Double-bracket quantum algorithms for diagonalization

Marek Gluza

School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, 637371 Singapore, Republic of Singapore

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This work proposes double-bracket iterations as a framework for obtaining diagonalizing quantum circuits. Their implementation on a quantum computer consists of interlacing evolutions generated by the input Hamiltonian with diagonal evolutions which can be chosen variationally. No qubit overheads or controlled-unitary operations are needed but the method is recursive which makes the circuit depth grow exponentially with the number of recursion steps. To make near-term implementations viable, the proposal includes optimization of diagonal evolution generators and of recursion step durations. Indeed, thanks to this numerical examples show that the expressive power of double-bracket iterations suffices to approximate eigenstates of relevant quantum models with few recursion steps. Compared to brute-force optimization of unstructured circuits double-bracket iterations do not suffer from the same trainability limitations. Moreover, with an implementation cost lower than required for quantum phase estimation they are more suitable for near-term quantum computing experiments. More broadly, this work opens a pathway for constructing purposeful quantum algorithms based on so-called double-bracket flows also for tasks different from diagonalization and thus enlarges the quantum computing toolkit geared towards practical physics problems.

A method for preparing on a quantum computer states of complicated materials.

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[1] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik. ``Noisy intermediate-scale quantum algorithms''. Rev. Mod. Phys. 94, 015004 (2022).

[2] Lennart Bittel and Martin Kliesch. ``Training variational quantum algorithms is np-hard''. Phys. Rev. Lett. 127, 120502 (2021).

[3] Daniel Stilck Franca and Raul Garcia-Patron. ``Limitations of optimization algorithms on noisy quantum devices''. Nature Physics 17, 1221–1227 (2021).

[4] Cornelius Lanczos. ``An iteration method for the solution of the eigenvalue problem of linear differential and integral operators''. Journal of Research of the National Bureau of Standards 45 (1950).

[5] Mario Motta, Chong Sun, Adrian TK Tan, Matthew J O’Rourke, Erika Ye, Austin J Minnich, Fernando GSL Brandao, and Garnet Kin Chan. ``Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution''. Nature Physics 16, 205–210 (2020).

[6] Christian Kokail, Christine Maier, Rick van Bijnen, Tiff Brydges, Manoj K Joshi, Petar Jurcevic, Christine A Muschik, Pietro Silvi, Rainer Blatt, Christian F Roos, et al. ``Self-verifying variational quantum simulation of lattice models''. Nature 569, 355–360 (2019).

[7] Stanisław D. Głazek and Kenneth G. Wilson. ``Renormalization of hamiltonians''. Phys. Rev. D 48, 5863–5872 (1993).

[8] Stanislaw D. Glazek and Kenneth G. Wilson. ``Perturbative renormalization group for hamiltonians''. Phys. Rev. D 49, 4214–4218 (1994).

[9] Franz Wegner. ``Flow-equations for hamiltonians''. Annalen der physik 506, 77–91 (1994).

[10] S Kehrein. ``The flow equation approach to many-particle systems''. Springer Tracts Mod. Phys. 217, 1–170 (2006).

[11] Franz Wegner. ``Flow equations and normal ordering: a survey''. Journal of Physics A: Mathematical and General 39, 8221 (2006).

[12] Percy Deift, Tara Nanda, and Carlos Tomei. ``Ordinary differential equations and the symmetric eigenvalue problem''. SIAM Journal on Numerical Analysis 20, 1–22 (1983).

[13] R.W. Brockett. ``Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems''. Linear Algebra and its Applications 146, 79–91 (1991).

[14] Moody T. Chu. ``On the continuous realization of iterative processes''. SIAM Review 30, 375–387 (1988). url: http:/​/​​stable/​2030697.

[15] Uwe Helmke and John B. Moore. ``Optimization and dynamical systems''. Springer London. (1994).

[16] Andrew M. Childs and Yuan Su. ``Nearly optimal lattice simulation by product formulas''. Phys. Rev. Lett. 123, 050503 (2019).

[17] Esteban A Martinez, Christine A Muschik, Philipp Schindler, Daniel Nigg, Alexander Erhard, Markus Heyl, Philipp Hauke, Marcello Dalmonte, Thomas Monz, Peter Zoller, et al. ``Real-time dynamics of lattice gauge theories with a few-qubit quantum computer''. Nature 534, 516–519 (2016).

[18] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami Barends, Sergio Boixo, Michael Broughton, Bob B Buckley, et al. ``Hartree-fock on a superconducting qubit quantum computer''. Science 369, 1084–1089 (2020).

[19] Frank HB Somhorst, Reinier van der Meer, Malaquias Correa Anguita, Riko Schadow, Henk J Snijders, Michiel de Goede, Ben Kassenberg, Pim Venderbosch, Caterina Taballione, JP Epping, et al. ``Quantum simulation of thermodynamics in an integrated quantum photonic processor''. Nature communications 14, 3895 (2023).

[20] Jeongrak Son, Marek Gluza, Ryuji Takagi, and Nelly HY Ng. ``Quantum dynamic programming'' (2024). arXiv:2403.09187.

[21] Alexander Streltsov, Gerardo Adesso, and Martin B. Plenio. ``Colloquium: Quantum coherence as a resource''. Rev. Mod. Phys. 89, 041003 (2017).

[22] Stavros Efthymiou, Sergi Ramos-Calderer, Carlos Bravo-Prieto, Adrián Pérez-Salinas, Diego García-Martín, Artur Garcia-Saez, José Ignacio Latorre, and Stefano Carrazza. ``Qibo: a framework for quantum simulation with hardware acceleration''. Quantum Science and Technology 7, 015018 (2021).

[23] Michael A Nielsen and Isaac L Chuang. ``Quantum computation and quantum information''. Cambridge University Press. (2010).

[24] JB Moore, RE Mahony, and U Helmke. ``Numerical gradient algorithms for eigenvalue and singular value calculations''. SIAM Journal on Matrix Analysis and Applications 15, 881–902 (1994).

[25] R Brockett. ``Dynamical systems that sort lists, solve linear programming problems and diagonalize symmetric matrices''. In Proc. 1988 IEEE Conference on Decision and Control, Linear Algebra Appl. Volume 146, pages 79–91. (1991).

[26] R Brockett. ``Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems''. Linear Algebra and its applications 146, 79–91 (1991).

[27] Steven Thomas Smith. ``Geometric optimization methods for adaptive filtering''. Harvard University. (1993).

[28] Christopher M Dawson and Michael A Nielsen. ``The solovay-kitaev algorithm''. Quantum Information & Computation 6, 81–95 (2006).

[29] Yu-An Chen, Andrew M. Childs, Mohammad Hafezi, Zhang Jiang, Hwanmun Kim, and Yijia Xu. ``Efficient product formulas for commutators and applications to quantum simulation''. Phys. Rev. Res. 4, 013191 (2022).

[30] Dave Wecker, Bela Bauer, Bryan K. Clark, Matthew B. Hastings, and Matthias Troyer. ``Gate-count estimates for performing quantum chemistry on small quantum computers''. Phys. Rev. A 90, 022305 (2014).

[31] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu. ``Theory of trotter error with commutator scaling''. Phys. Rev. X 11, 011020 (2021).

[32] Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. ``Simulating hamiltonian dynamics with a truncated taylor series''. Phys. Rev. Lett. 114, 090502 (2015).

[33] Guang Hao Low and Isaac L. Chuang. ``Hamiltonian Simulation by Qubitization''. Quantum 3, 163 (2019).

[34] John Watrous. ``The theory of quantum information''. Cambridge University Press. (2018).

[35] Pierre Pfeuty. ``The one-dimensional ising model with a transverse field''. Ann. Phys. 57, 79 – 90 (1970).

[36] Lin Lin and Yu Tong. ``Near-optimal ground state preparation''. Quantum 4, 372 (2020).

[37] Andrew M Childs and Robin Kothari. ``Limitations on the simulation of non-sparse hamiltonians''. Quantum Information & Computation 10, 669–684 (2010).

[38] Matthew B Hastings. ``On Lieb-Robinson bounds for the double bracket flow'' (2022). arXiv:2201.07141.

[39] Yichen Huang. ``Universal eigenstate entanglement of chaotic local hamiltonians''. Nuclear Physics B 938, 594–604 (2019).

[40] Elliott H Lieb and Derek W Robinson. ``The finite group velocity of quantum spin systems''. In Statistical Mechanics. Pages 425–431. Springer (1972).

[41] Bruno Nachtergaele, Robert Sims, and Amanda Young. ``Quasi-locality bounds for quantum lattice systems. i. lieb-robinson bounds, quasi-local maps, and spectral flow automorphisms''. Journal of Mathematical Physics 60, 061101 (2019).

[42] Tomotaka Kuwahara and Keiji Saito. ``Eigenstate thermalization from the clustering property of correlation''. Phys. Rev. Lett. 124, 200604 (2020).

[43] Fernando GSL Brandao, Elizabeth Crosson, M Burak Sahinoglu, and John Bowen. ``Quantum error correcting codes in eigenstates of translation-invariant spin chains''. Phys. Rev. Lett. 123, 110502 (2019).

[44] Álvaro M. Alhambra, Jonathon Riddell, and Luis Pedro García-Pintos. ``Time evolution of correlation functions in quantum many-body systems''. Phys. Rev. Lett. 124, 110605 (2020).

[45] Michael M. Wolf, Frank Verstraete, Matthew B. Hastings, and J. Ignacio Cirac. ``Area laws in quantum systems: Mutual information and correlations''. Phys. Rev. Lett. 100, 070502 (2008).

[46] David Pekker, Bryan K. Clark, Vadim Oganesyan, and Gil Refael. ``Fixed points of wegner-wilson flows and many-body localization''. Phys. Rev. Lett. 119, 075701 (2017).

[47] Steven J. Thomson and Marco Schirò. ``Local integrals of motion in quasiperiodic many-body localized systems''. SciPost Phys. 14, 125 (2023).

[48] Ryan LaRose, Arkin Tikku, Étude O’Neel-Judy, Lukasz Cincio, and Patrick J Coles. ``Variational quantum state diagonalization''. npj Quantum Information 5, 1–10 (2019).

[49] Jinfeng Zeng, Chenfeng Cao, Chao Zhang, Pengxiang Xu, and Bei Zeng. ``A variational quantum algorithm for hamiltonian diagonalization''. Quantum Science and Technology 6, 045009 (2021).

[50] Benjamin Commeau, Marco Cerezo, Zoë Holmes, Lukasz Cincio, Patrick J Coles, and Andrew Sornborger. ``Variational hamiltonian diagonalization for dynamical quantum simulation'' (2020). arXiv:2009.02559.

[51] Cristina Cirstoiu, Zoe Holmes, Joseph Iosue, Lukasz Cincio, Patrick J Coles, and Andrew Sornborger. ``Variational fast forwarding for quantum simulation beyond the coherence time''. npj Quantum Information 6, 82 (2020).

[52] Joe Gibbs, Kaitlin Gili, Zoë Holmes, Benjamin Commeau, Andrew Arrasmith, Lukasz Cincio, Patrick J Coles, and Andrew Sornborger. ``Long-time simulations for fixed input states on quantum hardware''. npj Quantum Information 8, 135 (2022).

[53] Roeland Wiersema and Nathan Killoran. ``Optimizing quantum circuits with riemannian gradient flow''. Phys. Rev. A 107, 062421 (2023).

[54] Emanuel Knill, Gerardo Ortiz, and Rolando D. Somma. ``Optimal quantum measurements of expectation values of observables''. Phys. Rev. A 75, 012328 (2007).

[55] David Poulin and Pawel Wocjan. ``Sampling from the thermal quantum gibbs state and evaluating partition functions with a quantum computer''. Phys. Rev. Lett. 103, 220502 (2009).

[56] Kristan Temme, Tobias J Osborne, Karl G Vollbrecht, David Poulin, and Frank Verstraete. ``Quantum metropolis sampling''. Nature 471, 87–90 (2011).

[57] Yimin Ge, Jordi Tura, and J Ignacio Cirac. ``Faster ground state preparation and high-precision ground energy estimation with fewer qubits''. Journal of Mathematical Physics 60, 022202 (2019).

[58] András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. ``Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics''. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing. Pages 193–204. (2019).

[59] Kok Chuan Tan, Dhiman Bowmick, and Pinaki Sengupta. ``Quantum stochastic series expansion methods'' (2020). arXiv:2010.00949.

[60] Yulong Dong, Lin Lin, and Yu Tong. ``Ground state preparation and energy estimation on early fault-tolerant quantum computers via quantum eigenvalue transformation of unitary matrices'' (2022). arXiv:2204.05955.

[61] Lin Lin and Yu Tong. ``Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers''. PRX Quantum 3, 010318 (2022).

[62] Ethan N Epperly, Lin Lin, and Yuji Nakatsukasa. ``A theory of quantum subspace diagonalization'' (2021). arXiv:2110.07492.

[63] A Yu Kitaev. ``Quantum measurements and the abelian stabilizer problem'' (1995). arXiv:quant-ph/​9511026.

[64] Lin Lin. ``Lecture notes on quantum algorithms for scientific computation'' (2022). arXiv:2201.08309.

[65] Gilles Brassard, Peter Hoyer, Michele Mosca, and Alain Tapp. ``Quantum amplitude amplification and estimation''. Contemporary Mathematics 305, 53–74 (2002).

[66] Robert M Parrish and Peter L McMahon. ``Quantum filter diagonalization: Quantum eigendecomposition without full quantum phase estimation'' (2019). arXiv:1909.08925.

[67] Nicholas H Stair, Renke Huang, and Francesco A Evangelista. ``A multireference quantum krylov algorithm for strongly correlated electrons''. Journal of chemical theory and computation 16, 2236–2245 (2020).

[68] Gene Golub and William Kahan. ``Calculating the singular values and pseudo-inverse of a matrix''. Journal of the Society for Industrial and Applied Mathematics, Series B: Numerical Analysis 2, 205–224 (1965).

[69] R.W. Brockett. ``Least squares matching problems''. Linear Algebra and its Applications 122-124, 761–777 (1989).

[70] Roger W Brockett. ``Smooth dynamical systems which realize arithmetical and logical operations''. Three Decades of Mathematical System Theory: A Collection of Surveys at the Occasion of the 50th Birthday of Jan C. WillemsPages 19–30 (2005).

[71] Anthony M Bloch. ``A completely integrable hamiltonian system associated with line fitting in complex vector spaces''. Bull. Amer. Math. Soc. (1985).

[72] Anthony Bloch. ``Estimation, principal components and hamiltonian systems''. Systems & Control Letters 6, 103–108 (1985).

[73] Anthony M Bloch. ``Steepest descent, linear programming and hamiltonian flows''. Contemp. Math. AMS 114, 77–88 (1990).

[74] Anthony M Bloch, Roger W Brockett, and Tudor S Ratiu. ``Completely integrable gradient flows''. Communications in Mathematical Physics 147, 57–74 (1992).

[75] Nic Ezzell, Bibek Pokharel, Lina Tewala, Gregory Quiroz, and Daniel A Lidar. ``Dynamical decoupling for superconducting qubits: a performance survey'' (2022). arXiv:2207.03670.

[76] Rajendra Bhatia. ``Matrix analysis''. Volume 169. Springer Science & Business Media. (1996).

[77] Steven T. Flammia and Yi-Kai Liu. ``Direct fidelity estimation from few pauli measurements''. Phys. Rev. Lett. 106, 230501 (2011).

[78] Marek Gluza. url:​marekgluza/​double_bracket_flow_as_a_diagonalization_quantum_algorithm.

[79] ``Scientific co2nduct''. url:

[80] Morris W Hirsch, Stephen Smale, and Robert L Devaney. ``Differential equations, dynamical systems, and an introduction to chaos''. Academic press. (2012).

Cited by

[1] Jeongrak Son, Marek Gluza, Ryuji Takagi, and Nelly H. Y. Ng, "Quantum Dynamic Programming", arXiv:2403.09187, (2024).

[2] Michael Kreshchuk, James P. Vary, and Peter J. Love, "Simulating Scattering of Composite Particles", arXiv:2310.13742, (2023).

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