Abstract
One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance (Landau et al. in Nat Phys, 2015) gave a polynomial time algorithm to compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unsolved, including whether there exist efficient algorithms when the ground space is degenerate (and of polynomial dimension in the system size), or for the polynomially many lowest energy states, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm, based on a rigorously justified RG type transformation, for finding low energy states for 1D Hamiltonians acting on a chain of n particles. In the process we resolve some of the aforementioned open questions, including giving a polynomial time algorithm for poly(n) degenerate ground spaces and an n O(log n) algorithm for the poly(n) lowest energy states (under a mild density condition). For these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time is \({\tilde{O}(nM(n))}\) , where M(n) is the time required to multiply two n × n matrices.
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Aharonov D., Arad I., Vazirani U., Landau Z.: The detectability lemma and its applications to quantum hamiltonian complexity. New J. Phys. 13(11), 113043 (2011)
Anshu A., Arad I., Vidick T.: Simple proof of the detectability lemma and spectral gap amplification. Phys. Rev. B 93(20), 205142 (2016)
Arad, I., Kitaev, A., Landau, Z., Vazirani, U.: An area law and sub-exponential algorithm for 1D systems. In: Proceedings of the 4th Innovations in Theoretical Computer Science (ITCS) (2013)
Arad, I., Kuwahara, T., Landau, Z.: Connecting global and local energy distributions in quantum spin models on a lattice. Technical report, J. Stat. Mech. 3, 033301 (2016)
Arad I., Landau Z., Vazirani U.: Improved one-dimensional area law for frustration-free systems. Phys. Rev. B 85, 195145 (2012)
Bravyi S., Gosset D.: Gapped and gapless phases of frustration-free spin-1 2 chains. J. Math. Phys. 56(6), 061902 (2015)
Bravyi S., Hastings M.B., Michalakis S.: Topological quantum order: stability under local perturbations. J. Math. Phys. 51(9), 093512 (2010)
Bridgeman J.C., Chubb C.T.: Hand-waving and interpretive dance: An introductory course on tensor networks. J. Phys. A Math. Theor. 50, 223001 (2017)
Chubb, C.T., Flammia, S.T.: Computing the degenerate ground space of gapped spin chains in polynomial time. Chic. J. Theor. Comput. 9, 1–35 (2016)
Dasgupta S., Gupta A.: An elementary proof of a theorem of Johnson and Lindenstrauss. Random Struct. Algorithms 22(1), 60–65 (2003)
de Beaudrap N., Osborne T.J., Eisert J.: Ground states of unfrustrated spin hamiltonians satisfy an area law. New J. Phys. 12(9), 095007 (2010)
Eisert J., Cramer M., Plenio M.B.: Colloquium: area laws for the entanglement entropy. Rev. Mod. Phys. 82(1), 277–306 (2010)
Evenbly G., Vidal G.: Tensor network renormalization yields the multiscale entanglement renormalization ansatz. Phys. Rev. Lett. 115(20), 200401 (2015)
Feynman R.P.: Simulating physics with computers. Int. J. Theor. Phys. 21(6), 467–488 (1982)
Hastings M.B.: Solving gapped hamiltonians locally. Phys. Rev. B 73(8), 085115 (2006)
Hastings M.B.: An area law for one-dimensional quantum systems. J. Stat. Mech. Theory Exp. 2007(08), P08024 (2007)
Huang, Y.: A polynomial-time algorithm for the ground state of one-dimensional gapped Hamiltonians (2014). arXiv:1406.6355
Huang, Y.: A simple efficient algorithm in frustration-free one-dimensional gapped systems (2015). arXiv:1510.01303
Keating J., Linden N., Wells H.: Spectra and eigenstates of spin chain hamiltonians. Commun. Math. Phys. 338(1), 81–102 (2015)
Kliesch, M., Gogolin, C., Kastoryano, J., Riera, M.A., Eisert, J.: Locality of temperature. Phys. Rev. X 4, 031019 (2014)
Kuwahara, T., Arad, I., Amico, L., Vedral, V.: Local reversibility and entanglement structure of many-body ground states. Quantum Sci. Technol. 2, 015005 (2017)
Landau, Z., Vazirani, U., Vidick, T.: A polynomial time algorithm for the ground state of one-dimensional gapped local Hamiltonians. Nat. Phys. (2015). doi:10.1038/nphys3345
Maldacena J.: Eternal black holes in anti-de sitter. J. High Energy Phys. 2003(04), 021 (2003)
Masanes L.: Area law for the entropy of low-energy states. Phys. Rev. A 80(5), 052104 (2009)
Molnar A., Schuch N., Verstraete F., Cirac J.I.: Approximating gibbs states of local hamiltonians efficiently with projected entangled pair states. Phys. Rev. B 91, 045138 (2015)
Pastawski, F., Yoshida, B., Harlow, D., Preskill, J.: Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence. JHEP 06, 149 (2015)
Roberts, B., Vidick, T., Motrunich, O.I.: Rigorous renormalization group method for ground space and low-energy states of local Hamiltonians. arXiv:1703.01994 (2017)
Schollwöck U.: The density-matrix renormalization group in the age of matrix product states. Ann. Phys. 326(1), 96–192 (2011)
Vershynin, R.: Introduction to the non-asymptotic analysis of random matrices. In: Eldar, Y. et al. (eds.) Compressed Sensing, Theory and Applications. Cambridge University Press. arXiv:1011.3027 (2012)
Verstraete F., Murg V., Cirac J.I.: Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems. Adv. Phys. 57(2), 143–224 (2008)
Vidal G.: Class of quantum many-body states that can be efficiently simulated. Phys. Rev. Lett. 101, 110501 (2008)
Vidal G.: Entanglement renormalization: an introduction. arXiv:0912.1651 (2009)
White S.R.: Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863–2866 (1992)
White S.R.: Density-matrix algorithms for quantum renormalization groups. Phys. Rev. B 48, 10345–10356 (1993)
Wilson K.G.: The renormalization group: Critical phenomena and the kondo problem. Rev. Mod. Phys. 47(4), 773 (1975)
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Communicated by M. M. Wolf
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Arad, I., Landau, Z., Vazirani, U. et al. Rigorous RG Algorithms and Area Laws for Low Energy Eigenstates in 1D. Commun. Math. Phys. 356, 65–105 (2017). https://doi.org/10.1007/s00220-017-2973-z
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DOI: https://doi.org/10.1007/s00220-017-2973-z