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We present algorithmic perturbative solution of $\mathcal{N}=4$ SYM quantumspectral curve in the case of twist 2 operators, valid to in principlearbitrary order in coupling constant. The latter treats operator spins asarbitrary integer values and is written in terms of special class of functions-- products of rational functions in spectral parameter with sums of Baxterpolynomials and Hurwitz functions. It is shown that this class of functions isclosed under elementary operations, such as shifts, partial fractions,multiplication by spectral parameter and differentiation. Also, it is fullysufficient to solve arising non-homogeneous multiloop Baxter and first orderdifference equations. As an application of the proposed method we present thecomputation of anomalous dimensions of twist 2 operators up to four loop order.

Anomalous dimensions of twist 2 operators and $\mathcal{N}=4$ SYM quantum spectral curve