Quantum Galois Machine: Exact Rational Dilations and Denominator Costs

When a quantum operation is specified exactly, irrational amplitudes in its unitary description can disappear from its action on density matrices, leaving an operation with rational coefficients whose implementation still requires additional resources. I develop a Quantum Galois Machine that determines whether an algebraically specified quantum channel can be executed using rational matrices and a finite environment initially prepared in a computational basis state. For every unitary channel with rational coefficients, the minimum dimension of an environment that may be discarded is exactly the least number of rational squares needed to represent a positive rational parameter extracted from the channel. This gives a complete classification with possible dimensions one, two, three and four, each attained by an explicit example. The associated square class multiplies under parallel and sequential composition, so two identical operations admit a joint implementation without an environment even when one operation requires four dimensions. Returning the environment to its initial basis state or retaining a specified coherent control imposes different arithmetic obstructions. Although unrestricted approximation removes this exact environment requirement, a bound on rational denominators restores a quantitative obstruction: a nonsquare output probability with reduced denominator S forces diamond-norm error at least 2/(SB^2) when every entry denominator of an environment-free implementation is at most B. A constructive Householder approximation supplies a complementary upper bound, giving logarithmic denominator bit cost in inverse precision for each fixed channel with nonsquare parameter. These results provide finite certificates that account for the coefficient field, phase convention, auxiliary systems and numerical description size.