Theory of Trotter Error with Commutator Scaling

Product formulas offer a powerful, simple approach to quantum simulation. A new theory quantifying their errors puts these algorithms on a rigorous foundation, showcasing their superiority over other methods.

[1]  Alexey V Gorshkov,et al.  Persistence of locality in systems with power-law interactions. , 2014, Physical review letters.

[2]  Alexandru Paler,et al.  Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity , 2018, Physical Review X.

[3]  Krysta Marie Svore,et al.  Quantum Speed-Ups for Solving Semidefinite Programs , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[4]  Mechthild Thalhammer,et al.  Defect-based local error estimators for splitting methods, with application to Schrödinger equations, Part II. Higher-order methods for linear problems , 2014, J. Comput. Appl. Math..

[5]  Edward Farhi,et al.  A Quantum Algorithm for the Hamiltonian NAND Tree , 2008, Theory Comput..

[6]  Xin Wang,et al.  Time-dependent Hamiltonian simulation with $L^1$-norm scaling , 2019, Quantum.

[7]  Dominic W. Berry,et al.  High-order quantum algorithm for solving linear differential equations , 2010, ArXiv.

[8]  de Hans Raedt,et al.  PRODUCT FORMULA METHODS FOR TIME-DEPENDENT SCHRODINGER PROBLEMS , 1990 .

[9]  M. Hastings,et al.  Gate count estimates for performing quantum chemistry on small quantum computers , 2013, 1312.1695.

[10]  Minh C. Tran,et al.  Locality and Digital Quantum Simulation of Power-Law Interactions , 2018, Physical review. X.

[11]  Bruno Nachtergaele,et al.  Lieb-Robinson Bounds and the Exponential Clustering Theorem , 2005, math-ph/0506030.

[12]  F. Alet,et al.  Many-body localization edge in the random-field Heisenberg chain , 2014, 1411.0660.

[13]  Yuan Su,et al.  Nearly optimal lattice simulation by product formulas , 2019, Physical review letters.

[14]  A. Ferris,et al.  Fourier transform for fermionic systems and the spectral tensor network. , 2013, Physical review letters.

[15]  Alán Aspuru-Guzik,et al.  Quantum Chemistry in the Age of Quantum Computing. , 2018, Chemical reviews.

[16]  Stuart Hadfield,et al.  Divide and conquer approach to quantum Hamiltonian simulation , 2017 .

[17]  Andrew M. Childs,et al.  Hamiltonian Simulation with Nearly Optimal Dependence on all Parameters , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[18]  Guang Hao Low,et al.  Hamiltonian simulation with nearly optimal dependence on spectral norm , 2018, STOC.

[19]  Alán Aspuru-Guzik,et al.  Quantum computational chemistry , 2018, Reviews of Modern Physics.

[20]  Robert I. McLachlan,et al.  On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods , 1995, SIAM J. Sci. Comput..

[21]  Mechthild Thalhammer,et al.  Convergence Analysis of High-Order Time-Splitting Pseudospectral Methods for Nonlinear Schrödinger Equations , 2012, SIAM J. Numer. Anal..

[22]  M. Suzuki,et al.  Decomposition formulas of exponential operators and Lie exponentials with some applications to quantum mechanics and statistical physics , 1985 .

[23]  Maria Kieferova,et al.  Simulating the dynamics of time-dependent Hamiltonians with a truncated Dyson series , 2018, Physical Review A.

[24]  I. Chuang,et al.  Optimal Hamiltonian Simulation by Quantum Signal Processing. , 2016, Physical review letters.

[25]  W. Auzinger,et al.  Local error structures and order conditions in terms of Lie elements for exponential splitting schemes , 2014 .

[26]  R. Cleve,et al.  Efficient Quantum Algorithms for Simulating Sparse Hamiltonians , 2005, quant-ph/0508139.

[27]  Andrew M. Childs,et al.  Simulating Hamiltonian dynamics with a truncated Taylor series. , 2014, Physical review letters.

[28]  S. Descombes,et al.  An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime , 2010 .

[29]  R. Somma.  A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation , 2015, 1512.03416.

[30]  Ilse C. F. Ipsen,et al.  Relative perturbation techniques for singular value problems , 1995 .

[31]  R. Feynman.  Simulating physics with computers , 1999 .

[32]  Hartmut Neven,et al.  Improved Fault-Tolerant Quantum Simulation of Condensed-Phase Correlated Electrons via Trotterization , 2019, Quantum.

[33]  F. Verstraete,et al.  Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space. , 2011, Physical review letters.

[34]  H. Neven,et al.  Low-Depth Quantum Simulation of Materials , 2018 .

[35]  Daniel A. Spielman,et al.  Exponential algorithmic speedup by a quantum walk , 2002, STOC '03.

[36]  Ilse C. F. Ipsen.  Relative perturbation results for matrix eigenvalues and singular values , 1998, Acta Numerica.

[37]  John Preskill,et al.  Quantum Algorithms for Quantum Field Theories , 2011, Science.

[38]  Guillaume Aubrun,et al.  Locally restricted measurements on a multipartite quantum system: data hiding is generic , 2015, Quantum Inf. Comput..

[39]  M. Hastings,et al.  Solving strongly correlated electron models on a quantum computer , 2015, 1506.05135.

[40]  D. W. Robinson,et al.  The finite group velocity of quantum spin systems , 1972 .

[41]  C. Lubich,et al.  Error Bounds for Exponential Operator Splittings , 2000 .

[42]  I. Chuang,et al.  Hamiltonian Simulation by Qubitization , 2016, Quantum.

[43]  Alán Aspuru-Guzik,et al.  On the Chemical Basis of Trotter-Suzuki Errors in Quantum Chemistry Simulation , 2014, 1410.8159.

[44]  P. Høyer,et al.  Higher order decompositions of ordered operator exponentials , 2008, 0812.0562.

[45]  Fernando Casas,et al.  A Concise Introduction to Geometric Numerical Integration , 2016 .

[46]  D. Aharonov,et al.  Fast-forwarding of Hamiltonians and exponentially precise measurements , 2016, Nature Communications.

[47]  Dmitri Maslov,et al.  Toward the first quantum simulation with quantum speedup , 2017, Proceedings of the National Academy of Sciences.

[48]  P. Zoller,et al.  Digital quantum simulation, Trotter errors, and quantum chaos of the kicked top , 2018, npj Quantum Information.

[49]  Alexey V Gorshkov,et al.  Nearly linear light cones in long-range interacting quantum systems. , 2014, Physical review letters.

[50]  Alán Aspuru-Guzik,et al.  Quantum Simulation of Electronic Structure with Linear Depth and Connectivity. , 2017, Physical review letters.

[51]  Simon Benjamin,et al.  Error-Mitigated Digital Quantum Simulation. , 2018, Physical review letters.

[52]  Tomotaka Kuwahara,et al.  Strictly Linear Light Cones in Long-Range Interacting Systems of Arbitrary Dimensions , 2019, 1910.14477.

[53]  A. Harrow,et al.  Quantum algorithm for linear systems of equations. , 2008, Physical review letters.

[54]  Andrew Lucas,et al.  Finite Speed of Quantum Scrambling with Long Range Interactions. , 2019, Physical review letters.

[55]  Mechthild Thalhammer,et al.  High-Order Exponential Operator Splitting Methods for Time-Dependent Schrödinger Equations , 2008, SIAM J. Numer. Anal..

[56]  Matthew B. Hastings,et al.  Spectral Gap and Exponential Decay of Correlations , 2005 .

[57]  Peter Zoller,et al.  Quantum localization bounds Trotter errors in digital quantum simulation , 2018, Science Advances.

[58]  B. Lanyon,et al.  Towards quantum chemistry on a quantum computer. , 2009, Nature chemistry.

[59]  Earl Campbell,et al.  Compilation by stochastic Hamiltonian sparsification , 2019 .

[60]  Andrew M. Childs,et al.  Exponential improvement in precision for simulating sparse Hamiltonians , 2013, Forum of Mathematics, Sigma.

[61]  Seth Lloyd,et al.  Universal Quantum Simulators , 1996, Science.

[62]  Michael Kastner,et al.  Interplay of soundcone and supersonic propagation in lattice models with power law interactions , 2015, 1502.05891.

[63]  Amnon Ta-Shma,et al.  Adiabatic quantum state generation and statistical zero knowledge , 2003, STOC '03.

[64]  Jeongwan Haah,et al.  Quantum Algorithm for Simulating Real Time Evolution of Lattice Hamiltonians , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[65]  Maris Ozols,et al.  Simulating Large Quantum Circuits on a Small Quantum Computer. , 2019, Physical review letters.

[66]  M. Troyer,et al.  Elucidating reaction mechanisms on quantum computers , 2016, Proceedings of the National Academy of Sciences.

[67]  E. Campbell.  Random Compiler for Fast Hamiltonian Simulation. , 2018, Physical review letters.

[68]  Jens Eisert,et al.  Lieb-Robinson bounds and the simulation of time evolution of local observables in lattice systems , 2013, 1306.0716.

[69]  Trygve Helgaker,et al.  Molecular Electronic-Structure Theory: Helgaker/Molecular Electronic-Structure Theory , 2000 .

[70]  David Gosset,et al.  Polynomial-time classical simulation of quantum ferromagnets , 2016, Physical review letters.

[71]  M. Suzuki,et al.  General theory of fractal path integrals with applications to many‐body theories and statistical physics , 1991 .

[72]  Bruno Nachtergaele,et al.  Propagation of Correlations in Quantum Lattice Systems , 2006, math-ph/0603064.

[73]  Charles R. Johnson,et al.  Matrix Analysis, 2nd Ed , 2012 .

[74]  Yuan Su,et al.  Faster quantum simulation by randomization , 2018, Quantum.