Quantum computing is expected to offer solutions to computational problems that are otherwise computationally intractable. Although the core technology is still being developed, quantum-inspired computing already has been offering practical advantages for several computationally challenging problems. Despite the promising potential of both quantum computing and quantum-inspired computing, applications in civil engineering remain underexplored. This study aims to lay the foundation for future adoption by introducing the fundamental principles of quantum computing and quantum-inspired computing and by conducting a multivocal literature review. The review provides insights into the current research landscape in civil engineering and offers a detailed analysis of potential use cases and application areas. The findings are expected to serve as a foundation for guiding future research endeavors and practical deployments of quantum computing and quantum-inspired computing in civil engineering, as these technologies continue to mature. • Introduction to quantum and quantum-inspired computing (QC & QiC). • Identifying the pros and cons of QC & QiC given civil engineering (CE) requirements. • Presenting a multivocal literature review on QC & QiC for CE. • Recommending high-potential use cases and QC & QiC approaches in CE • Identifying research topics to advance QC & QiC in CE for real-world impact.

Variational Quantum Algorithms (VQAs), such as the Quantum Approximate Optimization Algorithm (QAOA) of [Farhi, Goldstone, Gutmann, 2014], have seen intense study towards near-term applications on quantum hardware. A crucial parameter for VQAs is the \emph{depth} of the variational ``ansatz'' used -- the smaller the depth, the more amenable the ansatz is to near-term quantum hardware in that it gives the circuit a chance to be fully executed before the system decoheres. In this work, we show that approximating the optimal depth for a given VQA ansatz is intractable. Formally, we show that for any constant $ε>0$, it is QCMA-hard to approximate the optimal depth of a VQA ansatz within multiplicative factor $N^{1-ε}$, for $N$ denoting the encoding size of the VQA instance. (Here, Quantum Classical Merlin-Arthur (QCMA) is a quantum generalization of NP.) We then show that this hardness persists in the even ``simpler'' QAOA-type settings. To our knowledge, this yields the first natural QCMA-hard-to-approximate problems.

The diamond norm plays an important role in quantum information and operator theory. Recently, it has also been proposed as a regularizer for low-rank matrix recovery. The norm constants that bound the diamond norm in terms of the nuclear norm (also known as trace norm) are explicitly known. This paper provides a simple characterization of all operators saturating the upper and lower bounds.

In low-rank matrix recovery, one aims to reconstruct a low-rank matrix from a minimal number of linear measurements. Within the paradigm of compressed sensing, this is made computationally efficient by minimizing the nuclear norm as a convex surrogate for rank. In this paper, we identify an improved regularizer based on the so-called diamond norm, a concept imported from quantum information theory. We show that-for a class of matrices saturating a certain norm inequality-the descent cone of the diamond norm is contained in that of the nuclear norm. This suggests superior reconstruction properties for these matrices. We explicitly characterize this set of matrices. Moreover, we demonstrate numerically that the diamond norm indeed outperforms the nuclear norm in a number of relevant applications: These include signal analysis tasks, such as blind matrix deconvolution or the retrieval of certain unitary basis changes, as well as the quantum information problem of process tomography with random measurements. The diamond norm is defined for matrices that can be interpreted as order-4 tensors and it turns out that the above condition depends crucially on that tensorial structure. In this sense, this paper touches on an aspect of the notoriously difficult tensor completion problem.