This paper presents a method to double the efficiency of Hamiltonian simulation on a quantum computer by leveraging generalized quantum signal processing. The authors show that by using a controlled operation that selects between a unitary U and its inverse U†, the complexity of Hamiltonian simulation can be reduced by a factor of 2 compared to standard quantum signal processing methods. This improvement is achieved by modifying the controlled operations used in quantum signal processing to allow for a more efficient construction of the required unitary operations. The paper begins by reviewing the history of Hamiltonian simulation, noting that early methods used product formulae, while more advanced techniques utilized quantum walks and linear combinations of unitaries. These methods provided logarithmic complexity in the inverse error, but the joint scaling between the error and time was not optimal. The development of quantum signal processing by Low and Chuang improved this, but the true additive complexity was not achieved until the introduction of generalized quantum signal processing. The authors then describe how quantum signal processing works, using block encoding to represent a Hamiltonian as a unitary operation. They show that by using a controlled operation that selects between U and U†, the required eigenvalues for Hamiltonian evolution can be transformed more efficiently. This approach allows for the construction of the desired Hamiltonian evolution with fewer controlled operations, leading to a significant reduction in the overall complexity. The paper also addresses the challenge of ensuring that the sum of squares of the polynomials used in the construction is exactly equal to 1, which is necessary for the correct transformation of eigenvalues. This is achieved by using a factor slightly less than 1 to ensure the sum does not exceed 1, and by finding new functions that satisfy the required conditions. The authors conclude that by using a controlled operation that selects between U and U†, the number of controlled operations needed for Hamiltonian simulation can be halved, leading to a significant speedup in the simulation process. This approach is broadly applicable and can provide a factor of 2 speedup for Hamiltonian simulation in most practical cases.This paper presents a method to double the efficiency of Hamiltonian simulation on a quantum computer by leveraging generalized quantum signal processing. The authors show that by using a controlled operation that selects between a unitary U and its inverse U†, the complexity of Hamiltonian simulation can be reduced by a factor of 2 compared to standard quantum signal processing methods. This improvement is achieved by modifying the controlled operations used in quantum signal processing to allow for a more efficient construction of the required unitary operations. The paper begins by reviewing the history of Hamiltonian simulation, noting that early methods used product formulae, while more advanced techniques utilized quantum walks and linear combinations of unitaries. These methods provided logarithmic complexity in the inverse error, but the joint scaling between the error and time was not optimal. The development of quantum signal processing by Low and Chuang improved this, but the true additive complexity was not achieved until the introduction of generalized quantum signal processing. The authors then describe how quantum signal processing works, using block encoding to represent a Hamiltonian as a unitary operation. They show that by using a controlled operation that selects between U and U†, the required eigenvalues for Hamiltonian evolution can be transformed more efficiently. This approach allows for the construction of the desired Hamiltonian evolution with fewer controlled operations, leading to a significant reduction in the overall complexity. The paper also addresses the challenge of ensuring that the sum of squares of the polynomials used in the construction is exactly equal to 1, which is necessary for the correct transformation of eigenvalues. This is achieved by using a factor slightly less than 1 to ensure the sum does not exceed 1, and by finding new functions that satisfy the required conditions. The authors conclude that by using a controlled operation that selects between U and U†, the number of controlled operations needed for Hamiltonian simulation can be halved, leading to a significant speedup in the simulation process. This approach is broadly applicable and can provide a factor of 2 speedup for Hamiltonian simulation in most practical cases.