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).

Figure 1

Measurement-based implementation of the quantum operation $\mathcal{O}$ mapping $n=3$ qubits to $m=1$ qubit via its Jamiolkowksi state $E_{\mathcal{O}}$. The input qubits of the Jamiolkowski state are red, the output qubit is blue, and the ellipses indicate Bell measurements.

Figure 2

Construction and usage of the resource state for the concatenated quantum task entanglement purification at a logical level. The first figure illustrates the concatenation of several different quantum tasks. In this example, the quantum tasks decoding in the three-qubit bit-flip code $\left(T_1\right)$, one round of the entanglement purification protocol [18] $\left(T_2\right)$, and encoding in the three-qubit bit-flip code $\left(T_3\right)$ are concatenated. On the right-hand side, an LC-equivalent graph state for performing all three quantum tasks in one go is shown. This resource state implements one round of entanglement purification at a logical level, i.e., without decoding. The second figure depicts the usage of the resource state in a measurement-based implementation of the concatenated quantum task entanglement purification at a logical level.

Figure 3

Graphical illustration of the construction of the resource state for the concatenated quantum tasks $O^{\prime \otimes m}\circ O$. The ellipses indicate Bell measurements.

Figure 4

Illustration of the concatenation of single-qubit output with single-qubit input quantum tasks via a Bell measurement. Here, the single-qubit output task $O$ is concatenated with the single-qubit input task $O^{\prime }$, e.g., $O$ might be a decoding procedure in a quantum error correction and $O^{\prime }$ might be an encoding procedure into another quantum error correction code.

Figure 5

Measurement-based implementation of two rounds of the DEJMPS entanglement distillation protocol. The protocol operates on four input pairs and produces one output pair probabilistically.

Figure 6

Constructing the resource state for three rounds of entanglement purification from the resource state for one purification round.

Figure 7

Constructing the resource state for five rounds of entanglement purification from the resource state for three purification rounds.

Figure 8

Graph-state representation of resource states for one, two, and three rounds of entanglement purification using the DEJMPS protocol. The red circle indicates a $R_z(-\pi /2)$ rotation.

Figure 9

Quantum repeaters establish long-distance quantum communication by combining entanglement purification and entanglement swapping.

Figure 10

Resource state of a $[m_1,m_2]$-generalized Shor code. The red circles indicate that a Hadamard rotation needs to be applied at the end. The red vertex corresponds to the input qubit, the blue vertices to the output qubits.

Figure 11

Graph-state representation (up to local Clifford operations) for a code switcher between the three-qubit phase-flip code and the five-qubit cluster-ring code. Colors correspond to input and output qubits, respectively.

Figure 12

LC-equivalent graph state for syndrome readout for the three-qubit bit-flip code. The red vertices correspond to the input qubits whereas the blue vertices to the output qubits.

Figure 13

Construction of a resource state necessary for $n$ rounds of entanglement purification at a logical level.

Figure 14

The figure shows an LC-equivalent graph state for entanglement purification at a logical level, where the three-qubit bit-flip code and one round of entanglement purification using the DEJMPS protocol are considered. Input qubits are red and output qubits blue.

Figure 15

The figure shows LC-equivalent graph states for entanglement purification at a logical level for one (left) and two (right) purification rounds using a decoherence-free subspace encoding [62], i.e., the encoding $|0_L\rangle =|01\rangle$ and $|1_L\rangle =|10\rangle$. Input qubits are red and output qubits are blue.

Figure 16

Resource state of the five-qubit cluster-ring code. The red vertex is the input qubit.

Figure 17

Resource state of a $[m_1,m_2]$-generalized Shor code. The red circles indicate that a Hadamard rotation needs to be applied.

Figure 18

The resource state for the concatenated Shor code is local Clifford equivalent to a two-colorable graph state. The qubit on the very left is the input qubit. The dashed lines connect to other copies of the center block with 27 qubits.