  Quantum signal processing provides an optimal procedure for simulating
Hamiltonian evolution on a quantum computer using calls to a block encoding of
the Hamiltonian. In many situations it is possible to control between forward
and reverse steps with almost identical cost to a simple controlled operation.
We show that it is then possible to reduce the cost of Hamiltonian simulation
by a factor of 2 using the recent results of generalised quantum signal
processing.
