Hamiltonian Simulation by Qubitization

Guang Hao Low1 and Isaac L. Chuang2

1Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
2Department of Electrical Engineering and Computer Science, Department of Physics, Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


We present the problem of approximating the time-evolution operator $e^{-i\hat{H}t}$ to error $\epsilon$, where the Hamiltonian $\hat{H}=(\langle G|\otimes\hat{\mathcal{I}})\hat{U}(|G\rangle\otimes\hat{\mathcal{I}})$ is the projection of a unitary oracle $\hat{U}$ onto the state $|G\rangle$ created by another unitary oracle. Our algorithm solves this with a query complexity $\mathcal{O}\big(t+\log({1/\epsilon})\big)$ to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are $d$-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where $\hat{H}$ is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed any $\hat{H}$ in an invariant $\text{SU}(2)$ subspace. A large class of operator functions of $\hat{H}$ can then be computed with optimal query complexity, of which $e^{-i\hat{H}t}$ is a special case.

► BibTeX data

► References

[1] S. Lloyd, ``Universal Quantum Simulators,'' Science 273, 1073 (1996).

[2] D. Aharonov and A. Ta-Shma, ``Adiabatic quantum state generation and statistical zero knowledge,'' in Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03, STOC '03 (ACM Press, New York, New York, USA, 2003) p. 20.

[3] A. M. Childs and N. Wiebe, ``Hamiltonian Simulation Using Linear Combinations of Unitary Operations,'' Quantum Information & Computation 12, 901 (2012).

[4] D. W. Berry and A. M. Childs, ``Black-box Hamiltonian simulation and unitary implementation,'' Quantum Information & Computation 12, 29 (2012).

[5] S. Lloyd, M. Mohseni, and P. Rebentrost, ``Quantum principal component analysis,'' Nature Physics 10, 631 (2014).

[6] D. W. Berry, A. M. Childs, and R. Kothari, ``Hamiltonian Simulation with Nearly Optimal Dependence on all Parameters,'' in 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS '15 (IEEE, Washington, DC, USA, 2015) pp. 792–809.

[7] G. H. Low and I. L. Chuang, ``Optimal Hamiltonian Simulation by Quantum Signal Processing,'' Physical Review Letters 118, 010501 (2017a).

[8] A. W. Harrow, A. Hassidim, and S. Lloyd, ``Quantum Algorithm for Linear Systems of Equations,'' Physical Review Letters 103, 150502 (2009).

[9] A. M. Childs, R. Kothari, and R. D. Somma, ``Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Dependence on Precision,'' SIAM Journal on Computing 46, 1920 (2017).

[10] A. N. Chowdhury and R. D. Somma, ``Quantum algorithms for Gibbs sampling and hitting-time estimation,'' Quantum Information & Computation 17, 41 (2017).

[11] F. G. Brandao and K. M. Svore, ``Quantum Speed-Ups for Solving Semidefinite Programs,'' 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) , 415 (2017).

[12] M.-H. Yung, J. D. Whitfield, S. Boixo, D. G. Tempel, and A. Aspuru-Guzik, ``Introduction to Quantum Algorithms for Physics and Chemistry,'' in Quantum Information and Computation for Chemistry (John Wiley & Sons, Inc., 2014) pp. 67–106.

[13] D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings, and M. Troyer, ``Gate-count estimates for performing quantum chemistry on small quantum computers,'' Physical Review A 90, 022305 (2014).

[14] D. Poulin, M. B. Hastings, D. Wecker, N. Wiebe, A. C. Doherty, and M. Troyer, ``The Trotter step size required for accurate quantum simulation of quantum chemistry,'' Quantum Information & Computation 15, 361 (2015).

[15] M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, and M. Troyer, ``Elucidating reaction mechanisms on quantum computers,'' Proceedings of the National Academy of Sciences 114, 7555 (2017).

[16] R. Babbush, D. W. Berry, I. D. Kivlichan, A. Y. Wei, P. J. Love, and A. Aspuru-Guzik, ``Exponentially more precise quantum simulation of fermions in second quantization,'' New Journal of Physics 18, 033032 (2016).

[17] I. D. Kivlichan, N. Wiebe, R. Babbush, and A. Aspuru-Guzik, ``Bounding the costs of quantum simulation of many-body physics in real space,'' Journal of Physics A: Mathematical and Theoretical 50, 305301 (2017).

[18] P. J. J. O'Malley, R. Babbush, I. D. Kivlichan, J. Romero, J. R. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. Ding, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Jeffrey, E. Lucero, A. Megrant, J. Y. Mutus, M. Neeley, C. Neill, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, P. V. Coveney, P. J. Love, H. Neven, A. Aspuru-Guzik, and J. M. Martinis, ``Scalable Quantum Simulation of Molecular Energies,'' Physical Review X 6, 031007 (2016).

[19] R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O'Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, ``Superconducting quantum circuits at the surface code threshold for fault tolerance,'' Nature 508, 500 (2014).

[20] S. Debnath, N. M. Linke, C. Figgatt, K. A. Landsman, K. Wright, and C. Monroe, ``Demonstration of a small programmable quantum computer with atomic qubits,'' Nature 536, 63 (2016).

[21] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, ``Exponential improvement in precision for simulating sparse Hamiltonians,'' in Proceedings of the 46th Annual ACM Symposium on Theory of Computing - STOC '14, STOC '14 (ACM Press, New York, New York, USA, 2014) pp. 283–292.

[22] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, ``Simulating Hamiltonian Dynamics with a Truncated Taylor Series,'' Physical Review Letters 114, 090502 (2015b).

[23] A. M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, and D. A. Spielman, ``Exponential algorithmic speedup by a quantum walk,'' in Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03, STOC '03 (ACM Press, New York, New York, USA, 2003) p. 59.

[24] A. M. Childs, ``On the Relationship Between Continuous- and Discrete-Time Quantum Walk,'' Communications in Mathematical Physics 294, 581 (2010).

[25] R. Kothari, Efficient algorithms in quantum query complexity, Ph.D. thesis, University of Waterloo (2014).

[26] M. Szegedy, ``Spectra of Quantized Walks and a $\sqrt{\delta\epsilon}$ rule,'' arXiv preprint quant-ph/​0401053 (2004a).

[27] D. W. Berry and L. Novo, ``Corrected Quantum Walk for Optimal Hamiltonian Simulation,'' Quantum Information & Computation 16, 1295 (2016).

[28] S. Kimmel, C. Y.-Y. Lin, G. H. Low, M. Ozols, and T. J. Yoder, ``Hamiltonian simulation with optimal sample complexity,'' npj Quantum Information 3, 13 (2017).

[29] S. Chakraborty, A. Gilyén, and S. Jeffery, ``The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation,'' arXiv preprint arXiv:1804.01973 (2018).
arXiv:1804.01973 http://arxiv.org/abs/1804.01973

[30] R. D. Somma and S. Boixo, ``Spectral Gap Amplification,'' SIAM Journal on Computing 42, 593 (2013).

[31] M. Szegedy, ``Quantum Speed-Up of Markov Chain Based Algorithms,'' in 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS '04 (IEEE, Washington, DC, USA, 2004) pp. 32–41.

[32] A. Daskin and S. Kais, ``An ancilla-based quantum simulation framework for non-unitary matrices,'' Quantum Information Processing 16, 33 (2017).

[33] G. Meinardus, Approximation of Functions: Theory and Numerical Methods, Springer Tracts in Natural Philosophy, Vol. 13 (Springer Berlin Heidelberg, Berlin, Heidelberg, 1967).

[34] L. K. Grover, ``A fast quantum mechanical algorithm for database search,'' Proceedings of the twenty-eighth annual ACM symposium on Theory of computing - STOC '96 STOC '96, 212 (1996).

[35] T. J. Yoder, G. H. Low, and I. L. Chuang, ``Fixed-Point Quantum Search with an Optimal Number of Queries,'' Physical Review Letters 113, 210501 (2014).

[36] J. McClellan, T. Parks, and L. Rabiner, ``A computer program for designing optimum FIR linear phase digital filters,'' IEEE Transactions on Audio and Electroacoustics 21, 506 (1973).

[37] G. H. Low, T. J. Yoder, and I. L. Chuang, ``Methodology of Resonant Equiangular Composite Quantum Gates,'' Physical Review X 6, 041067 (2016).

[38] M. Abramowitz, I. A. Stegun, and Others, ``Handbook of mathematical functions,'' Applied mathematics series 55, 62 (1966).

[39] J. P. Boyd, ``Rootfinding for a transcendental equation without a first guess: Polynomialization of Kepler's equation through Chebyshev polynomial expansion of the sine,'' Applied Numerical Mathematics 57, 12 (2007).

[40] A. M. Childs and R. Kothari, ``Limitations on the Simulation of Non-sparse Hamiltonians,'' Quantum Information & Computation 10, 669 (2010).

[41] R. D. Somma, ``A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation,'' Journal of Mathematical Physics 57, 062202 (2016).

[42] G. H. Low, T. J. Yoder, and I. L. Chuang, ``Quantum Imaging by Coherent Enhancement,'' Physical Review Letters 114, 100801 (2015).

[43] A. Gilyén, Y. Su, G. H. Low, and N. Wiebe, ``Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics,'' in Proceedings of the 51st Annual ACM Symposium on Theory of Computing - STOC '19 (ACM Press, New York, New York, USA, 2019) pp. 193–204.

[44] J. Haah, M. Hastings, R. Kothari, and G. H. Low, ``Quantum Algorithm for Simulating Real Time Evolution of Lattice Hamiltonians,'' in 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), FOCS '18 (IEEE, Washington, DC, USA, 2018) pp. 350–360.

[45] A. M. Childs and Y. Su, ``Nearly optimal lattice simulation by product formulas,'' arXiv preprint arXiv:1901.00564 (2019).

[46] G. H. Low and I. L. Chuang, ``Hamiltonian Simulation by Uniform Spectral Amplification,'' arXiv preprint arXiv:1707.05391 (2017b).

[47] G. H. Low, ``Hamiltonian simulation with nearly optimal dependence on spectral norm,'' in Proceedings of the 51st Annual ACM Symposium on Theory of Computing - STOC '19 (ACM Press, New York, New York, USA, 2019) pp. 491–502.

[48] G. H. Low and N. Wiebe, ``Hamiltonian Simulation in the Interaction Picture,'' arXiv preprint arXiv:1805.00675 (2018).

[49] A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, and Y. Su, ``Toward the first quantum simulation with quantum speedup,'' Proceedings of the National Academy of Sciences 115, 9456 (2018).

[50] J. Haah, ``Product Decomposition of Periodic Functions in Quantum Signal Processing,'' arXiv preprint arXiv:1806.10236 (2018).

[51] L. J. Karam and J. H. McClellan, ``Chebyshev digital FIR filter design,'' Signal Processing 76, 17 (1999).

Cited by

[1] Alexander F. Shaw, Pavel Lougovski, Jesse R. Stryker, and Nathan Wiebe, "Quantum Algorithms for Simulating the Lattice Schwinger Model", Quantum 4, 306 (2020).

[2] Shantanav Chakraborty, Kyle Luh, and Jérémie Roland, "Analog quantum algorithms for the mixing of Markov chains", Physical Review A 102 2, 022423 (2020).

[3] Jessica Lemieux, Bettina Heim, David Poulin, Krysta Svore, and Matthias Troyer, "Efficient Quantum Walk Circuits for Metropolis-Hastings Algorithm", Quantum 4, 287 (2020).

[4] Andrew M. Childs, Aaron Ostrander, and Yuan Su, "Faster quantum simulation by randomization", arXiv:1805.08385, Quantum 3, 182 (2019).

[5] Joran van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf, "Quantum SDP-Solvers: Better upper and lower bounds", arXiv:1705.01843, Quantum 4, 230 (2020).

[6] Ian D. Kivlichan, Craig Gidney, Dominic W. Berry, Nathan Wiebe, Jarrod McClean, Wei Sun, Zhang Jiang, Nicholas Rubin, Austin Fowler, Alán Aspuru-Guzik, Hartmut Neven, and Ryan Babbush, "Improved Fault-Tolerant Quantum Simulation of Condensed-Phase Correlated Electrons via Trotterization", Quantum 4, 296 (2020).

[7] Riley W. Chien, Sha Xue, Tarini S. Hardikar, Kanav Setia, and James D. Whitfield, "Analysis of superfast encoding performance for electronic structure simulations", Physical Review A 100 3, 032337 (2019).

[8] Andrew Zhao, Andrew Tranter, William M. Kirby, Shu Fay Ung, Akimasa Miyake, and Peter J. Love, "Measurement reduction in variational quantum algorithms", Physical Review A 101 6, 062322 (2020).

[9] P A M Casares and M A Martin-Delgado, "A quantum interior-point predictor–corrector algorithm for linear programming", Journal of Physics A: Mathematical and Theoretical 53 44, 445305 (2020).

[10] Alessandro Roggero, Andy C. Y. Li, Joseph Carlson, Rajan Gupta, and Gabriel N. Perdue, "Quantum computing for neutrino-nucleus scattering", Physical Review D 101 7, 074038 (2020).

[11] Nicholas P. Bauman, Guang Hao Low, and Karol Kowalski, "Quantum simulations of excited states with active-space downfolded Hamiltonians", The Journal of Chemical Physics 151 23, 234114 (2019).

[12] Fan-Xu Meng, Xu-Tao Yu, and Zai-Chen Zhang, "Improved quantum algorithm for MMSE-based massive MIMO uplink detection", Quantum Information Processing 19 8, 267 (2020).

[13] Koen Groenland, Freek Witteveen, Kareljan Schoutens, and Rene Gerritsma, "Signal processing techniques for efficient compilation of controlled rotations in trapped ions", New Journal of Physics 22 6, 063006 (2020).

[14] Mahmoud Mahdian and H. Davoodi Yeganeh, "Incoherent quantum algorithm dynamics of an open system with near-term devices", Quantum Information Processing 19 9, 285 (2020).

[15] Alexander Engel, Graeme Smith, and Scott E. Parker, "Quantum algorithm for the Vlasov equation", Physical Review A 100 6, 062315 (2019).

[16] I. Meyerov, A. Liniov, M. Ivanchenko, and S. Denisov, "Modeling Complex Quantum Dynamics: Evolution of Numerical Algorithms in the HPC Context", Lobachevskii Journal of Mathematics 41 8, 1509 (2020).

[17] Michael P. Kaicher, Simon B. Jäger, Pierre-Luc Dallaire-Demers, and Frank K. Wilhelm, "Roadmap for quantum simulation of the fractional quantum Hall effect", Physical Review A 102 2, 022607 (2020).

[18] Rolando D Somma, "Quantum eigenvalue estimation via time series analysis", arXiv:1907.11748, New Journal of Physics 21 12, 123025 (2019).

[19] Dominic W. Berry, Andrew M. Childs, Yuan Su, Xin Wang, and Nathan Wiebe, "Time-dependent Hamiltonian simulation with L1-norm scaling", arXiv:1906.07115, Quantum 4, 254 (2020).

[20] Changpeng Shao, "Quantum speedup of Bayes’ classifiers", Journal of Physics A: Mathematical and Theoretical 53 4, 045301 (2020).

[21] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan, "Quantum computational chemistry", arXiv:1808.10402, Reviews of Modern Physics 92 1, 015003 (2020).

[22] William M. Kirby and Peter J. Love, "Contextuality Test of the Nonclassicality of Variational Quantum Eigensolvers", Physical Review Letters 123 20, 200501 (2019).

[23] Abhoy Kole and Indranil Sengupta, 2020 IEEE International Test Conference India 1 (2020) ISBN:978-1-7281-7458-7.

[24] Yuan Su, "Framework for Hamiltonian simulation and beyond: standard-form encoding, qubitization, and quantum signal processing", Quantum Views 3, 21 (2019).

[25] Jeongwan Haah, "Product Decomposition of Periodic Functions in Quantum Signal Processing", arXiv:1806.10236, Quantum 3, 190 (2019).

[26] Shantanav Chakraborty, Leonardo Novo, and Jérémie Roland, "Finding a marked node on any graph via continuous-time quantum walks", Physical Review A 102 2, 022227 (2020).

[27] Andrew M. Childs and Yuan Su, "Nearly Optimal Lattice Simulation by Product Formulas", Physical Review Letters 123 5, 050503 (2019).

[28] Yingkai Ouyang, David R. White, and Earl T. Campbell, "Compilation by stochastic Hamiltonian sparsification", Quantum 4, 235 (2020).

[29] Bela Bauer, Sergey Bravyi, Mario Motta, and Garnet Kin-Lic Chan, "Quantum Algorithms for Quantum Chemistry and Quantum Materials Science", Chemical Reviews acs.chemrev.9b00829 (2020).

[30] Ryan Babbush, Dominic W. Berry, Jarrod R. McClean, and Hartmut Neven, "Quantum simulation of chemistry with sublinear scaling in basis size", npj Quantum Information 5 1, 92 (2019).

[31] Suguru Endo, Jinzhao Sun, Ying Li, Simon C. Benjamin, and Xiao Yuan, "Variational Quantum Simulation of General Processes", Physical Review Letters 125 1, 010501 (2020).

[32] Patrick Rall, "Quantum algorithms for estimating physical quantities using block encodings", Physical Review A 102 2, 022408 (2020).

[33] Jingwei Wen, Guoqing Qin, Chao Zheng, Shijie Wei, Xiangyu Kong, Tao Xin, and Guilu Long, "Observation of information flow in the anti-𝒫𝒯-symmetric system with nuclear spins", npj Quantum Information 6 1, 28 (2020).

[34] Leonardo Novo, "Bridging gaps between random approaches to quantum simulation", Quantum Views 4, 33 (2020).

[35] Vijay Balasubramanian, Matthew DeCross, Arjun Kar, and Onkar Parrikar, "Quantum complexity of time evolution with chaotic Hamiltonians", Journal of High Energy Physics 2020 1, 134 (2020).

[36] Trevor Keen, Thomas Maier, Steven Johnston, and Pavel Lougovski, "Quantum-classical simulation of two-site dynamical mean-field theory on noisy quantum hardware", Quantum Science and Technology 5 3, 035001 (2020).

[37] Dominic W. Berry, Craig Gidney, Mario Motta, Jarrod R. McClean, and Ryan Babbush, "Qubitization of Arbitrary Basis Quantum Chemistry Leveraging Sparsity and Low Rank Factorization", Quantum 3, 208 (2019).

[38] Mark Steudtner and Stephanie Wehner, "Estimating exact energies in quantum simulation without Toffoli gates", Physical Review A 101 5, 052329 (2020).

[39] Matthew B. Hastings, "Classical and Quantum Algorithms for Tensor Principal Component Analysis", arXiv:1907.12724, Quantum 4, 237 (2020).

[40] Matthew B. Hastings, "Duality in Quantum Quenches and Classical Approximation Algorithms: Pretty Good or Very Bad", Quantum 3, 201 (2019).

[41] Alessandro Roggero and Joseph Carlson, "Dynamic linear response quantum algorithm", Physical Review C 100 3, 034610 (2019).

[42] A. Roggero and A. Baroni, "Short-depth circuits for efficient expectation-value estimation", Physical Review A 101 2, 022328 (2020).

[43] Carlos Outeiral, Martin Strahm, Jiye Shi, Garrett M. Morris, Simon C. Benjamin, and Charlotte M. Deane, "The prospects of quantum computing in computational molecular biology", WIREs Computational Molecular Science (2020).

[44] A. Roggero, "Spectral-density estimation with the Gaussian integral transform", Physical Review A 102 2, 022409 (2020).

[45] Mario Motta, Tanvi P. Gujarati, Julia E. Rice, Ashutosh Kumar, Conner Masteran, Joseph A. Latone, Eunseok Lee, Edward F. Valeev, and Tyler Y. Takeshita, "Quantum simulation of electronic structure with a transcorrelated Hamiltonian: improved accuracy with a smaller footprint on the quantum computer", Physical Chemistry Chemical Physics (2020).

[46] Vedran Dunjko and Hans J. Briegel, "Machine learning & artificial intelligence in the quantum domain: a review of recent progress", Reports on Progress in Physics 81 7, 074001 (2018).

[47] Ryan Babbush, Nathan Wiebe, Jarrod McClean, James McClain, Hartmut Neven, and Garnet Kin-Lic Chan, "Low-Depth Quantum Simulation of Materials", Physical Review X 8 1, 011044 (2018).

[48] Shantanav Chakraborty, András Gilyén, and Stacey Jeffery, "The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation", arXiv:1804.01973.

[49] Ryan Babbush, Craig Gidney, Dominic W. Berry, Nathan Wiebe, Jarrod McClean, Alexandru Paler, Austin Fowler, and Hartmut Neven, "Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity", Physical Review X 8 4, 041015 (2018).

[50] Earl Campbell, "Random Compiler for Fast Hamiltonian Simulation", Physical Review Letters 123 7, 070503 (2019).

[51] András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe, "Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics", arXiv:1806.01838.

[52] Joran van Apeldoorn and András Gilyén, "Improvements in Quantum SDP-Solving with Applications", arXiv:1804.05058.

[53] Guang Hao Low and Nathan Wiebe, "Hamiltonian Simulation in the Interaction Picture", arXiv:1805.00675.

[54] Vedran Dunjko and Hans J. Briegel, "Machine learning \& artificial intelligence in the quantum domain", arXiv:1709.02779.

[55] Mario Motta, Erika Ye, Jarrod R. McClean, Zhendong Li, Austin J. Minnich, Ryan Babbush, and Garnet Kin-Lic Chan, "Low rank representations for quantum simulation of electronic structure", arXiv:1808.02625.

[56] Andrew M. Childs, Dmitri Maslov, Yunseong Nam, Neil J. Ross, and Yuan Su, "Toward the first quantum simulation with quantum speedup", arXiv:1711.10980.

[57] Yudong Cao, Jonathan Romero, Jonathan P. Olson, Matthias Degroote, Peter D. Johnson, Mária Kieferová, Ian D. Kivlichan, Tim Menke, Borja Peropadre, Nicolas P. D. Sawaya, Sukin Sim, Libor Veis, and Alán Aspuru-Guzik, "Quantum Chemistry in the Age of Quantum Computing", arXiv:1812.09976.

[58] Jeongwan Haah, Matthew B. Hastings, Robin Kothari, and Guang Hao Low, "Quantum algorithm for simulating real time evolution of lattice Hamiltonians", arXiv:1801.03922.

[59] Patrick Rebentrost, Maria Schuld, Leonard Wossnig, Francesco Petruccione, and Seth Lloyd, "Quantum gradient descent and Newton's method for constrained polynomial optimization", arXiv:1612.01789.

[60] Guang Hao Low and Isaac L. Chuang, "Hamiltonian Simulation by Uniform Spectral Amplification", arXiv:1707.05391.

[61] David B. Kaplan and Jesse R. Stryker, "Gauss's Law, Duality, and the Hamiltonian Formulation of U(1) Lattice Gauge Theory", arXiv:1806.08797.

[62] Dominic W. Berry, Mária Kieferová, Artur Scherer, Yuval R. Sanders, Guang Hao Low, Nathan Wiebe, Craig Gidney, and Ryan Babbush, "Improved techniques for preparing eigenstates of fermionic Hamiltonians", npj Quantum Information 4, 22 (2018).

[63] Ryan Babbush, Dominic W. Berry, and Hartmut Neven, "Quantum simulation of the Sachdev-Ye-Kitaev model by asymmetric qubitization", Physical Review A 99 4, 040301 (2019).

[64] Guoming Wang, "Quantum algorithm for linear regression", arXiv:1402.0660, Physical Review A 96 1, 012335 (2017).

[65] Dominic W. Berry, Andrew M. Childs, Aaron Ostrander, and Guoming Wang, "Quantum Algorithm for Linear Differential Equations with Exponentially Improved Dependence on Precision", Communications in Mathematical Physics 356 3, 1057 (2017).

[66] Sathyawageeswar Subramanian, Stephen Brierley, and Richard Jozsa, "Implementing smooth functions of a Hermitian matrix on a quantum computer", Journal of Physics Communications 3 6, 065002 (2019).

[67] Danial Dervovic, Mark Herbster, Peter Mountney, Simone Severini, Naïri Usher, and Leonard Wossnig, "Quantum linear systems algorithms: a primer", arXiv:1802.08227.

[68] David Poulin, Alexei Kitaev, Damian S. Steiger, Matthew B. Hastings, and Matthias Troyer, "Quantum Algorithm for Spectral Measurement with a Lower Gate Count", Physical Review Letters 121 1, 010501 (2018).

[69] Nicholas P. Bauman, Eric J. Bylaska, Sriram Krishnamoorthy, Guang Hao Low, Nathan Wiebe, Christopher E. Granade, Martin Roetteler, Matthias Troyer, and Karol Kowalski, "Downfolding of many-body Hamiltonians using active-space models: Extension of the sub-system embedding sub-algebras approach to unitary coupled cluster formalisms", Journal of Chemical Physics 151 1, 014107 (2019).

[70] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu, "A Theory of Trotter Error", arXiv:1912.08854.

[71] Natalie Klco and Martin J. Savage, "Digitization of scalar fields for quantum computing", arXiv:1808.10378, Physical Review A 99 5, 052335 (2019).

[72] Guang Hao Low, "Hamiltonian simulation with nearly optimal dependence on spectral norm", arXiv:1807.03967.

[73] Guang Hao Low, Vadym Kliuchnikov, and Luke Schaeffer, "Trading T-gates for dirty qubits in state preparation and unitary synthesis", arXiv:1812.00954.

[74] Ian D. Kivlichan, Nathan Wiebe, Ryan Babbush, and Alán Aspuru-Guzik, "Bounding the costs of quantum simulation of many-body physics in real space", Journal of Physics A Mathematical General 50 30, 305301 (2017).

[75] Ryan Babbush, Nathan Wiebe, Jarrod McClean, James McClain, Hartmut Neven, and Garnet Kin-Lic Chan, "Low Depth Quantum Simulation of Electronic Structure", arXiv:1706.00023.

[76] Guang Hao Low, Nicholas P. Bauman, Christopher E. Granade, Bo Peng, Nathan Wiebe, Eric J. Bylaska, Dave Wecker, Sriram Krishnamoorthy, Martin Roetteler, Karol Kowalski, Matthias Troyer, and Nathan A. Baker, "Q# and NWChem: Tools for Scalable Quantum Chemistry on Quantum Computers", arXiv:1904.01131.

[77] Daniel Litinski, "A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery", arXiv:1808.02892.

[78] Guang Hao Low, Theodore J. Yoder, and Isaac L. Chuang, "Methodology of Resonant Equiangular Composite Quantum Gates", Physical Review X 6 4, 041067 (2016).

[79] Ammar Daskin and Sabre Kais, "A generalized circuit for the Hamiltonian dynamics through the truncated series", Quantum Information Processing 17 12, 328 (2018).

[80] Mária Kieferová, Artur Scherer, and Dominic W. Berry, "Simulating the dynamics of time-dependent Hamiltonians with a truncated Dyson series", Physical Review A 99 4, 042314 (2019).

[81] Yuval R. Sanders, Guang Hao Low, Artur Scherer, and Dominic W. Berry, "Black-box quantum state preparation without arithmetic", arXiv:1807.03206, Physical Review Letters 122 2, 020502 (2018).

[82] András Gilyén and Tongyang Li, "Distributional property testing in a quantum world", arXiv:1902.00814.

[83] Andrew M. Childs and Jin-Peng Liu, "Quantum Spectral Methods for Differential Equations", Communications in Mathematical Physics 375 2, 1427 (2020).

[84] Alessandro Roggero and Joseph Carlson, "Linear Response on a Quantum Computer", arXiv:1804.01505.

[85] Zhikuan Zhao, "Quantum Statistical Inference", arXiv:1812.04877.

[86] Teng Bian, Daniel Murphy, Rongxin Xia, Ammar Daskin, and Sabre Kais, "Quantum computing methods for electronic states of the water molecule", Molecular Physics 117 15-16, 2069 (2019).

[87] Suguru Endo, Qi Zhao, Ying Li, Simon Benjamin, and Xiao Yuan, "Mitigating algorithmic errors in a Hamiltonian simulation", arXiv:1808.03623, Physical Review A 99 1, 012334 (2019).

[88] Seth Lloyd and Reevu Maity, "Efficient implementation of unitary transformations", arXiv:1901.03431.

[89] Changpeng Shao, "An Improved Algorithm for Quantum Principal Component Analysis", arXiv:1903.03999.

[90] Yimin Ge, Jordi Tura, and J. Ignacio Cirac, "Faster ground state preparation and high-precision ground energy estimation with fewer qubits", arXiv:1712.03193, Journal of Mathematical Physics 60 2, 022202 (2017).

[91] Leonardo Novo and Dominic W. Berry, "Improved Hamiltonian simulation via a truncated Taylor series and corrections", arXiv:1611.10033.

[92] Ronald de Wolf, "Quantum Computing: Lecture Notes", arXiv:1907.09415.

[93] M. B. Hastings, "The Short Path Algorithm Applied to a Toy Model", arXiv:1901.03884.

[94] Sathyawageeswar Subramanian and Min-Hsiu Hsieh, "Quantum algorithm for estimating Renyi entropies of quantum states", arXiv:1908.05251.

[95] Jarrod R. McClean, Fabian M. Faulstich, Qinyi Zhu, Bryan O'Gorman, Yiheng Qiu, Steven R. White, Ryan Babbush, and Lin Lin, "Discontinuous Galerkin discretization for quantum simulation of chemistry", New Journal of Physics 22 9, 093015 (2020).

[96] Alex Parent, Martin Roetteler, and Michele Mosca, "Improved reversible and quantum circuits for Karatsuba-based integer multiplication", arXiv:1706.03419.

[97] Ian D. Kivlichan, Christopher E. Granade, and Nathan Wiebe, "Phase estimation with randomized Hamiltonians", arXiv:1907.10070.

[98] András Gilyén, Seth Lloyd, Iman Marvian, Yihui Quek, and Mark M. Wilde, "Quantum algorithm for Petz recovery channels and pretty good measurements", arXiv:2006.16924.

[99] François Fillion-Gourdeau, Steve MacLean, and Raymond Laflamme, "Efficient state initialization by a quantum spectral filtering algorithm", Physical Review A 95 4, 042331 (2017).

[100] Rui Chao, Dawei Ding, Andras Gilyen, Cupjin Huang, and Mario Szegedy, "Finding Angles for Quantum Signal Processing with Machine Precision", arXiv:2003.02831.

[101] Shalev Ben-David, Andrew M. Childs, András Gilyén, William Kretschmer, Supartha Podder, and Daochen Wang, "Symmetries, graph properties, and quantum speedups", arXiv:2006.12760.

[102] Minh C. Tran, Yuan Su, Daniel Carney, and Jacob M. Taylor, "Faster Digital Quantum Simulation by Symmetry Protection", arXiv:2006.16248.

[103] Bojia Duan, Jiabin Yuan, Chao-Hua Yu, Jianbang Huang, and Chang-Yu Hsieh, "A survey on HHL algorithm: From theory to application in quantum machine learning", Physics Letters A 384, 126595 (2020).

[104] Jinfeng Zeng, Chenfeng Cao, Chao Zhang, Pengxiang Xu, and Bei Zeng, "A variational quantum algorithm for Hamiltonian diagonalization", arXiv:2008.09854.

[105] Chenyi Zhang, Jiaqi Leng, and Tongyang Li, "Quantum Algorithms for Escaping from Saddle Points", arXiv:2007.10253.

The above citations are from Crossref's cited-by service (last updated successfully 2020-10-28 06:50:27) and SAO/NASA ADS (last updated successfully 2020-10-28 06:50:29). The list may be incomplete as not all publishers provide suitable and complete citation data.