Fast Bosonic Control via Multiphoton Qubit-Oscillator Interactions

TL;DR

This work tackles slow state preparation for bosonic codes on planar superconducting hardware by introducing multiphoton qubit-oscillator interactions that enable faster preparation of rotationally symmetric states. It develops a multiphoton generalization of the Law-Eberly protocol to realize $n$-photon swaps and an inversion procedure for rapid synthesis within $\mathcal{H}_{0,n}$, and it introduces Fine-Tune-Then-Populate (FTP) for arbitrary state synthesis using selective rotations and mixed interaction orders. The framework is extended to multiple oscillators and tested with circuit QED simulations that include spurious terms and decoherence, demonstrating robust preparation of cat and GKP states with substantial time reductions compared to linear schemes. The results suggest practical gains for bosonic codes on planar hardware, with implications for scalable fault-tolerant quantum computing and flexible control architectures in superconducting circuits.

Abstract

We present a protocol for preparing oscillator states with $n$-fold rotational symmetry, which include many logical codewords for bosonic quantum error correction codes. The protocol relies on a multiphoton interaction between the oscillator and an auxiliary qubit. Further, we achieve arbitrary control over the oscillator's Hilbert space by using a combination of different multiphoton interaction orders. We also discuss the preparation of rotationally-symmetric multi-oscillator states using a generalized variant of the protocol. We show that the use of multiphoton qubit-oscillator interactions can substantially reduce the state preparation time, in comparison to the linear qubit-oscillator interactions that are usually employed. Furthermore, we perform numerical simulations that take into account qubit and oscillator relaxation and dephasing using realistic planar superconducting circuit parameters that validate the robustness of our protocol. Our findings can significantly improve the performance of bosonic codes on planar superconducting hardware, which are an almost inevitable necessity for scalable bosonic fault-tolerant superconducting quantum computers.

Figures (6)

• Figure 1: Multiphoton control schematic. The vertical columns represent the subspaces $\mathcal{H}_{k,n}$, and each circle represents a Fock state with two bars; the lower and upper bars represent the qubit ground and excited states, respectively. The $n$JC operation, $\hat{Q}_{j}^{(n)}$ combined with qubit rotatations, $\hat{C}_{j}$, enable arbitrary state synthesis within one column. Other interaction orders different than $n$ can be used to transition and create superpositions between states in different subspaces (columns).
• Figure 2: Rotationally-symmetric state synthesis time for different interaction orders as a function of number of steps. In $(a)$, $\Omega=2\pi\times25\text{ MHz}$, and in $(b)$, $\Omega=2\pi\times200\text{ MHz}$. The blue dots represent $n=1$ with $\Omega=g_1$. In the case of $n=2$, the red dots have $g_2=g_1/4$, while the $g_2=g_1/8$ for the orange dots. The two-photon state preparation outperforms its one-photon counterpart for larger circuit depths (number of steps) in both regimes ($\Omega<g_1$ and $\Omega>g_1$). The two-photon time estimates support is exclusively on even steps since for each two steps using a linear interaction, we need one step using a two-photon interaction.
• Figure 3: Arbitrary state preparation punch card. For a given target state, a shaded circle indicates a nonzero Fock state contribution, while an unshaded circle means zero contribution. The number of steps required to prepare a target state is equal to the height in each column, where the height is determined by the largest shaded Fock state above the dotted line, and the additional steps used to generate the base state. The entries of the height vector, $\vec{h}^{(n)}=(h_0^{(n)},h_1^{(n)},...,h_{n-1}^{(n)})$, correspond to the height of each column. For example, the state in (a) has $\vec{h}^{(n)}=(0,3)$, and the state in (b) has $\vec{h}^{(n)}=(4,0,1).$
• Figure 4: Schematic for multiphoton control over the two-oscillator subspace $\mathcal{H}_{0,n_1}\otimes\mathcal{H}_{0,n_2}.$ Each circle represents a tensor product of Fock states in $\mathcal{H}_{0,n_1}\otimes\mathcal{H}_{0,n_2}.$ The lower and upper horizontal bars represent the qubit ground and excited states, respectively. The operators $\hat{{Q}}^{(n_1)}_{j}$ and $\hat{\mathfrak{Q}}^{(n_2)}_{j}$ represent $n_1$JC and $n_2$JC operations on oscillator 1 and 2, respectively.
• Figure 5: Wigner functions of a four-component cat state prepared by (a) an ideal system and (b) circuit QED system consisting of a tunable transmon coupled to an LC resonator via an asymmetric SQUID in the presence of qubit and resonator energy relaxation and dephasing.