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# Superconducting quantum computing

Superconducting quantum computing is a promising implementation of quantum information that involves nanofabricated superconducting electrodes coupled through Josephson junctions. As in a superconducting electrode, the phase and the charge are conjugate variables, there exists three families of superconducting qubits, depending if the charge, the phase or neither of the two are good quantum numbers. This refers respectively to charge qubits, flux qubits, and hybrid qubits.

## Theory

Unlike other physical implementations of a qubit which involve truly two-level systems, such as nuclear spin and photon polarization, the integrated quantum circuit involved in a superconducting qubit is a many-level system, of which only the first two levels are used as the computational basis. A basic requirement for such an implementation is that the energy levels are not uniformly spaced, so that photons of a particular frequency which cause transition between the 0 and 1 levels do not cause transitions from the first level to the higher levels as well. For electronic signals to be carried from one part of the circuit to another without energy loss and hence decoherence, the system also needs to be non-dissipative, i.e., the metallic parts involved should have zero resistance. These circuits need to be operated at low temperatures so that firstly superconductivity is realized, and secondly thermal fluctuations do not cause transitions between energy levels.

### Quantum LC Circuit

The simplest non-dissipative quantum circuit consists simply of an inductor $L$ and capacitor $C$, with the metallic wires connecting them being superconducting. Here the flux $\Phi$ through the inductor and the charge $Q$ in the capacitor are canonically conjugate variables which obey the commutation relation

$[\Phi, Q] = i\hbar$

The Hamiltonian associated with this system is

$H = \frac{\Phi^2}{2L} + \frac{Q^2}{2C}$

which gives rise to the energy levels

$E = \hbar \omega_0 (n + \frac{1}{2})$

where $\omega_0 = \frac{1}{\sqrt{LC}}$.