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: .. math:: \hat{H} = 4E_C \hat{n}^2 + \tfrac{1}{2} E_L \hat{\varphi}^2 + H_{\mathrm{WL}}(\hat{\varphi} - \varphi_{\mathrm{ext}}) where :math:`E_C` is the charging energy, :math:`E_L` is the inductive energy, :math:`\hat{n}` is the charge number operator, :math:`\hat{\varphi}` is the phase operator, and :math:`\varphi_{\mathrm{ext}} = 2\pi \Phi / \Phi_0` is the external flux. The weak link term is: .. math:: H_{\mathrm{WL}}(\hat{\varphi}) = \Gamma \cos(\hat{\varphi}/2)\,\hat{\sigma}_x - \delta\Gamma \sin(\hat{\varphi}/2)\,\hat{\sigma}_y + \varepsilon_r \,\hat{\sigma}_z Here :math:`\Gamma` is the coupling to superconducting leads, :math:`\delta\Gamma` is the coupling asymmetry, and :math:`\varepsilon_r` is the resonant level energy (Fermi detuning). The Pauli matrices :math:`\hat{\sigma}_{x,y,z}` act on the two-dimensional Andreev subspace spanned by the states :math:`|{-}\rangle` and :math:`|{+}\rangle`. The full Hilbert space is a tensor product of the Fock space (bosonic LC mode, dimension :math:`N`) and the Andreev subspace (dimension 2), giving a total dimension of :math:`2N`. Protection mechanism ~~~~~~~~~~~~~~~~~~~~ The FerBo qubit achieves simultaneous protection against both dominant noise channels: **Relaxation protection.** In the high-impedance regime (:math:`Z/R_Q \gg 1`), the ground state :math:`|0\rangle` and first excited state :math:`|1\rangle` have support in *different* Andreev sectors (:math:`|{-}\rangle` and :math:`|{+}\rangle` respectively). Because the charge operator :math:`\hat{n}` does not couple different Andreev sectors, the matrix element :math:`|\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 :math:`\varphi` when the impedance is large. This delocalization reduces the sensitivity to flux noise, suppressing the curvature :math:`\partial^2 E_{01}/\partial \varphi_{\mathrm{ext}}^2`. The protected regime is bounded by :math:`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 :py:meth:`~HybridSuperQubits.ferbo.Ferbo._berry_contribution` and can be toggled via the ``include_berry`` flag in :py:meth:`~HybridSuperQubits.ferbo.Ferbo.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 :math:`H_{\mathrm{WL}}(\hat{\varphi} - \varphi_{\mathrm{ext}})`. - ``"EL"``: External flux appears in the inductive term :math:`\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 :math:`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 :math:`|n\rangle` where :math:`n` runs from :math:`-n_{\mathrm{cut}}` to :math:`n_{\mathrm{cut}}`. Fluxonium --------- The fluxonium qubit is an LC circuit shunted by a Josephson junction: .. math:: \hat{H} = 4E_C \hat{n}^2 + \tfrac{1}{2}E_L \hat{\varphi}^2 - E_J \cos(\hat{\varphi} - \varphi_{\mathrm{ext}}) The large inductance :math:`E_L` lifts the ground-state degeneracy of the transmon, producing a highly anharmonic spectrum. In the heavy regime (:math:`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 :math:`\tau`, which is controlled by a gate voltage: .. math:: E_{\mathrm{ABS}}(\varphi) = -\Delta \sqrt{1 - \tau \sin^2(\varphi/2)} 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: .. math:: \hat{H} = 4E_C \hat{n}^2 + \tfrac{1}{2}E_L \hat{\varphi}^2 + \sum_i V_{\mathrm{ABS}}^{(i)}(\hat{\varphi} - \varphi_{\mathrm{ext}}) where the sum runs over the transmission channels with individual transmissions :math:`\tau_i`. Josephson junction array (JJA) ------------------------------ The JJA module models an array of :math:`N` Josephson junctions as a metamaterial. It computes: - Plasma frequency :math:`\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: .. math:: \hat{H} = \hbar \omega \left(\hat{a}^\dagger \hat{a} + \tfrac{1}{2}\right) 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** (:math:`T_1`): dielectric loss in the capacitor, parameterized by quality factor :math:`Q_{\mathrm{cap}}`. - **Inductive losses** (:math:`T_1`): resistive losses in the inductor, parameterized by :math:`Q_{\mathrm{ind}}`. - **Flux bias line coupling** (:math:`T_1`): coupling to an external flux line with mutual inductance :math:`M` and impedance :math:`Z`. - **1/f flux noise** (:math:`T_\varphi`): dephasing from low-frequency flux fluctuations with amplitude :math:`A_\Phi`. These are computed from Fermi's golden rule using the matrix elements of the relevant noise operators between eigenstates.