An efficient explicit implementation of a near-optimal quantum algorithm for simulating linear dissipative differential equations

TL;DR

This paper introduces a novel, efficient quantum algorithm for simulating dissipative differential equations by combining block-encoding techniques with a coordinate transformation, enabling exponential Hamiltonian simulations with high success probability.

Contribution

It presents a new block-encoding method for dissipative problems that simplifies quantum circuit implementation and improves efficiency over existing approaches.

Findings

Successfully simulated the advection-diffusion equation on a quantum emulator.

Achieved exponential scaling in Hamiltonian simulations with a single QSP circuit.

Proved the method's efficiency and error convergence compared to prior LCHS circuits.

Abstract

We propose an efficient block-encoding technique for the implementation of the Linear Combination of Hamiltonian Simulations (LCHS) for simulating dissipative initial-value problems. This algorithm approximates a target nonunitary operator as a weighted sum of Hamiltonian evolutions, thereby emulating a dissipative problem by mixing various time scales. We introduce an efficient encoding of the LCHS into a quantum circuit based on a simple coordinate transformation that turns the dependence on the summation index into a trigonometric function. Classically, this method is equivalent to the use of a highly accurate Fej\'er-Clenshaw-Curtis quadrature formula. Quantumly, this significantly simplifies block-encoding of a dissipative problem and allows one to perform an exponential number of Hamiltonian simulations by a single Quantum Signal Processing (QSP) circuit. The resulting LCHS circuit has high success probability and the selector scales logarithmically with the number of terms in the LCHS sum and linearly with time. Careful analysis of error convergence proves that this method is more efficient than other LCHS circuits that have recently appeared in the literature. We verify the quantum circuit and its scaling by simulating it on a digital emulator of fault-tolerant quantum computers and, as a test problem, solve the advection-diffusion equation. The proposed algorithm can be used for modeling a wide class of nonunitary initial-value problems including the Liouville equation with added dissipation and linear embeddings of nonlinear systems, such as the Koopman-von Neumann and Carleman embeddings.