Preparing Hamiltonians for quantum simulation: A computational framework for Cartan decomposition via Lax dynamics

Chu, Moody

doi:10.1090/mcom/4056

Preparing Hamiltonians for quantum simulation: A computational framework for Cartan decomposition via Lax dynamics
HTML articles powered by AMS MathViewer

by Moody T. Chu;

Math. Comp.

DOI: https://doi.org/10.1090/mcom/4056

Published electronically: February 20, 2025

HTML | PDF | Request permission

Abstract:

Quantum algorithms usually are described via quantum circuits representable as unitary operators. Synthesizing the unitary operators described mathematically in terms of the unitary operators recognizable as quantum circuits is essential. One such a challenge lies in the Hamiltonian simulation problem, where the matrix exponential of a large-scale skew-Hermitian matrix is to be computed. Most current techniques are prone to approximation errors, whereas the parametrization of the underlying Hamiltonian via the Cartan decomposition is more promising. To prepare for such a simulation, this work proposes to tackle the Cartan decomposition by means of the Lax dynamics. The advantages include not only that it is numerically feasible with no matrices involved, but also that this approach offers a genuine unitary synthesis within the integration errors. This work contributes to the theoretic and algorithmic foundations in three aspects: exploiting the quaternary representation of Hamiltonian subalgebras; describing a common mechanism for deriving the Lax dynamics; and providing a mathematical theory of convergence.

References

Scott Aaronson, Quantum computing since Democritus, Cambridge University Press, Cambridge, 2013. MR 3058839, DOI 10.1017/CBO9780511979309
G. Aguilar, S. Cichy, J. Eisert, and L. Bittel, Full classification of Pauli Lie algebras, arXiv:2408.00081, 2024.
I. M. Anderson, Cartan involutions and Cartan decompositions of a semi-simple Lie algebra, Tutorials on…in 1 hour or less, Paper 5, 2005.
Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 70 (1993), no. 13, 1895–1899. MR 1208247, DOI 10.1103/PhysRevLett.70.1895
D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Simulating Hamiltonian dynamics with a truncated Taylor series, Phys. Rev. Lett. 114 (2015), 090502.
Olivier Bournez, Daniel S. Graça, and Amaury Pouly, Polynomial time corresponds to solutions of polynomial ordinary differential equations of polynomial length, J. ACM 64 (2017), no. 6, Art. 38, 76. MR 3713796, DOI 10.1145/3127496
Jack Carr, Applications of centre manifold theory, Applied Mathematical Sciences, vol. 35, Springer-Verlag, New York-Berlin, 1981. MR 635782, DOI 10.1007/978-1-4612-5929-9
Andrew M. Childs and Nathan Wiebe, Hamiltonian simulation using linear combinations of unitary operations, Quantum Inf. Comput. 12 (2012), no. 11-12, 901–924. MR 3012819, DOI 10.26421/QIC12.11-12-1
Fernando Casas, Ander Murua, and Mladen Nadinic, Efficient computation of the Zassenhaus formula, Comput. Phys. Commun. 183 (2012), no. 11, 2386–2391. MR 2956602, DOI 10.1016/j.cpc.2012.06.006
Moody T. Chu, The generalized Toda flow, the $\textrm {QR}$ algorithm and the center manifold theory, SIAM J. Algebraic Discrete Methods 5 (1984), no. 2, 187–201. MR 745438, DOI 10.1137/0605020
Moody T. Chu, Asymptotic analysis of Toda lattice on diagonalizable matrices, Nonlinear Anal. 9 (1985), no. 2, 193–201. MR 777988, DOI 10.1016/0362-546X(85)90072-0
Moody T. Chu, A differential equation approach to the singular value decomposition of bidiagonal matrices, Linear Algebra Appl. 80 (1986), 71–79. MR 851933, DOI 10.1016/0024-3795(86)90278-8
Moody T. Chu, Matrix differential equations: a continuous realization process for linear algebra problems, Nonlinear Anal. 18 (1992), no. 12, 1125–1146. MR 1171601, DOI 10.1016/0362-546X(92)90157-A
Moody T. Chu, A list of matrix flows with applications, Hamiltonian and gradient flows, algorithms and control, Fields Inst. Commun., vol. 3, Amer. Math. Soc., Providence, RI, 1994, pp. 87–97. MR 1297988
Moody T. Chu, Linear algebra algorithms as dynamical systems, Acta Numer. 17 (2008), 1–86. MR 2436009, DOI 10.1017/S0962492906340019
Moody T. Chu and Gene H. Golub, Inverse eigenvalue problems: theory, algorithms, and applications, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2005. MR 2263317, DOI 10.1093/acprof:oso/9780198566649.001.0001
Moody T. Chu and Larry K. Norris, Isospectral flows and abstract matrix factorizations, SIAM J. Numer. Anal. 25 (1988), no. 6, 1383–1391. MR 972462, DOI 10.1137/0725080
Moody T. Chu, Lax dynamics for Cartan decomposition with applications to Hamiltonian simulation, IMA J. Numer. Anal. 44 (2024), no. 3, 1406–1434. MR 4755060, DOI 10.1093/imanum/drad018
M. T. Chu, On the commutative substructure within Cartan pairs of subalgebras in $\frak {su}(2^{n})$, submitted, 2024.
J. I. Cirac and P. Zoller, Goals and opportunities in quantum simulation, Nat. Phys. 8 (2012), no. 4, 264–266.
L. Clinton, J. Bausch, and T. Cubitt, Hamiltonian simulation algorithms for near-term quantum hardware, Nat. Commun. 12 (2021), no. 1, 4989.
G. Cybenko, Reducing quantum computations to elementary unitary operations, Comput. Sci. Eng. 3 (2001), no. 2, 27–32.
Mehmet Daǧlı, Domenico D’Alessandro, and Jonathan D. H. Smith, A general framework for recursive decompositions of unitary quantum evolutions, J. Phys. A 41 (2008), no. 15, 155302, 18. MR 2449580, DOI 10.1088/1751-8113/41/15/155302
Mehmet Dagli, Lie algebra decompositions with applications to quantum dynamics, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–Iowa State University. MR 2712021
Domenico D’Alessandro, Introduction to quantum control and dynamics, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, Chapman & Hall/CRC, Boca Raton, FL, 2008. MR 2357229
Timothée Goubault de Brugière, Marc Baboulin, Benoît Valiron, and Cyril Allouche, Synthesizing quantum circuits via numerical optimization, Computational science—ICCS 2019. Part II, Lecture Notes in Comput. Sci., vol. 11537, Springer, Cham, 2019, pp. 3–16. MR 3975385, DOI 10.1007/978-3-030-22741-8
David P. DiVincenzo, Quantum computation, Science 270 (1995), no. 5234, 255–261. MR 1355956, DOI 10.1126/science.270.5234.255
Byron Drury and Peter Love, Constructive quantum Shannon decomposition from Cartan involutions, J. Phys. A 41 (2008), no. 39, 395305, 13. MR 2439220, DOI 10.1088/1751-8113/41/39/395305
Artur K. Ekert, Quantum cryptography based on Bell’s theorem, Phys. Rev. Lett. 67 (1991), no. 6, 661–663. MR 1118810, DOI 10.1103/PhysRevLett.67.661
Richard P. Feynman, Simulating physics with computers, Internat. J. Theoret. Phys. 21 (1981/82), no. 6-7, 467–488. Physics of computation, Part II (Dedham, Mass., 1981). MR 658311, DOI 10.1007/BF02650179
A. Giambruno, D. La Mattina, and V. M. Petrogradsky, Matrix algebras of polynomial codimension growth, Israel J. Math. 158 (2007), 367–378. MR 2342471, DOI 10.1007/s11856-007-0017-7
Otfried Gühne and Géza Tóth, Entanglement detection, Phys. Rep. 474 (2009), no. 1-6, 1–75. MR 2523078, DOI 10.1016/j.physrep.2009.02.004
Jeongwan Haah, Matthew B. Hastings, Robin Kothari, and Guang Hao Low, Quantum algorithm for simulating real time evolution of lattice Hamiltonians, SIAM J. Comput. 52 (2023), no. 6, FOCS18-250–FOCS18-284. MR 4679966, DOI 10.1137/18M1231511
Thomas Hawkins, Emergence of the theory of Lie groups, Sources and Studies in the History of Mathematics and Physical Sciences, Springer-Verlag, New York, 2000. An essay in the history of mathematics 1869–1926. MR 1771134, DOI 10.1007/978-1-4612-1202-7
Masahito Hayashi, Quantum information theory, 2nd ed., Graduate Texts in Physics, Springer-Verlag, Berlin, 2017. Mathematical foundation. MR 3558531, DOI 10.1007/978-3-662-49725-8
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Second printing, revised. MR 499562
Silvana Ilie, Computational complexity of numerical solutions of initial value problems for differential algebraic equations, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)–The University of Western Ontario (Canada). MR 2710821
Arieh Iserles, Hans Z. Munthe-Kaas, Syvert P. Nørsett, and Antonella Zanna, Lie-group methods, Acta numerica, 2000, Acta Numer., vol. 9, Cambridge Univ. Press, Cambridge, 2000, pp. 215–365. MR 1883629, DOI 10.1017/S0962492900002154
S. Jin, X. Li, and N. Liu, Quantum simulation in the semi-classical regime, Quantum 6 (2022), 739.
N. Khaneja and S. J. Glaser, Cartan decomposition of ${SU}(2^n)$ and control of spin systems, Chem. Phys. 267 (2001), no. 1, 11–23.
Anthony W. Knapp, Lie groups beyond an introduction, 2nd ed., Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1920389
Efekan Kökcü, Thomas Steckmann, Yan Wang, J. K. Freericks, Eugene F. Dumitrescu, and Alexander F. Kemper, Fixed depth Hamiltonian simulation via Cartan decomposition, Phys. Rev. Lett. 129 (2022), no. 7, Paper No. 070501, 7. MR 4473613, DOI 10.1103/physrevlett.129.070501
Stein Krogstad, Hans Z. Munthe-Kaas, and Antonella Zanna, Generalized polar coordinates on Lie groups and numerical integrators, Numer. Math. 114 (2009), no. 1, 161–187. MR 2557873, DOI 10.1007/s00211-009-0255-1
Seth Lloyd, Universal quantum simulators, Science 273 (1996), no. 5278, 1073–1078. MR 1407944, DOI 10.1126/science.273.5278.1073
Guang Hao Low and Isaac L. Chuang, Optimal Hamiltonian simulation by quantum signal processing, Phys. Rev. Lett. 118 (2017), no. 1, 010501, 5. MR 3664821, DOI 10.1103/PhysRevLett.118.010501
G. H. Low and I. L. Chuang, Hamiltonian simulation by qubitization, Quantum 3 (2019), 163.
Robert I. McLachlan and G. Reinout W. Quispel, Explicit geometric integration of polynomial vector fields, BIT 44 (2004), no. 3, 515–538. MR 2106014, DOI 10.1023/B:BITN.0000046814.29690.62
Massimo Melucci, Introduction to information retrieval and quantum mechanics, The Information Retrieval Series, vol. 35, Springer, Heidelberg, 2015. MR 3496532, DOI 10.1007/978-3-662-48313-8
Cleve Moler and Charles Van Loan, Nineteen dubious ways to compute the exponential of a matrix, SIAM Rev. 20 (1978), no. 4, 801–836. MR 508383, DOI 10.1137/1020098
Cleve Moler and Charles Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev. 45 (2003), no. 1, 3–49. MR 1981253, DOI 10.1137/S00361445024180
H. Z. Munthe-Kaas, G. R. W. Quispel, and A. Zanna, Generalized polar decompositions on Lie groups with involutive automorphisms, Found. Comput. Math. 1 (2001), no. 3, 297–324. MR 1838757, DOI 10.1007/s102080010012
Michael A. Nielsen and Isaac L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge, 2000. MR 1796805
Amaury Pouly and Daniel S. Graça, Computational complexity of solving polynomial differential equations over unbounded domains, Theoret. Comput. Sci. 626 (2016), 67–82. MR 3476576, DOI 10.1016/j.tcs.2016.02.002
M. Ragone, B. N. Bakalov, F. Sauvage, A. F. Kemper, C. Ortiz Marrero, M. Larocca, and M. Cerezo, A Lie algebraic theory of barren plateaus for deep parameterized quantum circuits, Nat. Commun. 15 (2024), no. 1, 7172.
R. Raussendorf and H. J. Briegel, A one-way quantum computer, Phys. Rev. Lett. 86 (2001), 5188–5191.
David Riley and Hamid Usefi, Lie algebras with finite Gelfand-Kirillov dimension, Proc. Amer. Math. Soc. 133 (2005), no. 6, 1569–1572. MR 2120270, DOI 10.1090/S0002-9939-05-07618-5
Henrique N. Sá Earp and Jiannis K. Pachos, A constructive algorithm for the Cartan decomposition of $\textrm {SU}(2^N)$, J. Math. Phys. 46 (2005), no. 8, 082108, 11. MR 2165831, DOI 10.1063/1.2008210
J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian problems, Applied Mathematics and Mathematical Computation, vol. 7, Chapman & Hall, London, 1994. MR 1270017, DOI 10.1007/978-1-4899-3093-4
V. Shende, S. Bullock, and I. Markov, Synthesis of quantum-logic circuits, IEEE Trans. Comput. Aided Design Integrated Circuit Syst. 25 (2006), no. 6, 1000–1010.
Masuo Suzuki, On the convergence of exponential operators—the Zassenhaus formula, BCH formula and systematic approximants, Comm. Math. Phys. 57 (1977), no. 3, 193–200. MR 463973, DOI 10.1007/BF01614161
Andreas Čap and Jan Slovák, Parabolic geometries. I, Mathematical Surveys and Monographs, vol. 154, American Mathematical Society, Providence, RI, 2009. Background and general theory. MR 2532439, DOI 10.1090/surv/154
G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett. 91 (2003), 147902.
Arthur G. Werschulz, Computational complexity of one-step methods for systems of differential equations, Math. Comp. 34 (1980), no. 149, 155–174. MR 551295, DOI 10.1090/S0025-5718-1980-0551295-0
R. Wiersema, E. Kökcü, A. F. Kemper, and B. N. Bakalov, Classification of dynamical Lie algebras for translation-invariant 2-local spin systems in one dimension, arXiv:2309.05690, 2023.
Mark M. Wilde, Quantum information theory, 2nd ed., Cambridge University Press, Cambridge, 2017. MR 3645110, DOI 10.1017/9781316809976
Robert Zeier and Thomas Schulte-Herbrüggen, Symmetry principles in quantum systems theory, J. Math. Phys. 52 (2011), no. 11, 113510, 48. MR 2906580, DOI 10.1063/1.3657939

AMS Website Down

Mathematics of Computation

Preparing Hamiltonians for quantum simulation: A computational framework for Cartan decomposition via Lax dynamics
HTML articles powered by AMS MathViewer

Abstract: