Quantum Circuits for the heat equation with physical boundary conditions via Schrodingerisation

TL;DR

The paper tackles the problem of quantum-simulating PDEs with physical boundary conditions, where non-unitary dynamics hinder direct quantum simulation. It leverages Schrödingerisation to map linear PDEs to Schrödinger-type systems in an augmented dimension, enabling unitary time evolution. Two principal approaches are developed to handle inhomogeneous terms from time-dependent boundaries: (i) a Duhamel/Lcu-based method for the integral term, and (ii) augmentation to convert to a homogeneous, autonomous problem followed by non-autonomous-to-autonomous transformation. A detailed circuit construction is provided for the heat equation under various boundary conditions, including complexity analyses and a comparison with autonomisation against the LCU approach, with broader implications for quantum simulation of PDEs in bounded domains.

Abstract

This paper explores the explicit design of quantum circuits for quantum simulation of partial differential equations (PDEs) with physical boundary conditions. These equations and/or their discretized forms usually do not evolve via unitary dynamics, thus are not suitable for quantum simulation. Boundary conditions (either time-dependent or independent) make the problem more difficult. To tackle this challenge, the Schrodingerisation method can be employed, which converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schrodinger-type equations, via the so-called warped phase transformation that maps the equation into one higher dimension. Despite advancements in Schrodingerisation techniques, the explicit implementation of quantum circuits for solving general PDEs, especially with physical boundary conditions, remains underdeveloped. We present two methods for handling the inhomogeneous terms arising from time-dependent physical boundary conditions. One approach utilizes Duhamel's principle to express the solution in integral form and employs linear combination of unitaries (LCU) for coherent state preparation. Another method applies an augmentation to transform the inhomogeneous problem into a homogeneous one. We then apply the quantum simulation technique from [CJL23] to transform the resulting non-autonomous system to an autonomous system in one higher dimension. We provide detailed implementations of these two methods and conduct a comprehensive complexity analysis in terms of queries to the time evolution input oracle.

Figures (7)

• Figure 1: Quantum circuit for the select oracle given in Lemma \ref{['lem:selectH']}
• Figure 2: Quantum circuit for $V_{\text{heat}}(\tau)$
• Figure 3: Modified quantum circuit for the last two operators in $V_{\text{heat}}(\tau)$
• Figure 4: Modified quantum circuit for $V_{\text{heat}}(\tau)$
• Figure 5: Quantum circuit for $U_{\text{auto}}^{(1)}(\tau)$. The dashed box corresponds to $\tilde{U}_{\text{auto}}^{(1)} (\tau)$. $F_s$ is the quantum Fourier transform in the $s$ direction.

Theorems & Definitions (24)

• Lemma 3.1
• proof
• Definition 4.1
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Lemma 4.4