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Quantum-Based Optimization of Gas Throughput in Natural Gas Transmission Networks Under Hydraulic Constraints Using QAOA

This paper demonstrates an end-to-end proof of concept for optimizing natural gas network throughput under hydraulic constraints using the Quantum Approximate Optimization Algorithm (QAOA), successfully recovering valid solutions on both a simulator and the IonQ Forte-1 quantum processor with significantly fewer circuit layers than expected.

Original authors: Alex Ben Ishay, Yuval Eyal, Yuval Cohen, Nati Erez

Published 2026-09-02

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Alex Ben Ishay, Yuval Eyal, Yuval Cohen, Nati Erez

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

Imagine a vast, invisible river of gas flowing through a network of steel pipes buried beneath the earth, connecting powerful supply stations to the homes and factories that rely on them. Moving this gas is not as simple as turning a valve; the physics of the flow are complex and unforgiving. The gas must maintain specific pressures to move efficiently, yet it cannot be pushed so hard that it damages the pipes or fails to reach the customers at the end of the line. Engineers currently manage these networks by manually testing different pressure settings on sophisticated computer simulations, a process that is slow, relies heavily on human experience, and often stops short of finding the absolute best way to move the most gas possible. As these networks grow larger and more intricate, the number of possible pressure combinations becomes so vast that even the fastest classical computers struggle to find the perfect solution in a reasonable time.

This is where a new approach, blending the emerging power of quantum computing with traditional engineering, steps in. The researchers behind this study tackled the problem of maximizing gas throughput in a transmission network by treating the pressure at every junction as a variable to be optimized. They focused on a specific, widely used formula that describes how gas flows through pipes, known as the Panhandle-B equation, which relates the flow rate to the difference in pressure squared. Instead of trying to solve the continuous, fluid-like equations directly, the team broke the problem down into a series of discrete steps. They imagined the possible pressure levels at each node as a set of distinct choices, turning the physical challenge into a massive puzzle of finding the single best combination of choices that satisfies all the physical rules while delivering the most gas to customers.

To solve this puzzle, the team employed an algorithm called the Quantum Approximate Optimization Algorithm, or QAOA. This method uses the unique properties of quantum bits, which can exist in a state of superposition, to explore many different pressure configurations simultaneously. The researchers built a mathematical model that rewarded solutions delivering high gas volume and penalized those that violated physical laws, such as flow moving backward or pressure dropping too low. They first tested this model on a simulator, a digital environment that mimics a quantum computer. Using a network with six nodes and five pipeline segments, the simulation successfully identified the optimal operating point. The algorithm found the specific pressure settings that maximized the total gas delivered to customers, matching the results of a classical computer that had checked every single possible combination one by one.

The true test of this work, however, went beyond the safety of a simulation. The researchers took a simplified version of their problem and ran it on a physical quantum processor, a real machine using trapped ions to perform calculations. To make the problem small enough for today's hardware, they fixed the pressure at the customer endpoints and reduced the number of decision points to just two intermediate junctions. This reduced the complexity enough to fit on the machine, which operated with a very shallow circuit, using only two layers of quantum operations. In the noisy, imperfect environment of current quantum hardware, one might expect the results to be garbled or useless. Instead, the machine produced two distinct, valid solutions. These solutions were physically meaningful and, remarkably, they bracketed the true optimal answer found by classical methods. One solution was just slightly below the ideal pressure setting, and the other was just slightly above it.

This outcome suggests that even with the significant limitations of today's quantum devices, such as circuit depth restrictions and hardware noise, the approach can yield useful, interpretable results. The researchers found that the quantum processor could identify operating points that were within a single step of the ideal continuous solution, effectively narrowing down the search space for engineers. The work does not claim to replace the high-fidelity hydraulic simulations that engineers use for final verification, but rather to act as a powerful partner. The quantum method can rapidly scan the vast landscape of possible pressure settings to find the most promising candidates, which can then be refined and validated by classical tools. By demonstrating that a real quantum processor can successfully navigate a constrained, real-world engineering problem, the study provides a proof of concept that quantum-assisted optimization is moving from theory into practice, offering a glimpse of a future where complex network management is accelerated by hybrid computing.

Technical Summary: Quantum-Based Optimization of Gas Throughput in Natural Gas Transmission Networks

Problem Statement
Maximizing gas throughput in natural gas transmission networks under hydraulic and operational constraints is a combinatorial optimization problem. The complexity of finding an exact solution grows exponentially with network size, rendering traditional exhaustive search computationally intractable for large systems. Current engineering practice relies on iterative, manual adjustments of nodal pressures within high-fidelity classical hydraulic simulators. This workflow is time-consuming, dependent on operator expertise, and offers no guarantee of global optimality. The specific challenge addressed in this paper is to determine the optimal nodal pressure profile that maximizes total weighted customer delivery while satisfying the non-linear Panhandle-B flow equation, mass conservation at junctions, directional flow consistency, and minimum pressure requirements at customer endpoints.

Methodology
The authors formulate the problem as a search over discretized nodal-pressure assignments, mapping the continuous physical variables to a discrete state space suitable for quantum algorithms.

Mathematical Formulation: The problem is modeled on a directed graph $G=(N, E)$ . The objective is to maximize weighted customer throughput. Constraints include the Panhandle-B equation (relating flow to the 0.51 power of pressure differences), mass conservation at intermediate junctions, and flow directionality.
Discretization: Nodal pressures are discretized using $n$ qubits per node, converting the continuous pressure variable $\pi_i$ into an integer $q_i$ . This transforms the problem into a combinatorial integer program.
QUBO Conversion: The problem is encoded into a Quadratic Unconstrained Binary Optimization (QUBO) formulation. A cost Hamiltonian $H(q)$ is constructed to jointly encode the delivery objective and physical constraint penalties (mass conservation, flow direction). The non-linear Panhandle-B exponent (0.51) is approximated using a second-degree polynomial to maintain the quadratic structure required for QAOA.
Algorithm: The Quantum Approximate Optimization Algorithm (QAOA) is employed to search the solution space. The algorithm alternates between a cost layer (encoding the Hamiltonian) and a mixer layer (promoting exploration) over $p$ layers. A classical optimizer (COBYLA) tunes the variational parameters ( $\gamma, \beta$ ) to minimize the expected cost.
Implementation: The workflow is implemented using the Classiq quantum software platform, which handles the domain-specific modeling (Qmod), circuit synthesis, and optimization.

Key Contributions

Unified Formulation: The paper presents a unified combinatorial optimization framework for gas network throughput maximization, incorporating the Panhandle-B flow law, junction conservation, and pressure constraints into a single QUBO expression.
Novel Cost Hamiltonian: A specific cost Hamiltonian is developed that encodes both the maximization of weighted delivery and the satisfaction of physical constraints, enabling a single-expression QAOA formulation.
Simulation Validation: The method is validated on a representative six-node, five-pipe network. The QAOA implementation (using $p=30$ layers) successfully recovered the maximum-throughput valid operating point, consistent with classical exhaustive enumeration and hydraulic simulation reference solutions.
Hardware Demonstration: The work provides an end-to-end execution of a reduced problem instance on the IonQ Forte-1 trapped-ion quantum processor. This represents a significant step in demonstrating the feasibility of the approach on near-term hardware.

Results

Simulator Performance: In simulator-based experiments with $p=30$ layers, the QAOA algorithm identified the optimal operating point (maximizing flow to the highest-weighted customer, C3). The solution was consistent with classical reference solutions, with discrepancies attributed primarily to pressure discretization (using 3 qubits per node), differences in friction modeling (Panhandle-B vs. Darcy-Weisbach), and thermal modeling assumptions.
Hardware Performance: A reduced instance was executed on the IonQ Forte-1 processor. To accommodate hardware limitations, the circuit depth was reduced to $p=2$ $p = 2$ layers, and the problem size was minimized to two decision variables (six qubits) plus four auxiliary qubits.
- Despite the shallow circuit depth and hardware noise, the Quantum Processing Unit (QPU) produced physically valid candidate solutions.
- The two highest-probability valid states found on the hardware bracketed the continuous classical optimum, with each state lying within one pressure-discretization step of the true optimum.
- This demonstrates that meaningful optimization behavior can be extracted from significantly shallower circuits than those required for noiseless simulation.

Significance and Claims
The paper claims to provide a "proof of concept" for near-term quantum-assisted gas-network optimization. Its primary significance lies in demonstrating that a hybrid quantum-classical workflow can be successfully executed on current trapped-ion hardware to solve a relevant engineering problem. The authors emphasize that the approach does not aim to immediately replace high-fidelity classical hydraulic simulations but rather to serve as a complementary tool: QAOA performs an automated global search of the combinatorial space to identify promising operating points, which are then validated by classical engineering tools.

The results suggest that useful optimization behavior is achievable before deep QAOA circuits become practical. The authors note that as qubit resources grow and discretization becomes finer, the dominant source of error (pressure quantization) will diminish, potentially bringing quantum solutions closer to classical hydraulic simulation references. The work establishes a foundation for future extensions, including scaling to larger networks, incorporating transient flow dynamics, and developing production pipelines that integrate quantum optimization with classical simulation software.

Technical Summary: Quantum-Based Optimization of Gas Throughput in Natural Gas Transmission Networks

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