Adaptive Estimation of Drifting Noise in Quantum Error Correction

TL;DR

The paper addresses drifting noise in quantum error correction by learning time-dependent error rates directly from syndrome histories. It introduces a rigorous window-based framework, including sliding-window, iterative sliding-window, and relative-window estimators, to capture multiple drift frequencies and fast drifts, with a theoretical connection to low-pass filtering via Dirichlet kernels and an explicit optimal window guidance. Through extensive simulations on repetition and surface-code memories under phenomenological and circuit-level noise, the authors demonstrate accurate estimation of detector-edge probabilities and show decoding performance that closely tracks ground truth while outperforming static error models. The methods require no extra experimental cost beyond syndrome collection and lay the groundwork for drift-aware, adaptive decoding that can be integrated into existing QEC workflows and extended to more complex noise models.

Abstract

Advancing quantum information processors and building fault-tolerant architectures rely on the ability to accurately characterize the noise sources and suppress their impact on quantum devices. In practice, noise often drifts over time, whereas conventional noise characterization and decoding methods typically assume stationarity or provide only a time-average behavior of the noise. This treatment can result in suboptimal decoding performance. In this work, we present a rigorous analytical framework to capture time-dependent Pauli noise, by exploiting the syndrome statistics of quantum error correction experiments. We propose a sliding-window estimation method which allows us to recover the frequency components of the noise, by using optimal window sizes that we derive analytically. We prove the noise-filtering behavior of sliding windows, linking window size to spectral cutoff frequencies, and provide an iterative algorithm that captures multiple drift frequencies. We further introduce an overlapping window algorithm that enables us to capture rapid multi-frequency noise drifts in a single-pass fashion. Simulations for both phenomenological and circuit-level noise models validate our framework, demonstrating robust tracking of multi-frequency drift. The logical error rate obtained from our estimated models consistently align with the ground-truth logical error rate, and we find suppression of logical errors compared to static error models. Our window-based estimation methods and adaptive decoding offer new insights into noise spectroscopy and decoder optimization under drift using only syndrome data.

Figures (12)

• Figure 1: Schematic overview of our methods for estimating time-varying noise from syndrome data of memory QEC experiments. (a) Sliding window estimation: A window of fixed size slides along the temporal axis of the syndrome data. Time-varying error rates, $p_{ij}^{\text{est}}(t)$, are estimated from the syndromes within the window at each position. (b) Iterative sliding window estimation: The sliding window estimation process iterates over a range of window sizes. Starting with an initial window, the method sequentially uses the output of one iteration as a prior for the next, refining the estimate with each step. This method aggregates information across multiple windows to estimate the error rate signal $p_{ij}^{\text{est}}(t)$. (c) Relative window estimation: Two overlapping windows of sizes $W$ and $W+1$ slide simultaneously along the temporal axis. The differential information between the two windows is analyzed to estimate the time-varying error rates.
• Figure 2: Impact of window size on noise spectrum estimation. (a) Ratio of the estimated noise spectrum, $P_{\text{est}}(\omega_m)$, to the true noise spectrum, $P(\omega_m)$, across different frequencies, demonstrating the damping introduced by various window sizes $W$. (b) Log-log plot of the phase spectrum, $\phi(\omega_m)$, against frequency, $\omega_m$, for different window sizes. Window sizes are defined as fractions of the total syndrome extraction cycles, $N$.
• Figure 3: Estimating DEM error rates using the sliding window approach, for a distance-3 repetition code under phenomenological noise model. Data and ancilla qubits experience depolarizing noise per syndrome extraction cycle with parameters $g_0=0.1$, $g_1=0.05$, and frequency $\omega_1 = 2\pi/(10^4 \Delta t$). (a) Estimated DEM error rate of bulk edge averaged over all qubits, as a function of the syndrome extraction cycles, for a window size of $W=1500$. The red curve shows the ground truth error rate, and the blue curve the estimated error rate. (b) Same as in (a) for a window size $W=5000$. (c) Same as in (a) for a window size $W=12000$. For the estimated signal, we repeat the simulation five times to plot the average behavior and display the standard deviation with the error bars.
• Figure 4: Estimating X-DEM error rates using the sliding window approach, for a distance-3 rotated surface code under phenomenological noise model. Data and ancilla qubits experience depolarizing noise per syndrome extraction cycle with parameters $g_0=0.1$, $g_1=0.05$, and and frequency $\omega_1 = 2\pi/(10^4 \Delta t$). (a) Estimated X-DEM error rate of bulk edge averaged over all qubits, as a function of syndrome extraction cycle, for a window size of $W=1500$. The red curve shows the ground truth error rate, and the blue curve the estimated error rate. (b) Same as in (a) for a window size $W=5000$. (c) Same as in (a) for a window size $W=12000$. For the estimated signal, we repeat the simulation five times to plot the average behavior and display the standard deviation with the error bars.
• Figure 5: Low-pass and band-pass filtering behavior of sliding window estimation, for a $d=3$ repetition code under phenomenological noise. All data and ancilla qubits experience uniform depolarizing per syndrome extraction cycle, with time varying error rate $g(t)$ defined by $g_0 = 0.1$, $g_1 = 0.05$, $g_2 = 0.04$, and frequencies $\omega_1 = 2\pi/(10^4 \Delta t$), $\omega_2 = 2\pi/(2\times 10^3\Delta t)$. (a) Estimated DEM error rate of bulk edge (light blue), averaged over all qubits, as a function of syndrome extraction cycle, for a window size of $W=2200$. Only the low-frequency component (yellow) $\omega_1 = 2\pi/(10^4 \Delta t$) is captured, meaning that the sliding window acts as a low-pass filter. The red line shows the full ground-truth signal consisting of two frequencies and the dark blue line shows the more rapidly drifting component. (b) Same as in (a) for a window size of $W=500$, which is able to capture the full time signal. (c) DEM error rate obtained by subtracting the signals of (a) and (b) to isolate the component drifting with frequency $\omega_2 = 2\pi/(2\times 10^3\Delta t)$.