Quantum algorithms
Distributed Variational Quantum Linear Solver
A hybrid quantum-classical method that distributes a block-partitioned linear system across networked NISQ computers.
- Year
- 2026
- Status
- Submitted
- Topics
- Quantum computing · Distributed optimization · Linear systems
A conceptual illustration of the mathematical idea, not a plot of measured experimental results.
The question
Can several small near-term quantum computers cooperate to solve a linear system too large for any one of them?
Central insight
Partitioning the matrix is not sufficient: local quantum objectives must be coupled so that their solutions are consistent with one global linear system.
Approach
- 01
Partition the coefficient matrix into square blocks held by different quantum computers.
- 02
Run a local variational quantum linear solver at each computer.
- 03
Coordinate local variables through row- and column-neighbor communication graphs.
- 04
Use distributed classical optimization to enforce global consistency.
My contribution
- Co-developed the distributed problem formulation and algorithm.
- Connected local quantum cost functions with networked classical coordination.
- Validated the method through scalable numerical quantum simulations.
A closer look
Why a block partition needs coordination
Partitioning Ax = b creates smaller matrix blocks, but each block is still part of a coupled equation. An agent holding Aᵢⱼ cannot independently solve its own block and concatenate the answers. Agents in the same block column must agree on their shared solution subvector; agents in the same block row must coordinate contributions to that row’s residual.
The algorithm builds these two requirements into a distributed objective over connected row and column communication graphs. Local variational circuits represent vectors, while classical scale variables account for their magnitudes. Neighbor communication, gradient tracking, and adaptive updates coordinate the parameters without transferring quantum states between devices.
Reading the numerical evidence
The experiments track both global residual and agreement among agents. Agreement alone is insufficient: all agents could converge to the same inaccurate answer. One larger demonstration partitions a structured 51-qubit linear-system problem into four 50-qubit local systems, using matrix product states for classical simulation. Its low-entanglement structure makes that simulation tractable.
These are numerical quantum simulations, not a demonstration on a network of physical quantum processors. The variational optimizer and circuit ansatz influence performance, so the results do not supply a general worst-case speedup guarantee. Realistic hardware noise, communication delays, and asynchronous execution remain further evaluation questions.
Result
The solvable linear-system dimension scales with the number of participating quantum computers rather than the capacity of one device.
What remains
Study robustness to realistic hardware noise, communication limitations, and alternative block partitions.
Paper & resources
Distributed Variational Quantum Linear Solver
Tong Shen, Zeru Zhu, and Ji Liu
arXiv:2604.01426 · Submitted to IEEE CDC 2026 · Preprint / Submitted