Programmable generation of arbitrary continuous-variable anharmonicities and nonlinear couplings

TL;DR

This work addresses the challenge of implementing arbitrary non-Gaussian operations in continuous-variable quantum platforms, which is essential for universal CV quantum computation and realistic quantum simulations. The authors introduce a hybrid CV–DV scheme that achieves programmable non-Gaussian unitaries by performing a Hamiltonian-level Fourier decomposition of the target non-quadratic potentials and implementing each Fourier component with bosonic quantum signal processing using a discrete-variable ancilla. The approach extends beyond single-mode gates to multi-mode couplings, demonstrated via two examples: a one-mode anharmonic double-well potential and a two-mode nonlinear coupling, with systematic improvability controlled by the number of Fourier components and Trotter steps. The framework offers a scalable path toward richer CV circuit compilation and simulation capabilities on near-term CV–DV devices, potentially enabling efficient quantum simulations of lattice gauge theories, chemical dynamics, and quantum chaos, while also providing practical speedups over Lamb-Dicke-based implementations in trapped-ion settings.

Abstract

Harmonic oscillators are promising continuous-variable (CV) quantum resources because their infinite-dimensional Hilbert spaces allow for resource-efficient quantum computing and simulation. To reach their full potential, CV platforms need to be able to efficiently implement non-Gaussian operations. However, schemes for generating arbitrary non-Gaussian operations are restricted to single modes, i.e., the implementation of anharmonic potentials. Here, we introduce a method for implementing arbitrary non-Gaussian operations applicable to both single- and multi-mode systems, allowing the generation of both anharmonicities and nonlinear multi-mode couplings. Our method synthesizes a target Hamiltonian by decomposing it into a Fourier series whose terms are implemented via bosonic quantum signal processing, which uses a discrete-variable (DV) system to induce a nonlinearity in the CV system. Our hybrid CV-DV protocol allows for the direct simulation of a broad range of CV phenomena (such as those in lattice gauge theory, chemical dynamics, and quantum chaos) and provides a richer toolbox for CV circuit compilation.

Figures (3)

• Figure 1: Decomposition of an anharmonic Hamiltonian into CV-DV trigonometric gates. (a) A target Hamiltonian, the double-well potential $V(X) = 0.05X^4-0.7X^2+0.2X$ (blue), is approximated by the Fourier series $\sum_{m=1}^{N_F} A_{m}\cos(mX)+B_{m}\sin(mX)$ truncated at the 2nd or 8th orders. Dashed lines depict the domain of consideration ($L=12$). (b) Middle row: CV-DV circuit for generating arbitrary single-mode potentials. Each term in the Fourier series of $V(X)$ is implemented using the corresponding qubit-dependent trigonometric gate $G_{c,s}$. Postselection at the end of the circuit removes the dependence on the qubit, leaving the boson-only unitary $U\approx e^{-iV(X)\Delta t}$. Top and bottom rows: Bosonic QSP sequences for the gates $G_{c,s}$.
• Figure 2: Simulating quantum tunneling in an anharmonic potential. (a) Quartic double-well potential of Eq. \ref{['eq:double-well-hamiltonian']}, with $\xi_1/\omega=8\xi_0/\omega=0.35$, where $\Delta E_{n}$ is the energy difference between the $n$th eigenstate and the ground state. The system is initially in the coherent state $\ket{\alpha=-X_0/\sqrt{2}}$, centered at the left well. Although the energy of the initial state is below the barrier, the system can tunnel into the right well, with time scale inversely proportional to $\Delta E_{1}$. (b) Expected position $\langle X\rangle$ of the oscillator as the CV-DV system is propagated using Eq. \ref{['eq:tunnelling-unitary']} with different numbers of Fourier components $N_F$. The time step is $\Delta t=20\pi/(\omega r)$ with Trotter number $r=500$. As $N_F$ increases, so does the agreement with the exact dynamics. (c) The infidelity $1-\mathcal{F}$ of the final state with respect to the exact result decreases with $N_F$, especially at larger times where the accumulated influence of anharmonicity is greater.
• Figure 3: Simulating the dynamics of two nonlinearly coupled oscillators. (a) The coupled modes, with $\omega_2 = \omega_1/2$ and $\xi\ll \omega_1$. (b) Time evolution of the populations of the multi-mode Fock states $\ket{1,0}$ and $\ket{0,2}$, both exact and using Fourier expansions with $N_F$ of 3 or 8. (c) The infidelity $1-\mathcal{F}$ of quantum states $(U_{\rm trot}(\Delta t))^r\ket{\alpha_1,\alpha_2=0}$ with respect to the exact state $\exp(-iH_{\text{nl}}r\Delta t)\ket{\alpha_1,\alpha_2=0}$ improves as $N_F$ increases. For these simulations, $\xi\Delta t=1.715\times 10^{-3}$ and $r = 2500$.