Magic state distillation is a crucial component in the leading approaches toimplementing universal fault tolerant quantum computation, with existingprotocols for both qubit and higher dimensional systems. Early work focused ondetermining the region of distillable states for qubit protocols, yetcomparatively little is known about which states can be distilled and with whatdistillable region for d>2. Here we focus on d=3 and present new four-qutritdistillation schemes that improve upon the known distillable region, andachieve distillation tight to the boundary of undistillable states for someclasses of state. As a consequence of recent results, this implies that thereis a family of quantum states that enable universality if and only if theyexhibit contextuality with respect to stabilizer measurements. We also identifya new routine whose fixed point is a magic state with maximal sum-negativityi.e., it is maximally non-stabilizer in a specific sense.
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