Building Block For Universal Continuous Variables Computation In Superconducting Devices

Abstract

Continuous variable (CV) quantum computation offers an alternative to qubit-based computing by exploiting the infinite-dimensional Hilbert space of bosonic modes. Despite recent progress, superconducting platforms have yet to demonstrate a scalable architecture capable of universal computation. Here, we design and numerically simulate a two-layer superconducting architecture that implements all five interactions of the universal CV gate set (rotation, displacement, squeezing, Kerr, and beam splitter) within experimentally accessible regimes. To this end, we employ a DC-SQUID as the bosonic mode, a fluxonium qubit to mediate nonlinear interactions, and two ancillary qubits that enable Gaussian and multi-mode operations. By tuning fluxes and frequencies, we achieve high fidelities ($\geq 98\%$) across all gates within state-of-the-art parameter ranges. The modular nature of the design allows straightforward scaling, establishing a feasible pathway toward high-fidelity, universal CV quantum computation based on superconducting circuits.

Figures (3)

• Figure 1: (a) Schematic representation of the building block for universal CV quantum computation. The circuit is composed of a two-layer circuit. Layer 1 comprises a DC-SQUID ($M$) that will encode our continuous variables and the auxiliary qubits ($R$ and $B$) responsible for the rotation and beam splitter interactions. On Layer 2, a fluxonium qubit ($F$) is coupled via an external magnetic flux $\Phi_{\textnormal{ext}}$ to the DC-SQUID, allowing the tuning of the second-order interaction between these devices. The arms connected to each device stand for the control lines that apply the fields that control the device frequencies. (b) Circuit diagram of schematics in (a). Each superconducting element $i=\{m,f,r,b\}$ is composed of a superconducting ring with two or more Josephson junctions with Josephson energies $E_{i}$ and capacitors $C_{i}$. We also have capacitive coupling between the mode and the auxiliary qubits in Layer 1. We assume no capacitive coupling between $F$ and the auxiliaries $R$ and $B$ due to the physical separation of the elements. The application of each interaction depends on our ability to tune the devices and apply external pulses via the control lines. The energy configuration for enabling each interaction can be seen in (c) where the relevant pulses $\Omega_i$ and detunings $\Delta_i$ are controlled to generate the interactions. The frequency values in each diagram stand for the ones used in our simulations.
• Figure 2: Numerical simulation of the fidelity of each single-mode operation as a function of the relevant device parameters. In all simulations, the system evolves under the Hamiltonian of the corresponding circuit up to the respective interaction time $\tau_{i}$ and considering the initial state $\ket{\psi_0}=\ket{\nu}_m\otimes\ket{g}_f\otimes\ket{g}_r\otimes\ket{g}_b$, with the coherent state amplitude $\nu = 2$, except for the squeezing operation in which the initial state of $F$ is $\ket{+}_f = (\ket{e}_{f} + \ket{g}_{f})/\sqrt{2}$. Then, we compare the states $\ket{\psi(\tau_i)}$ with the ideal state of each interaction. The insets show the Wigner functions for the state of $M$ during the interaction time. (a) For the rotation, the fidelity depends on the detuning $\Delta_r$. In the inset, we can see that the state of $M$ rotates around the phase-space origin. (b) For the squeezing, a behavior similar to the one in the rotation, but now, the ratio between $\Omega_S$ and $g_{mf}$ is the key to augment the fidelity. (c) The Kerr interaction evolved until $\chi = \pi/2$ generates a cat state when applied in the initial coherent state and then returns to the initial state. The fidelity of the complete evolution depends on the condition $\omega_f \gg g_{mf}$.
• Figure 3: The beam splitter operation is achieved using a dispersive interaction between two modes mediated by the coupler $B$, see Fig. \ref{['fig:system']}(a). In (a) we plot the fidelity of the subspace $\ket{\psi}_{m_1}\otimes\ket{\psi}_{m_2}$ for different values of detuning between the coupler and the modes. Again, the fidelity depends on the condition $|\Delta_b| \gg g_{mb}$. In (b), we see the Wigner plot for the states of the two modes. As can be seen, the coherent state $\ket{\psi(0)}_{m_1} = \ket{\nu}_{m_1}$, with $\nu =2$, is transferred from $M_1$ to $M_2$ after a $\pi/2$ rotation.