QPU Hamiltonian

In all cases we will refer to \(H\) as being of the form

\[ H = -\sum_j\Delta_jn_j \ + \ \sum_j\Omega_j\sigma^x_j \ + \ H_{i} \]

where \(H_i\) is the interaction term in the Hamiltonian. Values of \(\Omega_j\) and \(\Delta_j\) respectively represent the amplitude and the detuning of the driving field applied to the qubit \(j\). Avoiding technical details we will refer to eigenstates of \(H\) (and in particular to the ground state) as equilibrium states.

Although the QPU currently only supports a Rydberg interaction, EMU-MPS supports both the Rydberg interaction term and the XY interaction.

The Rydberg interaction reads

\[ H_{rr} = \sum_{i>j} U_{ij} n_{i}n_{j} \]

where

\[ U_{ij} = \frac{C_{6}}{r_{ij}^{6}}, \]

and the XY interaction reads

\[ H_{xy} = \sum_{i>j} U_{ij} (\sigma^+_{i}\sigma^-_{j} + h.c.) \]

where

\[ U_{ij} = \frac{C_{3}(1-3 \cos^2(\theta_{ij}))}{r_{ij}^{3}}, \]

In these formulas, \(r_{ij}\) represents the distance between qubits \(i\) and \(j\), and \(\theta_{ij}\) represents a configurable angle (see here). Currently, Pasqal quantum devices only support Rydberg interactions, and different devices have different \(C_6\) coefficients and support for different maximum driving amplitudes \(\Omega\). Intuitively, under stronger interactions (rydberg-rydberg and laser-rydberg), bond dimension will grow more quickly (see here), thus affecting performance of our tensor network based emulator. For a list of the available devices and their specifications, please refer to the Pulser documentation (see here).