Oscillator-qubit generalized quantum signal processing for vibronic models: a case study of uracil cation

TL;DR

This work introduces oscillator–qubit generalized quantum signal processing (OQ-GQSP), a compiler that synthesizes arbitrary nonlinear bosonic phase gates for hybrid oscillator–qubit quantum processors and enables efficient simulation of anharmonic vibronic dynamics. By combining inverted unary electronic-state encoding, multi-controlled displacement gates for off-diagonal couplings, and OQ-GQSP for diagonal anharmonic potentials, the method preserves bosonic modes in their native space while incorporating complex potentials via Fourier/Laurent approximations. Numerical demonstrations on the uracil cation show accurate state preparation and nonadiabatic dynamics with controllable resource costs, highlighting advantages over fully discrete encodings and identifying trade-offs between circuit depth and shot overhead due to probabilistic success. The approach provides a constructive middle ground between analog and digital vibronic simulators, with potential extensions to multivariate QSP and hardware-aware error mitigation for near-term devices.

Abstract

Hybrid oscillator-qubit processors have recently demonstrated high-fidelity control of both continuous- and discrete-variable information processing. However, most of the quantum algorithms remain limited to homogeneous quantum architectures. Here, we present a compiler for hybrid oscillator-qubit processors, implementing state preparation and time evolution. In hybrid oscillator-qubit processors, this compiler invokes generalized quantum signal processing (GQSP) to constructively synthesize arbitrary bosonic phase gates with moderate circuit depth O(log(1/{\varepsilon})). The approximation cost is scaled by the Fourier bandwidth of the target bosonic phase, rather than by the degree of nonlinearity. Armed with GQSP, nonadiabatic molecular dynamics can be decomposed with arbitrary-phase potential propagators. Compared to fully discrete encodings, our approach avoids the overhead of truncating continuous variables, showing linear dependence on the number of vibration modes while trading success probability for circuit depth. We validate our method on the uracil cation, a canonical system whose accurate modeling requires anharmonic vibronic models, estimating the cost for state preparation and time evolution.

Figures (9)

• Figure 1: Quantum simulation workflow for nonadiabatic dynamics of the uracil cation. (a) Target system specified by the vibronic coupling Hamiltonian $\hat{H}_{\mathrm{Ura}^+}$. (b) Anharmonic potential energy surfaces $V(x)$ beyond the harmonic approximation. (c) Construction of nonlinear bosonic phase gates via oscillator–qubit generalized quantum signal processing (OQ–GQSP). (d) Implementation on a hybrid oscillator–qubit quantum processor. (e) Measurements yield nonadiabatic electronic population dynamics.
• Figure 2: Four-state vibronic coupling model Hamiltonian for uracil cation. $V_r^{(\alpha)}(\hat{Q}_r) = d_r^{(\alpha)}[e^{(a_r^{(\alpha)})(\hat{Q}_r-q_{r,0}^{(\alpha)})}-1]^2+e_{r}^{(\alpha)}$ is the Morse potential.
• Figure 3: Quantum circuit for OQ-GQSP. $\ket{\mathrm{osc}}$ is a quantum oscillator and $\ket{q}$ is a qubit register. The left half of the circuit generates positive powers, and the right half generates negative powers of the Laurent (Fourier) series terms.
• Figure 4: Quantum circuit for state-dependent nonlinear bosonic phase gate with OQ-GQSP. $\ket{\psi_D}$ is a electronic state encoded with unary code.
• Figure 5: Abstract quantum circuit for uracil cation Hamiltonian dynamics simulation with $N=4$ electronic states and $M=3$ vibrational modes. The circuit consists of three stages: (i) initialization in a product state, (ii) Trotterized time evolution alternating between off-diagonal couplings and diagonal potentials, and (iii) measurement of electronic populations.

Theorems & Definitions (12)

• Theorem 1: Generalized Quantum Signal Processing (GQSP) Motlagh2024
• Lemma 1: Oscillator-Qubit Generalized Quantum Signal Processing (OQ-GQSP)
• Lemma 2: OQ-GQSP to state-dependent nonlinear bosonic phase gate
• Theorem 2: Anharmonic vibronic dynamics with OQ–GQSP
• Lemma \ref{lem:GQSP_qubit_oscillator_main}
• proof
• Lemma \ref{lem:GQSP_qubit_oscillator_state_dependent}
• proof
• Lemma \ref{lem:GQSP_qubit_oscillator_state_dependent}: Fourier truncation for phase functions
• proof