Dynamic Optimization on Quantum Hardware: Feasibility for a Process Industry Use Case

TL;DR

This work tackles real-time dynamic optimization with differential-algebraic equations embedded, focusing on feasibility on quantum hardware. It reformulates the dynamic problem into a QUBO by transforming DAEs to an ODE form and binarizing variables, then benchmarks classical IP-based solvers, simulated annealing, quantum annealing on D-Wave, and hybrid solvers on a CSTR use case. The results show that current quantum annealing does not yet outperform state-of-the-art classical solvers, largely due to embedding and hardware limitations, though hybrid methods show promise to handle embedding and decomposition; hardware limitations in qubit connectivity and embedding times are main bottlenecks. The findings point to a path forward where hardware advances and constraint-aware QUBO formulations, plus effective problem decomposition, could unlock practical quantum-assisted optimization for process industries.

Abstract

The quest for real-time dynamic optimization solutions in the process industry represents a formidable computational challenge, particularly within the realm of applications like model-predictive control, where rapid and reliable computations are critical. Conventional methods can struggle to surmount the complexities of such tasks. Quantum computing and quantum annealing emerge as \textit{avant-garde} contenders to transcend conventional computational constraints. We convert a dynamic optimization problem, {characterized by an optimization problem with a system of differential-algebraic equations embedded}, into a Quadratic Unconstrained Binary Optimization problem, enabling quantum computational approaches. The empirical findings synthesized from classical methods, simulated annealing, quantum annealing via D-Wave's quantum annealer, and hybrid solver methodologies, illuminate the intricate landscape of computational prowess essential for tackling complex and high-dimensional dynamic optimization problems. Our findings suggest that while quantum annealing is a maturing technology that currently does not outperform state-of-the-art classical solvers, continuous improvements could eventually aid in increasing efficiency within the chemical process industry.

Figures (6)

• Figure 1: Flowsheet of the Continuous Stirred Tank Reactor with inflow rate $F^0$, temperature $T^0$ and concentration $c^0$. The coolant temperature is denoted by $T^\mathrm{c}$, the height of the reactor by $h$ and the outflow rate by $F$, concentration $c$ and temperature by $T$.
• Figure 2: Comparison of concentration (a), reactant temperature (b), and cooling temperature (c) across three distinct methodologies for the reactor case study: the classical Ipopt solution of the DOP-ODE (blue, solid line), the best simulated annealing result for the QUBO (orange, dashed line), and the Gurobi solution of the QUBO (green, dotted line). Subplot (d) showcases the optimization error in $T^c$ per timestep, visualized as a colormap, under varying conditions of simulated annealing events and binarization bits for $T^c$. Subplot (e) depicts the computational time necessary to achieve these results, providing a comprehensive performance comparison.
• Figure 3: Two representations of the optimization problem for a scenario with three timesteps and a binarization resolution of four bits for the cooling temperature $T^c$: (a) showcases the $Q$ matrix, encapsulating the linear and quadratic coefficients of the problem, while (b) illustrates the corresponding graph network, highlighting the interconnections between binary variables.
• Figure 4: Results for the CSTR model using D-Wave’s Advantage system. (a) Illustrates the number of embedded qubits (blue dots) as a function of the original QUBO problem size (number of binary variables), which showcases the increased complexity and qubit requirements for larger problems. Additionally, it shows the embedding time on classical hardware (red crosses). (b) Depicts the error in $T^c$ per timestep across various problem sizes (number of binaries per $T^c$, number of anneals, and anneal time, as well as different chain strengths between 50 and 5000), where the color intensity corresponds to the QPU access time; longer compute-times (darker) generally correlate with lower errors. (c) Provides a colormap representation of the error in $T^c$ per timestep for a set of fixed simulation parameters (chain strength fixed at 50, annealing time fixed at 15 $\mu$s) and varying the number of anneals to achieve different total QPU access times, which are grouped into the illustrated timeslots, demonstrating a trend towards improved results for longer computations and better representations (number of binaries for $T^c$).
• Figure 5: (a) displays a flowchart following Footnote \ref{['fn:dwave_hybrid']}, illustrating the Kerberos algorithm's iterative routine. An initial input undergoes simultaneous computation through various solvers: a classical interruptible tabu search addressing the full problem, running in tandem with branches that subdivide the problem employing solvers on quantum processors and additional setups of parallel simulated annealing and tabu search. Decomposers fragment the problem, samplers tackle each piece, and composers merge the partial solutions back into a whole. This concurrent activity yields an optimized sample collection for the next iteration. (b) A colormap demonstrating the error in coolant temperature $T^c$ across each time step in relation to the count of anneals and annealing times per cycle, revealing improved performance with more extended annealing durations. Blank fields indicate a lack of data due to incompatibility between QPU access time and total runtime.