Simulating the dynamics of time-dependent Hamiltonians with a truncated Dyson series

Abstract

We provide a general method for efficiently simulating time-dependent Hamiltonian dynamics on a circuit-model based quantum computer. Our approach is based on approximating the truncated Dyson series of the evolution operator, extending the earlier proposal by Berry et al. [Phys. Rev. Lett. 114, 090502 (2015)] to evolution generated by explicitly time-dependent Hamiltonians. Two alternative strategies are proposed to implement time ordering while exploiting the superposition principle for sampling the Hamiltonian at different times. The resource cost of our simulation algorithm retains the optimal logarithmic dependence on the inverse of the desired precision. I. INTRODUCTIONSimulation of physical systems is envisioned to be one of the main applications for quantum computers [1]. Effective modeling of the dynamics and its generated time evolution is crucial to a deeper understanding of many-body systems, spin models, and quantum chemistry [2], and may thus have significant implications for many areas of chemistry and materials sciences. Simulation of the intrinsic Hamiltonian evolution of quantum systems was the first potential use of quantum computers suggested by Feynman [3] in 1982. Quantum simulation algorithms can model the evolution of a physical system with a complexity logarithmic in the dimension of the Hilbert space [4] (i.e. polynomial in the number of particles), unlike classical algorithms whose complexity is typically polynomial in the dimension, making simulations for practically interesting systems intractable for classical computers.The first quantum simulation algorithm was proposed by Lloyd [5] in 1996. There have been numerous advances since then providing improved performance [6][7][8][9][10][11][12][13][14][15]. One advance was to provide complexity that scales logarithmically in the error, and is nearly optimal in all other parameters [13]. Further improvements were provided by quantum signal processing methodology [14,16] as well as qubitization [15], which achieve optimal query complexity.An important case is that of time-dependent Hamiltonians. Efficient simulation of time-dependent Hamiltonians would allow us to devise better quantum control schemes [17] and describe transition states of chemical reactions [18]. Furthermore, simulation of dynamics generated by time-dependent Hamiltonians is a key component for implementing adiabatic algorithms [19] and the quantum approximate optimization algorithm [20] in a gate-based quantum circuit architecture.The most recent advances in quantum simulation algorithms are for time-independent Hamiltonians. Techniques for simulating time-dependent Hamiltonians based on the Lie-Trotter-Suzuki decomposition were developed in [7,9], but the complexity scales polynomially with error. More recent advances providing complexity logarithmic in the error [11,13] mention that their techniques can be generalized to time-dependent scenarios, but do not analyze this case. The most recent algorithms [14,15] are not directly applicable to the time-dependent case. H...

Time-dependent Hamiltonian simulation withL1-norm scaling

“…irrespective of how this operation is implemented [30]. We consider the SM and LCU models for all the time-dependent simulation algorithms so that we can give a fair comparison of their complexity.…”

“…We now explain the meaning of L [41] does not have an explicit complexity analysis and its implementation is streamlined by continuous qDRIFT (see Appendix A). The fractionalquery algorithm [6] does not have an explicit implementation for Hamiltonians in the LCU model, and its implementation in the SM model is streamlined by the Dyson-series approach [7,30,37]. auxiliary information, must be accessed by the quantum simulation algorithm; we assume such quantities can be computed efficiently.…”

“…In this section, we show how the Dyson-series algorithm [7,30,37] can be rescaled to have L 1 -norm scaling. We address this first for the SM model of Hamiltonian access before handling the LCU model (see Section 2.3 for definitions of these models).…”

“…The fractionalquery algorithm of Berry, Childs, Cleve, Kothari, and Somma can also simulate time-dependent Hamiltonians [6], with an exponentially improved dependence on precision and only logarithmic dependence on the derivative of the Hamiltonian. A related quantum algorithm for time-dependent Hamiltonian simulation was suggested by Berry, Childs, Cleve, Kothari, and Somma based on the truncated Dyson series [7], which is analyzed explicitly in [30,37].…”

“…In the rescaled Schrödinger equation, the time-dependent Hamiltonian H(τ ) has the same norm at all τ ∈ [0, t], so the norm inequality t 0 dτ H(τ ) ≤ t max τ ∈[0,t] H(τ ) holds with equality. Using this principle, we show that the simulation algorithm based on the truncated Dyson series [7,30,37] can also be improved to have L 1 -norm scaling.…”