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Benchmarking Classical, Hybrid, and Quantum Annealing Workflows on a Structured Control QUBO

This paper benchmarks classical, hybrid, and quantum annealing workflows on a structured binary control QUBO derived from greenhouse heater scheduling, finding that while classical and simulated quantum methods consistently yield near-optimal solutions, current quantum hardware and hybrid solvers do not yet demonstrate a performance advantage over classical baselines.

Original authors: Hamze Alavirad, Maryam Bahrami Zanjani

Published 2026-07-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Hamze Alavirad, Maryam Bahrami Zanjani

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine you are trying to solve a massive, tangled knot of string. In the world of science, this is what we call an "optimization problem." You have a goal—like finding the shortest path home, packing a suitcase perfectly, or, in this case, keeping a greenhouse warm enough for plants without burning a hole in your electricity bill. For decades, we've used powerful classical computers to untangle these knots by testing millions of possibilities, a bit like a very fast, very tired librarian checking every book on a shelf to find the right one.

But recently, a new kind of machine has entered the library: the quantum computer. Instead of reading books one by one, these machines use the strange rules of quantum physics to "feel" their way through the knot, hoping to find the loose end instantly. The big question everyone is asking is: Do these quantum machines actually untangle the knot faster or better than our old, reliable librarians? To find out, scientists need a fair test. They can't just ask the quantum computer to solve a math problem on a piece of paper; they need to see if it can handle a real-world scenario, like controlling a complex system, and if the answer it gives actually makes sense in the real world. This is where the story of a "quantum greenhouse" begins.

The Great Greenhouse Heaters Race

In this study, researchers Hamze Alavirad and Maryam Bahrami Zanjani set up a high-stakes race to see who can best control the heaters in a virtual greenhouse. Imagine a greenhouse that needs to stay cozy for its plants. The temperature outside changes, the sun comes and goes, and the plants have a specific "happy zone" where they grow best. The challenge is to decide, hour by hour, whether to turn the heater on or off for a whole day (24 hours). Turn it on too much, and you waste energy; turn it off too much, and the plants get cold. It's a tricky balancing act.

To make this a fair test for computers, the researchers turned this heating schedule into a giant puzzle called a QUBO (Quadratic Unconstrained Binary Optimization). Think of this as translating the problem into a language of only zeros and ones, where "1" means "heater on" and "0" means "heater off." The goal is to find the perfect sequence of ones and zeros that keeps the plants happy and the energy bill low.

The researchers put four different "contestants" into the ring to solve this 24-hour puzzle:

The Exact Solver: This is the "gold standard." It's like checking every single possible combination of heater schedules (there are billions of them) to find the absolute perfect answer. It takes a long time, but it knows the truth.
Classical Simulated Annealing (SA): This is a smart, old-school algorithm. Imagine a hiker trying to find the lowest point in a foggy valley. The hiker takes random steps, sometimes going uphill to escape a local dip, hoping to eventually find the deepest valley.
Path-Integral Simulated Quantum Annealing (PIA): This is a "simulated" quantum computer running on a regular computer. It tries to mimic the spooky quantum behavior of the hiker, allowing them to "tunnel" through hills instead of climbing over them, hoping to find the bottom faster.
The Real Quantum Contenders: These are the actual quantum machines from D-Wave. They ran two types of tests: a "Hybrid" workflow (where a quantum chip works with a classical computer) and a "Direct" workflow (where the quantum chip tries to solve the whole thing alone, but only on smaller, easier versions of the puzzle).

The Results: Who Won the Race?

When the race was over for the full 24-hour day, the results were a bit surprising for the quantum fans.

The Classical Champions: Both the old-school "hiker" (SA) and the "quantum simulator" (PIA) did an excellent job. They found solutions that were almost perfect, very close to the "gold standard" answer. They managed to keep the greenhouse at the right temperature while saving energy, and they did this reliably every single time they ran the test.

The Hybrid Struggle: The D-Wave Hybrid workflow, which was expected to be a powerhouse, didn't quite keep up. Under the time limits tested (15 to 60 seconds), it found solutions that were feasible (the plants didn't freeze), but they weren't as good as the classical computers. The hybrid solutions used more energy and didn't grow the plants as well. Even when the researchers gave the hybrid solver more time (up to 60 seconds), it didn't magically get better. It seemed to get stuck in a "good enough" spot rather than finding the "best" spot.

The Direct Quantum Test: For the smaller, easier puzzles (representing 10, 12, or 14 hours instead of 24), the direct quantum processor showed some promise. It managed to find the perfect answer 5 times out of 10 for the 10-hour puzzle and 2 times out of 10 for the 12-hour puzzle. However, as the puzzle got slightly bigger (14 hours), the quantum machine stopped finding the perfect answer entirely. While it never gave a "bad" answer that would freeze the plants, it became less reliable at finding the best answer as the problem grew.

The Takeaway: No Magic Bullet (Yet)

The most important thing this paper tells us is that, for this specific type of problem, quantum computers did not beat the classical computers.

The researchers were very careful to say that this doesn't mean quantum computers are useless. It just means that for this specific "greenhouse heating" puzzle, the classical methods are still the champions. The quantum machines were able to find solutions that worked (the plants stayed warm), but they weren't as efficient or as consistent as the classical algorithms.

The study also highlights a tricky part of using quantum computers: it's not enough to just get an answer from the machine. You have to decode that answer back into the real world (checking if the heater schedule actually works). The researchers found that the quantum machines sometimes gave answers that looked okay on paper but weren't the best when you checked the real-world physics.

In short, this paper is a reality check. It shows that while quantum computers are exciting and can solve small versions of these problems, they aren't yet the "super-solvers" that will instantly replace our classical computers for complex tasks like managing a greenhouse. The classical "hikers" are still walking the path with the best map. The quantum machines are still learning how to navigate the terrain, and for now, they need a little more practice before they can claim the crown.

Technical Summary: Benchmarking Classical, Hybrid, and Quantum Annealing Workflows on a Structured Control QUBO

Problem Formulation
The paper addresses the challenge of evaluating annealing-based optimization workflows (classical, quantum-inspired, hybrid, and direct quantum) on structured control problems. The authors formulate a reduced greenhouse heater scheduling problem as a Quadratic Unconstrained Binary Optimization (QUBO) model. The objective is to determine a binary heater schedule ( $u_k \in \{0, 1\}$ ) over a time horizon $H$ that maximizes a growth proxy while minimizing energy costs and switching events, subject to hard air-temperature constraints.

The physical model tracks air temperature ( $T_{air}$ ) and body/canopy temperature ( $T_{body}$ ) using discrete-time linear dynamics driven by outdoor temperature and solar radiation. The growth proxy is a quadratic function of a weighted temperature, penalizing deviations from a preferred growth temperature. The optimization problem is encoded into a QUBO where:

$Q_{growth}$ : Maximizes the growth proxy.
$Q_{energy}$ : Minimizes electricity consumption based on time-of-use tariffs.
$Q_{switch}$ : Penalizes heater state changes.
$Q_{bound}$ : Enforces temperature bounds ( $T_{min} \le T_{air} \le T_{max}$ ) using quadratic penalties and binary slack variables ( $M=6$ bits per bound).

A critical distinction in this work is the evaluation metric: raw QUBO energy is not the final performance criterion. Instead, all candidate binary solutions are decoded back into physical control trajectories and evaluated against the original simulator-side objective and feasibility constraints.

Methodology
The study benchmarks four distinct workflows on a primary instance with a 24-hour horizon ( $H=24$ , $\Delta t = 1$ hour), resulting in a BQM with 312 binary variables (24 control + 288 slack) and 4,596 couplers.

Exact Enumeration: Serves as the ground-truth baseline, exhaustively searching all $2^{24}$ schedules to find the optimal decoded solution.
Classical Simulated Annealing (SA): Uses a thermal annealing sampler on the binary quadratic model.
Path-Integral Simulated Quantum Annealing (PIA): A classical algorithm simulating quantum tunneling effects via path integrals.
D-Wave Leap Hybrid BQM: A hybrid quantum-classical workflow that decomposes the logical BQM for internal processing.
Direct QPU Execution: Performed on reduced instances ( $H=10, 12, 14$ ) using a D-Wave Advantage2 system. These require minor-embedding of the logical model into the physical qubit graph.

All stochastic solvers were executed with 10 independent repetitions. Performance was measured by the decoded objective value ( $J_{total}$ ), total energy, total growth, number of temperature violations, and switching events.

Key Results

Main Benchmark ( $H=24$ ):
- Exact Enumeration provided the reference optimum ( $J_{total} = -145.14$ ).
- Classical Baselines (SA and PIA): Both produced feasible, near-optimal solutions in all 10 repetitions. SA achieved a best objective of $-150.11$ (gap $\approx 4.97$ ), and PIA achieved $-149.11$ (gap $\approx 4.03$ ).
- Hybrid Workflow: The D-Wave Leap Hybrid solver was less reliable, producing feasible solutions in only 7/10 runs. Its best feasible objective was $-174.32$ (gap $\approx 29.18$ ), significantly worse than the classical baselines. Increasing the requested solver time from 15s to 60s did not yield monotonic improvements in decoded performance.
- Observation: The hybrid workflow did not outperform classical annealing baselines under the tested time limits and formulation. The performance gap appeared linked to the trade-off between growth, energy, and switching rather than a failure to satisfy constraints.
Reduced Instances (Direct QPU):
- For $H=10$ and $H=12$ , the direct QPU execution recovered the exact optimum in 5/10 and 2/10 runs, respectively.
- For $H=14$ , no exact optima were recovered, though the best solution remained close to optimal.
- Trend: As the horizon increased, the reliability of the QPU (measured by exact-hit rate and mean decoded objective) decreased. However, all decoded QPU solutions remained feasible (zero temperature violations). The degradation was characterized by a loss of optimality reliability rather than a loss of feasibility.

Contributions and Significance
The authors explicitly state that the paper does not claim quantum advantage. Instead, the significance lies in providing a reproducible benchmark for evaluating annealing-based workflows on structured control QUBOs.

Decoded Evaluation: The work emphasizes that for application-motivated problems, solver performance must be judged on the decoded physical trajectory and simulator-side objective, not just raw BQM energy. This highlights how embedding, penalty scaling, and sampling behavior affect the relationship between the mathematical model and physical performance.
Baseline Comparison: The study demonstrates that for moderate-sized structured control problems ( $H=24$ ), classical annealing methods (SA and PIA) currently serve as strong baselines that outperform the tested hybrid quantum-classical workflow.
Hardware Diagnostics: The direct-QPU results on reduced instances show that while current quantum hardware can find feasible solutions and occasionally exact optima for small instances, its reliability degrades with problem size, and it does not yet surpass classical methods in mean performance for these specific formulations.

Limitations
The paper acknowledges several limitations: the greenhouse model is a simplified proxy (single heater, no ventilation/humidity/CO2); the $H=24$ instance is too small to test the full scalability of hybrid decomposition workflows; and the penalty scaling for constraints was chosen empirically without a systematic sensitivity study. The results are intended as diagnostic benchmarks rather than definitive proofs of superiority for any specific solver.

The Great Greenhouse Heaters Race

The Results: Who Won the Race?

The Takeaway: No Magic Bullet (Yet)

Technical Summary: Benchmarking Classical, Hybrid, and Quantum Annealing Workflows on a Structured Control QUBO

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