<jats:title>Abstract</jats:title> <jats:p>We have developed a highly tuned software library that accelerates the calculation of quadrupole terms in the Barnes–Hut tree code by using a SIMD instruction set on the x86 architecture, Advanced Vector eXtensions 2 (AVX2). Our code is implemented as an extension of the Phantom-GRAPE software library that significantly accelerates the calculation of monopole terms. If the same accuracy is required, the calculation of quadrupole terms can accelerate the evaluation of forces over using only monopole terms because we can approximate gravitational forces from closer particles by quadrupole moments rather than by only monopole moments. Our implementation can calculate gravitational forces about 1.1 times faster in any system than the combination of the pseudoparticle multipole method and Phantom-GRAPE. Our implementation allows simulating homogeneous systems up to 2.2 times faster than with only monopole terms; however, the speedup for clustered systems is not enough because the increase of approximated interactions is insufficient to negate the increased calculation cost by computing quadrupole terms. We have estimated that improvement in performance can be achieved by using a new SIMD instruction set, AVX-512. Our code is expected to be able to accelerate simulations of clustered systems up to 1.08 times faster in an AVX-512 environment than with only monopole terms.</jats:p>