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Adiabatic preparation of many-body quantum states: getting the beginning and ending right

Emil T. M. Pedersen, Freek Witteveen, Klaus Mølmer, Matthias Christandl

TL;DR

The authors address how to maximize fidelity in adiabatic state preparation of many-body systems by enforcing smooth boundary conditions on the time-dependent Hamiltonian. They prove a refined adiabatic theorem showing that having n vanishing derivatives at the timeline endpoints yields an end-to-end error scaling as O(ε^{n+1}), and they demonstrate this both numerically in a mixed-field Ising chain and experimentally on a Rydberg-atom chain implemented on the Aquila platform. Their scheduling constructions (beta_n-based and related smoothings) markedly reduce end-to-end infidelity in the polynomial regime without altering bulk dynamics, while still offering compatibility with other techniques like slow passage through minimal gaps. On real hardware, decoherence and measurement errors damp the gains but do not erase them, indicating practical usefulness for scalable quantum state preparation in noisy environments.

Abstract

We present numerical calculations, and simulations performed on a Rydberg atom quantum simulator, of the adiabatic evolution of many-body quantum systems around a quantum phase transition. We demonstrate that the end-to-end transfer error, for a given process duration and dissipative losses, can be suppressed by adopting smooth initial and final scheduling functions for the Hamiltonian. We consider a one-dimensional mixed-field Ising model, as well as a chain of Rydberg atoms, and compare numerical calculations and experimental results for the end-to-end transfer error with different schedule functions. We show, in particular, that if the time dependent Hamiltonian is $n$ times differentiable with vanishing $1^{st}$ to $n^{th}$ order derivatives in the beginning and in the end, the infidelity with respect to the final adiabatic eigenstate scales as $1/T^{n+1}$ when evolving for time $T$.

Adiabatic preparation of many-body quantum states: getting the beginning and ending right

TL;DR

The authors address how to maximize fidelity in adiabatic state preparation of many-body systems by enforcing smooth boundary conditions on the time-dependent Hamiltonian. They prove a refined adiabatic theorem showing that having n vanishing derivatives at the timeline endpoints yields an end-to-end error scaling as O(ε^{n+1}), and they demonstrate this both numerically in a mixed-field Ising chain and experimentally on a Rydberg-atom chain implemented on the Aquila platform. Their scheduling constructions (beta_n-based and related smoothings) markedly reduce end-to-end infidelity in the polynomial regime without altering bulk dynamics, while still offering compatibility with other techniques like slow passage through minimal gaps. On real hardware, decoherence and measurement errors damp the gains but do not erase them, indicating practical usefulness for scalable quantum state preparation in noisy environments.

Abstract

We present numerical calculations, and simulations performed on a Rydberg atom quantum simulator, of the adiabatic evolution of many-body quantum systems around a quantum phase transition. We demonstrate that the end-to-end transfer error, for a given process duration and dissipative losses, can be suppressed by adopting smooth initial and final scheduling functions for the Hamiltonian. We consider a one-dimensional mixed-field Ising model, as well as a chain of Rydberg atoms, and compare numerical calculations and experimental results for the end-to-end transfer error with different schedule functions. We show, in particular, that if the time dependent Hamiltonian is times differentiable with vanishing to order derivatives in the beginning and in the end, the infidelity with respect to the final adiabatic eigenstate scales as when evolving for time .

Paper Structure

This paper contains 12 sections, 3 theorems, 30 equations, 5 figures.

Table of Contents

  1. Adiabatic theorem
  2. Ising model
  3. Schedule functions for adiabatic evolution
  4. Numerical results
  5. Results on a quantum simulator
  6. Conclusion and outlook
  7. Acknowledgments
  8. Schedule functions
  9. Piecewise smooth schedule functions
  10. Rydberg reference schedule
  11. Proof of Theorem 1
  12. Estimation of first-order component

Key Result

Theorem 1

Consider a Hamiltonian $H(\tau)$ for $\tau \in [0,1]$ that is $n$ times differentiable in $\tau$, with the first $n$ derivatives vanishing at $\tau=0$ and $\tau=1$. Let $E(\tau)$ be an eigenvalue of $H(\tau)$ separated from the rest of the spectrum by a finite gap, and let $\ket{\Phi(\tau)}$ be the

Figures (5)

  • Figure 1: (a) Energy gap and phases of the Ising model \ref{['eq:MFIM']} with $L=21$ spins. The curve shows the path \ref{['eq:path']} from the ferromagnetic to the anti-ferromagnetic phase. (b) Schedule functions with linear, diverging and vanishing $n^{th}$ order derivative in $\tau=0,1$ - see detailed forms in \ref{['sec:schedule']}. On the right the full schedule on $[0,1]$, on the left just the start.
  • Figure 2: Time-evolution of the infidelity $\delta(\tau)$ during adiabatic passage of a length $L=11$ Ising chain for the linear, $s_{\beta_1}$, and $\beta_1$ schedules as shown in \ref{['fig:schedule']}. Note that the infidelity peak is slightly higher and occurs at an earlier time for $\beta_1$ because $\beta_1$ passes through the phase transition faster and at an earlier value of $\tau$. The color represents the total time $T$. Solid lines are the numerically simulated results. Dotted lines show the first-order component $\approx\varepsilon\gamma_0(\tau)/\Delta_{01}(\tau)$.
  • Figure 3: Ising model log-log plot of the final infidelity as a function of $\varepsilon = 1/T$ for a length $L=11$ chain. In black are the theoretical scalings for the polynomial and exponential regime, with prefactors fitted to numerical results. Results are shown for $s_{\beta_n}$ schedules with $n$ vanishing derivatives for $n=0,1,2$ and for the Sqrt-schedule.
  • Figure 4: Log-log plot of final infidelity as a function of $\varepsilon$, similar to \ref{['fig:final_fidelity_L11']}, for system size $L=11$ and $L=21$, to show dependence on system size. We show the $n=0$ schedule and the $\beta_n$ and $s_{\beta_n}$ schedule for $n=1$ (which have the same asymptotic scaling in $\varepsilon$, but with a different prefactor).
  • Figure 5: Rydberg model plot of final infidelity as a function of $\varepsilon=\frac{1}{T}$. (a) Numerical results for a (noiseless) simulation, with a schedule derived from \ref{['eq:IGapICosP05']} and \ref{['eq:igap_schedule']}, for a reference schedule with no vanishing derivatives, the $s_{\beta_1 , f}$ schedule with $n=1$ vanishing derivatives, and the Sqrt$_f$-schedule with diverging derivatives. (b) Noisy results. Here we consider a smaller range of $T=1.2$µs to $T=2.5$µs. Results from a numerical simulation, using the error model of wurtz2023aquila, as well as experiments on Aquila.

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 2: Lemma 1 of lidar2009adiabatic
  • Lemma 3: Generalization of Lemma 2.1 from hagedorn1989adiabatic
  • proof