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. 2019 Apr 12;5(4):eaau8342.
doi: 10.1126/sciadv.aau8342. eCollection 2019 Apr.

Quantum Localization Bounds Trotter Errors in Digital Quantum Simulation

Free PMC article

Quantum Localization Bounds Trotter Errors in Digital Quantum Simulation

Markus Heyl et al. Sci Adv. .
Free PMC article


A fundamental challenge in digital quantum simulation (DQS) is the control of an inherent error, which appears when discretizing the time evolution of a quantum many-body system as a sequence of quantum gates, called Trotterization. Here, we show that quantum localization-by constraining the time evolution through quantum interference-strongly bounds these errors for local observables, leading to an error independent of system size and simulation time. DQS is thus intrinsically much more robust than suggested by known error bounds on the global many-body wave function. This robustness is characterized by a sharp threshold as a function of the Trotter step size, which separates a localized region with controllable Trotter errors from a quantum chaotic regime. Our findings show that DQS with comparatively large Trotter steps can retain controlled errors for local observables. It is thus possible to reduce the number of gate operations required to represent the desired time evolution faithfully.


Fig. 1
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
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
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.


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