Guedong Park, Jinzhao Sun, Hyunseok Jeong (Mar 18 2025).

Abstract: Hypergraph states are important magic resources for realizing universal quantum computation and diverse non-local physical phenomena. However, noise detection for such states is challenging due to their large dimension and entanglement. This work proposes an efficient Clifford circuit-based scheme for tailoring and detecting noise in third-ordered hypergraph states generated by CCZ, CZ, and Z gates. The core part of our scheme is converting the noisy input state into a diagonal form and obtaining the convolution equation of noise rate via Clifford circuits. The depth of the Clifford circuit can be reduced to a constant, depending on the structure of the hypergraph state. After that, we decode it using the fast Hadamard-Walsh transform or some approximation method. The approximation with respect to the $l_2$-norm can be done efficiently by the number of qubits while keeping sufficient precision. Furthermore, the sparse noise assumption, which frequently holds in current experimental setups, enables $l_1$ approximation. Compared with state verification methods, our method allows us to attain richer information on noise rates and apply various noise-adapted error correction and mitigation methods. Moreover, it bridges the connection between the convolution equation's nonlinearity and the Clifford hierarchy of the hypergraph state inputs. Our results provide a deeper understanding of the nature of highly entangled systems and drive the interests of the research venues concerning magic state implementation.

Arxiv: https://arxiv.org/abs/2503.12870