Eigenenergies(first three levels,) of the qubit Hamiltonian as a function of the effective offset charge for different ratios. Energies are given in units of the transition energy, evaluated at the degeneracy point. The zero point of energy is chosen as the bottom of the level. The charge qubit (small, top) is normally operated at the "sweet spot" where fluctuations cause less energy shift and the anharmonicity is maximal. Transmon (large, bottom) energy levels are insensitive to fluctuations but the anharmonicity is reduced.
The transmon achieves its reduced sensitivity to charge noise by significantly increasing the ratio of theJosephson energy to the charging energy. This is accomplished through the use of a large shunting capacitor. The result is energy level spacings that are approximately independent of offset charge. Planar on-chip transmon qubits haveT1 coherence times approximately 30 μs to 40 μs.[5] Recent work has shown significantly improvedT1 times as long as 95 μs by replacing the superconductingtransmission line cavity with a three-dimensional superconducting cavity,[6][7] and by replacingniobium withtantalum in the transmon device,T1 is further improved up to 0.3 ms.[8] These results demonstrate that previousT1 times were not limited byJosephson junction losses. Understanding the fundamental limits on the coherence time insuperconducting qubits such as the transmon is an active area of research.
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 ratio. Here is theJosephson energy of the junction, and is the charging energy inversely proportional to the total capacitance of the qubit circuit. Transmons typically have (while for typical Cooper-pair-box qubits), which is achieved by shunting the Josephson junction with an additional largecapacitor.
The benefit of increasing the ratio is the insensitivity to charge noise—the energy levels become independent of the offset charge across the junction; thus thedephasing time of the qubit is prolonged. The disadvantage is the reduced anharmonicity, where is the energy difference betweeneigenstates and. 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 ofwith respect to tend to decrease exponentially with, the anharmonicity only has a weaker, algebraic dependence on as. 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 fromcircuit quantum electrodynamics also applicable toother superconducting qubits. Coupling to the resonators is done by placing a capacitor between the qubit and the resonator, at a point where the resonatorelectromagnetic field is greatest. For example, inIBM 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 similartransmission line resonators coupled to every transmon, becoming a part of the name. However, charge qubits operated at a similar regime, coupled to different kinds of microwave cavities are referred to as transmons as well.
Transmons are the default qubit in most large scale quantum processors, including Google'sWillow processor, a chip with 105 physical transmon qubits.[10] Other companies that use transmon qubits includeIBM,Rigetti, andIQM.
While most quantum computation systems are qubit-based, an alternative method is to use harmonic oscillator modes, or bosonic modes, as the logical subspace.[11][12] In such systems, transmons are used as ancillas for universal control of the cavity's bosonic modes.[13] This allows for physical realizations of oscillator-based error correction codes.[11] These codes are known as 'bosonic codes'.
Transmons have been explored for use asd-dimensionalqudits via the additional energy levels that naturally occur above thequbit subspace (the lowest two states). For example, the lowestthree levels can be used to make a transmonqutrit; in the early 2020s, researchers have reported realizations of single-qutritquantum gates on transmons[14][15] as well as two-qutritentangling gates.[16] Entangling gates on transmons have also been explored theoretically and in simulations for the general case of qudits of arbitraryd.[17]
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