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We generalize the binary quantum counting algorithm of Lesovik, Suslov, andBlatter [Phys. Rev. A 82, 012316 (2010)] to higher counting bases. Thealgorithm makes use of qubits, qutrits, and qudits to count numbers in a base2, base 3, or base d representation. In operating the algorithm, the number n <N = d^K is read into a K-qudit register through its interaction with a streamof n particles passing in a nearby wire; this step corresponds to a quantumFourier transformation from the Hilbert space of particles to the Hilbert spaceof qudit states. An inverse quantum Fourier transformation provides the numbern in the base d representation; the inverse transformation is fully quantum atthe level of individual qudits, while a simpler semi-classical version can beused on the level of qudit registers. Combining registers of qubits, qutrits,and qudits, where d is a prime number, with a simpler single-shot measurementallows to find the powers of 2, 3, and other primes d in the number n. We show,that the counting task naturally leads to the shift operation and an algorithmbased on the quantum Fourier transformation. We discuss possibleimplementations of the algorithm using quantum spin-d systems, d-well systems,and their emulation with spin-1/2 or double-well systems. We establish theanalogy between our counting algorithm and the phase estimation algorithm andmake use of the latter's performance analysis in stabilizing our scheme.Applications embrace a quantum metrological scheme to measure a voltage (analogto digital converter) and a simple procedure to entangle multi-particle states.

Quantum Abacus for counting and factorizing numbers