occupies a particular state, there is a non-zero probability of measuring a specific value only where the depicted wave function oscillates. All three rows are essentially different presentations of the same physical system. Single qubits The GHz energy gap between energy levels of a superconducting qubit is designed to be compatible with available electronic equipment, due to the terahertz gap (lack of equipment in the higher frequency band). The superconductor energy gap implies a top limit of operation below ~1THz beyond which Cooper pairs break, so energy level separation cannot be too high. On the other hand, energy level separation cannot be too small due to cooling considerations: a temperature of 1 K implies energy fluctuations of 20 GHz. Temperatures of tens of millikelvins are achieved in dilution refrigerators and allow qubit operation at a ~5 GHz energy level separation. Qubit energy level separation is frequently adjusted by controlling a dedicated bias current line, providing a "knob" to fine tune the qubit parameters. Single qubit gates A single qubit gate is achieved by rotation in the Bloch sphere. Rotations between different energy levels of a single qubit are induced by microwave pulses sent to an antenna or transmission line coupled to the qubit with a frequency resonant with the energy separation between levels. Individual qubits may be addressed by a dedicated transmission line or by a shared one if the other qubits are off resonance. The axis of rotation is set by quadrature amplitude modulation of microwave pulse, while pulse length determines the angle of rotation. More formally (following the notation of ) for a driving signal of frequency , a driven qubit Hamiltonian in a rotating wave approximation is , where is the qubit resonance and are Pauli matrices. To implement a rotation about the axis, one can set and apply a microwave pulse at frequency for time . The resulting transformation is . This is exactly the rotation operator by angle about the axis in the Bloch sphere. A rotation about the axis can be implemented in a similar way. Showing the two rotation operators is sufficient for satisfying universality as every single qubit unitary operator may be presented as (up to a global phase which is physically inconsequential) by a procedure known as the decomposition. Setting results in the transformation up to the global phase and is known as the NOT gate. Coupling qubits The ability to