Efficient parallelization of quantum basis state shift

TL;DR

This paper presents a parallelized quantum basis state shift algorithm that significantly reduces circuit depth and gate count, enabling more efficient quantum walk implementations and potential speedups.

Contribution

It introduces a parallelization technique for basis state shifts that achieves linear gate scaling, improving over the quadratic scaling of previous Fourier transform-based methods.

Findings

Reduces quantum circuit depth for basis state shifts

Achieves linear scaling of gates with qubits in the parallel method

Provides explicit gate count for one-dimensional array shifts

Abstract

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.