We present deep (265 ks) Chandra X-ray observations of PSO J352.4034$-$15.3373, a quasar at z=5.831 that, with a radio-to-optical flux ratio of R>1000, is one of the radio-loudest quasars in the early universe and is the only quasar with observed extended radio jets of kpc-scale at $z gtrsim 6$. Modeling the X-ray spectrum of the quasar with a power law, we find a best fit of $Gamma = 1.99^{+0.29}_{-0.28}$, leading to an X-ray luminosity of $L_{2-10} = 1.26^{+0.45}_{-0.33} times 10^{45} {rm erg} {rm s}^{-1}$ and an X-ray to UV brightness ratio of $alpha_{rm OX} = -1.36 pm 0.11$. We identify a diffuse structure 50 kpc (${sim}8^{primeprime}$) to the NW of the quasar along the jet axis that corresponds to a $3sigma$ enhancement in the angular density of emission and can be ruled out as a background fluctuation with a probability of P=0.9985. While with few detected photons the spectral fit of the structure is uncertain, we find that it has a luminosity of $L_{2-10}sim10^{44} {rm erg} {rm s}^{-1}$. These observations therefore potentially represent the most distant quasar jet yet seen in X-rays. We find no evidence for excess X-ray emission where the previously-reported radio jets are seen (which have an overall linear extent of $0.^{primeprime}28$), and a bright X-ray point source located along the jet axis to the SE is revealed by optical and NIR imaging to not be associated with the quasar.
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