Quantum Error Correction Without Encoding via the Circulant Structure of Pauli Noise and the Fast Fourier Transform

Abstract: This work introduces distribution error correction (DEC) theory, where we correct the output distribution of a quantum computer to the ideal distribution exactly. If the 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. We prove that this matrix has a recursive 2 by 2 block circulant structure. Thus, the noisy output distribution can be corrected to the ideal output distribution via a Fast Fourier Transform. Therefore, DEC for Pauli error channels does not require encoding of the logical qubits into more physical qubits and it avoids the encoding overhead of standard quantum error correction codes. Moreover, we introduce a DEC implementation that requires executions of only 2 logical circuits. 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. The correction process 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%.