Arbitrary Boundary Conditions and Constraints in Quantum Algorithms for Differential Equations via Penalty Projections

TL;DR

This paper introduces a quantum algorithm that efficiently enforces arbitrary boundary conditions in differential equations using penalty projections, enabling faster quantum simulations of physical phenomena with complex constraints.

Contribution

It proposes a novel penalty projection method for quantum algorithms to handle arbitrary boundary conditions in differential equations, with efficiency guarantees and error bounds.

Findings

Cost of enforcing constraints scales logarithmically with penalty strength

For heat equations, overhead is polylogarithmic in system size and time

Numerical experiments validate the effectiveness of the approach

Abstract

Complicated boundary conditions are essential to accurately describe phenomena arising in nature and engineering. Recently, the investigation of a potential speedup through quantum algorithms in simulating the governing ordinary and partial differential equations of such phenomena has gained increasing attention. We design an efficient quantum algorithms for solving differential equations with arbitrary boundary conditions. Specifically, we propose an approach to enforce arbitrary boundary conditions and constraints through adding a penalty projection to the governing equations. Assuming a fast-forwardable representation of the projection to ensure an efficient interaction picture imulation, the cost of to enforce the constraints is at most $O(\log\lambda)$ in the strength of the penalty $\lambda$ in the gate complexity; in the worst case, this goes as $O([\|v(0)\|^2\|A_0\| + \|b\|_{L^1[0;t]}^2)]t^2/\varepsilon)$, for precision $\varepsilon$ and a dynamical system $\frac{\rm d}{{\rm d}t} v(t) = A_0(t) v(t) + b(t)$ with negative semidefinite $A_0(t)$ of size $n^d\times n^d$. E.g., for the heat equation, this leads to a gate complexity overhead of $\widetilde O(d\log n + \log t)$. We show constraint error bounds for the penalty approach and provide validating numerical experiments, and estimate the circuit complexity using the Linear Combination of Hamiltonian Simulation.