Robust high-order quantum simulation using finite-width pulses

Abstract

We present a general framework for promoting first-order pulse sequences in quantum simulation to higher-order sequences that maintain robustness in the presence of finite pulse-width effects. Our approach maps a given first-order pulse sequence to a first-order Trotter formula, applies higher-order Trotter-formula constructions, and then compiles the resulting evolution back into physically implementable finite-width pulses via dynamically corrected gates. The resulting sequences achieve arbitrarily high-order error scaling with respect to the control cycle time of the underlying first-order sequence while maintaining robustness to finite pulse-width effects. The framework also enables the use of multi-product formulas for more efficient constructions. We apply the framework to several physically motivated quantum-simulation tasks and numerically verify the predicted error scalings.

Figures (10)

• Figure 1: Pulse sequence with $l$ control pulses. (a) Ideal pulses, where the $k$th pulse is instantaneous and implements $P_k$. (b) Finite-width pulses of duration $t_p$, where the $k$th pulse implements $\widetilde{P}_k$. $T_c$ denotes the control cycle time.
• Figure 2: Control-function variations for a smooth asymmetric pulse shape: (a) $f_k(t)$, (b) $f_k(t/c)/c$ with $c=2$, (c) $-f_k(t_p-t)$, and (d) $-f_k(t_p-t/c)/c$.
• Figure 3: Simulation error versus pulse width $t_p$ for engineering the target Ising Hamiltonian in Eq. \ref{['Ising_Htarg']} from the homogeneous Ising Hamiltonian in Eq. \ref{['Ising_H0']} using the pulse sequence of Ref. parrarodriguez2020digitalanalog. The blue curve shows the naive finite-width implementation of the base sequence, while the red and black curves show the corresponding first- and second-order DCG implementations.
• Figure 4: Simulation error versus $T$ for engineering the Heisenberg Hamiltonian in Eq. \ref{['target_CR']} from the cross-resonance Hamiltonian in Eq. \ref{['system_CR']} for $n=4,J=1$ and $t_p=10^{-4}$. Circle markers denote naive finite-width implementations of the first-, second-, and fourth-order Trotter sequences from Construction 1, while triangle markers denote the corresponding DCG-based implementations. The purple curve shows the fourth-order sequence with the negative-time segment included.
• Figure 5: Simulation error for the anisotropic Heisenberg-chain example with pulse width $t_p=10^{-4}$. We compare the first-, second-, and fourth-order sequence from Construction 2, and the naive fourth-order sequence obtained without the robust compilation step of Lemma \ref{['lemma_pulse_impl']}.

Theorems & Definitions (14)

• Theorem 1
• proof
• Lemma 1
• Theorem 2
• proof
• Proposition 1
• proof
• Lemma 2
• proof
• proof : Proof of Lemma \ref{['lemma_pulse_impl']}