couple qubits is essential for implementing 2-qubit gates. Coupling two qubits can be achieved by connecting both to an intermediate electrical coupling circuit. The circuit may be either a fixed element (such as a capacitor) or be controllable (like the DC-SQUID). In the first case, decoupling qubits during the time the gate is switched off is achieved by tuning qubits out of resonance one from another, making the energy gaps between their computational states different. This approach is inherently limited to nearest-neighbor coupling since as physical electrical circuit must be laid out between connected qubits. Notably, D-Wave Systems' nearest-neighbor coupling achieves a highly connected unit cell of 8 qubits in Chimera graph configuration. Quantum algorithms typically require coupling between arbitrary qubits. Consequently, multiple swap operations are necessary, limiting the length of quantum computation possible before processor decoherence. Quantum bus Another method of coupling two or more qubits is by way of a quantum bus, by pairing qubits to this intermediate. A quantum bus is often implemented as a microwave cavity modeled by a quantum harmonic oscillator. Coupled qubits may be brought in and out of resonance with the bus and with each other, eliminating the nearest-neighbor limitation. Formalism describing coupling is cavity quantum electrodynamics. In cavity quantum electrodynamics, qubits are analogous to atoms interacting with an optical photon cavity with a difference of GHz (rather than the THz regime of electromagnetic radiation). Resonant excitation exchange among these artificial atoms is potentially useful for direct implementation of multi-qubit gates. Following the dark state manifold, the Khazali-Mølmer scheme performs complex multi-qubit operations in a single step, providing a substantial shortcut to the conventional circuit model. Cross resonant gate One popular gating mechanism uses two qubits and a bus, each tuned to different energy level separations. Applying microwave excitation to the first qubit, with a frequency resonant with the second qubit, causes a rotation of the second qubit. Rotation direction depends on the state of the first qubit, allowing a controlled phase gate construction. Following the notation of, the drive Hamiltonian describing the excited system through the first qubit driving line is formally written , where is the shape of the microwave pulse in time, is resonance frequency of the second qubit, are the Pauli matrices, is the coupling coefficient between the two qubits via the resonator, is qubit detuning, is stray (unwanted) coupling between qubits, and is Planck's constant divided by .