This paper presents a method for enumerating all encoding operators in the Clifford group for a given stabilizer. Furthermore, we classify encoding operators into the equivalence classes such that EDPs (Entanglement Distillation Protocol) constructed from encoding operators in the same equivalence class have the same performance. By this classification, for a given parameter, the number of candidates for good EDPs is significantly reduced. As a result, we find the best EDP among EDPs constructed from [[4,2]] stabilizer codes. This EDP has a better performance than previously known EDPs over wide range of fidelity.
We show a class of encoding operators for a quantum stabilizer code such that the number of gates of the inverse encoding quantum circuit is O(nk), where k qubits are encoded into n qubits by the stabilizer code. Then, we demonstrate that the performances of two-way EDPs are improved by the different choice of inverse encoding operators from the class of inverse encoding operators proposed in this paper. As an example, we show the best EDP among currently known two-way EDPs for the wide range of fidelity.
We introduce an entanglement distillation protocol that utilizes an arbitrary
[[n,k,d]] stabilizer code to convert
n raw copies of an
N-qubit Greenberger-Horne-Zeilinger (GHZ) state into
k logical copies in the presence of Pauli noise. Our explicit formulation of the scheme on a star network topology is efficiently scalable to arbitrary
N, requires only local operations, and minimizes the impact of errors on idle physical qubits, enabling the distribution of high-fidelity entangled logical states between any number of quantum processors. We report the results of numerical experiments using the 5-qubit code, toric code, and
[[144,12,12]] bivariate bicyle code to protect against independent single-qubit depolarizing noise on all qubits. Our encoding scheme improves the final GHZ state fidelity relative to a single bare GHZ state sent over the same noisy channel for depolarizing rates up to or above
6% for all codes tested. Additionally, we use the stabilizer formalism to prove that our protocol can be applied to any
N-qubit Calderbank-Shor-Steane (CSS) stabilizer state, from which the GHZ state emerges as a special case. This result explicitly establishes the working principle of previous stabilizer-based Bell pair and GHZ state distillation schemes, and generalizes such protocols beyond the traditionally considered states. We also show that some stabilizer codes can distill any stabilizer state with our protocol.