Distributed Variational Quantum Linear Solver

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 to $4^n$ in the worst case for an $n$-qubit system. We address this computational bottleneck through two complementary strategies. First, we present a distributed VQLS (D-VQLS) framework (https://code.ornl.gov/olcf-qcfd/DVQLS.git), built on NVIDIA CUDA-Q, that enables asynchronous, scalable distribution of the $O(L^2)$ cost evaluations. Second, a fast Walsh--Hadamard transform (FWHT)-based Pauli decomposition with coefficient-amplitude pruning threshold $\tau=0.01$ curbs LCU growth for the structured Toeplitz family, reducing $L$ from $O(2^n)$ to 64 for $n>6$ and compressing the circuit complexity per optimizer iteration from $O(n4^n)$ to $O(n)$. We derive the exact top-$L$ Frobenius error and connect it to worst-case solution error. For a 10-qubit tridiagonal Toeplitz system, the $L=64$ pruning yields a $256\times$ reduction---from 23 million to 90k circuits per optimizer iteration. 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 more than 360k circuits per optimizer iteration (from larger-$L$ pruning) for the 10-qubit system. Systematic profiling identifies the optimal resource allocation for distributed quantum circuit workloads, yielding a $2.52\times$ speedup for the configurations studied.