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Quantum Random Feature Method for Solving Partial Differential Equations

Junpeng Hu, Shi Jin, Nana Liu, Lei Zhang

TL;DR

The paper proposes the Quantum Random Feature Method (QRFM) to solve high-dimensional PDEs by combining a classical random feature basis with quantum linear-algebra techniques. It develops a full quantum workflow: define oracles for collocation data, construct a random-feature map via LCU and QSVT, assemble block-encodings of differential and boundary operators, and solve the resulting quantum linear system to recover a state encoding the PDE solution. The authors prove a quadratic speedup over the classical RFM in the dominant linear-algebra step and illustrate the approach with a 1D Helmholtz test, including a dual kernel-regression interpretation. They also discuss extensions to local feature bases, quantum preconditioning, and nonlinear PDEs, highlighting QRFM’s potential to enable scalable, mesh-free quantum PDE solvers on future hardware.

Abstract

Quantum computing holds significant promise for scientific computing due to its potential for polynomial to even exponential speedups over classical methods, which are often hindered by the curse of dimensionality. While neural networks present a mesh-free alternative to solve partial differential equations (PDEs), their accuracy is difficult to achieve since one needs to solve a high-dimensional non-convex optimization problem using the stochastic gradient descent method and its variants, the convergence of which is difficult to prove and cannot be guaranteed. The classical random feature method (RFM) effectively merges advantages from both classical numerical analysis and neural network based techniques, achieving spectral accuracy and a natural adaptability to complex geometries. In this work, we introduce a quantum random feature method (QRFM) that leverages quantum computing to accelerate the classical RFM framework. Our method constructs PDE solutions using quantum-generated random features and enforces the governing equations via a collocation approach. A complexity analysis demonstrates that this hybrid quantum-classical algorithm can achieve a quadratic speedup over the classical RFM.

Quantum Random Feature Method for Solving Partial Differential Equations

TL;DR

The paper proposes the Quantum Random Feature Method (QRFM) to solve high-dimensional PDEs by combining a classical random feature basis with quantum linear-algebra techniques. It develops a full quantum workflow: define oracles for collocation data, construct a random-feature map via LCU and QSVT, assemble block-encodings of differential and boundary operators, and solve the resulting quantum linear system to recover a state encoding the PDE solution. The authors prove a quadratic speedup over the classical RFM in the dominant linear-algebra step and illustrate the approach with a 1D Helmholtz test, including a dual kernel-regression interpretation. They also discuss extensions to local feature bases, quantum preconditioning, and nonlinear PDEs, highlighting QRFM’s potential to enable scalable, mesh-free quantum PDE solvers on future hardware.

Abstract

Quantum computing holds significant promise for scientific computing due to its potential for polynomial to even exponential speedups over classical methods, which are often hindered by the curse of dimensionality. While neural networks present a mesh-free alternative to solve partial differential equations (PDEs), their accuracy is difficult to achieve since one needs to solve a high-dimensional non-convex optimization problem using the stochastic gradient descent method and its variants, the convergence of which is difficult to prove and cannot be guaranteed. The classical random feature method (RFM) effectively merges advantages from both classical numerical analysis and neural network based techniques, achieving spectral accuracy and a natural adaptability to complex geometries. In this work, we introduce a quantum random feature method (QRFM) that leverages quantum computing to accelerate the classical RFM framework. Our method constructs PDE solutions using quantum-generated random features and enforces the governing equations via a collocation approach. A complexity analysis demonstrates that this hybrid quantum-classical algorithm can achieve a quadratic speedup over the classical RFM.

Paper Structure

This paper contains 14 sections, 4 theorems, 49 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The Random Feature Method
  3. The Quantum Random Feature Method
  4. Oracles
  5. Random Feature Basis
  6. Linear System and Solution
  7. Analysis
  8. Numerical Experiments
  9. Dual formulation: quantum kernel regression
  10. Discussion
  11. Local Random Feature Function and Partition of Unity
  12. Construction of Oracle $U_x$
  13. Cyclic Matrix Representation
  14. Block-encoding of the Coordinate Operators

Key Result

Lemma 2

Let $U_B$ be an $(\alpha, a, 0)$-block-encoding of $B$, where $B$ is an $m$-qubit operator. Then is an $(\alpha, a+m, 0)$-block-encoding of $\mathop{\mathrm{diag}}\nolimits(B)$, the diagonal matrix formed by the diagonal entries of $B$. Within the $\text{CNOT}$ gate, the subscript index is used to denote the control qubit.

Figures (9)

  • Figure 1: The ansatz for constructing $U_w$ and $U_b$.
  • Figure 2: The quantum circuit of $U_\Phi$ using QSVT, where $CR_{\phi}$ is used to implement $e^{i\phi U_\Pi}$ using an ancilla qubit.
  • Figure 3: The quantum circuit of $U_{\tilde{A}}$ and $U_{\tilde{f}}$.
  • Figure 4: The quantum circuits of $U_{\sigma(wx+b)}$ using QSVT.
  • Figure 5: The quantum circuit for $U_{A}$ is constructed with reference to $U_{\mathcal{L}}$ (the block-encoding of the linear differential operator) and $U_{\mathcal{B}}$ (the block-encoding of boundary conditions). Additionally, the operator $SELECT_{\mathcal{B}}$ specifies the control mechanism when $x_i$ lies on the boundaries.
  • ...and 4 more figures

Theorems & Definitions (7)

  • Example 1
  • Lemma 2
  • Lemma 3: Quantum singular value transformation (QSVT)
  • Example 2
  • Lemma 4
  • Theorem 5
  • proof