This paper addresses the question of understanding quantum algorithms in terms of unitary operators. Many quantum algorithms can be expressed as applications of operators formed by conjugating so-called classical operators. The operators that are used for conjugation are determined by the problem and any additional structure possessed by the Hilbert space that is acted upon. We prove many commutative laws for these different operators, and we use those to phrase and analyze old and new problems and algorithms. As an example, we review the Abelian subgroup problem. We then introduce the problem of determining a group homomorphism, and we give classical and quantum algorithms for it. We also generalize Deutsch’s problem and improve the previous best algorithms for earlier generalizations of it.

• Received 4 September 1997

DOI:https://doi.org/10.1103/PhysRevA.59.3280