Doubling Efficiency of Hamiltonian Simulation via Generalized Quantum Signal Processing

Doubling Efficiency of Hamiltonian Simulation via Generalized Quantum Signal Processing

18 Jan 2024 | Dominic W. Berry, Danial Motlagh, Giacomo Pantaleoni, Nathan Wiebe
The paper "Doubling Efficiency of Hamiltonian Simulation via Generalized Quantum Signal Processing" by Dominic W. Berry, Danial Motlagh, Giacomo Pantaleoni, and Nathan Wiebe introduces a method to reduce the cost of Hamiltonian simulation on a quantum computer. The authors leverage generalized quantum signal processing, which uses block encoding of the Hamiltonian and controlled operations between forward and reverse steps to achieve this. They demonstrate that by controlling between the walk operator \( U \) and its adjoint \( U^\dagger \), the number of controlled operations required can be halved compared to standard quantum signal processing. This reduction in complexity allows for a factor of 2 improvement in the efficiency of Hamiltonian simulation. The key technical result is a theorem that shows how to construct polynomials \( P(U) \) and \( Q(U) \) using controlled operations between \( U \) and \( U^\dagger \), which can then be used to approximate the Hamiltonian evolution. The proof of this theorem involves detailed mathematical analysis and the use of Chebyshev polynomials. The authors also provide an appendix that proves the positivity of the polynomial and another appendix that shows the block form of the generalized quantum signal processing operator.The paper "Doubling Efficiency of Hamiltonian Simulation via Generalized Quantum Signal Processing" by Dominic W. Berry, Danial Motlagh, Giacomo Pantaleoni, and Nathan Wiebe introduces a method to reduce the cost of Hamiltonian simulation on a quantum computer. The authors leverage generalized quantum signal processing, which uses block encoding of the Hamiltonian and controlled operations between forward and reverse steps to achieve this. They demonstrate that by controlling between the walk operator \( U \) and its adjoint \( U^\dagger \), the number of controlled operations required can be halved compared to standard quantum signal processing. This reduction in complexity allows for a factor of 2 improvement in the efficiency of Hamiltonian simulation. The key technical result is a theorem that shows how to construct polynomials \( P(U) \) and \( Q(U) \) using controlled operations between \( U \) and \( U^\dagger \), which can then be used to approximate the Hamiltonian evolution. The proof of this theorem involves detailed mathematical analysis and the use of Chebyshev polynomials. The authors also provide an appendix that proves the positivity of the polynomial and another appendix that shows the block form of the generalized quantum signal processing operator.
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Understanding Doubling the efficiency of Hamiltonian simulation via generalized quantum signal processing