Kramers-Protected Error Correction With Andreev Spin Qubits

Kramers-protected error correction with Andreev spin qubits

Haoran Lu, Isidora Araya Day, Anton R. Akhmerov, Bernard van Heck, Valla Fatemi

June 9th 2026

Spin

Spin Qubit

Andreev Bound State

\[ U_{\sigma} = E_J \cos(\varphi) \]

Spin degenerate bound state!

Max Hays PhD thesis, 2021

Andreev Bound State + Spin Orbit

\[ U_{\sigma} = E_J \cos(\varphi) + E_{\sigma}(\vec{\epsilon}_{SO}\cdot \vec{\sigma})\sin(\varphi) \]

Superconducting Spin Qubits, C. Padurariu & Y. Nazarov, 2009

Andreev Spin Qubit

\[ \propto \frac{dE(\varphi)}{d\varphi} \]

Spin degree of freedom couples to supercurrent!

Andreev Spin Qubit Readout

\[\lvert 0\rangle\]

\[\lvert 1\rangle\]

Spin degree of freedom couples to supercurrent!

Thus, if we can measure supercurrent, we can do readout.

Coherent manipulation of an Andreev spin qubit, M. Hays et al. 2021

Inductance Couples To The ASQ Ground State

\[ E_L \propto \frac{1}{L} \propto \frac{d^2E(\varphi)}{d\varphi^2} \]

We can infer the ASQ state by measuring its inductance.

Andreev Spin Qubit Coupled To A Transmon

• Superconducting phase couples the 2 elements
• It allows readout of ASQ via transmon

Direct manipulation of a superconducting spin qubit strongly coupled to a transmon qubit, M. Pita-Vidal et al. 2022

Why Are ASQ Challenging?

Why are ASQ challenging?

Material challenges:

• InAs nanowires usually have nuclear spins (phase decoherence)
• Graphene has no spin-orbit coupling
• Other materials are not reliable

How To Make ASQ More Robust?

Looking ahead,

How to make ASQ more robust?

Can we take advantage of the lack of magnetic field?

Kramers’ Theorem

Eigenstates of a time-reversal symmetric system with half-integer spin come in degenerate pairs.

Kramers’ Theorem At TRS Flux

At fluxes \(\varphi = 0,\pi\), the eigenstates of the Hamiltonian are degenerate.

Therefore, we need a magnetic field or a drive to implement gates.

Design: ASQ In A Network Of Inductors

\[ \lvert 1_L\rangle,\quad \lvert 0_L\rangle \]

\[ H_{\sigma} = J_{12}\sigma_{1,z}\sigma_{2,z} + J_{23}\sigma_{2,z}\sigma_{3,z} + J_{13}\sigma_{1,z}\sigma_{3,z} \]

Kramers theorem forces a degenerate computational space at TRS flux.

Dispersive Readout Via Network Inductance

\[ H_{\sigma} = J_{12}\sigma_{1,z}\sigma_{2,z} + J_{23}\sigma_{2,z}\sigma_{3,z} + J_{13}\sigma_{1,z}\sigma_{3,z} \]

\[ H_r = \hbar\omega_{r,0}a^\dagger a + \sum_{i\neq j}\hbar\chi_{ij}\sigma_{i,z}\sigma_{j,z}a^\dagger a \]

A single readout resonator detects bit flips.

Dispersive Readout Via Network Inductance

\[ H_{\sigma} = J_{12}\sigma_{1,z}\sigma_{2,z} + J_{23}\sigma_{2,z}\sigma_{3,z} + J_{13}\sigma_{1,z}\sigma_{3,z} \]

\[ H_r = \hbar\omega_{r,0}a^\dagger a + \sum_{i\neq j}\hbar\chi_{ij}\sigma_{i,z}\sigma_{j,z}a^\dagger a \]

Logical Gates

• R_x(θ) using EDSR to drive each spin
• R_z(θ) using adiabatic flux detuning

\[ U_{\sigma} = E_J \cos(\phi) + E_{\sigma} \left( [\vec{\epsilon}_{SO} + \delta\vec{\epsilon}_{SO}\cos(\omega t)] \cdot\vec{\sigma} \right)\sin(\phi) \]

\[ \theta = \hbar^{-1}\int_0^T dt \left( E_{\lvert 0_L\rangle}(\Phi_t) - E_{\lvert 1_L\rangle}(\Phi_t) \right) \]

Conclusions

• Proof of principle of repetition error-correcting code with ASQ
• Materials with which to make the junctions remain a challenge