Divide and conquer approach to quantum Hamiltonian simulation

Hadfield, Stuart; Papageorgiou, Anargyros

doi:10.1088/1367-2630/aab1ef

Cited by 30 publications

(20 citation statements)

References 60 publications

(169 reference statements)

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“…Indeed, Jordan, Lee, and Preskill claimed that product formulas can simulate an n-qubit lattice system with (nt) 1+o(1) gates [6], but they did not provide rigorous justification and it is unclear how to formalize their argument. Subsequent work improves the analysis of the product-formula algorithm using information about commutation among terms in the Hamiltonian [3,4,19,35], the distribution of norms of terms [36], and by randomizing the ordering of terms [37,38]. However, none of these improvements can achieve the claimed gate complexity (nt) 1+o (1) for lattice simulation.…”

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confidence: 99%

Nearly Optimal Lattice Simulation by Product Formulas

2019

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We consider simulating an n-qubit Hamiltonian with nearest-neighbor interactions evolving for time t on a quantum computer. We show that this simulation has gate complexity (nt) 1+o(1) using product formulas, a straightforward approach that has been demonstrated by several experimental groups. While it is reasonable to expect this complexity-in particular, this was claimed without rigorous justification by Jordan, Lee, and Preskill-we are not aware of a straightforward proof. Our approach is based on an analysis of the local error structure of product formulas, as introduced by Descombes and Thalhammer and further simplified here. We prove error bounds for canonical product formulas, which include well-known constructions such as the Lie-Trotter-Suzuki formulas. We also develop a local error representation for time-dependent Hamiltonian simulation, and we discuss generalizations to periodic boundary conditions, constant-range interactions, and higher dimensions. Combined with a previous lower bound, our result implies that product formulas can simulate lattice Hamiltonians with nearly optimal gate complexity.

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confidence: 99%

Nearly Optimal Lattice Simulation by Product Formulas

2019

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“…Here, we show that our OPMF model can, in principle, be used for accurate, time-efficient inference when modelling arbitrarily large genomic regions. ‘Divide and conquer’ strategies have also previously been used to reduce the computational expense of large-scale simulations [ 50 , 51 ]. Here, a large simulation is split into smaller batches and batch results are aggregated to give overall estimates.…”

Section: Discussionmentioning

confidence: 99%

Cluster mean-field theory accurately predicts statistical properties of large-scale DNA methylation patterns

Kerr

Sproul

Grima

2022

J. R. Soc. Interface.

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The accurate establishment and maintenance of DNA methylation patterns is vital for mammalian development and disruption to these processes causes human disease. Our understanding of DNA methylation mechanisms has been facilitated by mathematical modelling, particularly stochastic simulations. Megabase-scale variation in DNA methylation patterns is observed in development, cancer and ageing and the mechanisms generating these patterns are little understood. However, the computational cost of stochastic simulations prevents them from modelling such large genomic regions. Here, we test the utility of three different mean-field models to predict summary statistics associated with large-scale DNA methylation patterns. By comparison to stochastic simulations, we show that a cluster mean-field model accurately predicts the statistical properties of steady-state DNA methylation patterns, including the mean and variance of methylation levels calculated across a system of CpG sites, as well as the covariance and correlation of methylation levels between neighbouring sites. We also demonstrate that a cluster mean-field model can be used within an approximate Bayesian computation framework to accurately infer model parameters from data. As mean-field models can be solved numerically in a few seconds, our work demonstrates their utility for understanding the processes underpinning large-scale DNA methylation patterns.

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“…Trotterization (and its alternative variants [23,25,31,49,56]) provides a simple approach to quantum simulation and is by far the only known approach that can exploit the commutativity of Hamiltonian. Indeed, in the extreme case where all the terms in the Hamiltonian commute, we can simultaneously diagonalize them and apply the first-order Lie-Trotter formula S 1 (t) without error.…”

Section: Combining Sparsity Commutativity and Initial-state Knowledgementioning

confidence: 99%

“…Hamiltonians arising in practice often have additional features beyond sparseness, such as locality [30,71], commutativity [24,25,66], and symmetry [32,72], that can be used to improve the performance of simulation. Besides, prior knowledge of the initial state [6,26,61,65] and the norm distribution of Hamiltonian terms [17,20,31,44,53] have also been proven useful for quantum simulation.…”

Section: Introductionmentioning

confidence: 99%

Nearly tight Trotterization of interacting electrons

Huang

Campbell

2021

Quantum

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We consider simulating quantum systems on digital quantum computers. We show that the performance of quantum simulation can be improved by simultaneously exploiting commutativity of the target Hamiltonian, sparsity of interactions, and prior knowledge of the initial state. We achieve this using Trotterization for a class of interacting electrons that encompasses various physical systems, including the plane-wave-basis electronic structure and the Fermi-Hubbard model. We estimate the simulation error by taking the transition amplitude of nested commutators of the Hamiltonian terms within the η-electron manifold. We develop multiple techniques for bounding the transition amplitude and expectation of general fermionic operators, which may be of independent interest. We show that it suffices to use (n5/3η2/3+n4/3η2/3)no(1) gates to simulate electronic structure in the plane-wave basis with n spin orbitals and η electrons, improving the best previous result in second quantization up to a negligible factor while outperforming the first-quantized simulation when n=η2−o(1). We also obtain an improvement for simulating the Fermi-Hubbard model. We construct concrete examples for which our bounds are almost saturated, giving a nearly tight Trotterization of interacting electrons.

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Divide and conquer approach to quantum Hamiltonian simulation

Cited by 30 publications

References 60 publications

Nearly Optimal Lattice Simulation by Product Formulas

Nearly Optimal Lattice Simulation by Product Formulas

Cluster mean-field theory accurately predicts statistical properties of large-scale DNA methylation patterns

Nearly tight Trotterization of interacting electrons

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