Universal quantum gate set

A universal quantum gate set can approximate every finite-qubit unitary operator arbitrarily accurately using finite circuits over the set.

Universality is a property of a set of quantum logic gates: finite circuits from the set approximate arbitrary unitary transformations.

If every generator of a universal quantum gate set has an exact finite implementation over another set, the latter is also universal. For a target unitary and tolerance $\epsilon >0$, first choose an approximating circuit over the universal set, then replace each generator by its exact implementation. The compiled circuit has the same operator and therefore the same approximation error. Overall global phases can be ignored under the usual physically projective convention. Exact synthesis of an already universal family is sufficient; it does not imply exact synthesis of every possible unitary with a finite alphabet of gates.

The largest pure-state norm distance between outputs of two unitary operators on the same normalized input. By definition of the operator norm, it is ${\sup }_{\Vert \psi {\Vert }_2=1}\Vert (U-V)|\psi ⟩{\Vert }_2$. If it is at most $\epsilon$, every input is approximated to that accuracy. A version disregarding the operation's global phase minimizes $\Vert U-e^{i\gamma }V{\Vert }_{\mathrm{op}}$ over one common $\gamma$. Error in a finite quantum circuit accumulates at most additively by the quantum circuit gate-error telescoping bound.

For a suitable finite inverse-closed universal quantum gate set, the Solovay--Kitaev theorem approximates a fixed-dimensional unitary to operator norm error $\epsilon$ with a gate sequence of length polynomial in $\log (1/\epsilon )$.