Higher order decompositions of ordered operator exponentials

By Nathan Wiebe, Dominic W. Berry, Peter Høyer, Barry C. Sanders. 2008

188 citations recorded by the source corpus.

We present a decomposition scheme based on Lie–Trotter–Suzuki product formulae to approximate an ordered operator exponential with a product of ordinary operator exponentials. We show, using a counterexample, that Lie–Trotter–Suzuki approximations may be of a lower order than expected when applied to problems that have singularities or discontinuous derivatives of appropriate order. To address this problem, we present a set of criteria that is sufficient for the validity of these approximations, prove convergence and provide upper bounds on the approximation error. This work may shed light on why related product formulae fail to be as accurate as expected when applied to Coulomb potentials.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/1854259aa8d28c67f5a859233a14d7982e014ca2

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In Computational Theory and Mathematics

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Citations in the corpus

Hamiltonian simulation using linear combinations of unitary operations (2012, methodology): “Lemma 9 gives Using the upper bound q k, ≤ 2 k/ 3 k from Appendix A of [20], we have This bound can be simpliﬁed using (4 k + 1)!”
Low-Depth Quantum Simulation of Materials (2017, background): “Even the most elaborate Trotter schemes scale subpolynomially but not polylogarithmically with respect to the reciprocal of the simulation error, 1=ε, [80,81].”
Efficient Variational Quantum Simulator Incorporating Active Error Minimization (2016, methodology): “For the lowest-order symmetric TrotterSuzuki decomposition [60,61], each time step in Eq.”
High-order quantum algorithm for solving linear differential equations (2010, background): “Therefore the amplitude would not drop below the amplitude on the eigenspace corresponding to the largest eigenvalue.”
Digitized adiabatic quantum computing with a superconducting circuit (2015, methodology): “S1 [28], In [29, 30], it is shown that such an evolution can be simulated with error (cid:15) by choosing M = T 2 a 2 max L 2 /(cid:15) where a max = max (cid:96),t { a (cid:96) ( t ) } .”
Elucidating reaction mechanisms on quantum computers (2016, methodology): “5), as observed in previous studies [17], and take the norm of the Hamiltonian to scale as O(N2) as anticipated asymptotically for a local basis [85] then the scaling that arises from using the second–order Trotter formula for time– dependent Hamiltonians [83] would be O ( h3/2N8.”
Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity (2018, methodology): “This is conceptually similar to a Trotter decomposition; however, as the error in this approximation can be made exponentially small by choosing the patches in Lemma 3 to be linearly far apart, the error can be controlled in a tighter fashion without requiring short time steps (unlike Trotter decomp”
Hamiltonian Simulation in the Interaction Picture (2018, background): “[3] Nathan Wiebe, Dominic Berry, Peter Hyer, and Barry C Sanders.”