 2
   of Equation 6, and a similar expression is used for term 1:


   $$
   \Delta E_{\text{loose} } =
   4 \sum_{\alpha} \lVert H_{\alpha} \rVert
   \left( \sum_{\beta}' \lVert H_{\beta} \rVert \right)^2 \Delta t^2

   + 4 \sum_{\alpha} \lVert H_{\alpha} \rVert^2
   \sum_{\beta}' \lVert H_{\beta} \rVert \Delta t^2
   $$


   Possible extensions of this function would be to get the
   expectation value of the error operator with the Hartree-Fock
   state or CISD state, which can scalably bound the error in
   the ground state but much more accurately than the triangle
   inequality.