Simulating Hamiltonian dynamics.

Ayaka Usui, Anna Sanpera, María García Díaz

Local Hamiltonians H k describe nontrivial k -body interactions in quantum many-body systems. Here, we address the dynamical simulatability of a k -local Hamiltonian by a simpler one H k ′ with k ′ < k , under the realistic constraint that both Hamiltonians act on the same Hilbert space. When it comes to exact simulation, we build upon known methods to derive examples of H k and H k ′ that simulate the same physics. We also address the most realistic case of approximate simulation. There, we upper-bound the error up to which a Hamiltonian can simulate another one, regardless of their internal structure, and show an example suggesting that the accuracy of a ( k ′ = 2 ) -local Hamiltonian to simulate H k with k > 2 increases with k . Finally, we propose a method to search for the k ′ -local Hamiltonian that simulates, with the highest possible precision, the short time dynamics of a given H k Hamiltonian. Published by the American Physical Society 2024

Osama M. Raisuddin, Suvranu De

Linyun Wan, Jie Liu, Zhenyu Li +1 more

Quantum dynamics simulations (QDSs) are one of the most highly anticipated applications of quantum computing. Quantum circuit depth for implementing Hamiltonian simulation algorithms is commonly time dependent so that long time dynamics simulations become impratical on near-term quantum processors. The Hamiltonian simulation algorithm based on Cartan decomposition (CD) provides an appealing scheme for QDSs with fixed-depth circuits while limited to time-independent case. In this work, we generalize this CD-based Hamiltonian simulation algorithm for studying time-dependent systems by combining it with variational Hamiltonian simulation. The time-dependent and time-independent parts of the Hamiltonian are treated with the variational approach and the CD-based Hamiltonian simulation algorithms, respectively. As such, only fixed-depth quantum circuits are required in this hybrid Hamiltonian simulation algorithm while still maintaining high accuracy. We apply this new algorithm to study the response of spin and molecular systems to $δ$-kick electric fields and obtain accurate spectra for these excitation processes.

Riccardo Manenti, Mário Motta

Abstract This chapter presents some quantum algorithms to simulate the dynamics of a quantum system, one of the most important applications of a quantum computer. We introduce the conditions for efficient simulation of Hamiltonian dynamics. We then present some Trotterization methods, in which the time evolution operator is approximated as a product of terms, each of which can be implemented by a quantum circuit at polynomial cost. We show high-order decompositions and illustrate the use of a Trotterization method to simulate sparse Hamiltonians. We discuss the Taylor series method and the qubitization method. The latter achieves optimal cost in simulation time and target accuracy. We conclude the chapter with a proof that simulating Hamiltonian dynamics is BQP-complete