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arXiv:2609.02405

Constructing and characterising anharmonic potentials in a harmonic oscillator

Clara Yun Fontaine, Mansi Somani, Kehui Yu, May Chee Loke, Jonathan Schwinger, Pak-Tik Fong, Ni-Ni Huang, Adrian Copetudo, Mustafa Bakr, Hoi-Kwan Lau, Tanjung Krisnanda, Yvonne Y. Gao

quant-phSubmitted Sep 2, 2026• Announced Sep 3, 2026• 0 citations• Revised Oct 21, 2026 (v2)
v1 · Sep 2, 2026v2 · Oct 21, 2026
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Abstract

Continuous-variable quantum systems offer a natural platform for quantum information processing and analogue quantum simulation of real-world processes. Realising these applications, however, requires anharmonic potentials with programmable functional forms, which are challenging to engineer on demand. Here, in a superconducting harmonic oscillator coupled to a transmon qubit, we construct programmable non-Gaussian phase gates $e^{-iV(\hat{X})}$, corresponding to the impulsive action of an anharmonic potential $V(\hat{X})$, and characterise them by reconstructing their corresponding force pointwise. Using bosonic quantum signal processing, we compile target potentials into modular circuits of qubit rotations interleaved with a fixed conditional-displacement primitive. We first demonstrate a cubic phase gate whose cubic nature is jointly validated by the resulting non-Gaussian states and the force reconstructed pointwise from the gate's action on coherent probe states. We then construct a family of double-well potentials and use the reconstructed force to witness the double-well topology and tunable asymmetry. Finally, we engineer an approximate Morse potential and obtain its force profile using position-compressed probes, which extend reliable reconstruction towards the steep repulsive wall. These results establish a programmable route to constructing the potential landscape of quantum harmonic oscillators and a practical method for reconstructing its spatial structure, providing a building block for quantum simulation of anharmonic dynamics.

Comments
Main text: 9 pages with 4 figures. Supplemental Material: 15 pages with 3 figures and 16 tables

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Continuous-variable quantum systems offer a natural platform for quantum information processing and analogue quantum simulation of real-world processes. Realising these applications, however, requires anharmonic potentials with programmable functional forms, which are challenging to engineer on demand. Here, in a superconducting harmonic oscillator coupled to a transmon qubit, we construct programmable non-Gaussian phase gates e−iV(X^)e^{-iV(\hat{X})}e−iV(X^), corresponding to the impulsive action of an anharmonic potential V(X^)V(\hat{X})V(X^), and characterise them by reconstructing their corresponding force pointwise. Using bosonic quantum signal processing, we compile target potentials into modular circuits of qubit rotations interleaved with a fixed conditional-displacement primitive. We first demonstrate a cubic phase gate whose cubic nature is jointly validated by the resulting non-Gaussian states and the force reconstructed pointwise from the gate's action on coherent probe states. We then construct a family of double-well potentials and use the reconstructed force to witness the double-well topology and tunable asymmetry. Finally, we engineer an approximate Morse potential and obtain its force profile using position-compressed probes, which extend reliable reconstruction towards the steep repulsive wall. These results establish a programmable route to constructing the potential landscape of quantum harmonic oscillators and a practical method for reconstructing its spatial structure, providing a building block for quantum simulation of anharmonic dynamics.
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The motional modes of trapped ions are promising candidates for continuous-variable (CV) quantum computation and simulation. However, to achieve a universal gate set for CV quantum computation, anharmonic nonlinear unitaries are required. Here, we experimentally demonstrate a protocol that allows us to approximate the cubic phase gate. By numerically optimizing the phase angles and amplitudes of a set of state-dependent force pulses, we demonstrate the cubic phase gate for both vacuum and low-ααα coherent state inputs and verify its performance. Our protocol can also be extended to approximate other nonlinear unitaries.
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We address the nonlinear properties of the double-Morse potential as a resource for single-mode quantum states due to its double-well structure and anharmonicity. We obtain analytical expressions for the ground-state wavefunction and the corresponding ground-state energy, using the asymmetry (width) parameter ααα as the primary control parameter. We then assess non-Gaussianity and nonclassicality as quantitative signatures of nonlinearity and quantumness, and we find that both increase monotonically with ααα. Furthermore, we analyze the metrological performance of the model for estimating the structural parameter ααα. By evaluating the corresponding Fisher information, we show that position measurements are optimal and can saturate the Cramér-Rao bound. In particular, the estimation of ααα is most precise in the shallow-well regime, where the quantum Fisher information is largest. For deep wells, enhanced sensitivity is instead obtained for the reparameterized control variable A=2e−αx0A=2e^{-αx_0}A=2e−αx0​, provided that x0x_0x0​ is independently calibrated. These results establish the double-Morse potential as a controllable source of non-Gaussianity and nonclassicality, with a metrological behavior that depends on the chosen estimation parameter. We highlight possible applications of this model in quantum sensing, continuous-variable quantum information, and quantum simulation.
Bosonic quantum error correction represents a route towards significant hardware overhead reduction for fault-tolerant quantum computation. However, controlling quantum harmonic oscillators that encode the quantum information comes with its own challenges. In many platforms such as superconducting circuits and trapped ions, the bosonic degree of freedom is controlled by an ancillary qubit whose decay during a qubit-oscillator gate propagates to the oscillator as a continuous error. Such errors generally lie beyond the scope of errors that the bosonic code is capable of correcting. We introduce the continuous-variable flag: an auxiliary oscillator coupled to the ancilla whose phase-space position encodes the continuous error corrupting the storage mode. After the gate, a heterodyne measurement of the flag estimates this continuous-variable error and a subsequent feedback control undoes it up to a bosonic code stabilizer. The scheme requires no new ingredients, as the flag is driven by the same dispersive coupling, displacements and rotations that implement the gates it is supposed to protect. We construct sequences protecting conditional rotations to all orders in ancilla decay with one flag, and conditional displacements to first order with one flag and to all orders with two. Simulations with realistic circuit-QED parameters predict more than three orders of magnitude fewer logical errors for the parity measurement of a four-component cat code, and a finite-energy GKP logical lifetime near its noiseless-ancilla bound. By discretizing continuous-variable errors, the flags introduced in this work are a useful tool for any hybrid oscillator-qubit processor, and open the route towards fault-tolerant simulation and computation with bosonic modes.