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Fixed Depth Hamiltonian Simulation via Cartan Decomposition

Efekan Kökcü, Thomas Steckmann, Yan Wang, J. K. Freericks, Eugene F. Dumitrescu, and Alexander F. Kemper
Phys. Rev. Lett. 129, 070501 – Published 9 August 2022
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    Abstract

    Simulating quantum dynamics on classical computers is challenging for large systems due to the significant memory requirements. Simulation on quantum computers is a promising alternative, but fully optimizing quantum circuits to minimize limited quantum resources remains an open problem. We tackle this problem by presenting a constructive algorithm, based on Cartan decomposition of the Lie algebra generated by the Hamiltonian, which generates quantum circuits with time-independent depth. We highlight our algorithm for special classes of models, including Anderson localization in one-dimensional transverse field XY model, where O(n2)-gate circuits naturally emerge. Compared to product formulas with significantly larger gate counts, our algorithm drastically improves simulation precision. In addition to providing exact circuits for a broad set of spin and fermionic models, our algorithm provides broad analytic and numerical insight into optimal Hamiltonian simulations.

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    • Received 3 May 2021
    • Revised 9 April 2022
    • Accepted 28 June 2022

    DOI:https://doi.org/10.1103/PhysRevLett.129.070501

    © 2022 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Quantum simulation
    1. Physical Systems
    Quantum spin models
    1. Techniques
    Abstract algebraGroup theorySpin lattice models
    Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

    Authors & Affiliations

    Efekan Kökcü1,*, Thomas Steckmann1, Yan Wang2, J. K. Freericks3, Eugene F. Dumitrescu2,†, and Alexander F. Kemper1,‡

    • 1Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA
    • 2Oak Ridge National Laboratory, Computational Sciences and Engineering Division, Oak Ridge, Tennessee 37831, USA
    • 3Department of Physics, Georgetown University, 37th and O Streets NW, Washington, D.C. 20057, USA

    • *ekokcu@ncsu.edu
    • †dumitrescuef@ornl.gov
    • ‡akemper@ncsu.edu

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    Issue

    Vol. 129, Iss. 7 — 12 August 2022

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    Images

    • Figure 1
      Figure 1

      Hamiltonian algebra dimensions of the nearest-neighbor Heisenberg, XY, TFXY models, and TFIM and dimension of full su(2n) for comparison to the generic case. The dimensions can exactly be calculated as |g(Heisenberg)|=4n−1−4, |g(TFIM)|=|g(TFXY)|=n(2n−1), and |g(XY)|=n(n−1).

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    • Figure 2
      Figure 2

      (a) Schematic relationship of the Hamiltonian algebra g(H) and its partitioning into a subalgebra k, its compliment m, and the Cartan subalgebra h. (b) KHK decomposition (Theorem 1) applied to a time-evolution operator generated by an element of m. (c) Hamiltonian algebra g(H) for the two-site TFIM and the Cartan decomposition generated by the involution θ(g)=−gT. Here we list the bases that span g(H) and its Cartan decomposition. (d) Decomposed time evolution for the two-site TFIM.

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    • Figure 3
      Figure 3

      (a) Circuit implementation of the given exponentials of Pauli strings and the compact arrow notation. The R gate shown here is Rx(π/2). (b) Unoptimized and (c) optimized circuit for K in an n=5-site TFXY model (this system size is chosen for illustrative purposes). The circuits have O(n3) (80) and O(n2) (20) controlled NOT (CNOT) gates, respectively.

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    • Figure 4
      Figure 4

      Displacement of the spin excitation N=⟨N^2⟩ and its absolute difference from the exact result |N−Nexact| in the TFXY model with a random Z field, for standard deviation σ=0 in (a) and σ=3 in (b). The excitation becomes trapped around its original position as σ increases. The localization is captured to within a small constant error by our Cartan algorithm (solid curves). The two Trotter decompositions use 180 (dotted) and 1332 (dashed) CNOTs, which correspond to the CNOT counts of the optimized and nonoptimized Cartan circuits, respectively.

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