  Randomization has been applied to Hamiltonian simulation in a number of ways
to improve the accuracy or efficiency of product formulas. Deterministic
product formulas are often constructed in a symmetric way to provide accuracy
of even order 2k. We show that by applying randomized corrections, it is
possible to more than double the order to 4k + 1 (corresponding to a doubling
of the order of the error). In practice, applying the corrections in a quantum
algorithm requires some structure to the Hamiltonian, for example the Pauli
strings as are used in the simulation of quantum chemistry.
