Low Depth Phase Oracle Using a Parallel Piecewise Circuit

TL;DR

This paper introduces a parallel piecewise circuit method for applying phase or rotation operations based on a function, achieving minimal depth and efficient implementation for quantum algorithms involving piecewise approximations.

Contribution

It presents a novel parallelization technique for phase and rotation oracles using piecewise functions, reducing circuit depth to one and optimizing resource usage.

Findings

Achieves circuit depth as low as O(log n + log S) for piecewise approximations.

Uses recursive catalyst towers for efficient elementary rotations.

Provides resource estimates for multiple oracle repetitions with T-count scaling.

Abstract

We explore the important task of applying a phase $\exp(i\,f(x))$ to a computational basis state $\left| x \right>$. The closely related task of rotating a target qubit by an angle depending on $f(x)$ is also studied. Such operations are key in many quantum subroutines, and frequently $f(x)$ can be well-approximated by a piecewise function; examples range from the application of diagonal Hamiltonian terms (such as the Coulomb interaction) in grid-based many-body simulation, to derivative pricing algorithms. Here we exploit a parallelisation of the piecewise approach so that all constituent elementary rotations are performed simultaneously, that is, we achieve a total rotation depth of one. Moreover, we explore the use of recursive catalyst `towers' to implement these elementary rotations efficiently. We find that strategies prioritising execution speed can achieve circuit depth as low as $O(\log{n}{+}\log{S})$ for a register of $n$ qubits and a piecewise approximation of $S$ sections (presuming prior preparation of enabling resource states), albeit total qubit count then scales with $S$. In the limit of multiple repetitions of the oracle, we find that catalyst tower approaches have an $O(S\cdot n)$ T-count.