  The ZX-calculus is a graphical language for reasoning about quantum computing
and quantum information theory. As a complete graphical language, it
incorporates a set of axioms rich enough to derive any equation of the
underlying formalism. While completeness of the ZX-calculus has been
established for qubits and the Clifford fragment of prime-dimensional qudits,
universal completeness beyond two-level systems has remained unproven until
now. In this paper, we present a proof establishing the completeness of
finite-dimensional ZX-calculus, incorporating only the mixed-dimensional
Z-spider and the qudit X-spider as generators. Our approach builds on the
completeness of another graphical language, the finite-dimensional ZW-calculus,
with direct translations between these two calculi. By proving its
completeness, we lay a solid foundation for the ZX-calculus as a versatile tool
not only for quantum computation but also for various fields within
finite-dimensional quantum theory.
