A universal gate set is a finite collection of quantum gates that, through repeated composition, can approximate any unitary operation on any number of qubits to arbitrary accuracy. The most commonly cited universal gate set consists of the Clifford gates (H, S, CNOT) plus the T gate. The Solovay-Kitaev theorem guarantees that any single-qubit unitary can be approximated to precision ε using O(log^c(1/ε)) gates from a universal set, where c is approximately 3.97 (improved to nearly 1 with optimal synthesis algorithms).
The concept of universality is analogous to classical computing, where the NAND gate alone is universal for Boolean logic. However, quantum universality has a richer structure because the continuous nature of quantum operations (arbitrary rotations) must be approximated by a discrete gate set. The key insight is that while Clifford gates alone are classically simulable, adding any non-Clifford gate (the T gate being the standard choice) promotes the set to universality.
Different quantum hardware platforms have different native gate sets. IBM's processors natively support {√X, Rz(θ), CNOT}, Google's support {√X, Rz(θ), CZ or √iSWAP}, and Quantinuum's support {Rz(θ), Ry(θ), ZZ(θ)}. Quantum compilers translate algorithm-level circuits into these native gate sets, optimizing for minimal gate count and circuit depth. For fault-tolerant computation, the practically universal gate set is {H, S, CNOT, T}, and circuit optimization focuses on minimizing the number of expensive T gates.