Multi-mode Hamiltonian Simulation by Quantum Signal Processing

Efficient Hamiltonian simulation of multiple interacting bosonic and fermionic modes is a key challenge in quantum computing. We present a framework for this problem on hybrid qubit-oscillator quantum processors with native gates including the qubit rotation, conditional displacement and beam splitter gates. Our method uses quantum signal processing (QSP) and quantum singular value transformation (QSVT), both of which are frameworks for realizing uni-variate polynomial transformations of a matrix. We first use QSP on $U(N)$ to realize a block encoding of the Hamiltonian $H$, i.e., a unitary operator of which the Hamiltonian is one matrix block, and then use QSVT to realize the Hamiltonian evolution $e^{-itH}$. The number of uses of the input gate for each variable scales as ${\mathcal{O}}(n'\log(|t|\alpha/\epsilon)(|t|\alpha+\log(\epsilon^{-1})))$ for an $n'$-local Hamiltonian, simulation time $t$, target accuracy $\epsilon$ and some scaling factor $\alpha$ that depends on the format of the Hamiltonian and initial state. Finally, we discuss the performance of our algorithm on some common bosonic Hamiltonians, and compare it with other existing quantum simulation algorithms on hybrid qubit-oscillator quantum processors.