Quantum algorithms

Distributed Variational Quantum Linear Solver

A hybrid quantum-classical method that distributes a block-partitioned linear system across networked near-term quantum processors.

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Year: 2026
Status: Submitted
Topics: Quantum computing · Distributed optimization · Linear systems

The project in brief

Can several small near-term quantum computers cooperate to solve a linear system too large for any one of them?

Can several small quantum processors represent a solution to a linear system too large for one of them? Here the matrix is partitioned across agents, and only classical messages pass between processors.

The central connection is a least-squares identity: a graph Laplacian removes disagreement between local residuals while preserving their average. This gives a distributed objective; quantum circuits then parameterize its unknown vectors. The identity is exact, whereas training those circuits remains a nonconvex optimization problem.

A block matrix distributed across a grid of quantum computers

A conceptual illustration, not a plot of experimental results.

Objective

One global problem, smaller local state spaces

$A$ is a real $2^n\times2^n$ matrix and $b\in\mathbb R^{2^n}$ . Each agent holds one matrix block and uses $q$ system qubits to encode a local vector. Here $n$ counts global system qubits; ancillary qubits for measurement circuits are additional.

\begin{aligned} x^\star &\in \operatorname*{arg\,min}_{x\in\mathbb R^{2^n}}\|Ax-b\|_2^2,\\ A&=[A_{ij}]_{i,j=1}^{m},\qquad A_{ij}\in\mathbb R^{2^q\times2^q},\\ m&=2^{n-q},\qquad \text{number of agents}=m^2. \end{aligned}

Manuscript · Problem Formulation

Question: Recover a global least-squares solution from block-local data, normalized quantum states, and classical communication.
Method: Laplacian residual splitting, quantum direction-plus-scale representations, and column-wise gradient tracking with Adam.
Progress: An exact reformulation lemma and three numerical studies: representation size, partitioning, and optimizer ablation. No hardware or quantum-speedup claim.

Explore the project

01 / Problem

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My contribution

Co-developed the distributed problem formulation and algorithm.
Connected local quantum cost functions with networked classical coordination.
Evaluated the construction in numerical simulations, including partition-size and optimizer comparisons.

Papers & resources