Research ArticlePHYSICS

Quantum localization bounds Trotter errors in digital quantum simulation

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Science Advances  12 Apr 2019:
Vol. 5, no. 4, eaau8342
DOI: 10.1126/sciadv.aau8342
  • Fig. 1 Trotterized time evolution and resulting error on local observables.

    (A) Gate sequence for the digital quantum simulation (DQS) of an Ising model. The desired evolution up to total simulation time t is split into n repeated sequences of length τ = t/n, each decomposed into fundamental quantum gates. The example shows a gate sequence for a four-qubit chain with Ising spin-spin interactions (ZZ) and transverse and longitudinal fields (simulated by single-qubit operations along the X and Z directions on the Bloch sphere). (B) Magnetization dynamics M(t)=N1l=1NSlz(t) in the DQS of the Ising model for N = 20 spins and different Trotter step sizes τ compared to the exact solution. The normalized deviation Δℳ(t)/(hτ)2 with Δℳ(t) = |ℳτ=0(t) − ℳ(t)| from the ideal dynamics shows a collapse of the error dynamics ℳτ=0(t) for sufficiently small τ.

  • Fig. 2 Localization and quantum chaos in the Trotterized dynamics of the quantum Ising chain.

    (A) Rate function λIPR of the IPR, normalized to the maximally achievable value λD describing uniform delocalization over all accessible states. A sharp threshold as a function of the Trotter step size τ separates a localized regime at small τ from a quantum chaotic regime at large τ. (B) The long-time limit ℱ of the OTO correlator also signals a sharp quantum chaos threshold. ℱ is normalized with respect to ℱ0 = 1/8, the theoretical maximum. Full scrambling is only achieved for large Trotter steps.

  • Fig. 3 Trotter errors for local observables in the infinite long-time limit for the Ising model.

    Both the magnetization ℳ (A) and simulation accuracy QE (C) exhibit a sharp crossover from a regime of controllable Trotter errors for small Trotter steps τ to a regime of strong heating at larger τ. The dashed line in (A) refers to the desired case of the ideal evolution. The Trotter error exhibits a quadratic scaling at small τ for both the deviation of the magnetization, Δℳ = ℳ − ℳτ=0, (B) and QE (D). The solid lines in (B) and (D) represent analytical results obtained perturbatively in the limit of small Trotter steps τ. These results indicate the controlled robustness of digital quantum simulation against Trotter errors, in the long-time limit and largely independent of N.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/4/eaau8342/DC1

    Section S1. Temporal buildup of the Trotter error threshold

    Section S2. Trotter errors in the Ising model for alternative initial condition

    Section S3. Trotter errors in the lattice Schwinger model

    Section S4. Imperfections

    Fig. S1. Temporal buildup of Trotter errors on transient time scales.

    Fig. S2. Simulation accuracy QE for initial Neel state.

    Fig. S3. IPR for the DQS of the lattice Schwinger model.

    Fig. S4. Trotter errors for the DQS of the lattice Schwinger model.

    Fig. S5. Timing errors in the dynamics of the simulation accuracy QE(t) for the Ising model.

    Reference (41)

  • Supplementary Materials

    This PDF file includes:

    • Section S1. Temporal buildup of the Trotter error threshold
    • Section S2. Trotter errors in the Ising model for alternative initial condition
    • Section S3. Trotter errors in the lattice Schwinger model
    • Section S4. Imperfections
    • Fig. S1. Temporal buildup of Trotter errors on transient time scales.
    • Fig. S2. Simulation accuracy QE for initial Neel state.
    • Fig. S3. IPR for the DQS of the lattice Schwinger model.
    • Fig. S4. Trotter errors for the DQS of the lattice Schwinger model.
    • Fig. S5. Timing errors in the dynamics of the simulation accuracy QE(t) for the Ising model.
    • Reference (41)

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