Doubling the efficiency of Hamiltonian simulation via generalized quantum signal processing

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.