Theory background
This page provides a brief overview of the physics behind the qubit types implemented in HybridSuperQubits. For a detailed treatment, see the companion paper:
J. J. Caceres et al., “FerBo: a noise resilient qubit hybridizing Andreev and fluxonium states”, arXiv:2604.01145 (2026).
The FerBo qubit
The FerBo (Fermionic-Bosonic) qubit is a superconducting circuit consisting of a large inductance, a small capacitor, and a highly transmitting Josephson weak link arranged in parallel. It can be viewed as a fluxonium qubit in which the standard tunnel junction is replaced by a semiconductor weak link hosting Andreev bound states.
The circuit Hamiltonian is:
where \(E_C\) is the charging energy, \(E_L\) is the inductive energy, \(\hat{n}\) is the charge number operator, \(\hat{\varphi}\) is the phase operator, and \(\varphi_{\mathrm{ext}} = 2\pi \Phi / \Phi_0\) is the external flux.
The weak link term is:
Here \(\Gamma\) is the coupling to superconducting leads, \(\delta\Gamma\) is the coupling asymmetry, and \(\varepsilon_r\) is the resonant level energy (Fermi detuning). The Pauli matrices \(\hat{\sigma}_{x,y,z}\) act on the two-dimensional Andreev subspace spanned by the states \(|{-}\rangle\) and \(|{+}\rangle\).
The full Hilbert space is a tensor product of the Fock space (bosonic LC mode, dimension \(N\)) and the Andreev subspace (dimension 2), giving a total dimension of \(2N\).
Protection mechanism
The FerBo qubit achieves simultaneous protection against both dominant noise channels:
Relaxation protection. In the high-impedance regime (\(Z/R_Q \gg 1\)), the ground state \(|0\rangle\) and first excited state \(|1\rangle\) have support in different Andreev sectors (\(|{-}\rangle\) and \(|{+}\rangle\) respectively). Because the charge operator \(\hat{n}\) does not couple different Andreev sectors, the matrix element \(|\langle 0|\hat{n}|1\rangle|^2\) is exponentially suppressed, protecting against charge-induced relaxation.
Dephasing protection. As in the heavy fluxonium, the wavefunctions delocalize across the phase variable \(\varphi\) when the impedance is large. This delocalization reduces the sensitivity to flux noise, suppressing the curvature \(\partial^2 E_{01}/\partial \varphi_{\mathrm{ext}}^2\).
The protected regime is bounded by \(Z/R_Q \gtrsim 2E_C / (\pi \varepsilon_r)\).
Berry phase correction
In the Andreev representation (where the Andreev basis rotates with the phase),
a scalar Berry phase correction arises. This is implemented as
_berry_contribution() and can be toggled
via the include_berry flag in potential().
Flux grouping schemes
The FerBo Hamiltonian supports two flux grouping modes controlled by the
flux_grouping parameter:
"ABS": External flux is absorbed into the weak link term \(H_{\mathrm{WL}}(\hat{\varphi} - \varphi_{\mathrm{ext}})\)."EL": External flux appears in the inductive term \(\tfrac{1}{2}E_L(\hat{\varphi} - \varphi_{\mathrm{ext}})^2\).
Both are physically equivalent but may offer numerical advantages in different parameter regimes.
Andreev pair qubit
The Andreev pair qubit describes pure Andreev bound states in a superconducting weak link, discretized in the charge basis. The Hamiltonian contains:
A charging term \(4E_C (\hat{n} - n_g)^2\)
The Josephson resonance level (JRL) potential from the weak link
The charge basis is a half-charge basis with states \(|n\rangle\) where \(n\) runs from \(-n_{\mathrm{cut}}\) to \(n_{\mathrm{cut}}\).
Fluxonium
The fluxonium qubit is an LC circuit shunted by a Josephson junction:
The large inductance \(E_L\) lifts the ground-state degeneracy of the transmon, producing a highly anharmonic spectrum. In the heavy regime (\(E_L \ll E_C\)), the wavefunctions delocalize in phase space, providing protection against flux noise.
Gatemon
The gatemon replaces the tunnel junction with a gate-tunable semiconductor weak link. The junction potential depends on the transmission coefficient \(\tau\), which is controlled by a gate voltage:
The Hamiltonian is discretized in the charge basis with a Fourier-expanded ABS potential.
Gatemonium
The gatemonium combines a gate-tunable weak link with an inductive shunt, analogous to the fluxonium but with a semiconductor junction. It supports multiple transmission channels, each contributing independently to the junction potential.
The Hamiltonian:
where the sum runs over the transmission channels with individual transmissions \(\tau_i\).
Josephson junction array (JJA)
The JJA module models an array of \(N\) Josephson junctions as a metamaterial. It computes:
Plasma frequency \(\omega_p = 1/\sqrt{L_J C_J}\)
Resonance mode frequencies from the dispersion relation
Group velocity
Kerr nonlinearity matrix
Resonator
A quantum LC resonator with the simple harmonic oscillator Hamiltonian:
Provides creation/annihilation operators, position/momentum (voltage/current) operators, and zero-point fluctuation scales.
Noise channels
All qubits inheriting from QubitBase have access to built-in noise analysis
methods:
Capacitive losses (\(T_1\)): dielectric loss in the capacitor, parameterized by quality factor \(Q_{\mathrm{cap}}\).
Inductive losses (\(T_1\)): resistive losses in the inductor, parameterized by \(Q_{\mathrm{ind}}\).
Flux bias line coupling (\(T_1\)): coupling to an external flux line with mutual inductance \(M\) and impedance \(Z\).
1/f flux noise (\(T_\varphi\)): dephasing from low-frequency flux fluctuations with amplitude \(A_\Phi\).
These are computed from Fermi’s golden rule using the matrix elements of the relevant noise operators between eigenstates.