Quantum Simulation-Based Optimization for Cooling System Design

Abstract

Engineering design processes involve iterative design evaluations requiring numerous computationally intensive numerical simulations. Quantum algorithms promise substantial speedups for specific tasks relevant to engineering simulations. However, these advantages quickly vanish when considering data input and output on quantum computers. The recently introduced Quantum Simulation-Based Optimization (QuSO) framework circumvents this limitation by treating simulations as subproblems within a larger optimization problem. Here we adapt and implement QuSO for a simplified cooling system design problem, validate correctness in statevector simulations, and present a detailed gate-level complexity analysis for a single QuSO iteration. We express the scaling in terms of problem parameters and QAOA depth and iterations. We show that the cost function can be coherently computed over a superposition of exponentially many configurations using circuits of polynomial complexity. This does not yield a speedup for a single simulation instance, but it enables potential advantages arising from the subsequent QAOA-based search over configurations. The study serves as a proof-of-concept for integrating fault-tolerant quantum subroutines with simulation-based optimization in engineering workflows, clarifying both promise and practical limitations.

Figures (14)

• Figure 1: Cooling system model. (a) Example cooling system with an engine and battery as heat sources, two coolers, and a pump. (b) The simplification of the same cooling system that only covers heat conduction. $x_{ij}$ is a binary variable defining if a connection exists between two nodes, $T_i$ is the temperature at each node, and $\dot{Q}_i$ is the heat transfer rate from either a heat or cooling source.
• Figure 2: QAOA circuit with variational parameters $\gamma$, $\beta$ and Hamiltonians $H_\mathrm{C}$ and $H_\mathrm{M}$. The blue section highlights a single QAOA layer which is repeated $p$ times with individual parameters. A final measurement of $\langle H_\mathrm{C} \rangle$ is used to classically optimize $\gamma$ and $\beta$. After several iterations of that procedure, the circuit prepares the desired ground state of $H_\mathrm{C}$.
• Figure 3: Quantum circuit to embed the LCU of Eq. \ref{['eq:x-1']} in the top-left block of an $8\times8$ zero-padded matrix. This corresponds to $[\mathbf{U}_{01}/2]_\mathrm{block}$, and d(2) represents the least significant qubit.
• Figure 4: Quantum circuit for the block-encoding $U_\mathbf{A}$ of $\mathbf{A}(\mathbf{x})$ (cf. Eq. \ref{['eq:UA']}). The block-encoding of $\mathbf{U}_{ij}/2$ is not only controlled on the $\texttt{l}$-register (indicated by the half-filled control node) but also on the respective qubit $\ket{x_{ij}}$ of the ${\texttt{c}\text{-register}}$. The operation within the blue shaded area is repeated for all connections $(i, j)\in\mathbf{E}$.
• Figure 5: Quantum circuit representation of the Quantum Singular Value Transformation (QSVT) operator. The unitary $U_\mathbf{A}$ denotes the block-encoding of the matrix $\mathbf{A}(\mathbf{x})$, and the blue-shaded areas represent projector-controlled phase-shift operators. For even values of $d$, the rightmost block-encoding operator $U_\mathbf{A}$ should be replaced with its Hermitian adjoint $U_\mathbf{A}^\dagger$. To improve readability, we have omitted the controls on the ${\texttt{c}\text{-register}}$.