Quantum-Fourier-transform-based quantum arithmetic with qudits

Abstract

We present some basic integer arithmetic quantum circuits, such as adders and multiplier-accumulators of various forms, which operate on multilevel qudits. The integers to be processed are represented in an alternative basis after they have been Fourier transformed. Several arithmetic circuits operating on Fourier-transformed integers have appeared in the literature for two-level qubits. Here we extend these techniques to multilevel qudits, as they may offer some advantages relative to qubit implementations. The arithmetic circuits presented here can be used as basic building blocks for higher level algorithms such as quantum phase estimation, quantum simulation, quantum optimization, etc. Detailed decomposition is given down to elementary two-level single- and two-qudit gates as such gates are the most appropriate for physical implementation. A complexity analysis is given after this decomposition step and it is shown that the depth of the circuits is linear in the number of qudits employed and quadratic in the dimension of each qudit while their quantum cost is quadratic in the number of the qudits and quadratic in the dimension.