High order schemes for solving partial differential equations on a quantum computer

TL;DR

The paper tackles the resource bottlenecks of quantum PDE solvers by combining higher-order central finite differences with a novel decomposition of d-band matrices into Pauli strings. It extends previous tridiagonal approaches to general d-band operators, organizing contributing Pauli terms into commuting subsets via a Walsh-based framework, and demonstrates this on the 1D wave equation mapped to a Schrödinger evolution. Numerical experiments show that higher-order discretization can reduce the required qubits for a given accuracy, but the overall precision is still limited by Trotterization unless more steps are used, highlighting a trade-off between discretization quality and gate count. The results provide a practical route to implement quantum PDE solvers with controlled error budgets, quantify circuit complexity, and guide resource planning for quantum simulations of wave propagation and related PDEs.

Abstract

We explore the utilization of higher-order discretization techniques in optimizing the gate count needed for quantum computer based solutions of partial differential equations. To accomplish this, we present an efficient approach for decomposing $d$-band diagonal matrices into Pauli strings that are grouped into mutually commuting sets. Using numerical simulations of the one-dimensional wave equation, we show that higher-order methods can reduce the number of qubits necessary for discretization, similar to the classical case, although they do not decrease the number of Trotter steps needed to preserve solution accuracy. This result has important consequences for the practical application of quantum algorithms based on Hamiltonian evolution.

Figures (6)

• Figure 1: The figure illustrates the decomposition discussed in Proposition \ref{['theor:sets']}. Rather than a single element, all potential Pauli strings that could contribute to the decomposition are represented as a set $S_{k,j}$. Each set corresponds to $\mathbf{x}_{k,j}$ as defined in \ref{['eq:unique_solutions']}. Sets corresponding to the same bandwidth $d$ are depicted in the same color.
• Figure 2: Comparison of $\vec{\phi}_V (t)$ - solution obtained as $e^{-i H_k t} \vec{\psi} (t=0)$ with $\vec{u} (t) = \vec{u}_{sw}(t)/\norm{\vec{u_0}}$ an analytical standing wave solution at $t=1$, with $l=5$ and $c=1$ for different accuracy orders $\kappa=2k$ and different number of qubits for discretization $n$ (the total number of qubits is given by $n+1$).
• Figure 3: Comparison of $\vec{\phi}_V (t)$ - solution obtained with proposed quantum algorithm with $\vec{u} (t) = \vec{u}_{sw}(t)/\norm{\vec{u_0}}$ with fixed Trotter errors of $10^{-3}$, $10^{-5}$, and $10^{-7}$, represented by different marker shapes. Results for $4$ and $5$ qubits are given with red and green respectively (the total number of qubits is given by $n+1$). The Trotter formula is set to order $p=2$; the wave equation parameters $l=5$, $t=1$, and $c=1$.
• Figure 4: The relationship between the number of Trotter steps $r$ and the number of qubits for a discretization $n$ (total qubits $n+1$), using various discretization orders $\kappa$, necessary to reach an error of $\epsilon_{\text{ns}} = 10^{-5}$. The Trotter formula is utilized at order $p=2$, with wave equation parameters set to $l=5$, $t=1$, and $c=1$.
• Figure :

Theorems & Definitions (20)

• Proposition 1: Decomposition of a $d$-band matrix
• Corollary 1: Decomposition of a symmetrized matrix
• Proposition 2: Number of sets in the decomposition
• Corollary 2: Commuting subsets
• Proposition 3: Error for solving wave equation
• Proposition 4
• Remark 1
• Lemma 1
• proof
• Lemma 2