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Distributed Variational Quantum Eigensolver: Embarrassingly Parallel strategies on NISQ

This paper evaluates three embarrassingly parallel strategies for the Variational Quantum Eigensolver on NISQ devices using the CUNQA emulation platform, analyzing their trade-offs between speedup and accuracy in the presence of heterogeneous noise.

Original authors: Marta Losada, Daniel Faílde, Andrés Gómez

Published 2026-08-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Marta Losada, Daniel Faílde, Andrés Gómez

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The world of quantum computing is currently navigating a difficult but promising phase known as the "noisy intermediate-scale" era. In this period, the machines we have are powerful enough to perform complex calculations that classical computers struggle with, yet they are still plagued by imperfections. These imperfections, or "noise," cause the machines to make mistakes, limiting how long a calculation can run before the results become unreliable. To overcome these limits, scientists are exploring a strategy called distributed quantum computing. Instead of relying on a single, massive machine that does not exist yet, this approach connects several smaller, imperfect machines to work together on a single problem. The challenge lies in figuring out how to split the work among these different devices without letting their individual flaws ruin the final answer.

One of the most important tasks for these machines is finding the lowest energy state of a physical system, a process known as the Variational Quantum Eigensolver. Imagine trying to find the deepest valley in a vast, foggy landscape. The quantum computer acts as a guide that can sense the terrain, while a classical computer acts as the navigator, adjusting the path based on what the guide reports. This process requires the quantum machine to run the same calculation thousands of times to get a reliable reading. Because this task involves so many repetitions, it is an ideal candidate for splitting the work across multiple devices. However, if those devices are not identical—if one is noisier than the others—the way the work is divided can significantly change the speed and accuracy of the result.

Researchers at the Galicia Supercomputing Center in Spain set out to test exactly how to divide this work most effectively. They did not use physical quantum computers for this study, as the technology is still evolving. Instead, they used a sophisticated simulation platform called CUNQA to create a virtual environment. In this digital world, they built a pool of virtual quantum processors, each programmed to mimic the specific errors and noise patterns of real machines from different manufacturers, including IBM and OQC. By running their experiments in this controlled, simulated setting, they could isolate the effects of noise and test different strategies without the unpredictability of real hardware.

The team compared three distinct ways to distribute the workload. The first method, called shot-level distribution, involves taking a single calculation and splitting the total number of required measurements into smaller batches. These batches are sent to different virtual machines, and the results are combined later. The second method, circuit-level distribution, assigns entire, separate calculations to different machines. This is useful when a single step of the problem requires evaluating many different parts of the system simultaneously. The third method, candidate-level distribution, is designed for a specific type of problem-solving approach that tests many different potential solutions at once. In this case, each potential solution is sent to a different machine to be evaluated independently.

The researchers found that the best strategy depends heavily on the type of problem solver being used and the quality of the machines available. When the goal was to simply split the measurements, the results showed that this method works well only when the total number of measurements is very large. If the number of measurements is small, the time spent sending data back and forth between the machines outweighs the time saved by running them in parallel. However, when the workload involved running many separate calculations, splitting the circuits across different machines proved much more efficient.

A critical finding emerged regarding the impact of noise. The simulations revealed that not all machines are created equal, and mixing them requires care. When the researchers used a method that relies on precise geometric information to guide the search, the presence of even one noisy machine in the group could destabilize the entire process, causing the results to wobble or fail to converge. In contrast, a method that tests many different solutions at once proved more resilient. Because this approach evaluates many possibilities simultaneously, it can simply ignore the results from the noisiest machines and focus on the better ones. The study also showed that rotating which machine handles which part of the job helps prevent any single device from consistently skewing the results, leading to slightly faster and more reliable outcomes.

Ultimately, the work demonstrates that there is no single "one-size-fits-all" solution for connecting quantum computers. The most effective way to distribute a task depends on the specific algorithm being used and the characteristics of the hardware available. For some problems, splitting the measurements is sufficient, while for others, assigning whole tasks to different machines is far superior. The study suggests that as quantum technology matures and more devices are linked together, the ability to intelligently manage these differences will be just as important as the power of the machines themselves. By understanding how noise interacts with different distribution strategies, scientists can better prepare for a future where quantum computing is a collaborative, distributed effort rather than a solitary race for a single perfect machine.

Technical Summary: Distributed VQE: Embarrassingly Parallel Strategies on NISQ

Problem Statement
The Variational Quantum Eigensolver (VQE) is a leading algorithm for the Noisy Intermediate-Scale Quantum (NISQ) era, yet it faces significant scalability challenges due to its high demand for circuit executions. While Distributed Quantum Computing (DQC) offers a path to scalability by utilizing multiple Quantum Processing Units (QPUs), current hardware limitations—specifically the lack of quantum communication between devices—necessitate strategies where independent tasks are executed separately and combined via classical post-processing. A critical complication in this distributed setting is hardware noise heterogeneity. Different QPUs possess varying error profiles (gate errors, decoherence, readout inaccuracies), which can skew optimization trajectories, degrade final accuracy, and impact execution efficiency when tasks are distributed without regard to these backend differences.

Methodology
The authors investigate three "embarrassingly parallel" strategies for distributing VQE workloads across heterogeneous pools of virtual QPUs (vQPUs). The study utilizes the CUNQA platform, a DQC emulator designed for High-Performance Computing (HPC) environments, which supports configurable virtual QPUs with backend-dependent noisy simulations.

The experimental setup involves:

Target Problem: The one-dimensional Transverse Field Ising Model (TFIM) with $N=6$ spins, solved using a Hamiltonian Variational Ansatz (HVA) with 3 layers.
Hardware Emulation: A heterogeneous pool of four vQPUs emulating real IBM (Berlin, Boston, Marrakesh) and OQC (Qmio) backends using calibration data from June 2026.
Optimizers: Three distinct optimizers representing different parallelization needs:
1. COBYLA: A local, gradient-free optimizer requiring a single cost evaluation per step.
2. Quantum Natural Gradient (QNG): A gradient-based optimizer requiring extensive circuit evaluations for energy, gradients, and the Quantum Geometric Tensor (QGT).
3. Differential Evolution (DE): A population-based, gradient-free global optimizer evaluating multiple candidate vectors per generation.

Three Parallelization Strategies Evaluated

Shot-level Distribution: The total shot budget ( $M$ ) for a single circuit execution is split into batches and distributed across available QPUs. Partial counts are aggregated to compute expectation values.
Circuit-level Distribution: Independent circuits required for a single optimization step (e.g., different Hamiltonian terms, gradient shifts, or QGT elements) are assigned to different QPUs.
Candidate-level Distribution: Specific to population-based optimizers (DE), where each candidate parameter vector in the population is assigned to a distinct QPU for full cost evaluation.

The study also evaluates the impact of task assignment permutation, where the mapping of tasks to QPUs is rotated across optimization steps to mitigate systematic bias from specific backend noise profiles.

Key Results

Optimization Paths and Accuracy:
- Shot-level: Results reflect the average quality of the device pool. All optimizers are similarly affected by the inclusion of lower-quality backends, as the noise is averaged across the aggregated counts.
- Circuit-level: This strategy shows high sensitivity to backend heterogeneity, particularly for QNG. The instability of QNG when local geometry is ill-conditioned is exacerbated by noise in the few QGT evaluations, leading to oscillatory convergence. DE remains resilient under this strategy because its selection mechanism can discard candidates whose cost estimates are degraded by poor backends.
- Candidate-level: Similar to circuit-level, DE benefits from the selection mechanism. Permutation strategies (rotating task assignments) provide slight improvements by preventing specific candidates from being persistently penalized by poor backends.
Speedup and Parallel Efficiency:
- Shot-level: Scales poorly at low sampling budgets due to distribution and aggregation overhead dominating execution time. Efficiency improves significantly at higher shot counts, suggesting this strategy is most beneficial in measurement-intensive regimes.
- Circuit-level: Performs well for QNG and DE, as these optimizers naturally require multiple independent circuit evaluations per step, allowing for effective parallelization. QNG shows slightly better scaling due to its higher circuit count per step.
- Candidate-level: Effectiveness increases with the population size of the DE optimizer, as more independent candidates can be evaluated concurrently.

Significance and Claims
The paper claims that the suitability of a distributed VQE strategy is not determined solely by the number of available QPUs but depends critically on the optimizer structure, task granularity, and backend noise heterogeneity.

Optimizer Dependency: Gradient-based methods like QNG are highly sensitive to noise in specific circuit evaluations (e.g., QGT terms), making them vulnerable in heterogeneous pools without careful scheduling. Population-based methods like DE exhibit greater resilience due to their inherent selection mechanisms.
Granularity Trade-offs: Shot-level parallelization is effective for reducing measurement time but suffers from overhead at low shot counts. Circuit and candidate-level parallelization offer better scaling for optimizers that naturally generate multiple independent tasks.
Noise Management: The study demonstrates that while permutation strategies (rotating task assignments) can mitigate systematic bias, they do not fully eliminate the negative impact of heterogeneous noise on convergence stability, particularly for sensitive gradient-based optimizers.

The work provides a systematic comparison of distribution levels using CUNQA, establishing that the optimal strategy must be chosen based on the specific interplay between the optimization algorithm's requirements and the characteristics of the available quantum hardware.

Technical Summary: Distributed VQE: Embarrassingly Parallel Strategies on NISQ

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