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We explore the connection between $K3$ categories and 0-cycles on holomorphicsymplectic varieties. In this paper, we focus on Kuznetsov's noncommutative$K3$ category associated to a nonsingular cubic 4-fold.  By introducing a filtration on the $\mathrm{CH}_1$-group of a cubic 4-fold$Y$, we conjecture a sheaf/cycle correspondence for the associated $K3$category $\mathcal{A}_Y$. This is a noncommutative analog of O'Grady'sconjecture concerning derived categories of $K3$ surfaces. We study instancesof our conjecture involving rational curves in cubic 4-folds, and verify theconjecture for sheaves supported on low degree rational curves.  Our method provides systematic constructions of (a) the Beauville-Voisinfiltration on the $\mathrm{CH}_0$-group and (b) algebraically coisotropicsubvarieties of a holomorphic symplectic variety which is a moduli sace ofstable objects in $\mathcal{A}_Y$.

K3 categories, one-cycles on cubic fourfolds, and the Beauville-Voisin filtration