The FE² methodology is a well-established framework for multiscale finite-element analysis, in which the macroscopic constitutive response is determined by solving a representative volume element (RVE) problem at each macroscopic integration point. Its overall computational cost is prohibitive given the large number of integration points in a macroscopic finite element discretisation. Recently, it has been demonstrated that exponential acceleration of periodic and non-periodic RVE calculations can be achieved with quantum computing [1,2]. This was achieved by formulating the RVE calculations using the quantum Fourier transform (QFT), following the pioneering work by Moulinec and Suquet [3] introducing FFT-based homogenisation. In this work, we show how a large ensemble of RVEs, such as those arising in FE², can be evaluated simultaneously by exploiting quantum entanglement and parallelism. To this end, the quantum RVE solver is regarded as a black box with low-dimensional input, i.e., the average strain of the RVE handed down from the integration points of the FE, and low-dimensional output, i.e., the average stress of the RVE to be handed up to the FE equilibrium equations. As is usually the case in FE², all the RVE problems are assumed to be identical with respect to their discretisation material properties. The RVEs differ only in their inputs and outputs. The RVEs are solved simultaneously by introducing log M qubits, where M is the total number of integration points, and creating an equal superposition state involving all RVEs. Subsequently, each RVE solver in the superposition is provided with its unique input via a controlled unitary. We provide gate-count estimates and numerical experiments in one and two spatial dimensions, and illustrate the O(M polylog M + polylog N) scaling of RVE computations in QAFE² compared to the O(MN log N) scaling of classical FE². [1] B. Liu, M. Ortiz and F. Cirak. Towards quantum computational mechanics, Computer Methods in Applied Mechanics and Engineering, Vol. 432, 117403, 2024. [2] E. Febrianto, Y. Wang, B. Liu, M. Ortiz and F. Cirak. A quantum spectral method for non-periodic boundary value problems, arXiv preprint arXiv:2511.11494, 2025. [3] H. Moulinec and P. Suquet, A numerical method for computing the overall response of nonlinear composites with complex microstructure, Computer Methods in Applied Mechanics and Engineering, Vol. 157, No. 1–2, pp. 69–94, 1998.