Generalizing Pauli Checks for Qudit-based Quantum Error Detection and Mitigation

Abstract

Pauli Check Sandwiching (PCS) is a quantum error detection (QED) technique that protects a quantum circuit by utilizing a pair of controlled Pauli operators, or checks, and detecting errors that anti-commute with the checks. Further, PCS can be used for quantum error mitigation (QEM) via post-selection based on the Pauli check syndrome values. Currently, PCS is leveraged in the qubit space. In this paper, we introduce a generalized approach for applying PCS-based QED and QEM to quantum information of arbitrary dimension in the Hilbert space. Each pair of these extended checks consists of a sequence of gates in the Heisenberg-Weyl operator set that extend Pauli operators into the qudit space. These qudit checks use at least one ancilla qudit to detect qudit errors that do not commute with the unitary selected for the check. We show that our proposed methods for qudit QED can detect errors of arbitrary dimensions. More specifically, we prove that an arbitrary Heisenberg-Weyl error maps deterministically to a unique ancilla readout, and further, post-selecting on the $|0 \rangle$ readout guarantees unit fidelity in the noiseless check limit. We validate these findings numerically across dimensions $d=2$ through $d=9$, achieving error-mitigated fidelities above $97.5\% $ under realistic depolarizing error rates.