Construction of optimal resources for concatenated quantum protocols

We consider the explicit construction of resource states for measurement-based quantum information processing. We concentrate on special-purpose resource states that are capable to perform a certain operation or task, where we consider unitary Clifford circuits as well as non-trace preserving completely positive maps, more specifically probabilistic operations including Clifford operations and Pauli measurements. We concentrate on $1 \to m$ and $m \to 1$ operations, i.e. operations that map one input qubit to $m$ output qubits or vice versa. Examples of such operations include encoding and decoding in quantum error correction, entanglement purification or entanglement swapping. We provide a general framework to construct optimal resource states for complex tasks that are combinations of these elementary building blocks. All resource states only contain input and output qubits, and are hence of minimal size. We obtain a stabilizer description of the resulting resource states, which we also translate into a circuit pattern to experimentally generate these states. In particular, we derive recurrence relations at the level of stabilizers as key analytical tool to generate explicit (graph-) descriptions of families of resource states. This allows us to explicitly construct resource states for encoding, decoding and syndrome readout for concatenated quantum error correction codes, code switchers, multiple rounds of entanglement purification, quantum repeaters and combinations thereof (such as resource states for entanglement purification of encoded states).