Block encoding bosons by signal processing

TL;DR

This paper explores efficient quantum block encoding methods for bosonic systems, demonstrating how signal processing techniques like QSVT, QETU, and LOVE-LCU can be used to improve quantum simulations of lattice bosons.

Contribution

The work introduces novel quantum block encoding techniques for bosonic Hamiltonians using signal processing methods, including a new approach called LOVE-LCU, and compares their efficiency.

Findings

QSVT provides optimal asymptotic gate count scaling.

LOVE-LCU outperforms other methods for operators up to ~11 qubits.

Quantum signal processing enables efficient bosonic Hamiltonian encoding.

Abstract

Block Encoding (BE) is a crucial subroutine in many modern quantum algorithms, including those with near-optimal scaling for simulating quantum many-body systems, which often rely on Quantum Signal Processing (QSP). Currently, the primary methods for constructing BEs are the Linear Combination of Unitaries (LCU) and the sparse oracle approach. In this work, we demonstrate that QSP-based techniques, such as Quantum Singular Value Transformation (QSVT) and Quantum Eigenvalue Transformation for Unitary Matrices (QETU), can themselves be efficiently utilized for BE implementation. Specifically, we present several examples of using QSVT and QETU algorithms, along with their combinations, to block encode Hamiltonians for lattice bosons, an essential ingredient in simulations of high-energy physics. We also introduce a straightforward approach to BE based on the exact implementation of Linear Operators Via Exponentiation and LCU (LOVE-LCU). We find that, while using QSVT for BE results in the best asymptotic gate count scaling with the number of qubits per site, LOVE-LCU outperforms all other methods for operators acting on up to $\lesssim11$ qubits, highlighting the importance of concrete circuit constructions over mere comparisons of asymptotic scalings. Using LOVE-LCU to implement the BE, we simulate the time evolution of single-site and two-site systems in the lattice $\varphi^4$ theory using the Generalized QSP algorithm and compare the gate counts to those required for Trotter simulation.