Efficient explicit gate construction of block-encoding for Hamiltonians needed for simulating partial differential equations

TL;DR

This paper presents an explicit, efficient quantum block-encoding method for Hamiltonians used in simulating PDEs, achieving polynomial and exponential speedups in spatial and dimensional complexity.

Contribution

It introduces a novel explicit construction of block-encoding for Hamiltonians of PDEs, enabling efficient quantum simulation with significant speedups.

Findings

Achieves squared logarithmic scaling in spatial partitioning

Provides polynomial speedup over classical methods

Extends to multi-dimensional PDEs with exponential acceleration

Abstract

One of the most promising applications of quantum computers is solving partial differential equations (PDEs). By using the Schrodingerisation technique - which converts non-conservative PDEs into Schrodinger equations - the problem can be reduced to Hamiltonian simulations. The particular class of Hamiltonians we consider is shown to be sufficient for simulating almost any linear PDE. In particular, these Hamiltonians consist of discretizations of polynomial products and sums of position and momentum operators. This paper addresses an important gap by efficiently loading these Hamiltonians into the quantum computer through block-encoding. The construction is explicit and efficient in terms of one- and two-qubit operations, forming a fundamental building block for constructing the unitary evolution operator for that class of Hamiltonians. The proposed algorithm demonstrates a squared logarithmic scaling with respect to the spatial partitioning size, offering a polynomial speedup over classical finite-difference methods in the context of spatial partitioning for PDE solving. Furthermore, the algorithm is extended to the multi-dimensional case, achieving an exponential acceleration with respect to the number of dimensions, alleviating the curse of dimensionality problem. This work provides an essential foundation for developing explicit and efficient quantum circuits for PDEs, Hamiltonian simulations, and ground state and thermal state preparation.