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Recently, the Lyman-$\alpha$ (Ly$\alpha$) forest flux auto-correlationfunction has been shown to be sensitive to the mean free path ofhydrogen-ionizing photons, $\lambda_{\text{mfp}}$, for simulations at $z \geq5.4$. Measuring $\lambda_{\text{mfp}}$ at these redshifts will give vitalinformation on the ending of reionization. Here we present the firstobservational measurements of the Ly$\alpha$ forest flux auto-correlationfunctions in ten redshift bins from $5.1 \leq z \leq 6.0$. We use a sample of35 quasar sightlines at $z > 5.7$ from the extended XQR-30 data set, this datahas signal-to-noise ratios of $> 20$ per spectral pixel. We carefully accountfor systematic errors in continuum reconstruction, instrumentation, andcontamination by damped Ly$\alpha$ systems. With these measurements, weintroduce software tools to generate auto-correlation function measurementsfrom any simulation. For an initial comparison, we show our auto-correlationmeasurements with simulation models for recently measured$\lambda_{\text{mfp}}$ values and find good agreements. Further work inmodeling and understanding the covariance matrices of the data is necessary toget robust measurements of $\lambda_{\text{mfp}}$ from this data.

Measurements of the $z > 5$ Lyman-$\alpha$ forest flux auto-correlation functions from the extended XQR-30 data set