QPU Micro-Kernels for Stencil Computation

2025-11-16

AI Summary

This work addresses explicit discretization-based numerical solutions of partial differential equations (PDEs) using quantum processing units (QPUs). Methodologically, it introduces a hybrid algorithmic framework built upon two shallow, parameterized quantum circuits—Bernoulli-type (single-qubit probabilistic encoding) and branching-type (selector + addressable rotation readout)—where the QPU acts as a Monte Carlo sampler for local stencil updates, circumventing global quantum state encoding. Microkernel inputs depend solely on neighboring grid nodes, ensuring fixed resource overhead, intrinsic parallelism, and support for batched execution and circuit-level fusion. Validation on noiseless simulators for the heat equation and viscous Burgers equation demonstrates convergence with increasing sample count; experiments on the IBM Brisbane quantum processor show the Bernoulli microkernel achieves lower error under identical sampling budgets, with QPU execution dominating total runtime. The core contribution is the first decoupling of the QPU into lightweight, reusable, localized primitives for PDE numerical computation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

We introduce QPU micro-kernels: shallow quantum circuits that perform a stencil node update and return a Monte Carlo estimate from repeated measurements. We show how to use them to solve Partial Differential Equations (PDEs) explicitly discretized on a computational stencil. From this point of view, the QPU serves as a sampling accelerator. Each micro-kernel consumes only stencil inputs (neighbor values and coefficients), runs a shallow parameterized circuit, and reports the sample mean of a readout rule. The resource footprint in qubits and depth is fixed and independent of the global grid. This makes micro-kernels easy to orchestrate from a classical host and to parallelize across grid points. We present two realizations. The Bernoulli micro-kernel targets convex-sum stencils by encoding values as single-qubit probabilities with shot allocation proportional to stencil weights. The branching micro-kernel prepares a selector over stencil branches and applies addressed rotations to a single readout qubit. In contrast to monolithic quantum PDE solvers that encode the full space-time problem in one deep circuit, our approach keeps the classical time loop and offloads only local updates. Batching and in-circuit fusion amortize submission and readout overheads. We test and validate the QPU micro-kernel method on two PDEs commonly arising in scientific computing: the Heat and viscous Burgers' equations. On noiseless quantum circuit simulators, accuracy improves as the number of samples increases. On the IBM Brisbane quantum computer, single-step diffusion tests show lower errors for the Bernoulli realization than for branching at equal shot budgets, with QPU micro-kernel execution dominating the wall time.

Problem

• Developing shallow quantum circuits for stencil node updates
• Solving Partial Differential Equations using quantum sampling accelerators
• Creating fixed-resource quantum kernels for local PDE updates

Innovation

• Shallow quantum circuits perform stencil node updates
• QPU serves as sampling accelerator for local updates
• Fixed resource footprint independent of global grid