Hardware-efficient formulation of molecular cavity-QED Hamiltonians

TL;DR

This work tackles simulating cavity-QED Hamiltonians on near-term quantum devices, where classical approaches are hindered by an exponentially scaling photonic Hilbert space. It introduces BosonicOp and MixedOp with two mappers in Qiskit Nature, and compares standing-waves versus a localized photonic basis under a first-order Lie-Trotter evolution to study a two-level system in a cavity. A key contribution is the hardware-aware localization that enforces a 1D qubit chain, enabling zero-noise extrapolation to recover early-time dynamics and demonstrating robustness to relaxing 1D connectivity constraints. The approach reduces quantum-resource requirements and offers a practical path toward scalable cavity-QED simulations with near-term hardware, with implications for polaritonic chemistry and materials engineering.

Abstract

Light-matter coupled Hamiltonians are central to cavity materials engineering and polaritonic chemistry, but are challenging to simulate with classical hardware due to the scaling of the Hilbert space with the number of quantum photon modes and matter complexity. Leveraging the fact that quantum computers naturally represent photonic modes efficiently, we present a novel approach to simulate quantum-electrodynamical (QED) systems on near-term quantum hardware. After developing the bosonic and mixed operators in the Qiskit Nature framework, we employ them to simulate a first-order Trotterized Hamiltonian for a spontaneous-emission problem of a two-level system in an optical cavity. We find that using a standing-waves photonic basis approach leads to fidelity issues due to hardware connectivity constraints and two-qubits gates errors. Hence, we propose using a localized photonic basis approach that enforces nearest-neighbor couplings, thanks to which we can map the Hamiltonian as a 1D qubit chain. We significantly reduce the noise and, by applying the zero-noise extrapolation error mitigation technique, we recover the accurate quantum dynamics. Finally, we also show that this approach is resilient when relaxing the 1D qubit chain approximation.

Figures (7)

• Figure 1: Schematics of a two-level fermionic system placed in the center of an optical cavity, coupled to a bath of cavity modes. a) Representation of a two-level atom placed in the center of an optical cavity. The direction of confinement is the $z$ direction. b) Representation of the coupling of the atom with the cavity modes, visualized as standing waves. Note that only the odd modes (i.e. the ones that are non-zero in the center) couple to the matter.
• Figure 2: Quantum dynamics of a two-level system placed in the center of an optical cavity. The two-level system was initially in the excited state, while all photonic modes started in the vacuum state. For $N_{\text{ph}} = 24$ modes, the total number of qubits is 14 (2 for the matter, 12 for the photon modes). For $N_{\text{ph}} = 36$ modes, the total number of qubits is 20 (2 for the matter, 18 for the photon modes). For both cases, we report the exact statevector simulation (where we can observe a full Rabi oscillation) as well as the noisy curve (which reaches the saturation around $t \approx 0.25$ a.u.)
• Figure 3: Schematics of the qubit connectivity for the standing-waves approach and the localized basis approach. Panels a), b) and c) represent the connectivity for an ideal hardware, while panels d), e) and f) show the connectivity after the circuit is transpiled to ibm_pittsburgh. a) Required connectivity for the standing-waves approach on an ideal hardware. All photonic qubits $q^{pw}$ are connected to the central matter qubit $q_1^m$. b) Required connectivity for the localized basis approach on an ideal hardware, assuming that the tensor $\tau$ in Eq. \ref{['eq:h_qed_loc']} is tridiagonal and $\sigma \neq 0$ only between $q_1^m$ and the central localized function $q_0^{lb}$. c) Required connectivity for the localized basis approach on an ideal hardware, assuming that the tensor $\tau$ in Eq. \ref{['eq:h_qed_loc']} is tridiagonal and $\sigma \neq 0$ between $q_1^m$ and the three central localized function $q_0^{lb}, q_1^{lb}, q_2^{lb}$. d) Connectivity of the standing-waves approach mapped onto ibm_pittsburgh. All of the qubits representing a mode are divided into two branches, and SWAP operations (represents by the x in the connectors) are introduced to allow them to interact with $q_1^m$. e) Connectivity of the localized basis approach on the ibm_pittsburgh. Since we enforced $\tau$ and $\sigma$ to match the hardware layout, no SWAP operations are required, which makes this panel identical to panel b). f) Connectivity of the localized basis approach on the ibm_pittsburgh, when $\sigma \neq 0$ for the three central localized functions. Since the required connectivity for the qubit $q_1^m$ is 4, SWAP operations are required (in particular, 4 SWAPs per time-step between $q_0^{lb}, q_1^{lb}, q_2^{lb}$).
• Figure 4: Quantum dynamics of a two-level system placed in the center of an optical cavity when the modes are described with the localized basis approach (c.f. Section \ref{['sec:localizedbasis']}), assuming that the tensor $\tau$ in Eq. \ref{['eq:h_qed_loc']} is tridiagonal and $\sigma \neq 0$ only for a few central localized functions (1 in panels a and b, 3 in panels c and d). The two-level system was initially in the excited state, while all cavity modes started in the vacuum state. Due to the constraints on $\tau$ and $\sigma$, the statevector simulation for the localized basis does not reproduce the standing-waves reference, but still represents a good approximation. Note that relaxing the constraint on $\sigma$ leads to a better approximation for the statevector simulation, as it can be seen by comparing panel a with c, or b with d. The zero-noise extrapolation (ZNE) dynamics is obtained using a linear fit. a) $N_{\text{ph}} = 24$ cavity modes, approximated with $N_{\text{loc}} = 13$ localized functions, using $N_q = 15$ qubits. $\sigma \neq 0$ only for the central localized function ($\sigma_7$). b) $N_{\text{ph}} = 36$ cavity modes, approximated with $N_{\text{loc}} = 19$ localized functions, using $N_q = 21$ qubits. $\sigma \neq 0$ only for the central localized function ($\sigma_{10}$). c) $N_{\text{ph}} = 24$ cavity modes, approximated with $N_{\text{loc}} = 13$ localized functions, using $N_q = 15$ qubits. $\sigma \neq 0$ for the three central localized functions ($\sigma_6, \sigma_7, \sigma_8$). d) $N_{\text{ph}} = 36$ cavity modes, approximated with $N_{\text{loc}} = 19$ localized functions, using $N_q = 21$ qubits. $\sigma \neq 0$ for three central localized functions ($\sigma_{9}, \sigma_{10}, \sigma_{11}$).
• Figure 5: Quantum dynamics of a two-level system placed in the center of an optical cavity when the modes are described with the localized basis approach, assuming that the tensor $\tau$ in Eq. \ref{['eq:h_qed_loc']} is tridiagonal and $\sigma \neq 0$ only for a few central localized functions (1 in panel a , 3 in panel b). The two-level system was initially in the excited state, while all cavity modes started in the vacuum state. The ZNE dynamics is obtained using a linear fit. $N_{\text{ph}} = 24$ cavity modes, approximated with $N_{\text{loc}} = 13$ localized functions, using $N_q = 15$ qubits. $\sigma \neq 0$ only for the central localized function ($\sigma_7$). b) $N_{\text{ph}} = 24$ cavity modes, approximated with $N_{\text{loc}} = 13$ localized functions, using $N_q = 15$ qubits. $\sigma \neq 0$ for the three central localized functions ($\sigma_6, \sigma_7, \sigma_8$).