Kramers-protected error correction with Andreev sp n qub ts

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

June 9th 2026

Spin

Spin qubit

Andreev bound state

Uσ=EJcos⁡(φ)U_{\sigma} = E_J \cos(\varphi)
U_{\sigma} = E_J \cos(\varphi)

Max Heys PhD thesis, 2021

Spin degenerate bound state!

Andreev bound state + spin orbit

Uσ=EJcos⁡(φ)+Eσ(ϵSO⃗⋅σ⃗)sin⁡(φ)U_{\sigma} = E_J \cos(\varphi) + E_{\sigma} (\vec{\epsilon_{SO}} \cdot \vec{\sigma}) \sin(\varphi)
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

Spin degree of freedom couples to supercurrent!

∝dE(φ)dφ\propto \frac{dE(\varphi)}{d\varphi}
\propto \frac{dE(\varphi)}{d\varphi}

Andreev spin qubit

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

∣0⟩\lvert 0 \rangle
\lvert 0 \rangle
∣1⟩\lvert 1 \rangle
\lvert 1 \rangle

Inductance couples to the ASQ ground state

EL∝1L∝d2E(φ)dφ2E_L \propto \frac{1}{L} \propto \frac{d^2E(\varphi)}{d\varphi^2}
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

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

Superconducting phase couples the 2 elements

and it allows e.g. readout of ASQ via transmon

 Material challenges:

- InAs nanowires usually have nuclear spins (phase decoherence)

- Graphene has no spin-orbit coupling

- Other materials are not reliable

Why are ASQ challenging?

How to make ASQ more robust?

Can we take advantage of the lack of magnetic field?

Looking ahead,

Kramers' theorem

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

Kramers' theorem

At fluxes                 the eigenstates of the Hamiltonian are degenerate

φ=0,π\varphi=0,\pi
\varphi=0,\pi

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

Design: ASQ in a network of inductors

Kramers theorem forces a degenerate computational space at TRS flux

∣0L⟩\lvert 0_L \rangle
\lvert 0_L \rangle
∣1L⟩\lvert 1_L \rangle
\lvert 1_L \rangle
Hσ=J12σ1,zσ2,z+J23σ2,zσ3,z+J13σ1,zσ3,zH_{\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_{\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}

A repetition error-correcting code

\lvert 0_L \rangle
\lvert 1_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}

Protection against bit flip errors because:

- A bit flip error is a non-local operation

- Flipping a spin requires breaking TRS

- Hamiltonian is composed of stabilizers

 

 

Dispersive readout via network inductance

Hr=ℏωr,0a†a+∑i≠jℏχijσi,zσj,za†aH_r = \hbar \omega_{r,0} a^\dagger a + \sum_{i \neq j} \hbar \chi_{ij} \sigma_{i,z}\sigma_{j,z}a^\dagger a
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
Hσ=J12σ1,zσ2,z+J23σ2,zσ3,z+J13σ1,zσ3,zH_{\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_{\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}

A single readout resonator detects bip flips

Dispersive readout via network inductance

Hr=ℏωr,0a†a+∑i≠jℏχijσi,zσj,za†aH_r = \hbar \omega_{r,0} a^\dagger a + \sum_{i \neq j} \hbar \chi_{ij} \sigma_{i,z}\sigma_{j,z}a^\dagger a
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
Hσ=J12σ1,zσ2,z+J23σ2,zσ3,z+J13σ1,zσ3,zH_{\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_{\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}

Logical gates

Uσ=EJcos⁡(ϕ)+Eσ([ϵSO⃗+δϵSO⃗cos⁡(ωt)]⋅σ⃗)sin⁡(ϕ)U_{\sigma} = E_J \cos(\phi) + E_{\sigma} ([\vec{\epsilon_{SO}} + \delta \vec{\epsilon_{SO}} \cos(\omega t)]\cdot \vec{\sigma}) \sin(\phi)
U_{\sigma} = E_J \cos(\phi) + E_{\sigma} ([\vec{\epsilon_{SO}} + \delta \vec{\epsilon_{SO}} \cos(\omega t)]\cdot \vec{\sigma}) \sin(\phi)

-              using EDSR to drive each spin

 

 

 

-               using adiabatic flux detuning

Rx(θ)R_x(\theta)
R_x(\theta)
Rz(θ)R_z(\theta)
R_z(\theta)
θ=ℏ−1∫0Tdt(E∣0L⟩(Φt)−E∣1L⟩(Φt))\theta = \hbar^{-1} \int_0^T dt (E_{\lvert 0_L \rangle}(\Phi_t) - E_{\lvert 1_L \rangle}(\Phi_t) )
\theta = \hbar^{-1} \int_0^T dt (E_{\lvert 0_L \rangle}(\Phi_t) - E_{\lvert 1_L \rangle}(\Phi_t) )

Conclusions

- Proof of principle of repetition error-correcting code with ASQ

- Materials with which to make the junctions remain a challenge