Kramers-protected hardware-efficient error correction with Andreev spin qubits

Abstract

We propose an architecture for bit-flip error correction of Andreev spins that is protected by Kramers' degeneracy. Specifically, we show that a coupling network of linear inductors and Andreev spin qubits results in a static Hamiltonian composed of the stabilizers of a bit-flip code. The electrodynamics of the many-body spin states also respect these stabilizers, and we show how reflectometry off a single coupled resonator can thereby accomplish their projective measurement. We further show how circuit-mediated spin couplings enable error correction operations and a complete set of single- and two-module logical quantum gates. The concept, which we dub the Ising molecule qubit (or Isene), is experimentally feasible and provides a path for compact noise-biased qubits.

Figures (6)

• Figure 1: (a) Circuit for minimal bit-flip encoding with Andreev spins. The field-effect gate lines on each spin (black) include DC gate voltages (not shown) and AC drive (colored). Control circuitry for flux $\Phi$ is not shown. Andreev spins are modeled by Eq. \ref{['eq:ASQ_EPR']}, while linear inductors by a quadratic energy-phase relation $U_L = (\varphi_0^2/2L) \varphi^2$, with $\varphi_0 = \hbar/2e$ the reduced flux quantum. (b) Schematic level structure as a function of flux, with the Kramers' point at zero flux. Colored arrows correspond to EDSR-induced single-spin transitions between the logical manifold (lowest) and the error manifolds while they remain degenerate.
• Figure 2: (a) Each spin can be driven locally via EDSR. The two tones depicted imply two possible frequencies for resonant transitions. The solid line is the higher frequency drive. (b) Operations that invoke three or more states that complete a loop in state space can accomplish logical $X(\theta)$ gates. Three sequential $\pi$ pulses at the transitions shown would accomplish a logical $X(\pi)$ gate. See main text for other $\theta$. We have chosen here a gauge where matrix elements for the symmetric eigenstates of $X$ are equal to the negative of the antisymmetric eigenstates.
• Figure 3: Tunable inductors (inductor symbols with arrows) can used to turn on/off a spin-spin interaction between the neighboring spins of two modules. The spin-spin interaction of the form $\sigma_{3,a,z}\sigma_{1,b,z}$ on the last spin spin of module $a$ and the first spin of module $b$ (indicated by the green boxes) translates to a logical $ZZ$ interaction.
• Figure 4: The circuit diagram labeling the individual inductors, external fluxes, and circuit fluxes at each node. The gates at the bottom (black T shapes) can be driven electrically at nonzero frequency (wavy arrows).
• Figure 5: (a) Dispersive shift $\chi_{12}$ and (b) inter-spin coupling $J_{12}$ at Kramers' point. Parameter used in the simulation: $E_{\sigma 1}/h = 0.4GHz$, $E_{\sigma 2}/h = 0.3GHz$, $E_{\sigma 3}/h = 0.2GHz$; $E_{0,1}/h = E_{0,2}/h = E_{0,3}/h = 0.4GHz$. $L_\mathrm{vertical} = L_\mathrm{12} = L_\mathrm{23}$ and $L_\mathrm{vertical} = L_\mathrm{1} = L_\mathrm{2} = L_\mathrm{3}$. For this configuration $J_{23}$ and $\chi_{23}$ are of similar scale, while $J_{13}$ and $\chi_{13}$ are much smaller.