Notes on Quantum Computing for Thermal Science

TL;DR

The paper investigates quantum computing as a tool to accelerate the discretized heat-conduction problem in Thermal Science, framing the challenge within the NISQ era and exploring both variational and fault-tolerant quantum approaches. It develops a VQE-based pipeline that maps the linear system from a fully implicit finite-difference discretization to a ground-state problem of a constructed observable, with normalization, observable design, a parametrized ansatz, and de-normalization steps. To address scalability, it also discusses diagonalization-based measurement strategies using QFT and Hadamard tests, and it reviews the HHL algorithm as a pathway to exponential speedups under ideal conditions, including its clock and ancilla-resource overhead. Simulated results on small qubit counts reveal the current limitations due to noise and circuit depth, while outlining practical strategies (Pauli grouping, efficient ansatz, and diagonalization) that could enable quantum-accelerated heat-transfer simulations as hardware matures. Overall, the work maps a concrete program for leveraging quantum computing in thermal analysis, highlighting both near-term challenges and principled avenues for achieving quantum advantage in engineering applications.

Abstract

This document explores the potential of quantum computing in Thermal Science. Conceived as a living document, it will be continuously updated with experimental findings and insights for the research community in Thermal Science. By experiments, we refer both to the search for the most effective algorithms and to the performance of real quantum hardware. Those are fields that are evolving rapidly, driving a technological race to define the best architectures. The development of novel algorithms for engineering problems aims at harnessing the unique strengths of quantum computing. Expectations are high, as users seek concrete evidence of quantum supremacy - a true game changer for engineering applications. Among all heat transfer mechanisms (conduction, convection, radiation), we start with conduction as a paradigmatic test case in the field being characterized by a rich mathematical foundation for our investigations.

Figures (25)

• Figure 1: Bloch sphere representation of the qubit state $\ket{\psi_q}$. More details about the construction of the Bloch sphere representation of a single qubit are reported in Appendix \ref{['hilbert-bloch']}.
• Figure 2: Schematic of a composite system of two qubits. (a) Separate states, (b) tensor product of the two individual states, (c) entangled state, and (d) an example where two marginal probabilities are fixed ($p_{00}$, $p_{10}$) and consistent with the previous case, while varying $p_{11}$. The measurement probabilities for each basis state are indicated using colored bars and computed using the projector $P_m = \ket{m}\bra{m}$, where $\ket{m} \in \{\ket{0}, \ket{1}\}$ for single-qubit states or $\ket{m} \in \{\ket{00}, \ket{01}, \ket{10}, \ket{11}\}$ for the two-qubit case.
• Figure 3: Efficient ansatz (3 qubit, 24 parameters = four layers with six parameters each or, equivalently, eight parameters per qubit). For clarity, horizontal lines represent quantum wires which correspond to qubits in the circuit, red squares are the $R_Y$ gates (see Eq. (\ref{['RY']})), blue squares are the $R_Z$ gates (see Eq. (\ref{['RZ']})), blue dots represent control points in controlled gates, $\oplus$ symbol is used for a controlled-$X$ ($CNOT$) gate. The latter gate explicitly guarantees the desired entanglement.
• Figure 4: One time-step update of the temperature profile according to heat conduction equation by quantum computing ($3$ qubits, BaseEstimatorV2 quantum simulator, COBYLA classical minimizer with tolerance for termination $1\times10^{-3}$). The blue line is the initial temperature profile (with mean equal to $1$), the orange dashed line is the new temperature profile at time $\Delta t$, computed by finite-difference method. The blue dots are the mesh node temperatures computed by the quantum simulator.
• Figure 5: One time-step update of the temperature profile according to heat conduction equation by quantum computing ($4$ qubits, BaseEstimatorV2 quantum simulator, COBYLA classical minimizer with tolerance for termination $1\times10^{-3}$). The blue line is the initial temperature profile (with mean equal to $1$), the orange dashed line is the new temperature profile at time $\Delta t$, computed by finite-difference method. The blue dots are the mesh node temperatures computed by the quantum simulator.