## Andreev qubit

We have accomplished a quantitative theory of the quantum dynamics of Andreev level qubits. This is a new type of super-conducting flux qubit where the switching between the two persistent current states in a SQUID is achieved by employing a true microscopic system formed by the two-level Andreev bound states in a super-conducting atomic-size quantum point contact (QPC) embedded in the SQUID (see the figure above). In this Andreev level qubit (ALQ), the quantum information is stored in the microscopic quantum system; the Andreev bound states, similar to non-super-conducting solid-state qubits like localized spins on impurities or quantum dots. Read-out of the Andreev level qubit is achieved by monitoring the macroscopic persistent current or the induced flux in the SQUID, similar to the conventional flux qubits.

The essential difference between the ALQ and conventional flux qubits based on the phenomenon of macroscopic quantum coherence (MQC) is that its operation requires almost transparent Josephson junctions, while MQC qubits employ tunnel junctions. Furthermore, the ALQ dynamics involves two interacting, bosonic and fermionic, quantum fields: the super-conducting phase, and the two-level Andreev system (in MQC qubits only super-conducting phase is important). This difference made it necessary to reconsider and substantially extend the "orthodox" MQC theory. This is done by incorporating exact boundary condition for the junction in the Feynman path integral describing the qubit dynamics.

During time evolution of the Andreev levels, the current through the QPC changes, and the QPC operates as a quantum switch, which controls the direction of the circulating current in the SQUID. To maintain the current switching, the intrinsic dynamics of the current in the SQUID must be sufficiently fast. Fidelity of read out of the Andreev level state by performing quantum measurement of the circulating current, or of the corresponding induced flux, requires the Andreev energy to be small compared to the plasma frequency of electromagnetic fluctuations. To prevent dissipation, both the Andreev level energy and plasma frequency must be small compared to the super-conducting energy gap. These inequalities determine the window of the circuit parameters where the ALQ operates. They are consistent with typical parameters of the flux qubit circuits. Strong interaction of the Andreev levels in QPC with plasma oscillations gives rise to effective suppression of generic contact reflectivity; hence, the Andreev level energy and therefore the frequency of the qubit operation can be controlled by varying the circuit parameters.

The interaction between ALQs is achieved via inductive coupling of the qubit SQUIDs, providing an effective qubit Hamiltonian including qubit-qubit interaction. To investigate the relaxation and dephasing of ALQ, we have also considered the interaction of Andreev levels with phonon modes in the superconductor, and derived the corresponding kinetic equation. Evaluation of the relaxation and dephasing rates have shown that the phonon mechanism does not impose any further limitations on the qubit operation compared to the common for flux qubits dephasing mechanisms such as external flux fluctuations, and qubit radiation.

ALQ theory is a general theory for the quantum behaviour of highly transparent Josephson junctions. Several aspects of the ALQ are important for all flux qubits. This concerns first of all the mechanism of qubit interaction with the circuit plasma modes. Another important aspect is the quality of the tunnel Josephson junctions employed for the MQC qubits: the presence of just few transparent conducting modes in the tunnel barrier may considerably affect the qubit behaviour, introducing the ALQ features, and large amount of such modes may severely dephase the qubit. The results of the work are summarized and detailed in our recent publications.

The essential difference between the ALQ and conventional flux qubits based on the phenomenon of macroscopic quantum coherence (MQC) is that its operation requires almost transparent Josephson junctions, while MQC qubits employ tunnel junctions. Furthermore, the ALQ dynamics involves two interacting, bosonic and fermionic, quantum fields: the super-conducting phase, and the two-level Andreev system (in MQC qubits only super-conducting phase is important). This difference made it necessary to reconsider and substantially extend the "orthodox" MQC theory. This is done by incorporating exact boundary condition for the junction in the Feynman path integral describing the qubit dynamics.

During time evolution of the Andreev levels, the current through the QPC changes, and the QPC operates as a quantum switch, which controls the direction of the circulating current in the SQUID. To maintain the current switching, the intrinsic dynamics of the current in the SQUID must be sufficiently fast. Fidelity of read out of the Andreev level state by performing quantum measurement of the circulating current, or of the corresponding induced flux, requires the Andreev energy to be small compared to the plasma frequency of electromagnetic fluctuations. To prevent dissipation, both the Andreev level energy and plasma frequency must be small compared to the super-conducting energy gap. These inequalities determine the window of the circuit parameters where the ALQ operates. They are consistent with typical parameters of the flux qubit circuits. Strong interaction of the Andreev levels in QPC with plasma oscillations gives rise to effective suppression of generic contact reflectivity; hence, the Andreev level energy and therefore the frequency of the qubit operation can be controlled by varying the circuit parameters.

The interaction between ALQs is achieved via inductive coupling of the qubit SQUIDs, providing an effective qubit Hamiltonian including qubit-qubit interaction. To investigate the relaxation and dephasing of ALQ, we have also considered the interaction of Andreev levels with phonon modes in the superconductor, and derived the corresponding kinetic equation. Evaluation of the relaxation and dephasing rates have shown that the phonon mechanism does not impose any further limitations on the qubit operation compared to the common for flux qubits dephasing mechanisms such as external flux fluctuations, and qubit radiation.

ALQ theory is a general theory for the quantum behaviour of highly transparent Josephson junctions. Several aspects of the ALQ are important for all flux qubits. This concerns first of all the mechanism of qubit interaction with the circuit plasma modes. Another important aspect is the quality of the tunnel Josephson junctions employed for the MQC qubits: the presence of just few transparent conducting modes in the tunnel barrier may considerably affect the qubit behaviour, introducing the ALQ features, and large amount of such modes may severely dephase the qubit. The results of the work are summarized and detailed in our recent publications.