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Toward simulating quantum field theories with controlled phonon-ion dynamics: A hybrid analog-digital approach

Zohreh Davoudi, Norbert M. Linke, and Guido Pagano
Phys. Rev. Research 3, 043072 – Published 26 October 2021
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  • INTRODUCTION
  • THE HYBRID ANALOG-DIGITAL BUILDING…
  • A YUKAWA THEORY: SCALAR FIELDS COUPLED…
  • U(1) LATTICE GAUGE THEORY COUPLED TO…
  • A QUALITATIVE COST ANALYSIS
  • CONCLUSION AND OUTLOOK
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Quantum field theories are the cornerstones of modern physics, providing relativistic and quantum mechanical descriptions of physical systems at the most fundamental level. Simulating real-time dynamics within these theories remains elusive in classical computing. This provides a unique opportunity for quantum simulators, which hold the promise of revolutionizing our simulation capabilities. Trapped-ion systems are successful quantum-simulator platforms for quantum many-body physics and can operate in digital, or gate-based, and analog modes. Inspired by the progress in proposing and realizing quantum simulations of a number of relativistic quantum field theories using trapped-ion systems, and by the hybrid analog-digital proposals for simulating interacting boson-fermion models, we propose hybrid analog-digital quantum simulations of selected quantum field theories, taking recent developments to the next level. On one hand, the semi-digital nature of this proposal offers more flexibility in engineering generic model interactions compared with a fully-analog approach. On the other hand, encoding the bosonic fields onto the phonon degrees of freedom of the trapped-ion system allows a more efficient usage of simulator resources, and a more natural implementation of intrinsic quantum operations in such platforms. This opens up ways for simulating complex dynamics of, e.g., Abelian and non-Abelian gauge theories, by combining the benefits of digital and analog schemes.

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    • Received 24 April 2021
    • Accepted 13 September 2021

    DOI:https://doi.org/10.1103/PhysRevResearch.3.043072

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Gauge theoriesLattice field theoryQuantum algorithmsQuantum computationQuantum electrodynamicsQuantum field theoryQuantum gatesQuantum simulation
    Atomic, Molecular & OpticalNuclear PhysicsParticles & FieldsQuantum Information, Science & Technology

    Authors & Affiliations

    Zohreh Davoudi1,*, Norbert M. Linke2, and Guido Pagano3

    • 1Maryland Center for Fundamental Physics and Department of Physics, University of Maryland, College Park, Maryland 20742, USA
    • 2Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 20742, USA
    • 3Department of Physics and Astronomy, Rice University, 6100 Main Street, Houston, Texas 77005, USA

    • *Also affiliated with the RIKEN Center for Accelerator-based Sciences, Wako, Japan, during the early stages of this work.

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    Vol. 3, Iss. 4 — October - December 2021

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    • Figure 1
      Figure 1

      The building blocks of the analog-digital circuits in a trapped-ion quantum simulator. The gates from left to right are defined in Eqs. (4), (7), (10), (16), (14), and (18), respectively.

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    • Figure 2
      Figure 2

      The degrees of freedom of the lattice-regularized scalar field theory coupled to staggered fermions in 1+1 D (top row) are mapped to those in a linear trapped-ion quantum simulator (bottom row) with individual laser-beam addressing. This scheme involves only normal modes of motion.

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    • Figure 3
      Figure 3

      The schematic of the analog-digital quantum circuit associated with the time evolution of a four-site Yukawa theory for a single Trotter step, as expressed in Eqs. (32, 33, 34, 35). The gate symbols are defined in Fig. 1.

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    • Figure 4
      Figure 4

      The overlap between an initial state with no fermion, no antifermion, and no boson, and the corresponding time-evolved state, |〈ψ(0)|ψ(t)〉|2, (in blue) along with the average number of bosons generated 〈Nd〉≡1N〈ψ(t)|∑k=−N/2N/2−1dk†dk|ψ(t)〉, (in red) as a function of time t using exact (solid curve) and Trotterized evolution (unfilled points) for the Yukawa theory. The corresponding model parameters are given in Table 1. The time t is in units of lattice spacing b, which is set to one.

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    • Figure 5
      Figure 5

      The degrees of freedom of the lattice-regularized Schwinger model in the HOBM (top row) are mapped to those in a linear trapped-ion quantum simulator involving local and normal modes of motion (bottom row).

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    • Figure 6
      Figure 6

      The schematic of the analog-digital quantum circuit associated with the time evolution of a four-site Schwinger model for a single Trotter step, as expressed in Eqs. (47, 48, 49). The fermion-gauge interaction block must be repeated four times with various phases for the spin-phonon gates and single-spin rotations, see Eq. (47). The gate symbols are defined in Fig. 1.

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    • Figure 7
      Figure 7

      The overlap between an initial state with no fermion, no antifermion, and M bosons, and the corresponding time-evolved state, |〈ψ(0)|ψ(t)〉|2, (in blue) along with the average number of phonons generated, 〈Nd〉≡1N〈ψ(t)|∑j=1Ndj†dj|ψ(t)〉, (in red) as a function of time using exact (solid curve) and Trotterized evolution (unfilled points) for the Schwinger model within HOBM. The corresponding model parameters are given in Table 2. The time t is in units of lattice spacing b, which is set to one.

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