Trotter errors on local observables from perturbation theory
This protocol is extracted from research article:
Quantum localization bounds Trotter errors in digital quantum simulation
Sci Adv, Apr 12, 2019; DOI: 10.1126/sciadv.aau8342

As mentioned before, the Trotter errors for local observables can be captured using time-dependent perturbation theory in the limit of sufficiently small τ. In the following, we outline how to obtain the analytical expressions for the coefficients qE and m for QE and ℳ, respectively. First, we consider the simulation accuracy QE and, afterward, the Trotter errors on the magnetization ℳ.

For the derivation of the corrections appearing in QE, we utilize the energy of the target Hamiltonian H, and therefore, the simulation accuracy QE exhibits a substantial overlap with the emergent conserved quantity HFHF(nτ)τ=HF= const.(18)

Here, O(nτ)τ=ψ0|eiHFnτOeiHFnτ|ψ0 denotes the full Trotterized time evolution with Trotter step size τ as in the main text. Moreover, we define the expectation values in the initial state via O=ψ0|O|ψ0 and under the ideal time evolution as O(t)=O(t)τ=0.

To obtain all corrections to the desired order, we first express HF using the Magnus expansion up to second order in the Trotter step sizeHF=H+τC1+τ2C2+O(τ3)(19)withC1=i2[HX,HZ],C2=112[HXHZ,[HX,HZ]](20)

For convenience, we restrict the presentation from now on to a sequence of two elementary gates within one period, as we have for the case of the simulated quantum Ising chain. Using the above expansion for HF in combination with the conservation of HF, one obtains for the energy deviationΔE(nτ)=H(nτ)τH,=τΔC1(nτ)+τ2ΔC2(nτ)(21)whereΔCν(nτ)=CνCν(nτ)τ,ν=1,2(22)

As a next step, we use time-dependent perturbation theory to determine the leading order in τ corrections of ΔCv(nτ). For this purpose, we writeeiHFt=eiHtW(t),W(t)=Tei0tdtV(t)(23)with T denoting the time-ordering prescription andV(t)=eiHtVeiHt,V=τC1+τ2C2(24)

For the corrections to ΔE(nτ) quadratic in τ, we need to perform the time-dependent perturbation theory to first order in τ for C1 and can neglect any τ-dependent contributions for C2.

Let us first consider ΔC1(nτ), which givesΔC1(nτ)=C1C1(nτ)iτ0nτdt[C1(t),C1(nτ)](25)

The time integral can be conveniently evaluated by recognizing thatC1=i2[HX,HZ]=i2[H,HZ](26)since H = HX + HZ and thusC1(t)=12ddtHZ(t)(27)

This givesΔC1(nτ)=C1C1(nτ)iτ2[HZ(nτ)HZ,C1(nτ)](28)

In the limit of n → ∞, we can use the general property that expectation values of operators are governed by the so-called diagonal ensemble (31)O(nτ)nλpλλ|O|λ(29)where pλ = |〈λ|ψ0〉|2 and the set of all |λ〉 denotes the eigenstates for the target Hamiltonian H. Using particular properties of the considered protocol, the above result for ΔC1(nτ) can be simplified considerably. We can use, for example, that C1=0 and [HZ,C1(nτ)]=0, because |ψ0〉 is an eigenstate for HZ, which lastly yieldsΔC1(nτ)nτ4λpλλ|[HZ,[HZ,HX]]|λ(30)

For the contributions to ΔE(nτ) that are second order in τ stemming from ΔC2(nτ), we can restrict to the zeroth order in time-dependent perturbation theory for C2(nτ)τ, i.e., we can replace C2(nτ)τC2(nτ). This yieldsΔC2(nτ)nC2λpλλ|C2|λ(31)

Collecting all contributions, we lastly obtainQE=ΔEET=E0=qE(hτ)2+O[(hτ)3](32)withqE=1J2E0[C2λpλλ|C2|λ14λpλλ|[HZ,[HZ,HX]]|λ](33)given that ET=∞ = 0. This expression can be evaluated using full diagonalization, which provides access to all eigenstates |λ〉. For the considered parameters of our simulations, we find qE = 0.18, which is consistent with the full dynamical calculation in the small Trotter step limit (see Fig. 3D).

For estimating the lowest-order corrections in τ for other observables such as the magnetization ℳ, we can not make direct use of the emergent conserved quantity HF as for the energy of the target Hamiltonian. Still, we can perform time-dependent perturbation theory, which we now have to carry out up to second order. Following the same steps as before, we obtain the following expression for the magnetizationΔM(nτ)=M(nτ)τM(nτ)==τ212[{HZ2(nτ),M(nτ)}EZ2M(t)]+iτ26[C1(nτ)C1,M(nτ)]5τ2120nτdtC1(t)HZ(t)M(nτ)+h.c.(34)

Here, {A, B} = AB + BA denotes the anticommutator, and EZ is given by HZ0〉 = EZ0〉. In the limit n → ∞, we can again use that expectation values can be evaluated in the diagonal ensemble. In addition, the expression involving the time integral can be formally solved using the Lehman representation. Last, we obtainΔM(nτ)nm(hτ)2+O[(hτ)3](35)withm=112J2λpλλ|{HZ2,M}EZ2M|λ16J2λpλRe[λ|[HX,M]HZ|λ]+16J2λ,λpλEλEλRe[λ|[HZ,HX]HZ|λλ|M|λ]+16J2λ,λλ|M|λEλEλRe[CλCλ*λ|[HZ,HX]HZ|λ](36)where Cλ = 〈λ|ψ0〉 and Eλ denotes the eigenenergies of the target Hamiltonian H corresponding to the eigenstate |λ〉. Using full diagonalization, we can again evaluate this expression yielding for our model a value of m = 0.05, which we used in Fig. 3B for the asymptotic small τ prediction and which matches well the result from the full dynamics. Notice that for the presented derivation of the perturbative corrections for the magnetization ℳ, we used explicitly that the initial state is an eigenstate of ℳ. Choosing different observables or different initial conditions might yield linearly in τ contributions as the leading-order corrections.

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