Transmon

Eigenenergies ${\displaystyle E_{m}}$(first three levels, ${\displaystyle m=0,1,2}$) of the qubit Hamiltonian as a function of the effective offset charge ${\displaystyle n_{g}}$ for different ratios ${\displaystyle E_{J}/E_{c}}$. Energies are given in units of the transition energy ${\displaystyle E_{01}}$, evaluated at the degeneracy point ${\displaystyle n_{g}=0.5}$. The zero point of energy is chosen as the bottom of the ${\displaystyle m=0}$ level. The charge qubit (small ${\displaystyle E_{J}/E_{c}}$, top) is normally operated at the ${\displaystyle n_{g}=0.5}$ "sweet spot", where fluctuations in  cause less energy shift, and the anharmonicity is maximal. Transmon (large ${\displaystyle E_{J}/E_{c}}$, bottom) energy levels are insensitive to  fluctuations, but the anharmonicity is reduced.

The transmon design is similar to the first design of the charge qubit[9] known as a "Cooper-pair box"; both are described by the same Hamiltonian, with the only difference being the ${\displaystyle E_{\rm {J}}/E_{\rm {C}}}$ ratio. Here ${\displaystyle E_{\rm {J}}}$ is the Josephson energy of the junction, and ${\displaystyle E_{\rm {C}}}$ is the charging energy inversely proportional to the total capacitance of the qubit circuit. Transmons typically have ${\displaystyle E_{\mathrm {J} }/E_{\mathrm {C} }\gg 1}$ (while ${\displaystyle E_{\mathrm {J} }/E_{\mathrm {C} }\lesssim 1}$ for typical Cooper-pair-box qubits), which is achieved by shunting the Josephson junction with an additional large capacitor.

The benefit of increasing the ${\displaystyle E_{\rm {J}}/E_{\rm {C}}}$ ratio is the insensitivity to charge noise—the energy levels become independent of the offset charge ${\displaystyle n_{g}}$ across the junction; thus the dephasing time of the qubit is prolonged. The disadvantage is the reduced anharmonicity ${\displaystyle \alpha =(E_{21}-E_{10})/E_{10}}$, where ${\displaystyle E_{ij}}$ is the energy difference between eigenstates ${\displaystyle |i\rangle }$ and ${\displaystyle |j\rangle }$. Reduced anharmonicity complicates the device operation as a two level system, e.g. exciting the device from the ground state to the first excited state by a resonant pulse also populates the higher excited state. This complication is overcome by complex microwave pulse design, that takes into account the higher energy levels, and prohibits their excitation by destructive interference. Also, while the variation of ${\displaystyle E_{10}}$with respect to ${\displaystyle n_{g}}$ tend to decrease exponentially with ${\displaystyle E_{\mathrm {J} }/E_{\mathrm {C} }}$, the anharmonicity only has a weaker, algebraic dependence on ${\displaystyle E_{\mathrm {J} }/E_{\mathrm {C} }}$ as ${\displaystyle \sim (E_{\mathrm {J} }/E_{\mathrm {C} })^{-1/2}}$. The significant gain in the coherence time outweigh the decrease in the anharmonicity for controlling the states with high fidelity.

Measurement, control and coupling of transmons is performed by means of microwave resonators with techniques from circuit quantum electrodynamics also applicable to other superconducting qubits. Coupling to the resonators is done by placing a capacitor between the qubit and the resonator, at a point where the resonator electromagnetic field is greatest. For example, in IBM Quantum Experience devices, the resonators are implemented with "quarter wave" coplanar waveguides with maximal field at the signal-ground short at the waveguide end; thus every IBM transmon qubit has a long resonator "tail". The initial proposal included similar transmission line resonators coupled to every transmon, becoming a part of the name. However, charge qubits operated at a similar ${\displaystyle E_{\rm {J}}/E_{\rm {C}}}$ regime, coupled to different kinds of microwave cavities are referred to as transmons as well.

• Charge qubit
• Anharmonicity
• Circuit quantum electrodynamics (cQED)
• Dilution refrigerator
• List of quantum processors
• Quantum harmonic oscillator
• Superconducting quantum computing