A Quantum Linear Systems Pathway for Solving Differential Equations

TL;DR

The paper develops a quantum linear-systems pathway for solving differential equations by reducing discretized models to linear systems $A x = b$ and solving them with block-encoded Quantum Singular Value Transformation (QSVT) to approximate $A^{-1}$ as a polynomial of the singular values. It demonstrates the workflow on a complex tridiagonal system and extends to CFD problems, including the heat equation with mixed Dirichlet/Neumann boundaries and the Carleman-linearized Burgers' equation, while analyzing how $\sigma_{\min}$ and the inverse polynomial degree $d$ drive circuit depth. The work highlights both the potential quantum speedups and key bottlenecks, such as reliable $\sigma_{\min}$ estimation for sparse matrices and depth-reduction strategies, and stresses the need for benchmarks against classical reachability. Overall, it lays a foundational pathway toward scalable quantum PDE solvers and large-scale quantum linear-system applications.

Abstract

We present a systematic pathway for solving differential equations within the quantum linear systems framework by combining block encoding with Quantum Singular Value Transformation (QSVT). The approach is demonstrated on a complex tridiagonal linear system and extended to problems in computational fluid dynamics: the heat equation with mixed boundary conditions and the nonlinear Burgers' equation. Our scaling analysis of the heat equation shows how discretization influences the minimum singular value and the polynomial degree required for QSVT, identifying circuit-depth overhead as a key bottleneck. For Burgers' equation, we illustrate how Carleman-linearized nonlinear dynamics can be efficiently block encoded and solved within the QSVT framework. These results highlight both the potential and limitations of current methods, underscoring the need for efficient estimation of minimum singular value, depth-reduction techniques, and benchmarks against classical reachability. This pathway lays a foundation for advancing quantum linear system methods toward large-scale applications.

Figures (8)

• Figure 1: Quantum linear systems pathway for solving differential equations using QSVT. Here $\sigma_{\text{min}}$ denotes the smallest singular value of the matrix, and $\phi$ denotes the phases used for approximating the inverse function (see \ref{['sec:qsp', 'sec:inverse_function']}).
• Figure 2: Quantum circuit for solving the linear system $|x\rangle = \text{QSVT}(\bar{A}^{\dag})|b\rangle$ using QSVT, with the rescaled solution given by \ref{['eq:LSE']}. For demonstration, a square matrix $\bar{A}$ is used, with phases $\{\phi_i\}_{i=0}^d$ in the reflection convention \ref{['sec:reflection_convention']} for an odd-degree $d$ inverse polynomial (see \ref{['sec:inverse_function', 'eq:U_phi_QSVT']}). The $|\text{QSP}\rangle$ qubit applies the phase sequence $\vec{\phi}$. The $|\text{Flag}\rangle$ register contains $m$ data qubits plus one delete qubit as in \ref{['fig:A3_Block_Encode']}. The $|\text{Matrix}\rangle$ register consists of $n$ qubits for preparing $|b\rangle$ and encoding $\bar{A}^{\dag}$. All qubits start in $|0\rangle$ state. Projector-controlled phase shift gates $\raisebox{0.3ex}{$\prod$}_{\phi_i}$ and $\raisebox{0.3ex}{$\widetilde{\prod}$}_{\phi_i}$ (identical for a square matrix) are implemented using MCX gates with controls on all $|\text{Flag}\rangle = 0$. The vertical dotted line indicates a barrier separating $|b\rangle$ from the rest of the circuit; the dots indicate a repeated pattern. Post-selection on the $|\text{QSP}\rangle$ and $|\text{Flag}\rangle$ is denoted by $\langle 0|$, and the final solution state is $|x\rangle$.
• Figure 3: Solution of the complex tridiagonal linear system \ref{['sec:tridiagonal_complex']}. \ref{['fig:3a_Complex']} Circuit for block encoding $A^{\dagger}$ of \ref{['eq:complex_matrix']}, where control values of the data item $-b\textrm{i}$ are denoted as $-\vec{b}\textrm{i}$. \ref{['fig:3b_Complex']} Inverse function approximation using Quantum Signal Processing (QSP) \ref{['sec:qsp', 'sec:inverse_function']}. Subnormalized $\sigma$ denote singular values of $A^{\dagger}/\alpha$, while optimal $\sigma$ denote singular values of $A^{\dagger}/||A^{\dagger}||_2$, with $||\cdot||_2$ the spectral norm. \ref{['fig:3c_Complex']} Parity plot comparing the true $(A^{+}\vec{y})$ with the QSVT solution $(A^{+}\vec{y})_{\text{QSVT}}$, showing real and imaginary parts separately.
• Figure 4: Solution of the heat equation \ref{['sec:heat_equation']}. \ref{['fig:4a_Heat']} Circuit for block encoding $A^{\dag}$ of matrix \ref{['eq:heat_A_numerical']}. \ref{['fig:4b_Heat']} Implicit solution with left Dirichlet and right Neumann boundary conditions via QSVT, compared with the true solution, for $t=[0, 100]$. \ref{['fig:4c_Heat']} Scaling of the minimum subnormalized singular value $(\sigma_{\text{min}})$ and inverse polynomial degree $d$ versus time step $\Delta t (\Delta x = 0.125)$; $\Delta t_{\text{exp}}^{\text{max}}$ indicates the threshold separating explicit and implicit schemes. \ref{['fig:4d_Heat']} Scaling with the number of matrix qubits $n=\{3, 4, 5, 6\}$ for $\Delta t = \{1, 2, 3\}$. A double exponential fit is shown and tested for $n = 7$. \ref{['fig:4e_Heat']} Estimated $\sigma_{\text{min}}$ and polynomial degree as $n$ scales, with the same legend as \ref{['fig:4d_Heat']}. Extrapolation can estimate circuit depth required for quantum advantage over a 50-qubit classical simulator.
• Figure 5: \ref{['fig:5a_Burgers']} Matrix representation of $L^{\dagger}$ for solving Burgers’ equation. Colored sub-blocks highlight distinct matrix components, while the uncolored block corresponds to $A_1^2 = 0$ in \ref{['eq:A_truncation']}. Data items are labeled $[a_i]_{i=0}^{13}$. Continuous colored lines indicate repeating $a_i$ values along diagonals with fixed offsets, and black square dots mark single, non-repeating entries. \ref{['fig:5b_Burgers']} Initial condition at $t=0$, and comparison of the QSVT implicit solution with the classical explicit solution at $t=0.3$.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2