Efficient parallelization of quantum basis state shift

Abstract

Basis state shift is central to many quantum algorithms, most notably the quantum walk. Efficient implementations are of major importance for achieving the quantum speedup for computational applications. We optimize the state shift algorithm by incorporating the shift in different directions in parallel. This provides a significant reduction in the depth of the quantum circuit in comparison to the currently known methods, giving a linear scaling in the number of gates versus working qubits in contrast to the quadratic scaling of the state-of-the-art method based on the quantum Fourier transform. For a one-dimensional array of size 2n for n > 4, we derive the total number of 15n + 74 two-qubit CX gates in the parallel circuit, using a total of 2n-2 qubits including an ancilla register for the decomposition of multi-controlled gates. We focus on the one-dimensional and periodic shift, but note that the method can be extended to more complex cases.