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

TL;DR

The paper addresses the prohibitive encoding overhead of quantum error correction by introducing distribution error correction (DEC) for Pauli noise, which corrects the output distribution exactly without encoding. It proves that the assignment matrix $\tilde{A}$ linking ideal and noisy distributions is a recursive 2$\times$2 block circulant, enabling exact correction via the Fast Walsh-Hadamard transform and a single-column characterization. A practical DEC workflow uses a vanilla Noise Estimation Circuit (NEC) and randomized compiling to bias errors toward Pauli channels, requiring only two logical circuits, with hardware demonstrations showing substantial fidelity improvements across GHZ, Dicke, QPE, and Grover circuits (e.g., $97.7\%$ corrected fidelity for a 30-qubit GHZ). The work situates DEC relative to existing error mitigation techniques, discusses open problems (Pauli-bias robustness, NEC improvements, non-Clifford behavior), and outlines paths for integrating DEC with QECC.

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%.

Figures (8)

• Figure 1: The NEC is constructed by replacing SX gates in the payload circuit with X gates (outlined in red). The NEC does not generate a superposition by construction and its ideal output when evolving from the ground state is a standard basis state $\ket{k}$ that we can efficiently determine classically.
• Figure 2: 30-qubit GHZ state preparation on quantum hardware. The x-axis uses decimal representation of the basis states.
• Figure 3: 20-qubit GHZ state preparation circuit on quantum hardware.
• Figure 4: Dicke $\ket{D^{20}_1}$ state preparation on quantum hardware.
• Figure 5: Dicke $\ket{D^{10}_1}$ state preparation on quantum hardware.

Theorems & Definitions (10)

• Proposition 1
• Proposition 2
• Corollary 1
• proof
• Corollary 2
• proof
• Theorem 1
• proof
• proof
• proof