Adjustable-depth quantum circuit for position-dependent coin operators of discrete-time quantum walks

TL;DR

This paper introduces an adjustable-depth quantum circuit for exact implementation of position-dependent coin operators in discrete-time quantum walks, balancing circuit depth and resource requirements.

Contribution

It proposes a novel adjustable-depth circuit that combines parallel and sequential operations to optimize exact implementation of position-dependent coins.

Findings

The circuit achieves exact implementation with controllable depth.

It balances linear and exponential depth components.

The approach reduces resource requirements compared to previous methods.

Abstract

Discrete-time quantum walks with position-dependent coin operators have numerous applications. For a position dependence that is sufficiently smooth, it has been provided in Ref. [1] an approximate quantum-circuit implementation of the coin operator that is efficient. If we want the quantum-circuit implementation to be exact (e.g., either, in the case of a smooth position dependence, to have a perfect precision, or in order to treat a non-smooth position dependence), but the depth of the circuit not to scale exponentially, then we can use the linear-depth circuit of Ref. [1], which achieves a depth that is linear at the cost of introducing an exponential number of ancillas. In this paper, we provide an adjustable-depth quantum circuit for the exact implementation of the position-dependent coin operator. This adjustable-depth circuit consists in (i) applying in parallel, with a linear-depth circuit, only certain packs of coin operators (rather than all of them as in the original linear-depth circuit [1]), each pack contributing linearly to the depth, and in (ii) applying sequentially these packs, which contributes exponentially to the depth.