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General, efficient, and robust Hamiltonian engineering

Pascal Baßler, Markus Heinrich, Martin Kliesch

TL;DR

The paper tackles the challenge of engineering arbitrary local Hamiltonians on quantum devices with an always-on native Hamiltonian $H_S$. It introduces a general LP-based framework that interleaves fast single-qubit pulses (Pauli or Clifford gates) with free evolution under $H_S$ to exactly or approximately realize a target $H_T$, with the number of contributing terms $r$ and the number of pulse layers $s$ determining complexity. By leveraging efficient relaxations (randomized column sampling) and robust extensions via average Hamiltonian theory and composite pulses, the method achieves robustness against finite pulse times and control errors, while maintaining favorable scaling (e.g., evolution time scales linearly or sublinearly with system size) and practical runtimes (e.g., solving LPs for up to hundreds of qubits on a laptop). Numerical simulations demonstrate the approach on 2D lattice Ising models and ion-trap-inspired Heisenberg Hamiltonians, achieving high fidelities (often well above 99.9%) and confirming robustness advantages over naive implementations. The work proposes a flexible trade-off between classical pre-processing and quantum evolution, enabling fast synthesis of multi-qubit interactions and offering a path toward practical digital-analog quantum computation and quantum simulation on near-term devices.

Abstract

Implementing the time evolution under a desired target Hamiltonian is critical for various applications in quantum science. Due to the exponential increase in the number of parameters with system size and experimental imperfections, this task can be challenging in quantum many-body settings. We introduce an efficient and robust scheme to engineer arbitrary local many-body Hamiltonians. To this end, our scheme applies single-qubit $π$ or $π/2$ pulses to an always-on system Hamiltonian, which we assume to be native to a given platform. These sequences are constructed by efficiently solving a linear program (LP) which minimizes the total evolution time. In this way, we can engineer target Hamiltonians that are only limited by the locality of the interactions in the system Hamiltonian. Based on average Hamiltonian theory and using robust composite pulses, we make our schemes robust against errors, including finite pulse time errors and various control errors. To demonstrate the performance of our scheme, we provide numerical simulations. In particular, we solve the Hamiltonian engineering problem on a laptop for arbitrary two-local Hamiltonians on a 2D square lattice with $196$ qubits in only $60$ seconds. Moreover, we simulate the engineering of general Heisenberg Hamiltonians from Ising Hamiltonians using imperfect single-qubit pulses for smaller system sizes and achieve a fidelity exceeding $99.9\%$, which is orders of magnitude better than non-robust implementations.

General, efficient, and robust Hamiltonian engineering

TL;DR

The paper tackles the challenge of engineering arbitrary local Hamiltonians on quantum devices with an always-on native Hamiltonian . It introduces a general LP-based framework that interleaves fast single-qubit pulses (Pauli or Clifford gates) with free evolution under to exactly or approximately realize a target , with the number of contributing terms and the number of pulse layers determining complexity. By leveraging efficient relaxations (randomized column sampling) and robust extensions via average Hamiltonian theory and composite pulses, the method achieves robustness against finite pulse times and control errors, while maintaining favorable scaling (e.g., evolution time scales linearly or sublinearly with system size) and practical runtimes (e.g., solving LPs for up to hundreds of qubits on a laptop). Numerical simulations demonstrate the approach on 2D lattice Ising models and ion-trap-inspired Heisenberg Hamiltonians, achieving high fidelities (often well above 99.9%) and confirming robustness advantages over naive implementations. The work proposes a flexible trade-off between classical pre-processing and quantum evolution, enabling fast synthesis of multi-qubit interactions and offering a path toward practical digital-analog quantum computation and quantum simulation on near-term devices.

Abstract

Implementing the time evolution under a desired target Hamiltonian is critical for various applications in quantum science. Due to the exponential increase in the number of parameters with system size and experimental imperfections, this task can be challenging in quantum many-body settings. We introduce an efficient and robust scheme to engineer arbitrary local many-body Hamiltonians. To this end, our scheme applies single-qubit or pulses to an always-on system Hamiltonian, which we assume to be native to a given platform. These sequences are constructed by efficiently solving a linear program (LP) which minimizes the total evolution time. In this way, we can engineer target Hamiltonians that are only limited by the locality of the interactions in the system Hamiltonian. Based on average Hamiltonian theory and using robust composite pulses, we make our schemes robust against errors, including finite pulse time errors and various control errors. To demonstrate the performance of our scheme, we provide numerical simulations. In particular, we solve the Hamiltonian engineering problem on a laptop for arbitrary two-local Hamiltonians on a 2D square lattice with qubits in only seconds. Moreover, we simulate the engineering of general Heisenberg Hamiltonians from Ising Hamiltonians using imperfect single-qubit pulses for smaller system sizes and achieve a fidelity exceeding , which is orders of magnitude better than non-robust implementations.

Paper Structure

This paper contains 35 sections, 22 theorems, 150 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Overview of main results
  3. Generality
  4. Efficiency
  5. Robustness
  6. Numerical results
  7. Generality and efficiency of the linear program approach
  8. Scaling of the total evolution time
  9. Evolution time vs. classical runtime
  10. Numerical Hamiltonian simulations
  11. Simulation of a 2D lattice model
  12. Simulation of Heisenberg Hamiltonians with an ion trap model
  13. Summary of numerical results
  14. Hamiltonian engineering by linear programming
  15. Pauli conjugation
  16. ...and 20 more sections

Key Result

Proposition 4.2

Let $\mathcal{W}(\boldsymbol{J}) \in \mathbb{R}^{r \times s}$ such that $\operatorname{ker} (\mathcal{W}(\boldsymbol{J})^T) = \{ \boldsymbol{0} \}$. If there exists $\boldsymbol{x} \in \mathbb{R}^s$ such that $\mathcal{W}(\boldsymbol{J}) \boldsymbol{x} = \boldsymbol{0}$ and $\boldsymbol{x} > \boldsy

Figures (8)

  • Figure 1: We engineer the target Hamiltonian $H_T$ by interleaving the natural dynamics of the quantum device, governed by the system Hamiltonian $H_S$, with layers of single-qubit $\pi$ or $\pi/2$ pulses, $\boldsymbol{S}_i = S_i^{(1)} \otimes \dots S_i^{(n)}$, as in \ref{['eq:approximate_evolution']}. The large red box highlights that $H_S$ is assumed to be always on. Our Hamiltonian engineering results are straightforward to implement: apply single-qubit pulse layers $\boldsymbol{S}_{i-1} \boldsymbol{S}_i^\dagger$, let the system evolve freely under $H_S$ for a duration $t \lambda_i$, and repeat.
  • Figure 2: Scaling of total evolution time $\sum_i \lambda_i = \boldsymbol{1}^T \boldsymbol{\lambda}$ with system size $n$ for randomly sampled target Hamiltonians. Each data point represents the average and error bars represent the sample standard deviation over $50$ uniform random samples of the target interaction strengths $\boldsymbol{A}$ from $[-1 , 1]^r$. The numbers of variables in the LP is varied from $s = 3r$ to $6r$ to demonstrate the reduction of the total evolution time when increasing the search space. Top (a-c): Results for the efficient Pauli conjugation LP with $\pi$ pulses, using 2-local (a), 3-local (b), and random many-body (c) system Hamiltonians. Bottom (d-f): Results for the Clifford conjugation LP with $\pi/2$ pulses, using similar 2-local (d) and 3-local (e) system Hamiltonians, and a randomly sampled 5-local Hamiltonian in (f). Moreover, for (d-e) randomly selected interaction terms in the system Hamiltonian are set to zero.
  • Figure 3: A 2-local system Hamiltonian on a 2D square lattice with $n = 4, \dots , 196$ qubits as in \ref{['eq:2D_lattice']} and a constant interaction strenght $J$. Each data point represents the average and error bars represent the sample standard deviation over $50$ uniform random samples of the target interaction strengths $\boldsymbol{A}$ from $[-1 , 1]^r$. Left: The evolution time over the number of qubits $n$ is shown. Right: The runtime for solving the Pauli conjugation LP over the number of qubits $n$ is shown.
  • Figure 4: Left: The sample mean and standard deviation of the average gate infidelities \ref{['eq:avg_infidelity']} for implementing the time evolution of $\mathrm{e}^{-\mathrm{i} t H_T}$ with $H_T$ from \ref{['eq:ising_target']} and $t=1\, \mathrm{s}$ over the number of Trotter cycles $n_{\mathrm{Tro}}$ is shown. The first-order Trotter formula from \ref{['eq:first_order_trotter_sim']} is used to approximate the time evolution. The sample mean and standard deviation are calculated over $50$ random samples of the target Hamiltonians $H_T$. Note, that each of the $n_{\mathrm{Tro}}$ Trotter cycles contains $\kappa r$ evolution blocks $U(t\lambda_{\boldsymbol{c}})$, where $\kappa = 8$ and $s = 17$, with different rotation directions for the $\pi$ pulses, see \ref{['sec:robust_pauli']} for more details. Top Right: Table indicating which error types are present for the different simulations. Bottom right: Example $2 \times 3$ 2D square lattice on $n=6$ qubits with the two-body interactions, solid black lines, and the three-body interactions for the lower left qubit, colored areas.
  • Figure 5: Top: The sample mean and standard deviation of the average gate infidelities \ref{['eq:avg_infidelity']} for implementing the time evolution of $\mathrm{e}^{-\mathrm{i} t H_T}$ with $H_T$ from \ref{['eq:heisenberg']} and $t=1\, \mathrm{s}$ over the number of Trotter cycles $n_{\mathrm{Tro}}$ is shown. The second-order Trotter formula from \ref{['eq:sec_order_trotter']} is used to approximate the time evolution. The sample mean and standard deviation are calculated over $50$ random samples of the Heisenberg Hamiltonians $H_T$ on $n=8$ qubits. Top Left: The Clifford conjugation robust against finite pulse time errors (dark green) is compared to the non-robust Clifford conjugation (red). Top Middle: The Clifford conjugation in combination with the SCROFULOUS pulse sequence Cummins2002 robust against finite pulse time errors and rotation angle errors (dark blue) is compared to the non-robust Clifford conjugation (red). Top Right: The Clifford conjugation in combination with the SCROBUTUS pulse sequence kukita2021 robust against finite pulse time errors, rotation angle errors and off-resonance errors (light blue) is compared to the non-robust Clifford conjugation (red). Bottom: Table indicating which error types are present for the different simulations.
  • ...and 3 more figures

Theorems & Definitions (42)

  • Definition 4.1
  • Proposition 4.2
  • Lemma 4.3: Farkas farkas1902
  • Lemma 4.4: Stiemke stiemke1915
  • proof : Proof of Prop. \ref{['thrm:feasibility']}
  • Definition 4.5
  • Lemma 4.6: Hayakawa2023, Proposition 4
  • Definition 5.0
  • Lemma 5.0
  • Theorem 5.0
  • ...and 32 more