<p>We take initial steps towards a general framework for constructing logical gates in general quantum CSS codes. Viewing CSS codes as cochain complexes, we observe that <i>cohomology invariants</i> naturally give rise to diagonal logical gates. We show that such invariants exist if the quantum code has a structure that relaxes certain properties of a differential graded algebra. We show how to equip quantum codes with such a structure by defining <i>cup products</i> on CSS codes. The logical gates obtained from this approach can be implemented by a constant-depth unitary circuit. In particular, we construct a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>-fold cup product that can produce a logical operator in the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>-th level of the Clifford hierarchy on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> copies of the same quantum code, which we call the <i>copy-cup gate</i>. For any desired <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\Lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>, we can construct several families of quantum codes that support gates in the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\Lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>-th level with various asymptotic code parameters.</p>