Binary representation of coordinate and momentum in quantum mechanics

To simulate a quantum system with continuous degrees of freedom on a quantum computer based on qubits, it is necessary to reduce continuous observables (primarily coordinates and momenta) to binary observables. We consider this problem based on expanding quantum observables in series in powers of two, analogous to the binary representation of real numbers. The coefficients of the series (“digits”) are therefore orthogonal projectors. We investigate the corresponding quantum mechanical operators and the relations between them and show that the binary expansion of quantum observables automatically leads to renormalization of some divergent integrals and series (giving them finite values). The binary decomposition of the quantum observables automatically leads to renormalization (assignment of finite values) of some divergent integrals and series.