Qubit-qubit interaction in quantum computers. II. Adder algorithm with diagonal and off-diagonal interactions

The effect of off-diagonal interactions between quantum bits is studied in a simple quantum computer algorithm designed to calculate the sum of two n-bit numbers. Scaling laws an derived numerically for the dependence of the error on the number n of bits, the strength of the interaction, and the calculation time. The case of diagonal interactions, previously studied in the context of the quantum Fourier transform algorithm, is also considered. No substantial differences are found between the off-diagonal and diagonal forms of the interaction: in both cases, the errors accumulate almost coherently, despite the fact that the system's free evolution is constantly being interrupted by the logical gates, which scramble the coefficients of the wave function. Some comments on the possibility of using "field-insensitive" atomic states in ion-chain quantum computers are also given.