Simulating Hamiltonian Dynamics with a Truncated Taylor Series

Abstract

We describe a simple, efficient method for simulating Hamiltonian dynamics on a quantum computer by approximating the truncated Taylor series of the evolution operator. Our method can simulate the time evolution of a wide variety of physical systems. As in another recent algorithm, the cost of our method depends only logarithmically on the inverse of the desired precision, which is optimal. However, we simplify the algorithm and its analysis by using a method for implementing linear combinations of unitary operations together with a robust form of oblivious amplitude amplification. DOI: 10.1103/PhysRevLett.114.090502 PACS numbers: 03.67.Ac, 89.70.Eg One of the main motivations for quantum computers is their ability to efficiently simulate the dynamics of quantum systems [1], a problem that is apparently hard for classical computers. Since the mid-1990s, many algorithms have been developed to simulate Hamiltonian dynamics on a quantum computer [2][3][4][5][6][7][8][9][10][11][12], with applications to problems such as simulating spin models [13] and quantum chemistry [14][15][16][17]. While it is now well known that quantum computers can efficiently simulate Hamiltonian dynamics, ongoing work has improved the performance and expanded the scope of such simulations.Recently, we introduced a new approach to Hamiltonian simulation with exponentially improved performance as a function of the desired precision [18]. Specifically, we presented a method to simulate a d-sparse, n-qubit Hamiltonian H acting for time t > 0, within precision ϵ > 0, using O(τ logðτ=ϵÞ= log logðτ=ϵÞ) queries to H and O(nτlog 2 ðτ=ϵÞ= log logðτ=ϵÞ) additional two-qubit gates, where τ ≔ d 2 ∥H∥ max t. This dependence on ϵ is exponentially better than all previous approaches to Hamiltonian simulation, and the number of queries to H is optimal [18]. (For simplicity, we refer to combinations of logarithms like those in the above expressions as logarithmic.) Roughly speaking, doubling the number of digits of accuracy only doubles the complexity.The simulation algorithm of [18] is indirect, appealing to an unconventional model of query complexity. In this Letter, we describe and analyze a simplified approach to Hamiltonian simulation with the same cost as the method of [18]. The new approach is easier to understand, and the reason for the logarithmic cost dependence on ϵ is immediate. The new approach decomposes the Hamiltonian into a linear combination of unitary operations. Unlike the algorithm of [18], these terms need not be self-inverse, so the algorithm is efficient for a larger class of Hamiltonians. The new approach is also simpler to analyze: we give a selfcontained presentation in four pages.The main idea of the new approach is to implement the truncated Taylor series of the evolution operator. Similar to previous approaches for implementing linear combinations of unitary operators [12,13], the various terms of the Taylor series can be implemented by introducing an ancillary superposition and performing controlled operations. The time e...

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