Distributed Variational Quantum Linear Solver

2026-04-15

AI Summary

This work addresses the scalability bottleneck of the Variational Quantum Linear Solver (VQLS), whose conventional LCU decomposition incurs O(L²) circuit evaluation overhead, hindering application to large-scale problems. The authors propose the first distributed asynchronous VQLS framework, integrating the fast Walsh–Hadamard transform (FWHT) with thresholding of LCU coefficients to compress an exponential number of Pauli terms down to a constant order, drastically reducing the number of circuits required per optimization round. Implemented via NVIDIA CUDA-Q for multi-GPU state-vector simulation, the approach reduces the number of circuits per iteration from 23 million to 90 thousand on a 10-qubit system while maintaining over 99.99% fidelity. On a 96-GPU cluster, it achieves 95.3% weak scaling efficiency and a 2.52× speedup, effectively overcoming the scalability limitations of VQLS.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

The Variational Quantum Linear Solver (VQLS), a hybrid quantum-classical algorithm for solving linear systems, faces a practical scalability bottleneck: the Linear Combination of Unitaries (LCU) decomposition requires O(L^2) circuit evaluations per optimizer iteration, where $L$ can grow as 4^n for n-qubit systems for the worst case scenario. We address this computational bottleneck through two complementary strategies. First, we present a distributed VQLS (D-VQLS) framework, built on NVIDIA CUDA-Q, that enables asynchronous, scalable distribution of the O(L^2) cost-function evaluations. Second, a fast Walsh--Hadamard transform (FWHT)-based Pauli decomposition with 1% coefficient thresholding curbs the exponential growth of LCU terms, reducing L from O}(2^n) to O(1) for n > 6 qubits and compressing the per-iteration circuit complexity from O(n * 4^n) to O(n) for sparse, structured matrices. For a 10-qubit tridiagonal Toeplitz system, this yields a 256x reduction, from 23 million to 90,112 circuits per iteration, while preserving over $99.99\%$ solution fidelity. Additionally, to inform feasibility on early fault-tolerant QPUs, the paper provides resource estimates -- gate counts, qubit requirements, and circuit evaluations per iteration -- for VQLS applied to arbitrary matrices. The D-VQLS framework is validated on the NERSC Perlmutter supercomputer using multi-node, multi-GPU ideal state-vector simulations, achieving over 99.99% fidelity against classical solutions on tridiagonal Toeplitz and Hele--Shaw flow benchmarks, with near-ideal strong scaling up to 24 GPUs and 95.3% weak scaling efficiency at 96 GPUs processing 360,448 circuits per iteration for a 10-qubit system. Systematic profiling identifies the optimal resource allocation for distributed quantum circuit workloads, yielding a 2.52x speedup for the configurations studied.

Problem

• Variational Quantum Linear Solver
• scalability bottleneck
• Linear Combination of Unitaries
• quantum circuit evaluation
• exponential growth

Innovation

• Distributed VQLS
• Fast Walsh–Hadamard Transform
• LCU Compression
• Scalable Quantum Simulation
• Hybrid Quantum-Classical Algorithm