Quantum Distribution Error Mitigation via the Circulant Structure of Pauli Noise

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%.

💡 Research Summary

The paper introduces Distribution Error Mitigation (DEM), a method that directly corrects the output probability distribution of a quantum circuit rather than merely adjusting expectation values of observables. The authors start by assuming that the composite noise affecting a circuit can be modeled as a Pauli channel. Under this assumption, the ideal output distribution vector x and the noisy distribution vector z are linked by a stochastic matrix A such that z = A x. They prove that A has a recursive 2 × 2 block‑circulant structure, which is equivalent to an XOR convolution between a “noise vector” a (the first column of A) and the ideal distribution: z_i = ∑j a{i⊕j} x_j. Because XOR‑circulant matrices are diagonalized by the Walsh‑Hadamard transform (WHT), the linear system can be solved efficiently using Fast Walsh‑Hadamard Transform (FWHT) operations: x = IFWHT(FWHT(z) ÷ FWHT(a)). This reduces the computational cost from O(2^{3n}) for direct matrix inversion to O(2^{n} log 2^{n}).

The central practical challenge is obtaining the noise vector a. The authors propose a simple tomography scheme using a single “Noise Estimation Circuit” (NEC). The NEC is built from the payload circuit by replacing every SX (√X) gate with a plain X gate, thereby eliminating superposition creation while preserving the circuit’s topology. The NEC’s ideal output is a known computational basis state |k⟩, which can be classically determined. By sampling the NEC on hardware, one obtains a noisy distribution b that corresponds to column k of A. Using the symmetry property A_{i,j}=a_{i⊕j}, the full noise vector is reconstructed as a = P_x b, where P_x is the appropriate Pauli‑X string that maps column k to column 0. Consequently, only one additional logical circuit execution (besides the payload) is required to estimate a.

With a in hand, DEM proceeds in two steps: (1) run the payload circuit enough shots to estimate z, (2) apply FWHT to z, element‑wise divide by FWHT(a) (treating zero entries with a pseudo‑inverse rule), and finally apply the inverse FWHT to obtain the corrected distribution x̂. This process is linear in the number of shots and does not require any extra quantum resources beyond the two circuits.

To make DEM scalable for larger qubit numbers, the authors discuss two complementary techniques. The “compression” approach assumes that the support of the ideal distribution is a subset of the noisy support; one can then restrict the linear system to a K × K submatrix A_S (K ≪ 2^n) and solve it directly or with iterative matrix‑free solvers. The “binning” heuristic pads and reorders the support set to the nearest power of two, enabling the use of the same FWHT‑based inversion on a reduced index space. Both methods mitigate memory and runtime bottlenecks while preserving the essential XOR‑circulant structure.

Error analysis treats DEM as a form of measurement‑error mitigation, fitting into the standard model M_exp = Λ M_ideal + Δ, where Λ corresponds to the stochastic matrix A and Δ captures non‑classical residuals. The authors show that shot noise in estimating a and z propagates as O(1/√M) to the corrected expectations, and that the doubly‑stochastic nature of A ensures that probability normalization is maintained.

Experimental validation is performed on IBM quantum hardware using a variety of circuits: 20‑ and 30‑qubit GHZ state preparation, 5‑qubit Grover search (582 CZ gates), 6‑ and 10‑qubit quantum phase estimation, and 10‑ and 20‑qubit Dicke‑1 state preparation. Raw output fidelities range from 10 % to 23 % for the most challenging cases. After applying DEM, fidelities improve dramatically—most notably, the 30‑qubit GHZ state’s fidelity rises from 23.2 % to 97.7 %, and the 5‑qubit Grover circuit’s fidelity jumps from 10.2 % to 74.9 %. These results demonstrate that (i) Pauli‑noise biasing via randomized compiling is effective on current devices, (ii) the NEC provides a reliable estimate of a, and (iii) the FWHT‑based correction robustly restores the full output distribution.

In summary, the paper leverages the algebraic structure of Pauli channels to formulate a fast, low‑overhead error mitigation technique that operates at the distribution level. By requiring only two logical circuit executions and using classical FFT‑like transforms, DEM offers a practical path to improve the quality of NISQ‑era quantum computations, especially for tasks where the full output histogram is needed (e.g., sampling‑based algorithms, variational algorithms, and quantum chemistry). The method’s compatibility with existing mitigation frameworks and its demonstrated scalability make it a promising addition to the quantum error mitigation toolbox.