Quantum Distribution Error Mitigation via the Circulant Structure of Pauli Noise

Abstract

This work introduces distribution error mitigation (DEM), which mitigates the error in the output distribution of a quantum circuit. We provide a rigorous theoretical foundation. If the composite noise affecting the circuit is a Pauli channel, the ideal output distribution and noisy distribution in the standard basis are related by a stochastic matrix. This system is described by a XOR convolution (the matrix is recursive 2 by 2 block circulant) between a noise vector and the ideal distribution. The noisy output distribution can be corrected to the ideal output distribution via a Fast Walsh-Hadamard Transform. We introduce a tomography method to approximate the noise vector, which requires sampling of only one logical circuit. The quantum overhead of DEM requires sampling of only two logical circuits. We provide techniques to scale the application of DEM efficiently. Accuracy bounds are provided. The approach is tested with quantum hardware executions consisting of 20-qubit and 30-qubit GHZ state preparation, 5-qubit Grover, 6-qubit and 10-qubit quantum phase estimation, and 10-qubit and 20-qubit Dicke state preparation circuits. DEM dramatically improves the accuracies of the output distributions for all demonstrations. For 30-qubit GHZ state preparation, a corrected distribution fidelity of 97.7% is achieved from an initial raw fidelity of 23.2%.