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We propose new quantum algorithms to solve the regulator and the principalideal problem in a real-quadratic number field. We improve the algorithmsproposed by Hallgren by using two different techniques. The first improvementis the usage of a period function which is not one-to-one on its period. Weshow that even in this case Shor's algorithm computes the period with constantprobability. The second improvement is the usage of reduced forms (a, b, c) ofdiscriminant D with a>0 instead of reduced ideals of the same discriminant.These improvements reduce the number of required qubits by at least 2 log D.

Quantum Algorithms for many-to-one Functions to Solve the Regulator and the Principal Ideal Problem