Coherence Limits in Interference-Based cos(2$\varphi$) Qubits

TL;DR

This work examines coherence limits for parity-protected cos(2\varphi) qubits formed by interference between two bi-harmonic Josephson elements in a SQUID loop. It develops a unified Hamiltonian description that applies across experimental platforms, analyzes energy relaxation (T1) and pure dephasing (T\varphi) channels, and reveals a fundamental trade-off between charge and flux noise protections. Through extensive numerical studies over circuit parameters (including $E_{J\Sigma_2}/E_C$, $\delta\Phi$, and junction asymmetry), the authors show that, with currently achievable parameters, T1 can exceed milliseconds while T\varphi remains limited to a few microseconds due to residual flux or charge noise, placing practical coherence limits on this qubit class. They also demonstrate that maximizing coherence requires balancing noise channels at intermediate parameter values, and discuss how dramatically larger $|E_{J\Sigma_2}/E_{J\Sigma_1}|$ could, in principle, push T2 into the tens of microseconds to millisecond range, albeit at the cost of demanding fabrication and control. Overall, the paper highlights fundamental limits of cos(2\varphi) qubits and motivates exploration of alternative designs or architectures to achieve simultaneous protection against relaxation and dephasing.

Abstract

We investigate the coherence properties of parity-protected $\cos(2\varphi)$ qubits based on interferences between two Josephson elements in a superconducting loop. We show that qubit implementations of a $\cos(2\varphi)$ potential using a single loop, such as those employing semiconducting junctions, rhombus circuits, flowermon and KITE structures, can be described by the same Hamiltonian as two multi-harmonic Josephson junctions in a SQUID geometry. We find that, despite the parity protection arising from the suppression of single Cooper pair tunneling, there exists a fundamental trade-off between charge and flux noise dephasing channels. Using numerical simulations, we examine how relaxation and dephasing rates depend on external flux and circuit parameters, and we identify the best compromise for maximum coherence. With currently existing circuit parameters, the qubit lifetime $T_1$ can exceed milliseconds while the dephasing time $T_\varphi$ remains limited to only a few microseconds due to either flux or charge noise. Our findings establish practical limits on the coherence of this class of qubits and raise questions about the long-term potential of this approach.

Figures (14)

• Figure 1: Electrical circuit. The circuit is made of two junctions A and B in parallel. Each junction has its own Josephson energies ($E_\mathrm{J_{i j}}$), where $i=A,B$ denotes the junction and $j=1,2$ is the order of the harmonic of the Josephson junction. A large shunt capacitance creates the charging energy $E_\mathrm{C}$. The circuit has three internal quantum operators: the charge number of the common metallic island $\widehat{n}$, in red, and the two phases of the junctions $\widehat{\varphi}_{A,B}$, in purple. It can be controlled via two classical external knobs shown in blue with the applied voltage, $n_\mathrm{g}$, and the applied magnetic flux, $\Phi$. The same control knobs couple to external noise densities denoted $S_{n_\mathrm{g}}$ and $S_\Phi$, electrical and magnetic, respectively.
• Figure 2: Magnetic flux dependence on the junctions harmonics energy. Magnetic flux dependence of the Josephson harmonics, see Eq. \ref{['eq:ham2']}. With a small junctions asymmetry, two regimes can be reached at $\delta \Phi \sim \pm 1/2$ ($\Phi / \Phi_0 \sim 0$), where the physics is similar to a transmon, and at $\delta \Phi \sim 0$, where the $\cos(2\varphi)$ is dominant. We took $E_{\mathrm{J}\Sigma_2}/E_{\mathrm{J}\Sigma_1}=-0.1$ and $d = 1%$.
• Figure 3: Impact of magnetic flux offset and junctions asymmetry on the wavefunctions. a) Pure $\cos(2\varphi)$. The lowest eigenstates are delocalized in the two wells due to the inter-well interaction energy $\kappa$. b) Influence of the offset magnetic flux $\delta \Phi$ with asymmetry set to zero. c) Influence of the junctions asymmetry with the offset magnetic flux $\delta \Phi = 10^{-3}$. For all figures, we have $E_{\mathrm{J}\Sigma_2}/E_{\mathrm{J}\Sigma_1}=-0.1$ and $E_{\mathrm{J}\Sigma_2}/E_\mathrm{C} = 20$.
• Figure 4: Eigenenergies $E_m$, $m \in \{0,1,2\}$, of the circuit Hamiltonian, see Eq. \ref{['eq:ham2']}, as a function of the offset charge $n_g$ for different ratios $E_{\mathrm{J}\Sigma_{\{1,2\}}}/E_\mathrm{C}$, linewise, and offset flux $\Phi$, columnwise. Energies are given in units of the transition energy $E_{01}$ of the pure $\cos (2 \varphi)$, evaluated at $n_g=0$. The zero point of energy is chosen as the bottom of the $m=0$ level. The left column shows the pure $\cos (2 \varphi)$ regime, at exactly $\delta \Phi = 0$ where only the second harmonics are present. The right column shows the interference-based $\cos (2 \varphi)$ case, where $\delta \Phi = 10^{-3}$. For all figures, we have $E_{\mathrm{J}\Sigma_2}/E_{\mathrm{J}\Sigma_1}=-0.1$.
• Figure 5: Effect of the junctions asymmetry. For $E_{\mathrm{J}\Sigma_2}/E_\mathrm{C} \in \{-1, -50, -150\}$, we show the effect of the junction asymmetry $d$ on the qubit frequency, a), and its relevant relaxation figure of merits, b), c), d). In b), we show in pink and yellow dash-dot lines the typical matrix element for the transmon Koch2007 and fluxonium Somoroff2023Earnest2018 qubit respectively. We see that for such a low asymmetry all figure of merits reach $<~10^{-2}$, confirming that the relaxation protection is well preserved for experimentally achievable asymmetries. For all figures, we have $E_{\mathrm{J}\Sigma_2}/E_{\mathrm{J}\Sigma_1}=-0.1$, $\delta \Phi = 10^{-5}$ and $n_\mathrm{g}=0.25$.