QPU Hamiltonian

We write the system Hamiltonian \(H\) in the form

\[ H = -\sum_j \Delta_j n_j \;+\; \sum_j \Omega_j \sigma^x_j \;+\; H_{\text{int}} \]

where:

• \(\Delta_j\) is the detuning applied to qubit \(j\) (energy offset).
• \(\Omega_j\) is the drive amplitude on qubit \(j\) (couples the two levels).
• \(n_j, \sigma^x_j\) are the usual number and Pauli-X operators on site \(j\).
• \(H_{\text{int}}\) collects all pairwise interaction terms.

Think of \(\Omega_j\) and \(\Delta_j\) as the pulse parameters that control each qubit. Eigenstates of \(H\) (in particular the ground state) are sometimes called equilibrium states.

emu-mps supports two types of pairwise interactions below. Pasqal QPUs currently exposes only the Rydberg interaction, but the emulator can handle both Rydberg and XY interactions.

Rydberg interaction

The Rydberg term is

\[ H_{rr} = \sum_{i>j} U_{ij}\, n_i n_j \qquad\text{with}\qquad U_{ij} = \frac{C_6}{r_{ij}^6}, \]

where

• \(r_{ij}\) is the distance between qubits \(i\) and \(j\),
• \(C_6\) is the device-dependent van der Waals coefficient.

This term penalizes having two nearby atoms both excited to the Rydberg state; it decays quickly with distance \((\varpropto 1/r^6)\).

XY interaction

The XY term (spin-exchange) is

\[ H_{xy} = \sum_{i>j} U_{ij}\,(\sigma^+_i \sigma^-_j + \sigma^-_i \sigma^+_j) \qquad\text{with}\qquad U_{ij} = \frac{C_3(1 - 3\cos^2\theta_{ij})}{r_{ij}^3}, \]

where

• \(C_3\) is a coupling constant,
• \(\theta_{ij}\) is the angle between the magnetic field and vector of the two atoms, see the Pulser XY tutorial.

The XY interaction mediates excitation hopping between sites and decays more slowly \((\varpropto 1/r^3)\) and can be anisotropic because of the angular factor.

Practical notes

• \(r_{ij}\) and \(\theta_{ij}\): distances and angles are determined by the register coordinates and the magnetic field; see Pulser XY tutorial for details.

• Device differences: different devices use different \(C_6\) (and \(C_3\)) values and support different maximum \(\Omega\). Check device specs in the Pulser devices and virtual devices tutorial.

• Performance impact: stronger interactions and stronger drives tend to increase entanglement, which raises the required MPS bond dimension and increases memory/CPU cost. See the MPS performance notes for details (mps/index.md).

• When modeling experiments: use the device parameters (\(C_6\), geometry, max \(\Omega\)) that match your target hardware for realistic simulations.

For more device details and examples, see the Pulser documentation: https://pulser.readthedocs.io/en/stable/tutorials/virtual_devices.html