The Representation of Numbers by States in Quantum Mechanics

The representation of numbers by tensor product states of composite quantum systems is examined. Consideration is limited to k-ary representations of length L and arithmetic modulo k^{L}. An abstract representation on an L fold tensor product Hilbert space H^{arith} of number states and operators for the basic arithmetic operations is described. Unitary maps onto a physical parameter based tensor product space H^{phy} are defined and the relations between these two spaces and the dependence of algorithm dynamics on the unitary maps is discussed. The important condition of efficient implementation by physically realizable Hamiltonians of the basic arithmetic operations is also discussed.