This article is a tutorial on the quantum treatment of superconducting electrical circuits. It is intended for new researchers with limited or no experience with the field, but shouldbe accessible to anyone with a bachelor’s degree in physics or similar. The tutorial has three parts. The first part introduces the basic methods used in quantum circuit analysis, starting from a circuit diagram and ending with a quantized Hamiltonian truncated to the lowest levels. The second part introduces more advanced methods supplementing the methods presented in the first part. The third part is a collection of worked examples of superconducting circuits. Besides the examples in the third part, the two first parts also includes examples in parallel with the introduction of the methods.
We implement a linear Heisenberg spin-1/2 chain with XXZ couplings, which in it self can be used as an analog quantum simulator, using superconducting circuits. Depending on the circuitthe spin chain can have arbitrary length. For a specific length of four qubits we show that the circuit can be used to implement a quantum spin transistor following the protocol proposed in Nature Communication 5 13070 (2016). We do this by finding experimentally realistic parameters for the circuit and proposing a chip design. The quantum transistor works similarly to its classical analogue allowing transfer or blockage depending on the state of the two gate qubits, but opens up a variety of possibilities when quantum mechanical superpositions are considered. The transistor is simulated under realistic decoherence and it is shown that it allows high-fidelity transfer when open, while it allows no transfer when closed. The main effect of the decoherence is faster leakage from the transistor. The transistor is also considered when it is in an superposition of open and closed. We obtain transition times less than 200ns, and rule out leakage to higher excited states in the superconducting circuit design. Finally, we discuss further spin models which can be obtained be altering the circuit in different ways.