  Basis state shift is central to many quantum algorithms, most notably the
quantum walk. Efficient implementations are of major importance for achieving a
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 $2^n$ 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.
