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Numerical methods and gate sequences
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Quantum localization bounds Trotter errors in digital quantum simulation

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The numerical data shown in this work was obtained for a quantum Ising chain with the Hamiltonian$H=HZ+HX$(11)where$HZ=J∑l=1N−1SlzSl+1z+h∑l=1NSlz,HX=g∑l=1NSlx$(12)

Many of the involved contributions in this model Hamiltonian mutually commute. Therefore, only a small set of elementary quantum gates is required to simulate the Trotterized dynamics. We used the following sequence of two gates$U(1)=U1U2,U1=e−iτHZ,U2=e−iτHX$(13)

For the presented simulations of observables, we computed the real-time evolution for 2 × 104 periods, except otherwise noted, using a Lanczos algorithm with full reorthogonalization. Because, for a finite-size system, observables still show remaining temporal fluctuations, we extracted the asymptotic long-time limit of the presented quantities by performing a stroboscopic time average over the last 104 periods. This large number of Trotter steps is far beyond realistic current-day implementations and serves here only as a worst-case scenario. However, when the Trotter errors on local observables can be controlled even in this idealized limit, one can expect the same to hold true on shorter times relevant for current experiments. In the Supplementary Materials, we illustrate in more detail how the Trotter error builds up on short time scales.

The IPR shown in Fig. 2 can, in principle, be obtained either by exact diagonalization or by use of a dynamical evolution. We chose the latter because it allows us to reach larger systems and is, in principle, an experimentally accessible approach. Dynamically, the IPR can be obtained by a stroboscopic mean(14)as one can prove by expanding $Pl$ in the eigenbasis of HF, followed by a summation of the resulting geometric series. Note that $Pl$ is the Loschmidt echo, a common indicator for quantum chaotic behavior in single-particle systems (27).

For the computation of the OTO correlator ℱ(t) defined in Eq. 6, we decompose ℱ(nτ) as$F(nτ)=〈ψ1(nτ)|ψ2(nτ)〉$(15)where the two states$|ψ1(nτ)〉=WeiHFnτVe−iHFnτ|ψ0〉$(16)$|ψ2(nτ)〉=eiHFnτVe−iHFnτW|ψ0〉$(17)can be obtained from forward and backward evolving the quantum many-body state with appropriate insertions of the W and V operators. Since the backward evolution has to be performed for every Trotter step n, the overall runtime of this approach scales proportional to n2. This limits the accessible total simulation time t = nτ. We used n = 103 for the data shown in Fig. 2B and we performed a stroboscopic average over the last 300 periods to obtain an estimate for the asymptotic long-time value.

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